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proceedings 12th International
BENCHMARK WORKSHOP
ON NUMERICAL ANALYSIS OF DAMS, 2nd - 4th October, 2013, Graz – AUSTRIA Edited by
Gerald Zenz and Markus Goldgruber
Heft / Volume 39
proceedings ICOLD 12th benchmark workshop
I C O L D
Austrian National Committee on Large Dams Stremayrgasse 10/II, A-8010 Graz, AUSTRIA, Phone: ++43/316/8861, [email protected], www.atcold.at
Published by the Austrian National Committee on Large Dams with the support of the Austrian Reservoir Commission (Staubeckenkommission)
PROCEEDINGS OF THE ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS 2nd - 4th OCTOBER, 2013, GRAZ – AUSTRIA
Edited by
Gerald Zenz Institute of Hydraulic Engineering and Water Resources Management Graz University of Technology
Markus Goldgruber Institute of Hydraulic Engineering and Water Resources Management Graz University of Technology
Published by
ATCOLD – AUSTRIAN NATIONAL COMMITTEE ON LARGE DAMS © 2014
All rights reserved
The authors are responsible for the content of their contributions. The texts of the various papers in this volume were set individually by typists under the supervision of each of the authors concerned.
Cover pictures copyright by Graz University of Technology Layout, Design, Cover Artwork by Institute of Hydraulic Engineering and Water Resources Management Printed by Graz University of Technology
Preface For the International Commission on Large Dams (ICOLD) the “Committee on Computational Aspects of Analysis and Design of Dams” is the general Organizer of Benchmark Workshops. This 12th Benchmark Workshop is held in the city of Graz, situated in the South of the Alps on both sides of the river Mur. Six universities with over 40,000 students are in addition responsible for the young and dynamic charm of our city. Since over 200 years Graz University of Technology is one of them. Advanced numerical tools with user friendly interfaces are available and widely used for structural analyses. Such numerical analyses require a solid theoretical background of the applicability of methods to be used. On the other hand, the results gained need a careful interpretation with respect to the underlying assumptions and their practical relevance. ICOLD Benchmark examples of generalized engineering problems are devoted to bridge the gap between numerical analyses, the interpretation of results and their theoretical as well as practical relevance. Since 1991 eleven benchmark workshops were organized for different numerical problems in the field of concrete and fill dams under static and dynamic loading conditions. The results of these benchmark workshops are made available to the dam engineering community on the internet and in proceeding. Results are published in ICOLD bulletins:
Bulletin N 94 Computer Software for Dams Bulletin N 122 Computational Procedures for Dam Engineering Bulletin N 155 Guidelines for Use of Numerical Models in Dam Engineering
This 12th Benchmark Workshop provides an excellent opportunity for engineers, scientists and operators to present and exchange their experiences and the latest developments related to the design, performance and monitoring of dams. There are three example topics and an open theme formulated and discussed: Theme A: Formulators:
Fluid Structure Interaction, Arch Dam - Reservoir at Seismic Loading Gerald Zenz, Markus Goldgruber
Theme B: Formulators:
Long Term Behavior of Rockfill Dams Camilo Marulanda, Joan Manuel Larrahondo
Theme C: Formulators:
Computational Challenges in Consequence Estimation for Risk Assessment Yazmin Seda-Sanabria, Enrique E. Matheu, Timothy N. McPherson
The support to this Benchmark Workshop from the Members of the Committee on Computational Aspects of Analysis and Design of Dams and especially from the Formulators is gratefully acknowledged. The contributions from the Core Organizing Team – Markus Goldgruber and Harald Breinhälter – are very much appreciated. Finally, I want to thank the participants for their scientific contributions herein. Gerald Zenz Chairman
Table of Content Conference Organizations
9
Acknowledgements
11
Overview of Contributions Theme A
13
Fluid Structure Interaction: Arch Dam ‐ Reservoir at Seismic Loading - Benchmark Problem Description
15
- Result Comparison of the Participants
27
- Papers
65
G. Maltidis and L. Stempniewski
67
W. Kikstra, F. Sirumbal and G. Schreppers
77
G. Faggiani and P. Masarati
87
A. Tzenkov, A. Abati and G. Gatto
99
M. Chambart, T. Menouillard, N. Richart, J.-F. Molinari and R. M. Gunn
111
A. Popovici, C. Ilinca and R. Vârvorea
123
R. Malm, C. Pi Rito, M. Hassanzadeh, C. Rydell and T. Gasch
139
M. Brusin, J. Brommundt and H. Stahl
149
S. Shahriari
165
A. Frigerio and G. Mazzà
167
A. Diallo and E. Robbe
177
Theme B
189
Long Term Behavior of Rockfill Dams - Benchmark Problem Description
191
- Papers
201
F. Ezbakhe and I. Escuder-Bueno
203
5
Table of Content Theme C
211
Computational Challenges in Consequence Estimation for Risk Assessment - Benchmark Problem Description
213
- Result Comparison of the Participants
227
- Papers
239
M. Davison, M. Hassan, O. Gimeno, M. van Damme and C. Goff
241
M.S. Altinakar, M.Z. McGrath, V.P. Ramalingam, D. Shen, Y. Seda Sanabria and E.E. Matheu 255 L. Mancusi, L. Giosa, A. Cantisani, A. Sole and R. Albano
271
O. Saberi, C. Dorfmann and G.Zenz
283
D. McVan, J. Ellis, G. Savant and M. Jourdan
293
B. A. Thames and A. J. Kalyanapu
309
D. Williams and K. Buchanan
325
Open Theme
339
Choice of Contributor - Papers
339
Behavior of an arch dam under the influence of creep, AAR and opening of the dam/foundation contact E. Robbe
341
Need for transient thermal models, with daily inputs, to explain the displacements of arch gravity dams I. Escuder, D. Galán and A. Serrano
349
The rehabilitation of Beauregard Dam: the contribution of the numerical modeling A. Frigerio and G. Mazzà
359
Earthquake Assessment of Slab and Buttress Dams H. B. Smith and L. Lia
369
Solution of dam-fluid interaction using ADAD-IZIIS software V. Mircevska, M. Garevski, I. Bulajic and S. Schnabl
379
Influence of Surface Roughness on Sliding Stability Tests and numerical modeling Ø. Eltervaag, G. Sas and L. Lia
389
Seismic Analysis of a Concrete Arch Dam P. Dakoulas
399
Earthquake safety assessment of arch dams based on nonlinear dynamic analyses S. Malla
411
7
Conference Organizations The conference was organized by Graz University of Technology, Graz, Austria Conference Chairman Gerald Zenz (Graz University of Technology, Austria) Committee on Computational Aspects of Analysis and Design of Dams Chairman I. Escuder‐Bueno (Spain) Vize‐Chairman G. Mazza (Italy) Technical Advisory Team RESTELLI, F. (Argentina) ZENZ, G. (Austria) CURTIS, D. (Canada) CHEN, S. (China) MARULANDA, C. (Colombia) VARPASUO, P. (Finland) TANCEV, L. (Form. Yug. Rep. of Macedonia) FROSSARD, E. (France) BEETZ, U. (Germany) DAKOULAS, P. (Greece) NOORZAD, A. (Iran)
MEGHELLA, M. (Italy) UCHITA, Y. (Japan) ANDERSEN, R. (Norway) POPOVICI, A. (Romania) GLAGOVSKY, V. (Russia) MINARIK, M. (Slovakia) HASSANZADEH, M. (Sweden) GUNN, R. (Switzerland) MATHEU, E. (United States) CARRERE, A. (Honorary Member) (France) FANELLI, M. (Honorary Member) (Italy)
Local Organizing Committee Graz University of Technology Institute of Hydraulic Engineering and Water Resources Management Head Gerald Zenz Organizer Markus Goldgruber
9
Acknowledgements The editors are grateful to the members of the following organizations for their support: ATCOLD ICOLD Benchmark Workshop Problem Formulation Teams International Water Power & Dam Construction
11
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
THEME A
13
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Theme A Fluid Structure Interaction Arch Dam – Reservoir at Seismic loading
Formulators: Dipl. Ing. Markus Goldgruber Univ. Prof. Dipl. Ing. Dr. techn. Gerald Zenz
Institute of Hydraulic Engineering and Water Resources Management Graz University of Technology
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Introduction Advanced numerical tools with user friendly interfaces are available for structural analyses. Such numerical analyses require a solid theoretical background of the applicability of methods to be used. On the other hand, the results gained need a careful interpretation with respect to the underlying assumptions and their practical relevance. ICOLD Benchmark examples of generalized engineering problems are devoted to bridge the gap between numerical analyses, the interpretation of results and their theoretical as well as practical relevance. Challenges of the analyses of concrete dams are always the definition of material parameters, the spatial discretization and the appropriate simulation of loading sequences. Additionally, specific attention is paid on the structural integrity and entire safety under seismic loading conditions. To account for this problem, the interaction of the dam and the reservoir is topic of this theme A. By means of the Finite Element Method linear and nonlinear analyses under dynamic excitation are carried out. However, for the required and appropriate simulation of the dam reservoir interaction different approaches are used. With respect to future nonlinear dynamic analyses, these simulations herein shall be in the time domain only. A common approach to take the dynamic water interaction into account is to use an added mass approach. A more sophisticated possibility is the use of Acoustic or Fluid Elements. The simulations of earthquake excitation of arch dams have shown that the analyzed stresses in the structure could vary significantly based on the interaction modeling. The added mass approach is still a widely used technique but tends to overestimate the stresses and therefore it is conservative in contrary to other techniques. This benchmark now intends to compare different modeling techniques and will show the amount of deviations. All investigations are carried out for an artificially generated symmetric arch dam and simplified loading and boundary conditions. Universities, engineering companies and regulatory bodies are invited to contribute to the benchmark and take part in the discussion of results gained. Focus of this benchmark example The focus of this benchmark is to carry out the Dynamic Fluid Structure Interaction for a large arch dam. Every participant may choose his own order of details in modeling. The main goal of this example is the application of different approaches like: Added mass technique (Westergaard, Zangar,…) Acoustic Elements (compressible, incompressible) Fluid Elements (compressible, incompressible) Further on, the usage of different Boundary Conditions is possible for: Reservoir - Foundation o Reflecting (on the bottom and the sides) o Non-reflecting (at the end of the reservoir) The modeling of the block joint opening – due to tensile stresses and nonlinear effects - is not focus of this benchmark example. However, to carry out this analysis in the time domain will provide the opportunity for further non-linear analyses.
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
General basic assumptions The following general basic assumptions and boundary conditions for the investigations should be used: Same spatial discretization (Model/Mesh) of the Structure, Foundation and Reservoir Same Material Parameters Acceleration-Time-History in X-,Y-,Z-Direction Reservoir is infinite in length (non-reflecting) Rayleigh Damping Results to be compared – Visualization Based on these basic assumptions and results gained the contributors are encouraged to intensify and focus their effort to achieve results with higher profound physical justification and explain the differences. (E.g.: different spatial discretization, more appropriate modeling of the interaction; different length of the reservoir; need for nonlinear effects). An interpretation of the evaluated results from an engineering point of view should be given.
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Model and Geometry An Arch Dam, Foundation and Reservoir Model layout for the benchmark has been generated and is available for downloading. Arch Dam Model Symmetric Geometry Total Height: 220 Meters Valley width (crest): ~ 430 Meters Valley width (bottom): ~ 80 Meters Arch Dam Geometry The Arch Dam Geometry has been generated with the Program “Arch Dam Design”, which was developed as part of the Master-Thesis by DI Manuel Pagitsch.
Arch Dam Model
Plan View
View from the upstream
Main Section
Figure 1: Arch dam geometry Foundation Model Symmetry is used for the foundation too. 500 Meters Height: Length: 1000 Meters Width 1000 Meters
18
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
500m
1000m
1000m Figure 2: Foundation geometry Reservoir Model Length: assumed minimum of 460 Meters (> 2x Height of the Dam) Modeling the interaction with Acoustic- or Fluid Elements
Figure 3: Reservoir geometry
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Acceleration Time History Transient Acceleration (amax ≈ 0.1g) X-,Y-,Z- Direction Artificially generated time history
Figure 4: Reservoir geometry
Material Parameters The Material properties are defined for isotropic and homogenous conditions. Rock mass Density: 0 kg/m3 Poisson - ratio: 0,2 Youngs - modulus: 25000 MPa Water Density: 1000 kg/m3 Bulk - modulus: 2200 MPa Dam
Density: 2400 kg/m3 Poisson - ratio: 0,167 Youngs - modulus: 27000 MPa
20
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Mesh Properties Two different Meshes of the entire system are provided for investigations, as these are a coarse and a fine mesh. If desired, the parts can also be provided as ABAQUS/CAE Model File, ACIS- or IGES-Files, if a specific mesh is intended to be discretized. Coarse Mesh Arch Dam Total number of nodes: 2083 Total number of elements: 356 312 quadratic hexahedral elements of type C3D20R (ABAQUS CAE) 44 quadratic wedge elements of type C3D15 (ABAQUS CAE) Foundation Total number of nodes: 11608 Total number of elements: 2340 quadratic hexahedral elements of type C3D20R (ABAQUS CAE) Reservoir Total number of nodes: 12493 Total number of elements: 2640 quadratic hexahedral elements of type C3D20R (ABAQUS CAE)
Figure 5: Coarse mesh of the dam, foundation and reservoir
21
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Fine Mesh Arch Dam Total number of nodes: 13733 Total number of elements: 2736 quadratic hexahedral elements of type C3D20R (ABAQUS CAE) Foundation Total number of nodes: 13298 Total number of elements: 2700 quadratic hexahedral elements of type C3D20R (ABAQUS CAE) Reservoir Total number of nodes: 12493 Total number of elements: 2640 quadratic hexahedral elements of type C3D20R (ABAQUS CAE)
Figure 6: Fine mesh of the dam, foundation and reservoir
22
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Elements and Node Numbering in ABAQUS/CAE The provided input-files are containing a list of the nodes and elements of the mesh and also predefined “node sets” for the different sections which should be investigated. The Node numbering of ABAQUS/CAE is plotted in the following figures.
Figure 7: Node numbering of wedge and brick elements in ABAQUS/CAE These figures are showing the node numbering for the two different element types which are used in the provided input-files.
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Loading The following loading sequence is intended to be used. Gravity Hydrostatic Water Pressure (full supply water level = Crest Height) Seismic Loading o Modal Superposition or o Direct time integration (Implicit/Explicit)
Results Following results should be evaluated and plotted. Eigenfrequencies (1 – 10) The evaluation of the first 10 Eigenfrequencies of the structure, including the interaction with the reservoir, should be provided. Mode Shapes (1 – 10) The evaluation and plotting of the first 10 Mode-Shapes of the structure, including the interaction with the reservoir, should be provided. Hoop Stresses, Vertical Stresses and Min./Max. Principal Stresses Evaluation of the different stresses should be done for Static Loads Seismic Loads (Min., Max.) 3 different sections (Main Section and ~45 degrees on the left and right hand side)
Left Section
Main Section
Right Section
Figure 8: Evaluation sections of the arch dam
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Figure 9: Evaluation examples for the stresses Radial Deformation Evaluation of the Radial Deformation should be done for Static Loads Seismic Loads (Min., Max.) Main section
Figure 10: Evaluation examples for the radial deformation
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
RESULT COMPARISON OF THE PARTICIPANTS THEME A
27
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Results Comparison Theme A Fluid Structure Interaction Arch Dam – Reservoir at Seismic loading Participants, Programs and Approaches Overall there are 11 participants from 9 different countries (Swiss, Netherlands, France, Germany, Sweden, Italy, Iran, Romania and Austria) who were contributing to the workshop and decided to solve the problem. Each participant had the opportunity to choose his preferred numerical program and modelling technique to account for the fluid structure interaction. The prevailing boundary conditions, which are the same for each participant, are defined in the section “Benchmark Problem Description Theme A”. The “Reference Solution” (REF) in the diagrams and tables doesn’t claim to be the optimum solution. It shows the results of the simulations done at the Institute of Hydraulic Engineering and Water Resources Management by Markus Goldgruber. The following table lists the participants and their used programs and approaches. The informations in the last column should point out some specific differences between the participants which may influence the results and are worth mentioning. Some of the participants have provided results of more than just one simulation, but for the comparison just one of these has been used. All the other results of the approaches and models can be found in their papers in the following section (Papers – Theme A) All participants had to evaluate eigenfrequencies, mode shapes, deformations and stresses. In the case of dynamic simulations one will get minimum and maximum values. Therefore, in the diagrams in the results section every participant has three lines, the minimum (left line) and the maximum line (right line) which indicate the minimum and maximum values out of the time history records and the line for static loading (middle line). This middle line indicates the static value out of the sum of the two load cases, gravity and hydrostatic water load. To retain the overview in the diagrams, the minimum, maximum and static values are not explicitly mentioned in the legend. The diagrams for the deformations and the upstream hoop stresses in the main section were plotted additionally for static and dynamic loading separately.
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Table 1: Participants, Programs and Approaches Finite Element Method Program Acoustic ABAQUS Elements
Mesh
A
G. Maltidis
B
W.Kikstra, F. Sirumbal, G. Schreppers
DIANA
Acoustic Elements
Coarse
C
G. Faggiani, P. Masarati
CANT-SD
Acoustic Elements
Coarse
D
A. Tzenkov, A. Abiati, G. Gatto
DIANA
Acoustic Elements
Coarse
E
M. Chambart
DIANA
Fine
Edyn = Esta * 1.25
F
A. Popovici, C. Ilinca, R. Vârvorea
Added mass (Westergaard)
Same as participant B, Construction steps for loadcase deadweight
ABAQUS
Added mass (Westergaard)
Coarse
G
R. Malm, C. Pi Rito, C. Hassanzadeh, C. Rydell, T. Gasch
ABAQUS
Acoustic Elements
Fine
H
M. Brusin
FENAS ECCON IPP
Added mass (Westergaard)
Fine
I
S. Shahriari
ANSYS
Added mass (Westergaard)
Coarse
J
A. Frigerio, G. Mazzà
COMSOL
Acoustic Elements
Coarse
A.Diallo, E. Robbe
Incompressible CODE_ASTER Finite Element Coarse added mass
K
REF M. Goldgruber
ABAQUS
Acoustic Elements
30
Coarse
Some Additional Informations 7.5% critical damping Compressible Fluid, Hybrid FrequencyTime Domain (HFTD) method
Coarse
Infinite Elements at the boundaries, Acceleration-TimeHistory applied on the bottom of the model Construction steps for loadcase deadweight Use of the full Westergaard formula (Period/Frequency dependent) Method to calculate the added mass matrices representing the fluid-structure interaction with a potential approach
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Results Eigenfrequencies
Figure 1: Eigenfrequencies 1 – 10(Column Chart) Table 2: Eigenfrequencies 1 – 10 (Table) Mode
Participant A B C D E F G H I J K REF
1 1.47 1.57 1.54 1.57 1.43 1.54 1.51 1.26 1.28 1.54 1.57 1.54
2 1.54 1.60 1.55 1.62 1.47 1.56 1.54 1.32 1.33 1.55 1.62 1.54
3 1.55 2.36 2.05 2.36 2.21 1.93 1.90 2.01 1.91 2.09 2.35 2.05
4 2.11 2.94 2.22 2.94 2.61 2.30 2.22 2.36 2.37 2.22 2.95 2.29
5 2.33 3.04 2.41 3.04 2.81 2.48 2.42 2.50 2.38 2.33 3.03 2.54
31
6 2.46 3.72 2.83 3.72 3.27 3.04 2.96 3.00 2.91 2.51 3.72 2.96
7 2.61 3.88 2.98 3.87 3.56 3.12 3.01 3.17 2.98 2.83 3.85 3.21
8 2.97 4.56 3.37 4.56 4.09 3.29 3.28 3.65 3.61 2.96 4.56 3.36
9 3.25 4.78 3.40 4.76 4.37 3.61 3.59 3.70 3.62 3.19 4.88 3.76
10 3.37 4.80 3.79 4.80 4.37 3.71 3.76 3.88 3.85 3.37 5.13 3.91
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Mode Shapes
Figure 2: Mode Shapes 1 – 10; Participants A – E 32
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Figure 3: Mode Shapes 1 – 10; Participants F – K
33
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HEIGHT (m)
Deformations – Main Section
220 200 180 160 140 120 100 80 60 40 20 0 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
DEFORMATION (m) A
B
C
D
E
REF
HEIGHT (m)
Figure 4: Deformation – Main Section (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
DEFORMATION (m) F
G
H
I
J
K
Figure 5: Deformation – Main Section (F – K)
34
REF
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Deformations (Only Static Load) – Main Section
220 200 180 160 140 120 100 80 60 40 20 0 0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
DEFORMATION (m) A
B
C
D
E
REF
HEIGHT (m)
Figure 6: Deformation (Only Static Load) – Main Section (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
DEFORMATION (m) F
G
H
I
J
K
REF
Figure 7: Deformation (Only Static Load) – Main Section (F – K)
35
0.09
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Deformations (Only Dynamic Load) – Main Section
220 200 180 160 140 120 100 80 60 40 20 0 -0.15 -0.13 -0.11 -0.09 -0.07 -0.05 -0.03 -0.01 0.01 0.03 0.05 0.07 0.09 0.11
DEFORMATION (m) A
B
C
D
E
REF
HEIGHT (m)
Figure 8: Deformation (Only Dynamic Load) – Main Section (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -0.15 -0.13 -0.11 -0.09 -0.07 -0.05 -0.03 -0.01 0.01 0.03 0.05 0.07 0.09 0.11
DEFORMATION (m) F
G
H
I
J
K
REF
Figure 9: Deformation (Only Dynamic Load) – Main Section (F – K)
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Hoop Stresses – Main Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -14.00 -12.00 -10.00 -8.00
-6.00
-4.00
-2.00
0.00
2.00
4.00
6.00
4.00
6.00
HOOP STRESS (MPa) A
B
C
D
E
REF
HEIGHT (m)
Figure 10: Hoop Stresses – Main Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -14.00 -12.00 -10.00 -8.00
-6.00
-4.00
-2.00
0.00
2.00
HOOP STRESS (MPa) F
G
H
I
J
K
REF
Figure 11: Hoop Stresses – Main Section – Upstream (F – K)
37
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Hoop Stresses (Only Static Load) – Main Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -8.00
-7.00
-6.00
-5.00
-4.00
-3.00
-2.00
-1.00
0.00
1.00
2.00
HOOP STRESS (MPa) A
B
C
D
E
REF
HEIGHT (m)
Figure 12: Hoop Stresses (Only Static Load) – Main Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -8.00
-7.00
-6.00
-5.00
-4.00
-3.00
-2.00
-1.00
0.00
1.00
2.00
HOOP STRESS (MPa) F
G
H
I
J
K
REF
Figure 13: Hoop Stresses (Only Static Load) – Main Section – Upstream (F – K)
38
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Hoop Stresses (Only Dynamic Load) – Main Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -10.00 -8.00 -6.00 -4.00 -2.00
0.00
2.00
4.00
6.00
8.00 10.00
HOOP STRESS (MPa) A
B
C
D
E
REF
HEIGHT (m)
Figure 14: Hoop Stresses (Only Dynamic Load) – Main Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -10.00 -8.00 -6.00 -4.00 -2.00
0.00
2.00
4.00
6.00
8.00 10.00
HOOP STRESS (MPa) F
G
H
I
J
K
REF
Figure 15: Hoop Stresses (Only Dynamic Load) – Main Section – Upstream (F – K)
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Hoop Stresses – Main Section – Downstream
220 200 180 160 140 120 100 80 60 40 20 0 -10.00
-7.50
-5.00
-2.50
0.00
2.50
5.00
HOOP STRESS (MPa) A
B
C
D
E
REF
HEIGHT (m)
Figure 16: Hoop Stresses – Main Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -10.00
-7.50
-5.00
-2.50
0.00
2.50
HOOP STRESS (MPa) F
G
H
I
J
K
REF
Figure 17: Hoop Stresses – Main Section – Downstream (F – K)
40
5.00
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Hoop Stresses – Left Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -10.00
-8.00
-6.00
-4.00
-2.00
0.00
2.00
4.00
6.00
HOOP STRESS (MPa) A
B
C
D
E
REF
HEIGHT (m)
Figure 18: Hoop Stresses – Left Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -10.00
-8.00
-6.00
-4.00
-2.00
0.00
2.00
4.00
HOOP STRESS (MPa) F
G
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Figure 19: Hoop Stresses – Left Section – Upstream (F – K)
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Figure 20: Hoop Stresses – Left Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -12.00
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Figure 21: Hoop Stresses – Left Section – Downstream (F – K)
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Figure 22: Hoop Stresses – Right Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -9.00
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Figure 23: Hoop Stresses – Right Section – Upstream (F – K)
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HEIGHT (m)
Hoop Stresses – Right Section – Downstream
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Figure 24: Hoop Stresses – Right Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -9.00
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Figure 25: Hoop Stresses – Right Section – Downstream (F – K)
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HEIGHT (m)
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220 200 180 160 140 120 100 80 60 40 20 0 -6.00
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Figure 26: Vertical Stresses – Main Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -6.00
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Figure 27: Vertical Stresses – Main Section – Upstream (F – K)
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Figure 28: Vertical Stresses – Main Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -12.00
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Figure 29: Vertical Stresses – Main Section – Downstream (F – K)
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HEIGHT (m)
Vertical Stresses – Left Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -4.00
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HEIGHT (m)
Figure 30: Vertical Stresses – Left Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -4.00
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Figure 31: Vertical Stresses – Left Section – Upstream (F – K)
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HEIGHT (m)
Vertical Stresses – Left Section – Downstream
220 200 180 160 140 120 100 80 60 40 20 0 -10.00
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Figure 32: Vertical Stresses – Left Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -10.00
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Figure 33: Vertical Stresses – Left Section – Downstream (F – K)
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HEIGHT (m)
Vertical Stresses – Right Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -4.00
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HEIGHT (m)
Figure 34: Vertical Stresses – Right Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -4.00
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VERTICAL STRESS (MPa) F
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Figure 35: Vertical Stresses – Right Section – Upstream (F – K)
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HEIGHT (m)
Vertical Stresses – Right Section – Downstream
220 200 180 160 140 120 100 80 60 40 20 0 -10.00
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Figure 36: Vertical Stresses – Right Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -10.00
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Figure 37: Vertical Stresses – Right Section – Downstream (F – K)
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HEIGHT (m)
Minimum Principal Stresses – Main Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -14.00
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Figure 38: Minimum Principal Stresses – Main Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -14.00
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Figure 39: Minimum Principal Stresses – Main Section – Upstream (F – K)
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Minimum Principal Stresses – Main Section – Downstream
HEIGHT (m)
220 200 180 160 140 120 100 80 60 40 20 0 -14.00
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HEIGHT (m)
Figure 40: Minimum Principal Stresses – Main Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -14.00
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MIN. PRINCIPAL STRESS (MPa) F
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Figure 41: Minimum Principal Stresses – Main Section – Downstream (F – K)
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HEIGHT (m)
Minimum Principal Stresses – Left Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -12.00
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HEIGHT (m)
Figure 42: Minimum Principal Stresses – Left Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -12.00
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MIN. PRINCIPAL STRESS (MPa) F
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Figure 43: Minimum Principal Stresses – Left Section – Upstream (F – K)
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Minimum Principal Stresses – Left Section – Downstream
220 200 180 160 140 120 100 80 60 40 20 0 -20.00 -18.00 -16.00 -14.00 -12.00 -10.00 -8.00 -6.00 -4.00 -2.00 0.00
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Figure 44: Minimum Principal Stresses – Left Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -20.00 -18.00 -16.00 -14.00 -12.00 -10.00 -8.00 -6.00 -4.00 -2.00 0.00
2.00
MIN. PRINCIPAL STRESS (MPa) F
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Figure 45: Minimum Principal Stresses – Left Section – Downstream (F – K)
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HEIGHT (m)
Minimum Principal Stresses – Right Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -10.00
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Figure 46: Minimum Principal Stresses – Right Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -10.00
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Figure 47: Minimum Principal Stresses – Right Section – Upstream (F – K)
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Minimum Principal Stresses – Right Section – Downstream
220 200 180 160 140 120 100 80 60 40 20 0 -20.00 -18.00 -16.00 -14.00 -12.00 -10.00 -8.00 -6.00 -4.00 -2.00 0.00
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HEIGHT (m)
Figure 48: Minimum Principal Stresses – Right Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -20.00 -18.00 -16.00 -14.00 -12.00 -10.00 -8.00 -6.00 -4.00 -2.00 0.00
2.00
MIN. PRINCIPAL STRESS (MPa) F
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REF
Figure 49: Minimum Principal Stresses – Right Section – Downstream (F – K)
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HEIGHT (m)
Maximum Principal Stresses – Main Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -3.00
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MAX. PRINCIPAL STRESS (MPa) A
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HEIGHT (m)
Figure 50: Maximum Principal Stresses – Main Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -3.00
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MAX. PRINCIPAL STRESS (MPa) F
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Figure 51: Maximum Principal Stresses – Main Section – Upstream (F – K)
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HEIGHT (m)
Maximum Principal Stresses – Main Section – Downstream
220 200 180 160 140 120 100 80 60 40 20 0 -2.00
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E
HEIGHT (m)
Figure 52: Maximum Principal Stresses – Main Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -2.00
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Figure 53: Maximum Principal Stresses – Main Section – Downstream (F – K)
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HEIGHT (m)
Maximum Principal Stresses – Left Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -3.00
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Figure 54: Maximum Principal Stresses – Left Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -3.00
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Figure 55: Maximum Principal Stresses – Left Section – Upstream (F – K)
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Maximum Principal Stresses – Left Section – Downstream
HEIGHT (m)
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Figure 56: Maximum Principal Stresses – Left Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -2.00
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Figure 57: Maximum Principal Stresses – Left Section – Downstream (F – K)
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HEIGHT (m)
Maximum Principal Stresses – Right Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -3.00
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Figure 58: Maximum Principal Stresses – Right Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -3.00
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Figure 59: Maximum Principal Stresses – Right Section – Upstream (F – K)
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HEIGHT (m)
Maximum Principal Stresses – Right Section – Downstream
220 200 180 160 140 120 100 80 60 40 20 0 -2.00
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Figure 60: Maximum Principal Stresses – Right Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -2.00
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Figure 61: Maximum Principal Stresses – Right Section – Downstream (F – K)
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Discussion of Results Eigenfrequencies It was to calculate the first 10 Eigenfrequencies of the structural response, including the interaction with the reservoir under full reservoir conditions. The results are summarized in Figure 1 to Figure 3 and Table 2. The criteria to compare the results are only the Eigenfrequency – no shape deformation, nor mass contribution to the different modes and directions are considered. For commenting in detail, this information would be valuable. The lower frequencies compared to the others of the first two modes of participants H and I are due to the use of the added mass approach. The overestimation of the additionally excited mass leads to slightly lower Eigenfrequencies, while the remaining are around 1.5 Hz. Noticeable is that the participants who used the program Diana (B, D and E) are getting higher frequencies starting from the third. This counts also for the participant K, who used the Open Source software Code_Aster. All other participants are getting more or less the same Eigenfrequencies for the first 10 modes. Deformation The comparison of the deformations is done for the main (middle) section only. The static loading accounts for dead weight and water loading together. No temperature loading is accounted for. The static deformation reveals, that many results show almost the same behavior, except those from participants D, E, H and K. The higher static deformation of participants D, H and K are due to modelling of the construction stages. The dynamic deformations are, as expected, varying in a wider range. Especially worth mentioning is also the result by participant G, he, as the only one, used infinite elements on the vertical boundary and applied the acceleration-time-history record on the bottom of the model, which could be the reason for in general higher values. The lower values of radial deformation of participant E are a result of the higher young’s modulus used. Stresses The comparison of the stresses of each of the participant and each of the diagrams are focused on essential aspects. Therefore, the discussion is kept general and just the quality of some graphs and values is discussed, but not the quantity in detail. Every participant has used his own preferred program, modelling technique and approach, so it’s to await that different results are gained. As it was up to the participant to use immediate or stepped construction sequences, the stress distribution differs. A 0.5[MPa] difference for the static loading, at a stress level of 6[MPa], one might accept, but not larger (Figure 12 and Figure 13). Worth mentioning are the results by participant G. He used, as already mentioned in the discussion of the deformations, infinite elements and applied the acceleration time history just on the bottom of the model. So the same as for the deformation counts here, the stresses in contrast to the others are far the highest in almost each diagram and beyond awaited results. Participants B and D used the program Diana with the Hybrid Frequency-Time Domain Method (HFTD-Method), which takes frequency dependent properties, such as compressibility of fluid, reservoir-bottom absorption and far-field reflection, into account. Both of them got similar results compared to the others, which prove the usability of this sophisticated analysis method on the one hand – but shows the applicability of less elaborated models, under these assumptions, too.
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Participant K was the only one who used the Open Source software Code_Aster. Such software, which is mostly used at research facilities and universities, is license free, but often more difficult to apply than commercial ones. Nevertheless, the provided results are matching with the results from the other participants. It is worth mentioning, that the added mass approach (according to Westergaard) is able to provide, comparable results, under the circumstances of this benchmark. All of the three contributors (E, H and I) using this approach are in the range of expected results.
Conclusion The comparison of all participants’ shows, that despite of the same boundary conditions (Model of the Geometry, Finite Element Mesh, Load cases, Material properties, Linear analysis, etc.) still assumptions are taken and required to carry out the analyses. These additional assumptions are starting with the application of the construction sequence, increase of material properties for dynamic loading, abutment boundary properties, application of dynamic loading and some specific assumptions based on the program used. Best practice examples and recommendations are published in ICOLD Bulletins and are available for engineers. However, for any specific problem to be solved, the assumptions for an analysis applied need to be reconfirmed in the light of the entire problem. In general it is astonishing, to see the large differences between the results of individual. Everybody had the opportunity to choose his preferred modelling technique to account for the fluid structure interaction, but most of the contributors used either added mass technique or acoustic elements. In practice it is still common to use an added mass approach according to Westergaard. Normally this assumption yields conservative results in contrary to modelling with acoustic elements. The solution of participant e.g. I, who has used Westergaards’ formula with its fully, frequency dependent extension, shows very similar results to those analyzed using “higher” constitutive models. According to results of the participants using either the coarse or fine mesh (described in the former section) had just a marginal influence on the frequencies, deformations and stresses within the structure. The purpose of choosing this arch dam example (220m in height, totally symmetric) wasn’t just for evaluating the influence of different modelling techniques, but also for engineers, scientists and operators to have a kind of reference solution. The diagrams and tables of the results of all participants should help to quantify and compare frequencies, deformations and stresses of such a structure. Concluding, everybody should be aware of the fact, that results of such simulations should be treated critically, because mistakes in modelling and application cannot be excluded. Usually, reference solutions from former comparable projects for validation should be used to proof the results for plausibility.
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PAPERS THEME A
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Fluid Structure Interaction Arch Dam – Reservoir at Seismic Loading G. Maltidis1 and L. Stempniewski2 1
Federal Waterways Engineering and Research Institute, Kußmaulstrasse 17, 76187 Karlsruhe, GERMANY 2 Karlsruhe Institute for Technology, Institute of Concrete Structures and Building Materials, Department of Concrete Structures, Gotthard-Franz-Str. 3, 76131, Karlsruhe, GERMANY E-mail: [email protected]
Abstract The fluid structure interaction is an important issue that must be taken into account for the analysis and design of hydraulic structures. Since the first attempts to calculate the hydrodynamic pressures on structures analytically (Westergaard, von Kármán, Mononobe, Housner, Chwang, Zangar) the engineers and researchers have the last years a very useful tool, the finite and boundary element method, in order to analyze complicated structures taking into account different sophisticated phenomena. However, even nowadays, the common praxis is to use the early developed techniques, because of their simplicity and capability of implementation in the most finite element codes.
Introduction Since 1933, the hydrodynamic pressures on oscillating structures, which are in contact with water, are taken into account with the simplified assumption that the water is incompressible and the structure is star using the added mass approaches, first proposed by Westergaard for vertical star surfaces and later extended by Zangar for inclined surfaces. Although these approaches apply under conditions which hardly are met, they are widely used also nowadays because of their simplicity in incorporating them in finite element codes. However, the result of analysis with the added mass approach may come out to be very conservative leading to wrong decisions. The modeling of the water with finite solid element around the 1980’s gave the opportunity for the analyst to take account some phenomena, as the water compressibility but raised other numerical problems as such type of modeling of water is suffering many times of hourglass making the analysis instable. The use of acoustic elements seems to be the more beneficial, as there are hardly numerical problems, and most of the phenomena, which take place for a dynamic fluid structure interaction can be modeled. With acoustic elements the analyst can consider the water compressibility, the wave absorption at the infinite end of the reservoir and the impedance of wave radiation at the reservoir sediments.
Hydrodynamic Pressures Added mass approaches The most well-known added mass approach is the one of Westergaard (1933)[1]. Westergaard proposed the following formula for the computation of hydrodynamic pressures as added masses under the restrictions that the reservoir is infinite, the upstream surface of the dam is vertical and the dam is rigid: γ 7 m Hy w A (1) 8 g
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where m the added mass, H and y the height and the depth of the reservoir respectevily, γw the density of the water, g the gravitational acceleration and A the contributing area around the node.
Figure 1: Graphical representation of Westergaard’s and Zangar’s calculation models. Zangar (1952)[2], using an electric analogue, extended the added mass approach of Westergaard for inclined upstream surfaces of the dam, introducing reductive factors dependent on the angle of inclination. y y y y γ m 0,5 H Cm (2 ) (2 ) w A (2) H H H g H where Cm a coefficient based on the angle of inclination and the other parameters as Westergaard’s formula. Fluid Elements The fluid elements are solid elements to which the characteristics of the water are applied. The incompressibility or the water as well as the null shear resistance are introduced with a Poisson number equal with 0,5 or close to this value for the finite element programs. The bulk modulus of the water is K=2,2 GPa. The modeling of the water with solid elements causes numerical instabilities because of the introducing of zero energy modes (hourglass modes). This effect can be mitigated with the use of hourglass control and by applying the free surface boundary condition for the vertical node displacements [9]. Moreover, a nonlinear material behavior with tension cut off or a contact interaction which allows only compression to be transmitted will avoid unrealistic tension stress of the dam caused by the water. Acoustic Elements The acoustic elements are used to model the fluid behavior of the air. They have no shear and tension resistance and they transmit only pressures [10]. With assignment of the water bulk modulus they model the water behavior very good. Numerous boundary conditions can be assigned to the acoustic elements, which model natural phenomena such as wave absorption at the far end of the reservoir, sloshing of the free surface, wave impedance at the reservoir’s bottom due to sediments etc. For the acoustic elements no special numerical care has to be taken except for assigning the boundary conditions. Model aspects For this benchmark two models (one with coarse mesh and one with fine mess) are investigated. The mesh of the reservoir is the same for both cases. The foundation was considered massless, so no further care was taken for wave absorption or deconvolution of the seismic motion. Because of the massless foundation with no radiation absorption of the 68
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seismic waves and due to the lack of further non-linearities of the dam’s material and of the contact interfaces, a big enough structural damping is applied. As presented in [6] for a big range of frequencies the total Rayleigh damping is between 8 and 10 %. Due to the linear finite element analysis a 10% viscous damping is used by [6]. Here, because of the small peak ground acceleration of 0,1g a value of 7,5% of structural damping was chosen in order to determine the Rayleigh stiffness damping factor a and the Rayleigh mass factor β. For the reservoir hydrodynamic pressures, two added mass approaches and one reservoir modeling with acoustic elements were investigated. The generalized Westergaard’s [11] and the Zangar’s added mass approaches were used. The added masses were given via a user subroutine which defines user elements in Abaqus [5]. The two models with acoustic elements differ only in the wave absorption’s method of the far field. The first uses acoustic infinite element whereas the latter impedance boundary condition. The impedance condition can be given either as element based or as surface based condition. Moreover a boundary condition is given at the reservoir free surface constraining the dynamic acoustic pressures to be zero. The surfaces of the rock and the dam are tied with the surfaces of the reservoir.
Figure 2: The model with two different meshes and two reservoir modeling approaches: left fine mesh and acoustic elements and right coarse mesh and added mass elements.
Tied with the dam
Free surface condition
Tied with the rock Boundary impedance or acoustic infinite elements
Figure 3: The reservoir with its boundary conditions.
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Table 1: Material Parameters Density (kg) Poisson Ratio Young Modulus (MPa) Bulk Modulus (MPa)
Rock 0 0,2 25000 -
Water 1000 2200
Dam 2400 0,167 27000 -
Analysis’ methods The seismic analyses were carried out using the time history with direct integration and the modal time history. From computational time the modal time history is a little bit faster than the time history with direct integration. The time histories were gives as nodal acceleration to the boundaries of the rock. A baseline correction offered by Abaqus was also applied to them. Results The results of the analyses are presented at the next tables and diagrams. The first table shows the ten first modes for the dam, with empty and with full reservoir modeled by the different methods described before. The diagrams show due to lack of space only some of the results containing the minimal and maximal vertical, minimum principal and maximum principal stresses of the dam for the different reservoir models and for the different dam mesh. The results are given for the paths along the height of the dam, for the upstream and the downstream sections. For convenience abbreviations were introduced to the diagrams (e.g. “dti” refers to direct time integration, “West” to Westergaard’s added mass, “ac” to acoustic element, “imp” to impedance boundary condition for the acoustic elements, “inf” to acoustic infinite elements, “modal” to modal dynamic analysis). The results for the fine mesh model are given with dashpot line in order to differ easier than the ones of the coarse mesh model. The analysis with the infinite elements had more computational cost than the analysis with the impedance condition. In order to obtain similar results to the impedance boundary condition with the use of infinite acoustic elements, care must be given in the definition of the infinite elements’ thickness. There are trivial differences when the analyst uses the improved rather than the planar non-reflecting condition offered by Abaqus. The results of the fine model with acoustic elements for the reservoir gave too conservative results. The author believes that these results for the given meshes of dam and reservoir are not correct due to violation of the contact condition, according to which the slave surface nodes must be finer than the master surface nodes.
Figure 4: The frequencies for the two models (coarse left, fine right) and for the four reservoir models.
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Table 2: Ten first modes for the coarse model Mode Nr.
Empty
Acoustic
Westergaard f T (Hz) (sec)
f (Hz)
T (sec)
f (Hz)
T (sec)
f=1,92
T=0,52
f=1,47
T=0,68
f=1,29
f=2,03
T=0,49
f=1,54
T=0,65
f=2,91
T=0,34
f=1,55
f=3,59
T=0,28
f=3,63
Zangar
f (Hz)
T (sec)
T=0,78
f=1,39
T=0,72
f=1,32
T=0,76
f=1,44
T=0,69
T=0,65
f=1,96
T=0,51
f=2,12
T=0,47
f=2,11
T=0,47
f=2,30
T=0,43
f=2,49
T=0,40
T=0,28
f=2,33
T=0,43
f=2,44
T=0,41
f=2,65
T=0,38
f=4,29
T=0,23
f=2,46
T=0,41
f=2,86
T=0,35
f=3,07
T=0,33
f=4,50
T=0,22
f=2,61
T=0,38
f=3,08
T=0,32
f=3,33
T=0,30
f=4,80
T=0,21
f=2,97
T=0,34
f=3,57
T=0,28
f=3,82
T=0,26
f=5,19
T=0,19
f=3,25
T=0,31
f=3,73
T=0,27
f=4,05
T=0,25
f=5,52
T=0,18
f=3,37
T=0,30
f=3,77
T=0,27
f=4,07
T=0,25
1
2
3
4
5
6
7
8
9
10
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Table 3: Ten first modes for the fine model Mode Nr.
Empty
Acoustic
Westergaard f T (Hz) (sec)
f (Hz)
T (sec)
f (Hz)
T (sec)
f=1,91
T=0,52
f=1,47
T=0,68
f=1,30
f=2,03
T=0,49
f=1,54
T=0,65
f=2,90
T=0,35
f=1,54
f=3,57
T=0,28
f=3,62
Zangar
f (Hz)
T (sec)
T=0,77
f=1,41
T=0,71
f=1,33
T=0,75
f=1,45
T=0,69
T=0,65
f=1,98
T=0,50
f=2,14
T=0,47
f=2,00
T=0,50
f=2,32
T=0,43
f=2,50
T=0,40
T=0,28
f=2,29
T=0,44
f=2,48
T=0,40
f=2,68
T=0,37
f=4,27
T=0,23
f=2,46
T=0,41
f=2,91
T=0,34
f=3,11
T=0,32
f=4,48
T=0,22
f=2,53
T=0,40
f=3,12
T=0,32
f=3,37
T=0,30
4,78
T=0,21
f=2,96
T=0,34
f=3,62
T=0,28
f=3,85
T=0,26
f=5,17
T=0,19
f=3,13
T=0,32
f=3,81
T=0,26
f=4,12
T=0,24
f=5,49
T=0,18
f=3,27
T=0,31
f=3,86
T=0,26
f=4,13
T=0,24
1
2
3
4
5
6
7
8
9
10
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The next diagrams give some representative comparisons between results for the different reservoir models and analysis’ methods.
Figure 6: The vertical stresses for the different reservoir models at the downstream main section.
Figure 7: The vertical stresses for the different reservoir models at the upstream main section.
Figure 8: Comparison between the coarse and fine model for the vertical stresses.
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Figure 9: The hoop stresses for the different reservoir models at the downstream main section.
Figure 10: The hoop stresses for the different reservoir models at the upstream main section.
Figure 11: The radial deformations for the different reservoir models at the main section.
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Figure 12: Comparison between the modal dynamic analysis and the direct time integration for the coarse model with the Zangar’s approach at the downstream (left) and upstream (right) main section.
Conclusion The earthquake analysis of an arch dam-reservoir-foundation system was performed with different modeling aspects according to the formulators’ directions. The results show very near values for the two added mass approaches, with the one of Zangar to be a little bit more favorable than the one of Westergaard. The acoustic elements models with the two nonreflecting approaches give identical results. The coarse and fine models differ only in the base stresses due to the coarser mess of the coarse model and some deviations are noticed at the added mass models. Although the modal dynamic analysis is much faster than the direct time integration, delivers conservative results.
Acknowledgements This paper consists a part of the research project „Earthquake Analysis and Design of Hydraulic Structures“, which is funded by the Federal Waterways and Research Institute of Germany in cooperation with the Institute of Concrete Structures of the Karlsruhe Institute for Technology. The contribution of both participated Institutes and persons involved is highly acknowledged.
References [1] Westergaard, H. M. (1933). Water pressures on dams during earthquakes. Transactions of the American Society of Civil Engineers, American Society of Civil Engineers, New York, New York, Paper 1835, 1933. [2] Zangar, C. N. (1952). Hydrodynamic Pressures on dams due to horizontal earthquake effects. U.S. Departmant of Interior, Bureaus of Reclamation, Engineering Monographs No.11 [3] Chwang, A.T., Housner, G.W. (1978). Hydrodynamic pressures on sloping dams during earthquakes. Part 1. Momentum Method. Journal of Fluid Mechanics, vol. 2, part 2, pp. 335-341 [4] Chwang, A.T. (1978). Hydrodynamic pressures on sloping dams during earthquakes. Part 2. Exact Theory. Journal of Fluid Mechanics, vol. 2, part 2, pp. 343-348 [5] ABAQUS (2011),User’s manual, Version 6.11. Dassault Systèmes Simulia Corporation, Providence RI, USA.
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[6] US Bureau of Reclamation (2006), State-of-Practice for the NonlinearAnalysis of Concrete Dams at the Bureau of Reclamation, USBR Report, Colorado, USA. [7] United States Society on Dams (2008). Numerical models for seismic evaluation of concrete dams. Review, evaluation and interpretation of results. USSD, Denver, USA. [8] Zienkiewicz, O. C., Bettess, P. (1978). Fluid-Structure dynamic interaction and wave forces. An introduction to numerical treatment. International Journal For Numerical Methods In Engineering, Vol. 13, 1-16. [9] Wilson, E.D., Khalvati,M. (1983). Finite elements for the dynamic analysis of fluid-solid systems. International Journal For Numerical Methods In Engineering, Vol. 19, 16571668. [10] Matthew Muto, Nicolas von Gersdorff, Zee Duron, Mike Knarr (2012). Effective Modeling of Dam-Reservoir Interaction Effects Using Acoustic Finite Elements, in Proceedings of Innovative Dam and Levee Design and Construction for Sustainable Water Management, 32nd Annual USSD Conference, New Orleans, Louisiana, April 2327, 2012, Pages 1161-1168. [11] Kuo, James Shaw-Han, (1982). Fluid-structure interactions: added mass computations for incompressible fluid. UCB/EERC-82/09, Earthquake Engineering Research Center, University of California, Berkeley, 1982-08.
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HFTD Analysis of an arched dam at seismic loading W. Kikstra1, F. Sirumbal1 and G. Schreppers1 1
TNO DIANA BV, Delftechpark 19a, 2628 XJ Delft, NETHERLANDS E-mail: [email protected]
Abstract Hybrid Frequency-Time Domain (HFTD) method is applied to analyze the response of a damfoundation-reservoir system. With this method the effect of frequency dependent properties such as compressibility of fluid, reservoir-bottom absorption and far-field reflection can be considered. HFTD results are compared with transient Newmark-type results and effect of frequency dependent properties is found to reduce amplitudes and stresses in the dam for the chosen bottom absorption in the reservoir.
Introduction and analysis procedure The benchmark case study of the arch dam-reservoir interaction at seismic loading was modeled with DIANA software. The formulation used by DIANA to couple the Finite Element equations of motion for Fluid-Structure Interaction (FSI) problems is a mixed displacement – scalar potential approach, which defines the solid variables in terms of displacement degrees of freedom (DOF), and the fluid variables in terms of pressure DOF. This definition of the fluid domain using Acoustic Finite Elements is called the Eulerian pressure formulation. One of the advantages of this type of formulation is the simple description of the fluid domain using a single scalar pressure variable ( ). This reduces considerably the number of variables of the system since only one DOF per node is required to describe the motion of the fluid domain. Taking into account that for dam-reservoir interaction problems the fluid motion is not substantial but small, considerable simplifications can be made in its equation of motion formulation. Those simplifications are a consequence of the following hypotheses assumed in DIANA FSI models: Small displacement amplitudes Small velocities (convective effects are omitted) Inviscid (viscous effects are neglected) Small compressibility (variation of density is small) No body forces in the fluid
Based on these hypotheses, the scalar fluid wave equation of motion is defined by Eq.(1).The wave speed (c), defined in terms of the fluid density ( ) and bulk modulus ( ), is defined by Eq.(2). ̈
(1) (2)
√
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In the same way, the FSI condition expressed in Eq. (3) relates the fluid gradient pressure in the normal direction ( ) to the interface surface (ΓI) with the structure acceleration vector ( ̈ ). ̈
(3)
on ΓI
In addition to the fluid-structure interface (ΓI) condition defined in Eq. (3), Figure 1 shows three types of boundary conditions which can be defined in DIANA FSI models.
Figure 1: Fluid and solid domains, fluid-structure interface and boundary conditions It is possible to specify two types of boundary conditions for the free surface of the reservoir (Γs). The first and simplest one is a consequence of neglecting the effect of the surface waves, prescribing a pressure equal to zero in the horizontal top free surface, as expressed in Eq. (4). This essential or Dirichlet type of boundary condition is the one used in the benchmark case study. Additionally, in DIANA it is possible to define a second type of boundary condition which takes into account the pressure caused by the free surface gravity waves. (4)
on Γs
Two types of infinite extent boundary condition (Γe) are considered in the benchmark case study. The first one defined by Eq. (5.a) prescribes the hydrodynamic pressure equal to zero at a distance equal to the reservoir length. The second one defined by Eq. (5.b) is a radiation boundary theoretically located at an infinite distance from the dam, which ensures that no incoming waves enter into the system (only outgoing waves). on Γe
(5.a)
̇ on Γe
(5.b)
Finally, two types of bottom boundary condition (Γb) are considered in the benchmark case study. The first one is setting the gradient of the pressure in the normal direction equal to zero, as expressed in Eq. (6.a). The second one defined by Eq. (6.b) is radiation boundary which takes into account the energy absorption of the bottom materials in terms of the wave 78
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reflection coefficient ( ), defined as the ratio between the amplitudes of the incident pressure wave over the reflective pressure wave. on Γb
(6.a) ̇ on Γb
(6.b)
In this paper, the Hybrid Frequency-Time Domain method is applied to solve the fluid equation of motion in the frequency domain. In this way, the hydrodynamic pressure amplitude vector is obtained in terms of the structural displacement amplitude vector. On the other hand, the structure non-linear equation of motion defined in the time domain is formulated based on relative displacement. In this way, the earthquake ground acceleration vector is introduced as external loading. The non-linear internal force of the structure is defined as the difference between the linear internal force and an unknown pseudo force vector, which is substituted into the non-linear equation to obtain the HFTD pseudo-linear equation of motion of the dam in the time-domain. Nevertheless, the HFTD method solves the pseudo-linear equation of motion in the frequency domain, and therefore it is required to use the Discrete Fourier Transform (DFT) to pass from one domain to the other. The fluid contribution is defined by additional mass and additional damping which both are defined in the frequency domain, and thus can be defined as frequency dependent properties. As a consequence the HFTD method can account for the effects of non-linear material behavior, compressibility of fluid, radiation at infinite extend and reservoir-bottom absorption, whereas standard transient analysis with Newmark-type of time-integration schemes cannot consider these effects together.
Model definition Because in the mesh that has been provided by the benchmark formulators some node connection incompatibilities were found, a new mesh has been defined using the provided geometry specifications as shown in Figure 2.
Figure 2: Components and geometrical characteristics of the foundation-dam-reservoir interaction system
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Element with second order displacement interpolation were used for foundation and dam and element with linear fluid-pressure interpolation were used for the reservoir. Mesh characterstics are listed in Table 1 and material characteristics are listed in the Tables 2-4. Component
Table 1: Characteristics of the Finite Element Model Mesh Type
Dam
Solid 3D
Foundation
Solid 3D
Reservoir
Flow 3D
Parameter Modulus of elasticity
DIANA element name CHX60 CTP45 CHX60 CTP45 CTP15H CHX20H
Number of Elements
Number of Nodes
712
3601
4896
23339
2670
11950
Table 2: Material parameters for the concrete dam
Poison modulus
DIANA variable name YOUNG
Value / Type 2.7 x 1010
Units N / m2
POISON
1.67 x 10-1
-
3
Density
DENSIT
2.4 x 10
Rayleigh damping
RAYLEI
5.71199 x 10-1 1.447 x 10-3
Parameter Conductivity
Table 3: Material parameters for the reservoir fluid
kg / m3 -
DIANA variable name CONDUC
Value / Type 1
Units -
Sonic speed
CSOUND
1.483 x 103
m/s
Density
DENSIT
1.0 x 103
kg / m3
Wave reflection coefficient for infinite extent boundary Wave reflection coefficient for bottom absorption boundary
ALPHAB
0
-
ALPHAB
5.0 x 10-1
-
Table 4: Material parameters for the foundation soil Parameter Modulus of elasticity Poison modulus Density
DIANA variable name YOUNG POISON
Value / Type 2.5 x 1010 2.0 x 10-1
Units N / m2 -
DENSIT
0
kg / m3
The transient responses of the two analysis cases shown in Figure 3 are determined and studied. Case I corresponds to the frequency independent system, for which the fluid of the reservoir is assumed to be incompressible ( ) and the infinite extent and bottom boundary conditions of the reservoir are defined by Eqs. (5.a) and (6.a), respectively. This frequency independent analysis case is solved with both HFTD and Newmark methods, with the objective of assessing the accuracy of HFTD method. On the other hand, Case II corresponds to a frequency dependent system in which fluid compressibility and reservoir radiation boundary conditions are included. Reservoir bottom absorption ( ) and radiation boundary of infinite extent, defined by Eqs. (6.b) and (5.b), respectively, are 80
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included in the analysis. As it was previously explained, the transient analysis of this frequency dependent system cannot be solved by the standard Newmark method, only by the HFTD method. Non-linear material behavior of the dam-structure is not considered in these cases.
Figure 3: Foundation-dam-reservoir interaction analysis cases
Results Prior to the time domain analysis, eigenvalues of the dam-foundation system including interaction with the fluid reservoir were determined. The first 10 mode-shapes and corresponding eigenvalues are presented in Figure 4. Case I was solved using both Newmark time integration and HFTD analysis. Figure 5 shows the amplitude of the displacement of the crest at the main section. Agreement is so close that the vertical stress, hoop stress and radial displacement results for Case I will only be displayed for the Newmark time stepping analysis. The envelopes for hoop stresses, vertical stresses and radial displacements along with their static values are given in figures 6 to 8, respectively. On the other hand, the frequency dependent properties of the reservoir and reservoir boundaries of Case II introduce more damping in the system which should lead to lower responses to the earthquake loading. Figure 9 shows that the crest amplitude has considerably lower peaks compared to Case I. The same magnitude reduction is observed in the envelopes of hoop stresses, vertical stresses and radial displacements given in Figures 10 to 12, respectively. A 2D study to discriminate the effects of fluid compressibility, radiation boundary and bottom absorption shows that reservoir fluid compressibility increases the response, that there is minimal influence of the radiation boundary if sufficient reservoir length is modeled and that bottom absorption damps the response of the dam-reservoir system [3]. For all the analysis results presented (Eigen-analysis, Newmark and HFTD), the effect of selfweight gravity and hydrostatic pressure loads were taking into account as initial conditions.
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Figure 4: Mode-shapes and eigenvalues of the dam-foundation-reservoir interaction system
Figure 5: HFTD vs Newmark. Crest amplitude [m] at main section 82
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Figure 6: Vertical stress [Pa] against elevation [m] for left, main and right dam sections at upstream and downstream dam face for incompressible reservoir.
Figure 7: Hoop stress [Pa] against elevation [m] for left, main and right dam sections at upstream and downstream dam face for incompressible reservoir.
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Figure 8: Radial displacement [m] against elevation [m] for left, main and right dam sections at upstream and downstream dam face for incompressible reservoir.
Figure 9: Compressible vs Incompressible. Crest amplitude [m] at main section
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Figure 10: Vertical stress [Pa] against elevation [m] for left, main and right dam sections at upstream and downstream dam face for compressible reservoir.
Figure 11: Hoop stress [Pa] against elevation [m] for left, main and right dam sections at upstream and downstream dam face for incompressible reservoir.
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Figure 12: Radial displacement [m] against elevation [m] for left, main and right dam sections at upstream and downstream dam face for compressible reservoir.
Conclusion The given foundation-dam-reservoir system was analyzed for the full duration of the earthquake. With the HFTD method implemented in the standard version of DIANA for the case of frequency independent properties the same results could be reproduced as with implicit time stepping with Newmark’s method. With HFTD the effect of frequency dependent properties such as compressibility of fluid, reservoir bottom absorption and infinite extend reflection have been analyzed and quantified, resulting in an interesting method to be applied specially in the dynamic analysis of dam-reservoir interaction models.
References [1] Veletsos, A.S., and Ventura, C.E. (1985). Dynamic analysis of structures by the DFT method. J. Struct. Eng. ASCE, Vol.111, 2625-2642. [2] TNO DIANA BV (2011). DIANA User’s Manual – Release 9.4.4, Delft, The Netherlands. [3] Sirumbal, F. (2013) Numerical modeling of dam-reservoir interaction seismic response using the Hybrid Frequency-Time Domain ( HFTD ) method Masters thesis, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands. [4] Darbre, G.R. (1996). Nonlinear dam-reservoir interaction analysis. Proc. Eleven World Conf. on Earthquake Engineering, Acapulco, Paper No. 760. [5] Zienkiewicz, O.C., and Bettes, P. (1978). Fluid-structure dynamic interaction and wave forces. An introduction to numerical treatment. Int J Numer Meth Eng, Vol. 13, 1-16.
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Finite Element Modelling of Seismic Fluid-Structure Interaction for a large Arch Dam G. Faggiani1 and P. Masarati1 1
Ricerca sul Sistema Energetico - RSE SpA, via R. Rubattino 54, 20134 Milano, ITALY E-mail: [email protected]
Abstract The linear dynamic fluid-structure interaction at seismic loading for the artificially generated large arch dam provided for the Theme A was modelled using the approach of acoustic compressible elements, with both the coarse and fine meshes provided by the formulators of the 12th ICOLD Benchmark Workshop. The effects of incompressible fluid (theoretical hypothesis of the added mass models) and of the partial absorption of the hydrodynamic pressure waves at the reservoir boundary (bottom and sides) were also investigated. Simulations were carried out using the RSE in-house FEM code CANT-SD, specifically designed for dynamic linear and non-linear analyses of dam-reservoir systems. The coarse and fine meshes showed not dissimilar results, except at dam-foundation interface. The results of the analyses confirmed that incompressible models could result relatively conservative and highlighted the benefits of the approach of acoustic elements, mainly the possibility to take into account the damping effect on the fluid boundary.
Introduction Seismic safety assessment of large arch dams is actually a very important matter. Both dam-reservoir interaction and non-linear mechanisms due to the contraction joint opening and sliding could greatly affect the mechanical behaviour of the dam: therefore they must be properly considered in order to obtain reliable numerical simulations under strong earthquakes. Theme A of the 12th ICOLD Benchmark Workshop on Numerical Analysis of Dams [1] is aimed at comparing the different available approaches to model the dynamic fluid-structure interaction: to this purpose the structural response to a seismic loading of an artificially generated large arch dam has to be investigated. The analyses have been carried out using the RSE in-house FEM code CANT-SD [2], specifically designed for linear and non-linear dynamic (seismic) analyses of dam-reservoir systems. This code is currently used at RSE for safety assessment of concrete dams, and it was adopted to deal with some themes proposed in previous ICOLD Benchmark Workshops [3] [4] [5] [6] [7] [8]. Regarding the main simulation options required to effectively address the present Theme A, CANT-SD models the fluid reservoir by means of acoustic elements and solves the transient dynamic coupled problem using an implicit direct time integration method.
Geometrical and physical model The two FEM parabolic meshes, Coarse and Fine, considered in the simulations are reported in Figure 1 and 2. Only the dam meshes provided in the Theme A have been adopted unchanged. The fluid domain was obtained by extruding the upstream face of the dam mesh for a length 3 times the total height of the dam (220 m): the resulting mesh was only joined to 87
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the upstream dam face, not to the rock. Foundation meshes were modified in order to obtain coincident meshes at concrete-rock interface as necessary for the proper operation of CANTSD.
Figure 1: Arch dam FEM model – Coarse mesh
Figure 2: Arch dam FEM model – Fine mesh A monolithic behaviour of the dam body has been considered, as no construction joints have been modelled. Dam concrete and foundation rock, assumed to behave linear-elastically, were characterized by the physical-mechanical parameters provided by the formulators and reported in Table 1 along with the properties of the fluid. A 5% structural damping ratio (ξ) was assumed in the analyses: damping matrix is expressed as linear combination of mass and stiffness matrices according to Rayleigh formulation. The calibration of structural damping ratio was based on the frequency range resulting from modal analysis, such that the damping was almost constant in this range (Figure 3). Table 1: Material properties Parameter Modulus of elasticity (MPa) Bulk modulus (MPa) Poisson’s ratio (-) Density (kg/m3)
Rock mass 25000 0.2 0
88
Water 2200 1000
Dam concrete 27000 0.167 2400
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0.150
ξstiffness ξmass ξRayleigh
Damping ratio [-]
0.125 0.100 0.075
0.050 0.025 0.000
0
1
2
3
4
5
6
7
8
9 10 11
Frequency [Hz]
Figure 3: Rayleigh viscous damping The dynamic fluid-structure interaction, modelled in CANT-SD following the classic acoustic approach [9], is here briefly summarized. The hydrodynamic pressure in the compressible fluid of the reservoir is governed by the wave equation: 2 p 2 p 2 p 1 2 p x 2 y 2 z 2 C 2 t 2
(1)
where C = √(k/) is the velocity of sound in the fluid, k the bulk modulus and the density. The boundary conditions are (Figure 4): free surface p0
upstream face of the dam
open boundary
̈ ̇
bottom and sides ̇ where n is the outward normal, ̈ the normal acceleration and q the damping coefficient on the bottom and sides. The upstream face of the dam results the only surface of interaction between structure and fluid: the accelerations at the dam face represent the “actions” of the dam on the reservoir, which in turn “reacts” through the hydrodynamic pressures exerted on the dam face. The last boundary condition accounts for the partial absorption of hydrodynamic pressure waves [10] [11]: this damping phenomenon is mainly caused by the layer of sedimentary material possibly deposited in the reservoir, but could be significant also in cases of few or no accumulated sediments.
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Upstream face of the dam Free surface
Bottom and sides Open boundary
Figure 4: Boundaries of reservoir Finite element subdivision of the reservoir leads to the discretized form of the outlined acoustic problem, i.e. a system to be coupled with the structural one (that includes the hydrodynamic pressure loads). The fluid-structure coupled problem is governed by the following system: {
̈ ̈
̇ ̇
(2) ̈
The first subsystem governs the mechanical behaviour of the dam and the (unknown) pressure p represents an applied load; the second governs the acoustic behaviour of the reservoir and the (unknown) acceleration ü represents an assigned boundary condition. The seismic analysis proposed in Theme A was performed for three physical models of the reservoir behaviour, differing in fluid compressibility and/or boundary absorption. The first physical model (Base Case) fully respects the requirements of the formulators: the provided value of the bulk modulus of water (2200 MPa) was adopted, boundary absorption was neglected (q=0 on bottom and sides, i.e. reflecting condition) and the non-reflecting condition was considered at the end of the reservoir. The other physical models have the aim to examine, for the particular given scenario, the following interesting aspects: the effect of the incompressibility hypothesis and the importance of the boundary absorption. A brief hint of both aspects follows, along with the necessary choices for the two additional simulations. The assumption of fluid incompressibility is connected with the “added mass models”, which assimilate the action of the fluid to that of “some kind of mass” (physical mass or a mass matrix) attached to the upstream face of the dam. It is worth noting that the comparison reported in this paper just concerns the incompressibility hypothesis, and can be therefore only applied to the rigorous “added mass matrix” model, clearly defined in [9] and referred to as “Finite Element Added Hydrodynamic Mass Model” in [12]. No investigation was made about the effects of any possible change on this matrix (i.e. for computational convenience), nor any comparison was made with Westergaard-type techniques, however strongly discouraged in [12] for arch dams. 90
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A key simplified parameter [13] that determines the significance of water compressibility is the ratio Ωr=f1r/f1 of the fundamental frequency of the reservoir to that of the dam-foundation system without water: the higher this ratio, the less the importance of fluid compressibility. Although the limit Ωr=2 is valid only for gravity and not for arch dams, as clearly stated in [13], it has often been used, at least as a broad clue, for arch dams too [12]. The fundamental frequency f1r of the reservoir can be roughly evaluated as C/4H, where C is the velocity of sound in water and H is the water depth: for the analysed system the value is f1r=1.7 Hz. The fundamental frequencies of the dam-foundation system without water were calculated and resulted 1.9 Hz for anti-symmetric mode shape and 2.0 Hz for symmetric mode shape. The value of the ratio Ωr, about 0.9, indicates that compressibility should probably be important. The physical model to investigate the effects of compressibility (Incompressible Case) was developed using a value of the bulk modulus one hundred times greater than that of the Base Case (k=2200×100 MPa) and assuming the reflecting boundary condition at the end of the reservoir. Two modal analyses were carried out, the first with k=2200×100 MPa and the second with k=2200×10000 MPa to verify that the value of the bulk modulus assumed in dynamic analysis was great enough to simulate incompressibility: no significant differences were found in the results. The coupled mechanical-acoustic approach allows taking into account the effect of the partial absorption of hydrodynamic pressure waves on the boundary of the reservoir [10] [11], impossible to simulate using any added mass model. To assign the boundary absorption, the damping coefficient q must be quantified: to this purpose it is convenient to express it by means of the wave reflection coefficient α (the ratio of the amplitude of the reflected hydrodynamic pressure wave to the amplitude of the incident one) [11]: 1 (1 ) q (3) C (1 ) The wave reflection coefficient α represents a more physically meaningful description of the phenomenon: α=1 corresponds to a reflecting (rigid) boundary, α=0 corresponds to a nonreflecting (transmitting) boundary, -1<α<0 corresponds to an even major damping behaviour. The value of α can be determined on the basis of field investigations [12] [14] [15]. The results reported in [14] indicate values of α, measured at seven concrete dam sites, varying over a range from -0.55 to 0.66: three of these values were negative, due to thick layers of soft sediments. A value of 0.82 was determined at the dam site investigated in [15]: a rock site with very little or no accumulated sediments. Making reference to the reported results of field investigation, a value α=0.5 was considered sensible to examine the importance of the boundary absorption in the seismic analysis: the corresponding physical model (Damped Case) was obtained from the Base Case using a damping coefficient q=0.000225. Table 2 summarizes all the simulations: Base Case (B) was performed both with the coarse and fine meshes, while Incompressible (I) and Damped (D) Cases only with the coarse mesh. Table 2: Summary of simulations Case Base Incompressible Damped
Coarse mesh B I D
91
Fine Mesh B -
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Loadings The numerical analyses simulated the effects of the following loadings/actions: 1. dead weight 2. hydrostatic pressure with water level at crest height 3. seismic loading The seismic loading was provided in the Theme by means of artificially generated acceleration time histories in the three coordinate directions X (upstream), Y (cross-stream) and Z (vertical): the elastic response spectra (5% damping ratio) are reported in Figure 5. It’s worth noting that the given time histories represent a quite moderate intensity earthquake. The acceleration time histories were only assigned to the bottom and sides of the foundation, not to the boundary of the reservoir too. The transient dynamic coupled problem was solved using an implicit direct time integration method (HHT) [16]; an integration step of 0.002 s was chosen, in order to well represent frequencies up to 25 Hz. 0.35
upstream
0.3
cross-stream vertical
Acceleration [g]
0.25 0.2
0.15 0.1 0.05
0
0
0.5
1
1.5 2 2.5 Period [s]
3
3.5
4
Figure 5: Elastic response spectra
Results The following sections report and discuss the results of Base Case with both coarse and fine meshes and of Incompressible and Damped Cases. The stress state is represented either by diagrams showing vertical and hoop static and dynamic (envelope) stresses, or by contour plots showing principal stresses envelopes, expressed in megapascal, positive if tensile. The displacements of the dam, expressed in metres, are positive if directed downstream. Eigenfrequencies and mode shapes The modal analysis allowed the computing of the natural resonant frequencies of the damreservoir system and the corresponding mode shapes. Table 3 reports the first four eigenfrequencies for both the coarse and the fine mesh. The values obtained with the two meshes were almost identical: as expected, the refined discretization exhibited a slightly more flexible behaviour than the coarse one. These frequencies corresponded to the first anti92
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symmetric and the first three symmetric mode shapes, shown in Figure 6 and 7 for the coarse and fine mesh respectively. The first four eigenfrequencies occurred in a range of periods of about 0.45÷0.65 s, matching the response spectra in their descending branch. Table 3: Natural frequencies Mode 1 2 3 4
Eigenfrequencies [Hz] Coarse Mesh Fine Mesh 1.547 1.540 1.551 1.549 2.052 2.050 2.229 2.222
Figure 6: Mode Shapes (Base Case, Coarse Mesh)
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Figure 7: Mode Shapes (Base Case, Fine Mesh) Seismic simulation with Coarse and Fine Meshes The transient dynamic analysis supplied the stress-strain state of the system due to the application of the seismic loading. The simulations developed for the Base Case with Coarse (CB) and Fine (FB) meshes showed how the use of a more refined mesh does not lead to significant differences in the results, though involving quite a higher computational effort (about 5 times). Vertical and hoop stresses in dam main section are reported in Figure 8 and 9 for upstream and downstream faces respectively, both for fine and coarse meshes: stresses were evaluated for static and seismic loads. Vertical stresses resulted always compressive upstream, except at dam foundation interface, while downstream tensile vertical stresses, up to 1 MPa, occurred in the upper part of the dam, above 610 m a.s.l., caused by the seismic loading. Due to the seismic loading, a variation of vertical stress of about 1÷1.5 MPa was observed both upstream and downstream. Figure 8 highlights that vertical stresses calculated with the coarse mesh are underestimated near the dam-foundation interface, where tensile stress concentrations are usual in many arch dams. Hoop stresses both on upstream and downstream faces were compressive, confirming that the arch effect was activated and the dam behaved according to its monolithic scheme. The effect of the seismic loading, greater in the upper part of the dam, involved a stress variation of about 4 MPa upstream (at 675 m a.s.l.) and 2 MPa downstream (at about 695 m a.s.l.). Figure 10 and Figure 11 report the tensile stress envelope and the dynamic maximum displacement contour plots for the two analysed meshes. Tensile stress on the downstream face was essentially vertically oriented: due to the seismic loading the upper central part of the dam exhibited stresses up to 1.6 MPa. Figure 11 shows a downstream displacement of about
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715 C Static 695 F Static 675 CB Max FB Max 655 CB Min 635 FB Min 615 595 575 555 535 515 495 6.00 4.00 2.00 0.00 -2.00 -4.00 -6.00 -8.00 Vertical Stresses [MPa]
Height [m a.s.l.]
Height [m a.s.l.]
8 cm, due to the seismic loading. A similar upstream displacement was observed for the minimum envelope. The contour plots allowed the overall comparison between simulations with coarse and fine meshes and confirmed that the spatial trend of the stress-strain state was comparable in the whole dam body. 715 695 675 655 635 615 595 C Static 575 F Static 555 CB Max FB Max 535 CB Min 515 FB Min 495 1.00 -1.00 -3.00 -5.00 -7.00 -9.00 -11.00 Hoop Stresses [MPa]
715 695 675 655 635 615 595 C Static 575 F Static 555 CB Max FB Max 535 CB Min 515 FB Min 495 6.00 4.00 2.00 0.00 -2.00 -4.00 -6.00 -8.00 Vertical Stresses [MPa]
Height [m a.s.l.]
Height [m a.s.l.]
Figure 8: Base Case - Vertical (left) and hoop (right) stresses on the upstream surface 715 695 675 655 635 615 595 C Static 575 F Static 555 CB Max FB Max 535 CB Min 515 FB Min 495 1.00 -1.00 -3.00 -5.00 -7.00 -9.00 -11.00 Hoop Stresses [MPa]
Figure 9: Base Case - Vertical (left) and hoop (right) stresses on the downstream surface
Figure 10: Base Case - Maximum principal stress – Coarse (left) and Fine (right) meshes, downstream view. 95
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Figure 11: Base Case - Maximum dynamic displacement – Coarse (left) and Fine (right) meshes, upstream view
715 695 675 655 635 615 595 575 555 535 515 495 4.00
Static B Max B Min I Max I Min D Max D Min
2.00 0.00 -2.00 -4.00 Vertical Stresses [MPa]
Height [m a.s.l.]
Height [m a.s.l.]
Incompressible fluid and absorption effects Based on the comparison discussed in the previous section, the effects of incompressible fluid (Incompressible Case - I) and of the reservoir boundary absorption (Damped Case – D) on the dynamic response of the dam were investigated only with the coarse mesh. The results of the analyses are summarized and compared with those of the Base Case (B) in Figure 12 and 13, where vertical and hoop stresses in dam main section are reported for upstream and downstream face respectively. The general trend of stresses for the Base Case was already illustrated in the previous section. The curves representing stresses for Incompressible and Damped Case generally laid respectively outside and within those of the Base Case, confirming that the incompressible model could result relatively conservative and that the reservoir boundary absorption could reduce the earthquake response of the dam. In the analysed situation, relevant to an earthquake of very moderate severity, the difference among the three models could look worthless. However these differences could become highly significant if a greater seismic loading were considered. Referring to the hoop stresses, a 2 times amplified earthquake would result in the occurrence of tensile stresses in the arcs (Figure 14) involving the transition from a monolithic to an independent cantilever separated by vertical joints behaviour, if these stresses act in the dam for a significant height (starting from the crest). Figure 14 illustrates that for incompressible model tensile stresses resulted about 3÷5 times higher than for the damped model, involving the upper 80 m of the dam, a double height if compared with damped model.
-6.00
715 695 675 655 635 615 595 575 Static B Max 555 B Min I Max 535 I Min 515 D Max D Min 495 1.00 -1.00 -3.00 -5.00 -7.00 -9.00 -11.00 Hoop Stresses [MPa]
Figure 12: Comparison among Base, Incompressible and Damped Case - Vertical (left) and hoop (right) stresses on the upstream surface 96
715 695 675 655 635 615 595 575 555 535 515 495 4.00
Height [m a.s.l.]
Height [m a.s.l.]
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Static B Max B Min I Max I Min D Max D Min
2.00 0.00 -2.00 -4.00 Vertical Stresses [MPa]
-6.00
715 695 675 655 635 615 595 Static 575 B Max B Min 555 I Max 535 I Min D Max 515 D Min 495 1.00 -1.00 -3.00 -5.00 -7.00 -9.00 -11.00 Hoop Stresses [MPa]
715 695 675 655 635 615 595 B Max 575 I Max 555 D Max B-ampli2 535 I-ampli2 D-ampli2 515 Static 495 6.00 4.00 2.00 0.00 -2.00 -4.00 -6.00 -8.00 Hoop Stresses [MPa]
Height [m a.s.l.]
Height [m a.s.l.]
Figure 13: Comparison among Base, Incompressible and Damped Case - Vertical (left) and hoop (right) stresses on the downstream surface 715 695 675 655 635 615 595 B Max 575 I Max D Max 555 B-ampli2 535 I-ampli2 D-ampli 515 Static 495 8.00 6.00 4.00 2.00 0.00 -2.00 -4.00 -6.00 -8.00 Hoop Stresses [MPa]
Figure 14: Comparison among Base, Incompressible and Damped Cases for an amplified earthquake- Hoop stresses on upstream (left) and downstream (right) surface
Conclusion Theme A of the 12th ICOLD Benchmark Workshop on Numerical Analysis of Dams, dealing with the linear dynamic fluid-structure interaction at seismic loading, has been approached by using CANT-SD, a RSE in-house FEM code for dynamic linear and non-linear analyses of dam-reservoir systems. The dynamic fluid-structure interaction was modelled with the approach of compressible acoustic elements. The analyses with the reflecting boundary condition on the bottom and sides of the fluid domain, performed with both coarse and fine meshes to test the effects of different spatial discretization, showed that the use of a more refined mesh does not lead to significant differences in the results, though involving quite a higher computational effort. Simulations considering incompressible acoustic elements or reservoir boundary absorption were also performed: the results of these analyses confirmed that the incompressible model could result relatively conservative and that reservoir boundary absorption could significantly reduce the earthquake response of the dam. The use of an incompressible model instead of a compressible one, capable to account for damping effect on the fluid boundary too, speeded up the transition from a monolithic structural scheme to a different one with independent cantilevers and joints. 97
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Acknowledgements This work has been financed by the Research Fund for the Italian Electrical System under the Contract Agreement between RSE S.p.A. and the Ministry of Economic Development General Directorate for Nuclear Energy, Renewable Energy and Energy Efficiency in compliance with the Decree of March 8, 2006.
References [1] Zenz G., Goldgruber M. (2013). Theme A formulation. Fluid Structure Interaction. Arch Dam – Reservoir at Seismic loading. 12th ICOLD Benchmark Workshop on Numerical Analysis of Dams, Graz, Austria. [2] Masarati P., Meghella M. (2000). The FEM computer code CANT-SD for non-linear static and dynamic analysis of dams. Enel.Hydro rep. n. 6045, Milano, Italy. [3] Faggiani G., Frigerio A., Masarati P., Meghella M.(2011). Finite element modelling of concrete swelling effects on Kariba dam. 11th ICOLD Benchmark Workshop on Numerical Analysis of Dams, Valencia, Spain. [4] Meghella M., Masarati P. (2007). FEM analyses for the interpretation of the structural behaviour of La Aceña dam. 9th ICOLD Benchmark Workshop on Numerical Analysis of Dams, St. Petersburg, Russia. [5] Meghella M., Frigerio A., Masarati P. (2005). Evaluation of AAR on the behaviour of Poglia dam adopting two different approaches. 8th ICOLD Benchmark Workshop on Numerical Analysis of Dams, Wuhan, Hubei, P.R. China. [6] Meghella M., Mazzà, G. (2003). Safety evaluation against sliding of a gravity dam with curved shape. 7th ICOLD Benchmark Workshop on Numerical Analysis of Dams, Bucharest, Romania. [7] Bon E., Chillè F., Masarati P., Massaro C. (2001). Analysis of the effects induced by alkali-aggregate reaction (AAR) on the structural behaviour of Pian Telessio dam. 6th ICOLD Benchmark Workshop on Numerical Analysis of Dams, Salzburg, Austria. [8] Bolognini L., Masarati P., Bettinali F. Galimberti C. (1994). Non-linear analysis of joint behaviour under thermal and hydrostatic loads for an arch dam. 3rd ICOLD Benchmark Workshop on Numerical Analysis of Dams, Paris, France. [9] Zienkiewicz O. C. (1977). The Finite Element Method, 3rd Edition, McGraw-Hill. [10] Fok K.L., Chopra A.K. (1986). Earthquake analysis of arch dams including dam-water interaction, reservoir boundary absorption and foundation flexibility. Earthquake Engineering and Structural Dynamics, Vol. 14, pp. 155-184. [11] Fenves G., Chopra A.K. (1983). Effects of reservoir bottom absorption on earthquake response of concrete gravity dams. Earthquake Engineering and Structural Dynamics, Vol. 11, pp. 809-829. [12] US Army Corps of Engineers – USACE (2003). Engineering and Design - Time-History Dynamic Analysis of Concrete Hydraulic Structures. EM 1110-2-6051. [13] Fok K.L., Chopra A.K. (1986). Frequency response functions for arch dams: hydrodynamic and foundation flexibility effects. Earthquake Engineering and Structural Dynamics, Vol. 14, pp. 769-795. [14] Ghanaat Y., Redpath B.B. (1995). Measurement of reservoir-bottom reflection coefficient at seven concrete dam sites. QUEST Structure Report No. QS95-01. [15] Ghanaat Y., Hall R.L., Redpath B.B. (2000). Measurement of dynamic response of arch dams including interaction effects. Proc. of 12WCEE2000. [16] Hilber H.M., Hughes T.J.R., Taylor R.L. (1977). Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Engineering and Structural Dynamics, Vol. 5, pp. 283-292. 98
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ICOLD 12-th International Benchmark Workshop on Numerical Analysis of Dams Theme A: Fluid Structure Interaction Arch Dam ‐ Reservoir at Seismic Loading A. Tzenkov1, A. Abati1 and G. Gatto1 1
STUCKY SA, Rue du Lac 33, PO Box, 1020 Renens, SWITZERLAND E-mail: [email protected]
Abstract One of the main concerns regarding the numerical dynamic analysis of arch dams is the proper modelling of the fluid-structure interaction between the dam and the impounded water. There are several approaches to this, which enables accounting for the hydrodynamic pressures on the upstream face of the dam with different precision and, respectively, with different computing effort. This work investigates the impact of the hydrodynamic approach opted for on the computed stresses and displacements of an example 220-m high doublecurvature arch dam. It is shown that, for this particular benchmark problem, it is important to consider the compressibility of water.
Introduction The hydrodynamic phenomena occurring on the interface between a dam and the impounded water may have significant effect on the structural response of the dam. The structure (the dam wall and its foundation) and the fluid (the impounded water) are two different physical systems that interact with each other and thus present a coupled problem. According to the classification given in [1], the latter is a Class I coupled problem in which the coupling occurs on the interfaces between the domains. A milestone procedure to account for the hydrodynamic effects on dams was established by Westergaard in 1933, [2] by introducing the concept of added mass. Although Westergaard’s approach was limited within the assumptions of rigid dams with vertical upstream faces, infinitely long reservoirs, and incompressible fluids, it enabled accounting for the hydrodynamic effects in the everyday engineering practice. Using the electric analog method, Zangar [3] improved Westergaard’s approach by establishing a family of parabolas by means of which it is possible to compute the hydrodynamic pressure on rigid dams with sloping upstream face. With the advent of the computer and the increased utilization of the numerical methods, it became possible to account for the effects due to (1) the fluid compressibility, (2) the hydrodynamic pressure waves’ partial absorption by deposited sediments, (3) the foundation inertia and damping, as well as for non-linear dam behaviour (4). Frequency domain [4][5], time domain [6], and hybrid frequency-time domain (HFTD) [7] procedures have been developed and used over the past decades. An inherent limitation of the frequency domain approach is that it presupposes linear structural behaviour; on the other hand, it enables readily considering effects (1), (2) and (3). In contrast, in the time-domain, nonlinear structural behaviour can be accounted with reasonable computational effort, but it is more difficult to simulate the other hydrodynamic phenomena. This work investigates the differences of the stresses and the displacements computed by 99
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means of the FEM in using three different approaches for modelling the hydrodynamic effects. The analyses are performed on the current Benchmark example 220 m high doublecurvature arch dam by means of the computer program DIANA [8]. First, two linear seismic analysis are carried out in the time domain by using added masses computed by the Westergaard formula and by means of a fluid-structure interaction analysis in the assumption of incompressible fluid. Next, a linear analysis is performed by means of the HFTD approach and in modelling the fluid’s compressibility. In addition, a nonlinear seismic analysis is performed in the time-domain, by means of a fluid-structure interaction analysis for incompressible fluid. Details of the procedures utilized, the results of the computations, and a discussion on the results are presented in the following sections.
Finite Element Model The finite element model of the example arch dam – foundation – reservoir system is created based on the geometry and the spatial discretization given by the Formulator of the current Theme A [9]. The only difference is that the reservoir model’s length is 5 times the dam height. Following the conditions of [9], the investigations have been done on two meshes: a coarse one, and a finer one. The structural system of the coarse mesh is modelled by 2516 hexahedral and wedge isoparametric solid finite elements (356 for the dam and 2160 for the foundation). The reservoir is modelled by 1956 3-D flow elements, while the dam-reservoir interface is represented by 177 fluid-structure interface elements. All the elements are based on quadratic interpolation. The total number of nodes of the coarse mesh is 21547. The fine mesh has 78426 nodes; the dam and the foundation are modelled by 2848 and 10080 elements respectively; however, it yields almost the same result regarding the structural response as the coarse mesh. Translational supports in the three global directions are specified as structural boundary conditions on the bottom and side surfaces of the foundation model. The material parameters of the dam and the foundation are the same as the ones prescribed in [9]. The sonic wave velocity =1483 m/s is associated with the reservoir fluid elements. Finally, water density of 1000 kg/m3 is specified for the fluid-structure interface elements. Westergaard Added Mass (WG) The added masses computed by means of the Westergaard formula [2] are applied on the nodes of the upstream face of the dam by means of CQ24TM boundary surface elements [8]. 178 such elements are defined for the coarse mesh; they are 712 for the finer model of the system. A distributed translational mass material model is associated to the boundary surface elements, which allows precise automatic calculation of the added masses. The total mass assembled for the coarse mesh finite element model is TM=0.14E+11 kg (for massless rock). Without added mass, TM=0.471E+10 kg. Fluid - Structure Interaction with Incompressible Fluid (FSI) As already mentioned, the solution is conducted in the time domain. To render the system frequency independent, the fluid is specified as incompressible by setting tending to infinity, assigning hydrodynamic pressure at the far-field and at the free surface of the reservoir, and setting the hydrodynamic pressure gradient equal to zero in the normal direction of the reservoir bottom. In this case, the hydrodynamic effect is represented by a consistent mass matrix that is added to the mass matrix of structural system. Fluid Structure Interaction with Compressible Fluid (HFTD) In this case, the system is frequency dependent. Sonic wave velocity c=1483 m/s is associated with the reservoir fluid elements and radiation boundary condition is specified for the fluid far-field, Equation (1): 100
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(1) ̇
Fully reflecting boundary is assumed for the reservoir bottom, Equation (2): (2) The hydrodynamic pressure is set
at the free surface of the reservoir.
Analysis Performed The load-cases specified in [9] are the self-weight of the dam, the hydrostatic pressure and the seismic loading. The present study is performed by means of phased analysis, which allows modelling the loading history. Thus, first are computed the stresses due to the self-weight of the dam. The hydrostatic pressure is applied next. Finally, the seismic loading is applied in the three directions of the supports using the base acceleration time-histories given in [9] and multiplied by a factor of 0.1 , where is the gravity acceleration. Static Analysis The dam construction stages are modelled in an approximate, but realistic way by means of phased activation of the dam elements. It is done in order to obtain correct strain and stress state due to the self-weight of the dam body prior to the application of the hydrostatic loading. In the linear analyses, the hydrostatic pressure is activated in a single phase as an instantaneous loading, whereas in the nonlinear analysis it is applied at ten steps corresponding to ten consecutive levels of filling of the dam reservoir. Eigenvalue Analysis The results of the computed eigenfrequencies for the cases of empty and full reservoir with added masses defined by the Westergaard formula (coarse and fine mesh) and by a FEM incompressible fluid-structure interaction analysis are presented in Table 1. Note that the general coordinate system axes are as follows: X-axis is the stream direction (from u/s to d/s), Y-axis is the cross-stream direction (from right to left), and Z-axis is the vertical direction. Table 1: Eigenrequencies and Effective Mass Percentage (Empty, WG, WG fine mesh, FSI) Mode 1 2 3 4 5 6 7 8 9 10
Empty Reservoir , Hz X, % Y, % 1.931 0.0 18.4 2.040 30.6 0.0 2.929 15.4 0.0 3.623 0.1 7.5 3.643 13.2 0.0 4.313 0.0 36.2 4.550 3.1 0.0 4.824 4.5 0.0 5.203 0.0 20.8 5.578 0.0 0.1 Σ 66.9 83.0
Z, % 0.0 1.0 0.7 0.1 15.0 0.0 3.4 61.7 0.0 0.0 81.8
Full Reservoir, WG WG FM , Hz X, % Y, % Z, % , Hz 1.314 0.0 8.5 0.0 1.305 1.340 26.4 0.0 0.2 1.336 2.004 12.4 0.0 0.1 1.992 2.362 15.7 0.0 1.1 2.352 2.519 0.0 2.4 0.0 2.498 2.961 0.0 6.6 0.0 2.944 3.184 0.8 0.0 0.1 3.153 3.703 1.4 0.0 0.1 3.680 3.903 0.0 0.7 0.0 3.860 3.938 7.1 0.0 1.6 3.914 Σ 63.8 18.1 3.3
101
FSI , Hz 1.572 1.622 2.362 2.944 3.040 3.719 3.869 4.555 4.764 4.803
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Mode 1,
1.527 Hz
Mode 2,
1.622 Hz
Mode 3,
2.362 Hz
Mode 4,
2.944 Hz
Mode 5,
3.040 Hz
Mode 6,
3.719 Hz
Mode 7,
3.869 Hz
Mode 8,
4.555 Hz
Mode 9,
4.764 Hz
Mode 10,
4.803 Hz
Figure 1: Dam Mode Shapes, Incompressible Fluid-Structure Interaction Analysis 102
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Envelopes of Min
80 Domain Analysis Hybrid Frequency Time The HFTD analysis is performed assuming linear structural behaviour and for only one time segment comprising the100whole duration of the seismic input. Envelopes of Max
120
Results
140
The following line types are used to designate the type of analysis carried out: 160
Static WG with Reservoir Damping
180
WG w/o Reservoir Damping FSI with Incompressible Fluid
200
FSI with Compressible Fluid 220 2
4
6
FSI with Incompressible Fluid and NONLIN Joint
8 10 12 14 16 18 20
Figure 2: Analysis Type Designation
Radial Displacement (cm)
Hoop Stresses and Cantilever Stresses 715
0
695
20
675
40
655
Envelopes of Max
60
Envelopes of Min
635
80
Envelopes of Max
615
100
595
120
575
140
555
160
Envelopes of Min
535
180
515
200
495
220
-14 -12 -10
-8
-6
-4
-2
0
2
4
6 -14 -12 -10
-8
-6
-4
-2
0
Hoop Stress (MPa)
Hoop Stress (MPa)
Figure 3: Main Section Hoop Stresses (U/S on left; D/S on right)
103
2
4
6
Reservoir Depth (m)
0
Elevation (m asl)
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Discussion and Conclusion The maximum static compressive stress is the hoop stress at ¾ the dam height on the u/s face of the main section. It amounts to approximately 7 MPa, which is within the admissible limits regarding the compressive strength of concrete. The maximum static radial displacement reaches 8.2 cm at the top of the central cantilever. The maximum compressive stresses during the ground motion reach approximately 12 MPa in the hoop direction on the u/s face. Seismic tensile hoop stresses varying between 1 MPa and 2 MPa are computed at the top part of u/s face of the main section; they disappear if the contraction joint opening/closing is modelled. The maximum seismic tensile vertical stresses exceed 2 MPA at ¾ the dam height on the upstream face of the main section. The maximum compressive and tensile stresses occurring during the earthquake are below the admissible limits. The maximum amplitude of the dynamic vibrations is about 8 cm with respect to the initial displaced shape of the dam. The hydrodynamic effect modelling approaches investigated in the present study lead to similar results regarding the structural response of the example arch dam. In general, the Westergaard added masses approach yields higher compressive and tensile stresses, as well as higher radial displacements. The compressible fluid analysis results in lower stresses with respect to the incompressible fluid assumption, which is due to the increased damping of the coupled system. In fact, according to [11], if the natural dam frequencies are significantly lower than this of the dam reservoir, , the behaviour of the latter is similar to the behaviour of incompressible fluid. In our case 1.98 Hz; this explains to a certain extent the obtained differences in the results with compressible and incompressible fluid.
References [1] Zienkiewicz, O.C., Taylor R.L. (2000). The Finite Element Method, Fifth edition, Vol. 1: The Basis, Butterworth-Heinemann, Oxford. [2] Westergaard, H.W., (1933). Water pressures on dams during earthquakes. American Society of Civil Engineers (ASCE), Proceedings. [3] Zangar, C.N., (1952). Hydrodynamic pressures on dam due to horizontal earthquake effects. Engineering Monograph No.11, U.S. Bureau of Reclamation. [4] Tan, H., and Chopra, A.K., (1995). Earthquake analysis of arch dams including damwater foundation rock interaction. Earthquake Eng. Struct. Dyn., 24(11), 1453–1474. [5] Wang, J. T., and Chopra, A. K. (2010). Linear analysis of concrete arch dams including dam-water-foundation rock interaction considering spatially varying ground motions. Earthquake Eng. Struct. Dyn., 39(7), 731–750. [6] Kuo, J. (1982). Added mass computations for incompressible fluid. Report UCB/EERC82/09, Earthquake Engineering Research Center, University of California at Berkeley. [7] Chavez, J.W., Fenves, J.L. (1993). Earthquake analysis and response of concrete gravity dams including base sliding. Report UCB/EERC-93/07, Earthquake Engineering Research Center, University of California at Berkeley. [8] TNO DIANA (2013). DIANA User’s Manual, Release 9.4.4. Delft, the Netherlands. [9] Graz University of Technology, Institute of Hydraulic Engineering and Water Resources Management (2013). 12-th International Benchmark Workshop on Numerical Analysis of Dams, Theme A. [10] Hughes, T. (1987). The Finite Element Method – Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Inc. [11] Sécurité des ouvrages d’accumulation (2003). Documentation de base pour la vérification des ouvrages d’accumulation aux séismes, rapports de l’OFEG, série Eaux, version 1.2.
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Dynamic Analysis of an Arch dam with FluidStructure Interaction Use of an Open source code AKANTU M. Chambart1, T. Menouillard1, N. Richart2, J.-F. Molinari2 and R. M. Gunn1 1
STUCKY SA, Rue du Lac 33, CH-1020 Renens, SWITZERLAND 2 EPFL ENAC LSMS, CH-1015 Lausanne, SWITZERLAND E-mail: [email protected], [email protected]
Abstract In this contribution, the fluid-structure interaction is modeled using the added mass technique and the incompressible fluid model. Results show that the added mass technique is more conservative. Different meshes are compared with this later method, demonstrating that the problem is not mesh dependent. Finally, the computation times obtained with two different software are compared, showing the efficiency of the new open-source software Akantu
Introduction One of the exercises proposed by the ICOLD for 12TH INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS consists in the dynamic analysis of an arch dam under a seismic loading. Since the geometry, the material properties and the loading are imposed to the participants, the focus is put on another modelling aspect which is the fluid-structure interaction. The methods proposed to model the fluid-structure interaction starts with the simplest ones, the added mass technique to the most sophisticated ones where the fluid is explicitly modeled as a compressible body. In our practice, as an engineering company, our choice for one method versus another is often governed by the gain in accuracy versus the loss in time. Most of the time in projects, the added mass approach is used because it is the fastest and usually sufficient to fulfill the authorities demands. This benchmark is an interesting opportunity to compare different methods in terms of results but also in terms of computation time. The comparison is carried out using the finite element software DIANA [1], simply using some of the proposed methods (added mass and incompressible fluid). As a second step, the results obtained with DIANA using the added mass technique are compared with the ones given by a new open-source software AKANTU [2] developed at the Swiss Ecole Polytechnique Fédérale de Lausanne. Akantu is a quite innovative object-oriented program written in C++. The objective using this latter software is to optimize the computation time, in order to get reliable results in shorten time, or run sophisticated computations in an acceptable amount of time. In order to estimate the mesh dependency of the problem, the results with two meshes are compared with the added mass technique. Therefore the need in future for High Performance Computing can be assessed.
Numerical Modelling Model and Mesh Figure 1 presents different meshes used with the Westergaard theory for dealing with the hydrodynamic pressure, and also the mesh with the fluid. There are a coarse and a fine mesh, whose numbers of elements are respectively 2874 and 22280.
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Figure 1: View of the meshes: coarse and fine, and with the fluid. Materials In this problem, three different materials are involved: the concrete of the dam, the rock of the foundation, and the water of the reservoir. Their density and stiffness are the valuable data in linear elastic problem. Table 1 presents the materials parameters. The density of the rock is numerically taken to be zero in order to avoid wave propagation at the boundary of the rock foundation. In addition, a dynamic Young’s modulus is used, whose value is 125% the static one [4]. Table 1: Materials properties. Properties Concrete Rock Water Density 2400 kg/m3 0 kg/m3 1000 kg/m3 Poisson ratio 0.167 0.2 Young’s modulus 27 GPa 25 GPa Loadings Static loadings. The self-weight of the concrete is given by the density of the concrete and the gravity. The hydrostatic pressure on the upstream face of the dam is given by the water level, i.e. 715 m asl. Dynamics loadings. The acceleration history of the seismic input is given through three accelerograms presented in Figure 2, and these three accelerograms are simultaneously applied at the boundary of the rock foundation. Their corresponding absolute displacement histories are also presented in Figure 2. Depending on the FEM program used, the boundary conditions may have to be given in displacement or in acceleration.
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Figure 3: Westergaard theory on the upstream face of a dam. Second, a method [3] where the fluid, considered incompressible, is explicitly meshed as a 3D part of the system is also considered. Solid elements are quadratic, whereas fluid elements are only linear. Therefore special interface elements are required on the upstream face of the dam 113
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to link the fluid to the solid domains. In addition, special boundary elements are involved to deal with the bottom absorption, the free surface and the far field surface of the reservoir. A method dealing with compressible fluid is not developed in this paper.
Vibration modes The vibration modes have been evaluated with different meshes and the Westergaard method for describing the hydrodynamic pressure. A very good agreement between the results of the different meshes was observed. The reason is that the linear elastic material model describes similar dynamic behavior of the structure, not depending on the mesh size. Figure 4 presents the deformed shape of the first 10 modes within the coarse mesh.
Figure 4: Deformed shapes of the first 10 vibrations modes. Table 2 presents the vibration frequencies and the mass percentage of the dam in each direction obtained with the coarse mesh and Westergaard theory. Mode 1 is a left-right motion with 20% of the mass of the dam. Mode 2 is the major vibration with 35% of the mass in the upstream-downstream direction. Table 2: Frequencies and mass percentage for the coarse mesh with Westergaard. Mode Frequency Mass percentage Mass percentage Mass percentage US/DS Vertical Left-Right [Hz] 1.48 1% 0% 20% 1 114
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1.50 2.27 2.67 2.92 3.38 3.73 4.25 4.54 4.64
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A damping effect of 5% is taken through Rayleigh coefficients. They are evaluated from the knowledge of two modes (1 and 13) containing 80% of total mass. The Rayleigh damping matrix C is defined by two parameters and as: (2) C=M+K where M is the mass matrix, K the stiffness. With the selected modes (1 to 13), the coefficients are: =0.726286 and =0.002346. Figure 5 shows the damping as a function of the frequency; the crosses on the graphs are the different vibration modes, and the damping of frequencies 1 and 13 are exactly 5%. As it has been selected, 80% of the mass of the dam has a damping between 4 and 5%. The modes with high frequency, i.e. greater than 10 Hz, are minor modes in terms of mass percentage. 10 9 8
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Results Table 3 presents the different computations run and the corresponding mesh size. With DIANA, two simulations with Westergaard and one with the fluid were run, whereas only two simulations were run with AKANTU. All simulation had 2000 steps for the dynamics. Table 3: Table of the different computations run with the two codes. Mesh Nodes Elements Elapsed time CPU time [s]
DIANA Westergaard Coarse Fine 3614 25057 2874 22280 1h10 7h00 13’291 s 40’752 s
With Fluid Coarse 14478 3632 4h00 34’841 s
AKANTU Westergaard Coarse Fine 3614 25057 2874 22280 0h13 5h40 2’819 s 74’000 s
The computations are run for both codes in parallel with the same number of processors available. However for Akantu the computation is performed on a powerful laptop while for Diana a workstation optimized to numerical simulations is used. The results show that Akantu
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is more efficient since the computation time is divided by 5 for the coarse mesh and by 1.25 with the fine mesh. The evaluation of the hoop, vertical stresses and radial displacement are performed on the three following sections denoted by Left, Mid and Right sections as shown in Figure 6.
Figure 6: View of the different sections for post-processing the results. Vertical stresses The vertical stress is a relevant result within the dam. Thus Figure 7 presents the vertical stress on the upstream and downstream faces of the Left section of the dam: static, maximum and minimum results are presented with the coarse and fine mesh and the Westergaard theory, with Diana and Akantu. Both meshes give similar results too for static, maximum and minimum vertical stresses. The vertical stress is always negative in static, but its maximum reaches positive value in some parts during the earthquake. All four results obtained with the Westergaard theory agree well. However the stress envelops obtained with the most sophisticated computation, i.e. incompressible meshed fluid, is quite smaller. The Westergaard theory seems conservative.
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Figure 7: Vertical stresses in the different sections. Hoop stresses The hoop stress is the other relevant results in terms of stress in an arch dam. Figure 8 presents the hoop stress at the different sections on the upstream and downstream faces of the dam: static, minimum and maximum stresses are presented for the different computations. There is a good agreement between the results obtained by both meshes and the Westergaard theory. One observes that the hoop during the earthquake is mainly negative, so the arch dam is in compressive stress state in static and also during the earthquake.
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Figure 8: Hoop stresses in the different sections. Principal maximum stresses Figure 9 presents the maximum principal stress at the different sections on the upstream and downstream faces of the dam: static, minimum and maximum stresses are presented for the different computations. There is a good agreement between the results obtained by both meshes and the Westergaard theory. One observes that the minimum of the maximum principal stress during the earthquake is close to the static maximum principal stress. Indeed the maximum principal stress can only increase due to the dynamic loading from the static case. This maximum principal stress behaves like the vertical stress in the arch dam during dynamics.
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Figure 9: Principal maximum stresses in the different sections. Principal minimum stresses Figure 10 presents the minimum principal stress at the different sections on the upstream and downstream faces of the dam: static, minimum and maximum stresses are presented for the different computations. There is a good agreement between the results obtained by both meshes and the Westergaard theory. This minimum principal stress behaves like the hoop stress in the arch dam during dynamics.
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Figure 10: Principal minimum stresses in the different sections. Radial displacements The radial displacement quantifies the motion of the dam, in static and during the Earthquake. Figure 11 presents the radial displacement of the different sections: static, minimum and maximum obtained with the coarse and fine mesh and the Westergaard theory with both software and also the displacement obtained with the incompressible fluid. There is a good agreement between these two results in static and in dynamic. The static radial displacement at the top of the Mid section is about 8 cm, whereas the maximum during the Earthquake is about 16 cm, and the minimum about 3 cm.
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Figure 11: Radial displacements.
Conclusion This benchmark allowed to investigate the field of how, in term of numerical method, to deal with the hydrodynamic pressure on the upstream face of an arch dam. This has then shown that the Westergaard theory, which consists in simply adding mass on nodes of the upstream face of the dam, is clearly not mesh dependent. Second, the method dealing with the mesh of the incompressible fluid, gives smaller envelops in terms of maximum and minimum of displacement and stresses of the arch dam. In addition, the development of the Westergaard theory in an open source code did not show any relevant difficulties and the computation time can be significantly decreased.
References [1] Water pressures on Dams during Earthquakes, H.M. Westergaard, Transactions, ASCE 98:418-472, 1933. [2] Fluid-Structure Interactions: added mass computations for incompressible fluid, James Shaw-Haw Kuo, University of California, Berkeley, August 1982. [3] Hybrid Frequency Time Domain – Validation, TNO DIANA Report, 2008-DIANAR003, 2010. [4] Sécurité des ouvrages d’accumulation – Documentation de base pour la verification des ouvrages d’accumulation aux séismes, rapports de l’OFEG, série Eaux, version 1.2, mars 2003. 121
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Study on Arch Dam– Reservoir Seismic Interaction A. Popovici1, C. Ilinca1 and R. Vârvorea1 1
Technical University of Civil Engineering, 124 Lacul Tei Bd. 020396-Bucharest, ROMANIA E-mail: [email protected]
Abstract The arch dam – reservoir seismic interaction is investigated using ABAQUS 6.11 and DESARC 3.1 software. DESARC computer code offers the advantage of simplicity and computation speed due to the degrees of freedom based on the stresses (Ritter modified method) being very recommended for arch dams preliminary structural analysis. The coarse mesh given by formulator was used for investigation in ABAQUS and 12 arches equally spaced on dam height were used in DESARC. The water effect was considered according to added mass procedure as well as acoustic elements. All analyses were performed in the linear elastic field. The results are presented in compliance with formulator requests: eigenfrequencies and mode shapes, hoop stresses, vertical stresses, min./max. principal stresses and radial displacements in three different sections for static and seismic loads. A special attention is paid to compare the results concerning arch dam – reservoir seismic interaction in different hypotheses applying two software.
Introduction The effects of different hypotheses on arch dam - reservoir seismic interaction are investigated in this paper based on data provided in Theme A by formulator, Graz University of Technology – Institute of Hydraulic Engineering and Water Resources Management [1]. The arch dam is a symmetrical structure with the followings main characteristics: maximum height 220 m chord length at dam’s crest 430 m valley width at bottom 80 m The analyses are carried out using ABAQUS 6.11 [2] and DESARC 3.1 [3],[4] software. The finite element mesh of the arch dam structure-foundation-reservoir system used in ABAQUS corresponds with alternative coarse mesh given by formulator (Figure 1). The main features of the coarse mesh are as follows: arch dam – total number of nodes 2083 total number of elements 356 (312 C3D20R and 44 C3D15) foundation – total number of nodes 11608 total number of elements 2340 C3D20R reservoir – total number of nodes 12493 total number of elements 2640 C3D20R
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Figure 1: Finite element mesh of the arch dam structure-foundation-reservoir system used in ABAQUS code Other hypotheses used in analyses performed with ABAQUS were the followings: the dead weight were applied on monolithic structure the reservoir influence on dam seismic response was considered by added mass procedure excepting the free vibration analysis for full reservoir using acoustic elements C3D20A for reservoir mesh; the water was considered as incompressible with shear modulus tends to zero; the seismic response was evaluated for empty and full reservoir by modal superposition and direct time integration; all analyses were based on assumption of linear elastic behavior of materials; three accelerograms in direct time integration were applied on the faces of foundation, respectively x, y and z directions; the fraction of critical damping in direct time integration was 5%, α and β coefficients in linear Rayleigh model were computed for ω1 and ω2 , resulting α=0.3900 and β=0.0064. DESARC is an interactive, fast and reliable computer code very recommended for preliminary fast static and dynamic structural analysis of arch dams. The simplicity and computation speed of DESARC is due to the degrees of freedom based on the stresses (Ritter modified method and not on displacements. The program can handle symmetric arch dams both with circular and parabolic midline, rotationally symmetric structure with liquid inside or outside, cooling towers etc. All the geometrical parameters (thickness, radii, half lengths and cantilever shape) are expressed under assumption of parabolic variation with elevation. Accordingly the real arch dam shape provided by formulator was equated as assumptions presented above (Figure 2)
Figure 2: Equating real dam shape under assumptions of DESARC
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A number of 12 arches of the dam was considered under analysis (Figure 3). In may be remark from figures 2 and 3, resulting structure under DESARC assumptions has the shape very close to the actual dam.
Figure 3: Geometrical features of the arch dam equated under DESARC assumptions Other hypotheses used in analyses performed with DESARC were the followings: the dam-foundation interaction was considered by Vogt coefficients (Er/ Ec=0.90, Er=25000 MPa and Ec=27000 Mpa); the reservoir influence on dam seismic response was considered by added mass procedure; the hydrodynamic pressures collinear with the direction of horizontal upstream – downstream earthquake (Phd,0) were evaluated with Westergaard relationship. For other directions (Phd,α) was applied the following relation: (1)
Phd,α = Phd,0 . cos α
α being the angle between earthquake direction and the normal to the surface at the point considered; the dead weight were applied on isolated cantilevers and monolithic structure, too.
Some aspects concerning mathematical models The effect of the water in the reservoir, under the assumption of an incompressible ideal fluid, is usually calculated by use of the added mass procedure. This is a mathematical artifice used in order to simplify the analysis of the structure-liquid seismic interaction. The added mass is determined from the hydrodynamic forces {Ph(t)} and is attached to the mass of the structure. Unlike the dead mass of a structure, the added mass acts only on hydrodynamic force direction, namely the direction of the normal to the surface it is applied. Assuming that the directions of hydrodynamic forces, of earthquake and of degrees of freedom of the structure are similar (Figure 4a), the added mass matrix [Mh] is determined with relation: {
}
[
] { ̈
̈ }
(2)
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̈ } is total acceleration response to normal direction at surface in the considered where { ̈ point. In this case the analysis of the structure-liquid seismic interaction is performed as for usual structure according to formula: ] { ̈ }
[
[ ] { ̇ }
[ ] { }
[
] { ⁄ } ̈
(3)
In the general case when the directions of the normal to surface, of the earthquake and of the structure degrees of freedom are different, the added masses computed according to (2) relationship must be projected successively on the normal to surface direction and on the degrees of freedom of the structure (Figure 4b).
Figure 4: Assessment of added masses The dynamic equilibrium equations which include the structure-liquid seismic interaction are written in this general case as follows: [ [
] [ ] [ ] [ ] [
] { ̈ } [ ] { ̇ } ] { } ̈
[ ] { } (4)
where [rc,n] has dimensions equal with the number of the degrees of freedom of the system and contains on diagonal the cosine directors between the normal to surface in the nodes of the system mesh and earthquake direction and [rn,x,y,z] has dimensions corresponding to the number of the degrees of freedom of the system and contains on diagonal the cosine directors between the normal to surface in the nodes of the system mesh and directions of the dynamic degrees of freedom of the system.
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Results of analyses The results are provided as requested by the formulator. In the Table 1 are presented the natural circular frequencies (ω, rad/s) and respectively natural frequencies (f, cycles/s ) values of the first 10 mode shapes including interaction with the reservoir modeled with acoustic elements C3D20A and added mass procedure computed with ABAQUS and the value of the frequency of the fundamental mode shape including interaction with the reservoir by added masses computed with DESARC. It may remark that natural frequencies computed with ABAQUS by added mass procedure are smaller with about 2…21 % relative to their counterparts having reservoir interaction modeled with acoustic elements. In figure 5 is illustrated the fundamental mode shape of the arch dam in hypothesis of the full reservoir with water elevation at the dam crest. Table 1: Eigenfrequencies ABAQUS
DESARC Eigenfrequency-added mass (Rad/s) (Hz) (Rad/s) (Hz) 7.756 1.235 10.557 1.681 7.837 1.248 12.139 1.933 14.865 2.367 15.047 2.396 16.460 2.621 17.942 2.857 19.430 3.094 19.594 3.120 22.935 3.652
Eigenfrequency-acoustic elements Mode no. (Rad/s) (Hz) 1 9.726 1.548 2 9.843 1.567 3 12.160 1.935 4 14.458 2.301 5 15.584 2.480 6 19.109 3.041 7 19.573 3.115 8 20.695 3.294 9 22.687 3.611 10 23.280 3.705
Figure 5: Fundamental mode shape view from downstream Figure 6 illustrates three accelerograms provided by formulator for earthquake analyses (maximum acceleration 0.1g) and respectively figure 7 response spectra corresponding to accelerograms used for modal superposition analyses (spectral analyses).
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Figure 6: Accelerograms provided by formulator for earthquake analyses
Figure 7: Response spectra (x,y,z) computed from the accelerograms provided by formulator and used in spectral analyses Radial displacements on x, y and z directions in dam central sections at crest and base levels computed with ABAQUS in direct time integration method are illustrated in figure 8. Hoop stresses, vertical stresses and radial displacements in the dam central section downstream/upstream faces generated by combined dead weight + hydrostatic pressures are illustrated in figures 9a,b. The results in spectral analysis and direct time integration, in the dam central section, downstream/upstream faces, full reservoir case are comparatively presented in figures 10a,b. The min./max stresses in the dam central section, downstream/upstream faces resulted during direct time integration under combined actions of dead weight + hydrostatic pressures + three-dimensional earthquake 0.1g are illustrated in figures 11a,b
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Figure 8: Radial displacements in dam central section at crest (top) and base (bottom) computed with ABAQUS
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Figure 9a: Combined dead weight + hydrostatic pressures – Hoop and vertical stresses, radial displacements on downstream face, central section
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Figure 9b: Combined dead weight + hydrostatic pressures – Hoop and vertical stresses, radial displacements on upstream face, central section
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Figure 10a: Spectral analysis and direct time integration, central section, downstream face, full reservoir
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Figure 10b: Spectral analysis and direct time integration, central section, upstream face, full reservoir.
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Figure 11a: Min./max stresses, direct time integration, dead weight+ hydrostatic pressure + three-dimensional earthquake 0.1g
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Figure 11b: Min./max., direct time integration, dead weight+ hydrostatic pressure + threedimensional earthquake 0.1g
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Comments on results of analyses In compliance with some results illustrated in the above captions and other ones resulted from the analyses carried out the followings comments may be pointed out: DESARC3.1 a very simply, friendly and fast computer code in all applications led to results close to those provided by much more sophisticated ABAQUS6.11 computer code. As a result DESARC computer code is very recommended for arch dams preliminary structural analysis. Usually, in engineering practice dead loads are applied on arch dam isolated cantilevers, taking into account that during dam construction the contraction joints are not grouted. However, even under these conditions, some of the dead weight of cantilevers is transferred to arches, this quota depending of the dam shape and valley opening. Coming back to the arch dam provided by formulator it may remark from stresses and displacements to dead loads and hydrostatic pressures computed with DESARC in both hypotheses of isolated cantilevers and monolithic structure (fig.8a,b). that the effects of isolated cantilever hypothesis in relation with monolithic structure are not important. Accordingly, the analyses with ABAQUS, for simplicity, were carried out only in case of monolithic structure. The dam displacements due to dead weight reach about 1 cm to upstream at the crest level. Dead weight is mainly transferred on cantilever. At the bottom of the central section the vertical stresses reach -7 MPa compression at upstream toe and -1.5 MPa compression at downstream heel. The hypothesis with isolated cantilever is a conservative one. On arches, the maximum stress reaches -2 MPa compression at the crest level. Hydrostatic pressure as independent load generates maximum radial displacement of 7.5 cm in central section, dam crest elevation. The vertical stresses vary between 1 MPa tension and 4 MPa compression at downstream face and, respectively between 8 MPa tension at the dam upstream toe and -2 MPa compression at upstream face. The hoop stresses vary between 0.5 MPa tension, -3.7 MPa compression at downstream face, respectively between 0.5 MPa tension, -7 MPa compression at upstream face. The big vertical tensile stress generated by hydrostatic pressure at the dam upstream toe is reduced to 1.5 MPa by vertical compressive stress in the same point due to dead weight. Based on results concerning natural periods values computed in full reservoir hypothesis (Table 1) (fundamental period 0.809 s ABAQUS, added mass, 0.645 ABAQUS, acoustic elements)it may conclude the hydrodynamic forces computed with Westergaard formula to generate added masses are higher than hydrodynamic forces developed by acoustic elements. A comparison between seismic responses computed in the full reservoir hypothesis by spectral analysis and direct time integration with ABAQUS points out a good correlation between correspondent displacements response their maximum reaching about 13 cm but generally significant differences between correspondent stresses. For instance in the dam central section at downstream face the principal stresses vary between 1.50 MPa (tension) and -4 MPa (compression) in direct time integration and respectively 0,9 MPa and -9.50 MPa in spectral analysis. Maximum displacement obtained in direct time integration at combination static loads (dead weight + hydrostatic pressures) + earthquake (full reservoir) reaches 18 cm to downstream at the crest level in central section. Max/min vertical stresses on downstream face varies between 0.8 MPa and -6.0 MPa . On upstream face at the dam upstream toe the vertical stress reaches tension of 7 MPa. It is a vulnerable point for cracking. On horizontal direction in the same point the hoop stresses vary between 3.5 MPa tension and -2.5 MPa compression. It may conclude that excepting dam upstream toe where exists cracking risk, the dam withstand earthquake action without notable incident. As a general conclusion based on data presented above, the seismic response parameters of arch dams, especially seismic stress state differ significantly function of method of analysis.
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Unfortunately, the scarcity of recordings concerning arch dams behavior during strong earthquakes make difficult to validate a method of analysis using field recordings from.
References [1] Graz University of Technology THEME A Fluid Structure Interaction Arch Dam – Reservoir at Seismic Loading. 12-th International Benchmark Workshop on Numerical Analysis of Dams, Graz – Austria, 2013. [2] ABAQUS 6.11. Abaqus / CAE User's Manual. United States of America: Abaqus Inc, 2009. [3] Fanelli M., Giuseppetti G., Rabagliati U. Il metodo di Ritter Modificato. L’Energia Eletrica, No.3, 1978. [4] Fanelli M., Fanelli A. Designer’s Guide DESARC 3.1 for Windows ISMES S.p.A., Bergamo, 1995. [5] Popovici A., Popescu C. Dams for water storage (in Romanian) Editura Tehnica, Bucharest, 1992. [6] Priscu R., Popovici A., Stematiu D., Stere C. Earthquake Engineering for Large Dams Editura Academiei (Romania) and John Willey & Sons Ltd. (U.K.), Chichester, 1985. [7] Popovici A., Abdulamit A., Toma I. Moldoveanu T. .Assessment of the seismic safety of dams based on „in site” measurements and back analyses. Proceedings XXI-st International Congress on Large Dams, Montreal, 2003.
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Concrete arch dam at seismic loading with fluid structure interaction R. Malm1,2, C. Pi Rito2, M. Hassanzadeh3,4, C. Rydell1,3 and T. Gasch1,3 1
KTH Royal Institute of Technology, Concrete Structures, Stockholm, Sweden. 2 SWECO Infrastructure, Hydropower and Dams, Stockholm, Sweden 3 Vattenfall Engineering, Stockholm, Sweden. 4 Lund University, Building Materials, Lund, Sweden. E-mail: [email protected]
Abstract A concrete arch dam have been analyzed during seismic loading with a model based on acoustic elements to describe the water and infinite elements as quiet boundaries to prevent wave reflection. The results have also been compared with a simplified model based on Westergaards added mass approach. The simplified model is only used, in this study, for comparison with the more advanced model with acoustic elements. Therefore the results from this simplified model are just used as a rough estimate of the induced stresses and displacements. Despite this, the simplified Westergaard model gives similar results compared to the more advanced model with acoustic elements for the water and infinite elements for the boundaries. The largest difference between the models often occurs at the nodes in the base of the arch dam, which may be due to poor discretization. Generally, the Westergaard added mass gives higher maximum principal stresses at the base on the upstream side than the acoustic model, while often underestimating the min principal stresses at the base on the downstream side. Both models show high tensile stresses near the base of the arch dam that would result in cracks.
Finite element models The studied geometry was given by [1], and in this paper, the mesh denoted as fine mesh in [1] has been used. All numerical analyses in this paper have been performed with the commercial software Abaqus ver.6.12. The model consists of a foundation with dimensions 1000 x 1000 x 500 m built up of 2700 second order brick elements with reduced integration. In one of the models presented in this paper, infinite elements have been used for the outer surfaces of the foundation in order to remove reflecting waves on the model edges, see Figure 1 a). The arch dam is 220 m high, with a width of 430 m at the crest and 80 m at the base. The arch dam consists of 2736 second order brick elements with reduced integration (8 integration points). The geometry of the reservoir is 460 m long (i.e. two times the dam height). In the following analyses, two different models to account for the water will be presented. In a simplified model, the reservoir has been replaced with nodal masses corresponding to the Westergaard added mass approach [2]. In the more advanced model, the reservoir has been included by means of acoustic elements and infinite acoustic elements to account for an infinite long reservoir. Loads and boundary conditions All analyses have been performed in two steps; static analysis of response due to gravity and hydrostatic water pressure, and dynamic implicit analysis (transient dynamic analysis) to calculate the response from induced ground accelerations due to the seismic load.
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In all analyses, the gravity load has only been assigned to the elements that constitute the arch dam. The rock is according to the instructions, [1], defined with zero density. In the simplified model, based on the Westergaard added mass approach, additional nodal masses are introduced on the upstream surface of the arch dam. However, it is important that the nodal masses are not assigned a gravity load since that would affect the static behavior of the dam. The hydrostatic water pressure has been assigned to the upstream surface of the arch dam, and to all rock surfaces subjected to the water pressure, as seen in Figure 1 b). The maximum hydrostatic pressure is defined as 2.1582 MPa (220 m x 9.81 m/s2 x 1000 kg/m3). In the more advanced model, with acoustic elements, an additional boundary condition with zero acoustic pressure on the free surface of the water has been defined.
a) b) Figure 1: a) Geometry and element types of the model, b) applied hydrostatic water pressure. The ground accelerations are illustrated in Figure 2. In the dynamic analysis, a constant time step of Δt = 0.01 s have been used. The upper frequency that can be captured in the analysis, i.e. the Nyquist frequency, is thereby 50 Hz. According to the frequency analyses that were performed, the cumulative mass is nearly 100 % for frequencies up to 30 Hz and thereby the chosen time increment is considered sufficient. In addition, the time history signal of the earthquake is sampled with Δt = 0.01 s and thereby the highest reproducible frequency in the sample is 50 Hz. Acceleration [G]
0.05 0 -0.05
Acceleration [G]
0.05 0 -0.05
Acceleration [G]
Horizontal 1 EQ
0.05 0 -0.05
0
2
4
6
8
10 12 Time [s] Horizontal 2 EQ
14
16
18
20
0
2
4
6
8
14
16
18
20
0
2
4
6
8
14
16
18
20
10 12 Time [s] Vertical EQ
10 Time [s]
12
Figure 2: Transient time history signals of the ground accelerations. In the static analysis, all nodes on the bottom surface of the foundation were constrained for all translations, i.e. in x-, y- and z-direction. In the seismic analysis, these boundary conditions are replaced, where all nodes on the bottom surface were assigned prescribed ground accelerations according to Figure 2. The sides of the foundation have not been assigned boundary conditions in neither the static nor the dynamic analysis. The reason for this is that 140
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these surfaces will be defined with infinite elements in the model with acoustic elements. It is not possible to define prescribed conditions in the form of accelerations, velocities or displacements on nodes that are part of infinite elements if the infinite elements are to function as quiet boundaries. One approach to overcome this would for instance be to calculate corresponding stresses from the induced accelerations. However, considering the large geometry of the rock, it is likely that seismic wave incoherence would influence the results. It is therefore unlikely to assume that the ground vibrations would be identical on all sides. In addition, the rock is defined with zero mass, and therefore it will not produce mass inertia forces. Hence, the approximation adopted here, where the boundary is applied only to the bottom nodes, is considered to be valid for this case. Material properties and damping All material properties have been defined by [1]. Rayleigh damping have also been used in all presented analyses. The damping ratio for concrete has been assumed equal to 4 % according to Regulatory Guide 1.61, [3]. The damping of the water is usually assumed to be 0.5 %; this has been judged to be negligible and therefore not included in the analyses. The lower frequency of the Rayleigh damping have been defined corresponding to the frequency where 5 % of the cumulative effective mass is active, which in this case is f1 = 1.27 Hz. This ensures minimal underestimation of response in the low frequency range according to [4] and [5]. The upper frequency have been defined corresponding to the frequency where 80 % of the cumulative effective mass is active, which in this case is f2 = 9.76 Hz. The corresponding damping values ξ1 and ξ2 have been calculated so that the minimum damping in the interval f1 – f2 is 1 % less than the target damping value, i.e. ξmin = 3 %. Based on this approach, the obtained Rayleigh damping coefficients are α = 0.66314 and β= 0.0013503 with the corresponding curve illustrated in Figure 3. The average damping in the interval f1 – f2 is 3.6 %, i.e. slightly lower than the target value of 4 % and thereby conservative. 0.1
Damping
0.08 0.06 0.04
m = 5 %
0.02 0
0
5
m = 80 % 10
15 20 Frequency (Hz)
25
30
Figure 3: Rayleigh damping curve (α = 0.66314 and β= 0.0013503) The Rayleigh damping coefficients have been defined in the material definition for concrete but also for the rock. Since the rock have zero density, only the stiffness proportional part of the Rayleigh damping (β= 0.0013503) will be active. The Rayleigh damping curve has been verified against a modal analysis with uniform damping of 4 %, where the Rayleigh damping gave slightly conservative result. Model 1: Westergaard added mass approach According to Westergaard [6], the hydrodynamic forces exerted on a dam due to earthquake ground motions are equivalent to inertia forces of a volume of water attached to the dam moving back and forth with the dam while the rest of the reservoir water remains inactive. The influence of the reservoir on a 2D rigid monolith with vertical upstream face is included by introducing the impulsive mass of the water and thereby, altering the dynamic properties of 141
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the structure. Kuo [7], further developed the method to also account for the curvature of an arch dam, this have however not been considered in this paper. The reason for this is that the simplified added mass model is only used, here, for comparison with the more advanced model with acoustic elements. Therefore, the results from this simplified model are just considered as a rough estimate of the induced stresses etc. A MatLab script have been developed to calculate the tributary surface area for each node on the upstream surface, and based on this the nodal mass has been calculated according to [6]. The calculated nodal masses are illustrated in Figure 4, where the colors in the figure represent the added weight (in kg) for each node on the upstream surface of the arch dam. 750
Westergaard added mass
4.26e+006
700 3.55e+006 650 2.84e+006 600 2.13e+006 550 1.42e+006 500
450
710000
-200
-150
-100
-50
0
50
100
150
200
Figure 4: Nodal masses based on Westergaard added masses Model 2: Acoustic and infinite elements In the more advanced model, the water has been included by means of 2640 second order acoustic elements. These elements are based on an acoustic formulation with the fluid wave velocity, i.e. the variation in pressure, as the independent variable. These elements do not include any terms for body forces and therefore, the hydrostatic pressure needs to be included as a pressure load on the structure, as previously shown in Figure 1 b). The finite element mesh of the acoustic elements is stationary at all nodes except at the boundaries of the fluid domain, i.e. the fluid-structure interface. For boundaries adjacent to a structural domain, the nodes of the acoustic medium can be prescribed to follow the nodes of the structural domain, giving a pressure change in the acoustic medium. A pressure boundary which prescribes zero acoustic pressure on the free surface has been defined, due to the lack of displacement DOF. This gives no actual displacement of the free surface but is correct in the sense of wave propagation in the medium. In order to account for an infinite long reservoir, 264 additional infinite acoustic elements are defined on the upstream side of the reservoir. In order to prevent reflecting waves on the model edges, 510 infinite elements have been defined for the surfaces on the foundation rock as illustrated in Figure 1 a). The infinite elements provide quiet boundaries and are based on Lysmer and Kuhlemeyer [8] for dynamic response. The damping constants for longitudinal and shear waves are calculated as (1) d p c p 2G
d s cs G
(2)
E E G 2 1 1 1 2 , where, and E is the elastic modulus, ν is poisons ratio and cp and cs are the longitudinal and shear wave speed. The infinite rock elements were defined with the same properties as rock (see [1]), with the addition of a defined density of 2600 kg/m3, required for the damping. The bottom surface
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was defined without infinite elements, due to the fact that the seismic ground motion was applied in the base. The length of the infinite rock elements are 1000 m, i.e. chosen as the total length of the original foundation. The infinite elements are active in static analyses, by providing horizontal stiffness to the foundation.
Eigenfrequencies and modes
143
Figure 5: The 10 first eigenmodes obtained from the acoustic model.
g) Mode 7 f) Mode 6
a) Mode 1
b) Mode 2
c) Mode 3
h) Mode 8
i) Mode 9
d) Mode 4
e) Mode 5
j) Mode 10)
The 10 first mode shapes obtained from the acoustic model are illustrated in Figure 5.
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In some cases, the acoustic model gives two closely spaced frequencies with the same mode shape. In Table 1, these are therefore presented together as one mode. In addition, some modes in the acoustic model are mainly relevant to the water, and hence these are not presented here. It can however be mentioned that the first mode corresponding to the water is about 0.7 Hz. The frequencies for each mode is also presented from the Westergaard added mass approach and the original model without water for comparison in Table 1. Table 1: Calculated eigenfrequencies for the ten first modes. Frequency (Hz) Mode
Acoustic elements
Added mass
W.o. water
1
1.5114
1.3730
2.0002
2
1.5431
1.3291
1.9012
2.054
2.8785
3
1.9056
1.9850
4
2.2255
2.3534
5
2.4254
2.4665
6
2.9626
2.9997
2.5633
3.5491
7
3.0084
3.1169
2.5782
3.5998
8
3.2818
3.3150
2.8773
4.1744
3.2224
4.4779
9
3.5886
10
3.7748
As seen in the table, the acoustic model gives frequencies that are in-between the case without water and the case with Westergaard added mass. Notable is also that the first and second mode have change place in the acoustic model compared to the others models.
Stresses Stresses are presented for three sections, Left section (-45°), Main section (0°) and Right section (+45°) from the center of the arch, as defined by [1]. The following stresses are presented; maximum and minimum principal stresses, hoop stresses and vertical stresses. For each of these stresses, the max and min envelope for each node on the upstream (US) or downstream (DS) surface are presented. This means that the presented stresses are not occurring at the same time; instead these are the maximum or minimum stresses that occur for the whole time period for each node on the upstream or downstream side of each of the studied sections. As seen in Figure 6 there is quite similar results between the simplified Westergaard added mass method and the method with acoustic and infinite elements. The minimum principal and hoop stresses show, generally, better resemblance between the two models than the max principal and vertical stresses. The largest difference between the models often occurs at the nodes in the base of the arch dam, which may be due to poor discretization. Generally, the Westergaard added mass gives higher maximum principal stresses at the base on the upstream side than the acoustic model, while underestimating the min principal stresses at the base on the downstream side. Both models show high tensile stresses near the base of the arch dam that would result in cracks. It can from this comparison not be drawn any conclusion that the simplified Westergaard added mass approach would be more conservative than the method with acoustic and infinite elements. The analyses also show that, despite the
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symmetric shape of the arch dam, there is a difference between the results at the left and right section, as seen in Figure 7.
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Figure 6 Stresses in the main section (0°) according to the model with acoustic elements and the simplified Westergaard model, on US and DS surface respectively.
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Figure 7: Stresses in the left (-45°) and right section (+45°) according to the model with acoustic elements on US and DS surface respectively.
Displacements The max and min envelope of the relative radial displacements are presented in Figure 8 for the Main section (0°). In figure, the relative displacement presented i.e. the rigid body displacement due to the earthquake has been removed. In the figure, a comparison between the acoustic model and the simplified Westergaard model is shown. The models give similar results, but the acoustic model gives larger deflections. x-dir (radial) upstream 700 680 660
Elevation (m)
640 620 600 580 560
Max (Westergaard) Min (Westergaard) Max (Acoustic) Min (Acoustic) Static
540 520 500 -0.05
0
0.05 0.1 Deflection (m)
0.15
Figure 8: Displacements in the main section (0°) on US surface.
Discussion A concrete arch dam have been analyzed during seismic loading with a model based on acoustic elements to describe the water and infinite elements to describe quiet boundaries to prevent wave reflection. The results from this model have also been compared to a simplified model based on Westergaards added mass approach. The simplified model is only used for comparison and should therefore be considered as a rough estimate of the induced stresses and displacements. The performed analyses have showed that there are several factors that have been assumed in this study that influence the results, such as Rayleigh damping – the choice of damping ratio for the different materials and especially the choice of corresponding frequencies influence the results. Seismic excitation – in these analyses, only the bottom of the foundation is subjected to the prescribed seismic excitation. Effects such as seismic wave incoherence have not been considered. Infinite boundaries – in the analyses infinite boundaries have defined for both the foundation but also for the reservoir. The results from the two models are quite similar, where the minimum principal and hoop stresses generally are more similar between the two models than the maximum principal and 147
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vertical stresses. The largest difference between the models often occurs at the nodes in the base of the arch dam, which may be due to poor discretization of the mesh at these points. Generally, the Westergaard added mass gives higher maximum principal stresses at the base on the upstream side than the acoustic model, while often underestimating the min principal stresses at the base on the downstream side. Both models show high tensile stresses near the base of the arch dam that would result in cracks. The analyses also show that, despite the symmetric shape of the arch dam, there is a difference between the results at the left and right section.
Acknowledgements The research presented was carried out as a part of "Swedish Hydropower Centre - SVC". SVC has been established by the Swedish Energy Agency, Elforsk and Svenska Kraftnät together with Luleå University of Technology, The Royal Institute of Technology, Chalmers University of Technology and Uppsala University. www.svc.nu. The presented work was financially supported by SVC, Sweco Infrastructure, KTH Royal Institute of Technology and Vattenfall Engineering. The authors would like to thank these organizations for making our participation in this benchmark workshop possible.
References [1] ICOLD, 2013. 12th International Benchmark Workshop on Numerical Analysis of Dams. Theme A – Fluid Structure Interaction. Arch Dam – Reservoir at Seismic loading. Graz University of Technology. [2] Westergaard, H. M. (1933). Water Pressures on Dams During Earthquakes. Transactions, American Society of Civil Engineers, Vol 98. [3] US NRC (2007). Damping values for seismic design of nuclear power plants. Regulatory Guide 1.61. U.S. Nuclear Regulatory Commission. [4] ASCE 4-98, 1998, “Seismic Analysis of Safety-Related Nuclear Structures and Commentary,” American Society of Civil Engineers, Reston, Virginia. [5] ASCE/SEI 43-05, 2005, “Seismic Design Criteria for Structures, Systems, and Components in Nuclear Facilities,” American Society of Civil Engineers, Reston, Virginia. [6] EM1110-6051 (2003). Time-History Dynamic Analysis of Concrete Hydraulic Structures. US Army Corps of Engineers. [7] Kuo, J.S.-H. (1982). Fluid-structure Interactions: Addedd Mass Computations for Incompressible Fluid. Report No. UCB/EERC-82/09. Earthquake Engineering Research Center, University of California, Berkeley. [8] Lysmer, J., and R. L. Kuhlemeyer, “Finite Dynamic Model for Infinite Media,” Journal of the Engineering Mechanics Division of the ASCE, pp. 859–877, August 1969.
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Fluid Structure Interaction Arch Dam – Reservoir at Seismic loading A solution using FEnas M. Brusin1, Dr. J. Brommundt1 and H. Stahl1 1
AF-Consult Switzerland Ltd., Täfernstrasse 26, CH-5405 Baden, SWITZERLAND E-mail: [email protected]
Abstract This paper represents a contribution to the ICOLD – 12th International Benchmark Workshop on Numerical Analysis of Dams, THEME A - Fluid Structure Interaction Arch Dam – Reservoir at Seismic loading. The formulated problem, modeling of a 220 m high and 430 m wide arch dam under static and seismic loads, was addressed by the finite element code FEnas. FEnas is a Swiss code originally developed at the Imperial College London. The interaction between the dam and the water in the reservoir was implemented using the added mass technique in the general form following Westergaard. The simulations performed included a comparative analysis of two meshes with different spatial discretization. The computed results show that the impact of the discretization is rather small. Moreover, the results of the analyses have been compared to typical deformations observed at Swiss dams and similar studies on existing dams. All results here are in good agreement with preceding studies of this kind.
Introduction Problem Definition The problem to be analyzed and solved here is exhaustively described in the problem description provided by the workshop organizers [1], who are referenced as the “formulator” in the following. The stresses and deformations of a dam interacting with the water in its completely filled reservoir under earthquake load shall be simulated. As the title of the benchmark workshops says, this interaction is the focus of this study. Consequently, typical further challenges in studies of this type are handled in a pragmatic way, e.g. all simulations assume linear elastic material behavior. The earthquake is represented by three correlated time series of acceleration in the three dimensions, lasting 20 seconds in total. Two FE meshes are provided, one in coarse and one in fine resolution. Finite Element Code FEnas The finite element analysis of the dam was performed using FEnas ECCON IPP/ Version 2.9.3 2012/08/24. FEnas which was initially developed at the Imperial College London and was continuously enhanced and integrated with a graphical user interface by Walder & Trueb Engineering AG, Switzerland [2]. FEnas is capable of solving a wide range of static and dynamic load problems including consideration of thermal impact. The dimension of the problems solvable in FEnas is constrained by aspects of computation only. Nonlinearity in terms of material behavior and geometry can be considered. The typical field of application of FEnas is in design of buildings and concrete structures, including tunnels and dams. The FEnas element library includes all element types necessary for this type of analyses. However, acoustic or fluid elements are not included and cannot be modeled.
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Modelling Concept and Implementation General Assumptions and Approach The software package FEnas allows to analyze the problem of interaction of fluids and structures under seismic impact using the added mass technique. This technique is considered to be a conservative approach and gives satisfactory results in engineering practice [3]. The mass of water which oscillates along with the structure is added as a discrete mass at all element nodes of the upstream face of the arch dam as described below. All analyses were carried out for the two meshes provided by the formulator, i.e. a coarse and a fine mesh. For each mesh the following three calculations were performed: Linear static analysis for the self-weight load and hydrostatic pressure load. Calculation of eigenvectors of oscillation for the first 10 tones, including the added mass of water. Linear dynamic analysis by using direct time integration, also including the added mass of water. For further analysis of the results a superposition of the results of static and dynamic analyses was performed as described below. All approaches to the modeling, loads, boundary conditions, and material parameters are identical for the simulations with the two meshes. Considering this, if not stated otherwise, the following discussions refer to the fine mesh; references to the coarse mesh are explicitly marked. Mesh Properties The mesh was delivered by the formulator of the benchmark competition. Two adaptations were performed: 4. The discretization of the terrain of the coarse mesh did not coincide with the discretisation of the dam at their mutual interface. One line of elements had to be corrected to create a consistently meshed interface. 5. The discretization of the terrain of the fine mesh required the same adaptation. Moreover, the discretisation of the terrain was not adapted to the refined discretisation of the dam. Therefore, the topmost layer of elements of the terrain mesh was refined to gain a consistent discretisation of dam and terrain at their interface. We consider the impact of these adaptations as minor. Their big advantage is that no code internal treatment and definition of the interfaces with different discretisation is needed but the meshes can be used as such. Due to the selected approach with the added mass, the developed finite element mesh for the reservoir provided by the formulator was not taken into account. The table below provides information about the mesh parameters. Figures of the mesh discretization are given further below. Table 1: Mesh parameters of the adapted meshes Total number of nodes elements Hexahedron-quadratic elements with 20 nodes wedge-quadratic elements with 15 nodes
Coarse mesh Dam Terrain 2083 11608 356 2340 312 2340 44 0
Fine mesh Dam Terrain 13733 15062 2736 3120 2736 2700 0 420
Material Parameters and Boundary Conditions The formulator provided the following linear elastic isotropic material characteristics. 150
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Table 2: Material parameters Parameter
Density Poisson – ratio Young’s – modulus
Dam 2400 kg/m3 0.167 27000 MPa
Rock Mass (Terrain) massless 0.2 25000 MPa
Considering that the outer limits of the modeled terrain are sufficiently far away from the dam, the boundary conditions at the outer surface and the bottom of the terrain were determined as springs with large stiffness that practically prevents any movement in all three directions (quasi fixed bearing). Modeling approach for added mass technique The mass and its distribution are relevant when calculating the natural frequencies of a structure as well for the dynamic analysis during the earthquake. The specific mass is calculated from density, and is in FEnas automatically taken into account in a physical manner. In order to take into account the interaction of fluids and structure, it is necessary to add a mass of water which oscillates together with the dam. Water joined to the body dam is added to the points of the upstream face as an additional concentrated mass according to Westergaard’s approach in its generalized form [3, 4]. Westergaard showed that the hydrodynamic pressures exerted on the face of the dam due to the earthquake ground motion is equivalent to the inertia forces of a body of water attached to the dam and moving back and forth with the dam while the rest of reservoir water remains inactive. He suggested a parabolic shape for this body of water with a base width equal to 7/8 of the height. In the generalized Westergaard method [3, 4] normal hydrodynamic pressure Pn at any point on the curved surface of the dam is proportional to the total normal acceleration shown in figure below, i.e.: (1) ̈ i
7 w H i (H i Zi ) 8
(2)
Figure 1: Westergaard hydrodynamic pressure [3] Where: ̈ - Total normal acceleration at point i i - Westergaard pressure coefficient at point i 151
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w - Mass density of water H i - Depth of water at vertical section that includes point i Z i - Height of point i above the base of the dam The normal pressure Pn at each point is then converted to an equivalent normal hydrodynamic force by multiplying by the tributary area associated with that point. This procedure is semiautomated in FEnas, the user has to define the surface load according to Westergaard, while the conversion from surface pressure to added mass at nodes is done automatically by the program. Since the Westergaard coefficient depends on the depth of water in front of the upstream face, which varies with dam side, the value H is taken in accordance with the colors on the next figure, one average value of H correspond for each color.
Figure 2: Westergaard added masses technique in FEnas (left: zones of similar H, right, hydrodynamic forces as vectors, colours show magnitude of force in kN/m2)
Modelling Approach for Self-Weight and Hydrostatic Load Deadweight or self-weight refer to the same load case. Application of this load case generally mimics the construction schedule. Here, a pragmatic two step modeling approach is implemented. Superposition of the two steps represents deadweight. In the first step, every second cantilever is modeled massless with a very low Young’s modulus, whereas all other cantilevers are modeled by their correct properties (cf. Table 2). In the second step, this procedure is repeated exchanging the material parameters between the two sets of cantilevers. For the further analysis, the joints are considered as grouted and the entire construction retains hydrostatic pressure as arch dam. Hydrostatic load has been defined as surface load perpendicular to the surface of all elements at the upstream surface. The magnitude of the surface load depends on the water level with respect to the vertical position of the element. Modelling Approach for Determining the Eigenfrequencies The formulator required calculation of the first 10 eigenfrequencies of the structure; this includes the interaction with the reservoir, i.e. the eigenfrequencies of the dam together with the added mass of water are to be calculated. The lowest natural frequencies are determined by resolving the characteristic value or eigenvalue problem. Here, only iterative methods can be used to solve eigenvalues problems. FEnas uses the “subspace iteration method”. The modes of oscillation are estimated for a number of start vectors (the subspace) and then iteration is used for continuous improvements. The minimum number of start vector required is equal to the number of requested eigenvalues; the maximum number is equal to the number of degrees of freedom of the entire FE structure.
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Modelling Approach for Transient Seismic Loads Linear dynamic analysis was performed using direct time integration (time history analysis). As method for integration of the dynamic equations of motion we have used the Newmark method. The accelerograms given by the formulator were used as such and are plotted in the following figure.
Figure 3: Accelerograms in x, y, z direction The acceleration values are given on the interval of 0.01 s. All performed simulation used time steps with the same duration. With a total of 2000 time steps of 0.01 s, the total duration of the earthquake is 20 s. The dynamic analysis is taking into account the Rayleigh damping effect. Raleigh dumping is based on assumption that damping is proportional to the stiffness and mass matrix of the structure. Here, a damping of 5% for the first two eigenfrequencies was considered.
Results Eigenfrequencies Table 3: Eigenfrequencies Mode No. 1 2 3 4 5 6 7 8 9
Coarse mesh Fine mesh Frequency [Hz] Period [s] Frequency [Hz] Period [s] 1.25 0.8000 1.26 0.7937 1.34 0.7463 1.32 0.7576 2.05 0.4878 2.01 0.4975 2.33 0.4292 2.36 0.4237 2.47 0.4049 2.50 0.4000 3.01 0.3322 3.00 0.3333 3.15 0.3175 3.17 0.3155 3.66 0.2732 3.65 0.2740 3.70 0.2703 3.70 0.2703 153
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10
3.88
0.2577
3.88
0.2577
Mode Shape
Figure 4: Radial deformation [mm] with mode shapes of the eigenfrequencies exaggerated by a factor of 17000 Stress Analysis All evaluations of results of stresses were done for vertical stresses, hoop stresses, and where considered useful for principal stresses 1 and 3. Note that the vertical stresses shown are not 154
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vertical with respect to a global coordinate system but vertical with respect to the element coordinate system. This means that the stresses shown are parallel to the element and dam surface. Stresses due to Static loads Stresses of static loads are presented separately for the load of self-weight and for the combination of self-weight and hydrostatic pressure load. They are presented for the upstream and downstream surface allowing checking if the model results are plausible.
Figure 5: Vertical and hoop stresses [MPa] due to self-weight
Figure 6: Vertical and hoop stresses [MPa] due to self-weight and hydrostatic pressure 155
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Seismic stress-analysis depending on the time Monitoring was performed for the vertical and hoop stresses for selected elements on the upstream and downstream face of the dam. There were selected three elements of both surfaces in three different heights at main cross section defined by the formulator, which is considered later (cf. Figure 10). On the figures below you can see the position of the elements at which stresses are monitored. Each of the figures shown below includes minimum (MIN) and maximum (MAX) stress observed at the selected elements (each of the selected elements includes 20 nodes). Note that these are the stress due to seismic loading only, self-weight and water load are not directly considered, water to a certain extend as the seismic loading affects the Westergaard masses as well.
Figure 7: Monitored elements on main cross section
Figure 8: Time series of stresses observed at monitored elements on upstream surface
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Figure 9: Time series of stresses observed at monitored elements on downstream surface Evaluation of the stresses for three different cross sections For three cross-sections, one in the center and two at each side halfway between the center and the right respectively left end of the dam, stresses over dam height are shown for the upstream and downstream surface. The three examined cross-sections are marked on the figure below.
Left section
Main section
Right section
Figure 10: Layout of analyzed cross sections The following loads respectively load combinations are plotted: Self-weight load - [SW] Self-weight + hydrostatic pressure load – Normal operating conditions – [NOC] Maximum of self-weight + hydrostatic pressure + seismic load – [MAX] Minimum of self-weight + hydrostatic pressure + seismic load – [MIN]
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It should be noted that the last two evaluations, maximum and minimum, represent envelopes of stresses, i.e. this state of stress is artificial, because the maximum / minimum do not occur at all nodes at the same time.
Figure 11: Stresses for main cross section
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Figure 12: Stresses for left cross section
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Figure 13: Stresses for right cross section Radial Deformation The evaluation of results of radial deformation was done considering the following sequences: Radial deformation due to the static loads Seismic radial deformation-analysis depending on the time Radial deformation analysis for three cross-section (cf. above) Radial deformation of the static loads Radial deformations of the static loads are presented separately for the load of self-weight and for the combination of self-weight and hydrostatic pressure load.
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Figure 14: Radial deformation [mm] – self-weight (left), self-weight + hydrostatic pressure (right) Seismic radial deformation-analysis depending on the time Monitoring was performed for the radial deformation for selected nodes on the upstream face of the dam. There were selected three nodes of upstream surfaces in three different heights at the three cross section defined by the formulator (see Figure 10). On the figures below you can see the position of the nodes at which radial deformation are monitored. Note that these are the radial deformations due to seismic loading only, self-weight and water load are not directly considered, water to a certain extend as the seismic loading affects the Westergaard masses as well.
Figure 15: Monitored nodes on upstream surface
Figure 16: Radial deformation for left, main and right cross section respectively
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Radial deformation at the three cross sections For the three cross-sections the radial deformations over height are plotted. Evaluation of the radial deformation is given for the following loads respectively combination of loads: - Self-weight load - [SW] - hydrostatic pressure load – Normal operating conditions – [NOC] - Maximum of hydrostatic pressure + seismic load – [MAX] - Minimum of hydrostatic pressure + seismic load – [MIN] These combinations mimic the observations of a pendulum.
Figure 17: Radial deformation – main cross section
Figure 18: Radial deformation – left cross section
Figure 19: Radial deformation – right cross section
Discussion and Summary Stress-strain analysis of arch dam during seismic activity including reservoir-structure interaction using added masses technique according to generalized Westergaard approach is presented. Interaction of an arch dam with the impounded water leads to an increase in the dam vibration periods. The fact that water moves with the dam increases the total mass that is in motion. This added mass increases the natural periods of the dam, which in turn affects on the effective earthquake inertia forces. The Westergaard method usually gives the largest 162
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added-mass values, which is evident by its increasing the vibration periods the most. However, this does not automatically give the largest stresses, because response of the dam also depends on the characteristics of the earthquake ground motion. Although Westergaard approach is widespread in practice, for different variants of quasi-static and dynamic analysis of seismic effects, the above-mentioned facts lead to the conclusion that the results must be taken with caution. After comparison of two different sizes of spatial discretization of meshes we can conclude, that the coarser mesh also gives satisfactory results which can be used in engineering practice. In fact, the differences are small. Whereas the performed simulations with linear material behavior are not very costly in terms of computation time and required computer power for both meshes, the coarser mesh could be very useful as soon as more complex processes, e.g. non-linear material behavior, are to be considered. Maximum earthquake stresses are located in the central upper portion of the dam as well along the dam-foundation contact zone. Maximum radial deformation occurs in the crest, and decreasing towards lateral sides, as well as towards fixed end. The vertical stresses observed at the upstream surface show some artefacts for the load cases with the earthquake as they are supposed to be zero at the top of the dam like it is observed for self-weight and self-weight with hydrostatic pressure. These non-zero values are considered to be artefacts. The same artefact in a much smaller magnitude is seen at the top of the downstream surface. Stress time histories curves shows several significant tensile stress cycles that can lead to open the contraction joints, especially in crown zone, while on the time histories of radial deformation in addition to timing of deformation could be noted as dominant periods of the first few natural frequencies. The results that are shown have been compared to typical deformations observed at Swiss dams. The overall behavior of the dam here is in good agreement with what AF has seen in reality. The comparison of the results of the earthquake simulations with previous studies of this type performed at AF was positive. We can conclude that the added mass technique in combination with the direct dynamic analysis relatively quickly and easy produce results useful in engineering practice.
Acknowledgements The FEnas developer and support team around Dr. Urs Trüeb provided valuable support in the conversion of the meshes into the FEnas mesh format.
References [1] Graz University of Technology, Institute of Hydraulic Engineering and Water Resources Management, 12th International Benchmark Workshop on Numerical Analysis of Dams, 2.-4. OCTOBER, 2013, GRAZ – AUSTRIA, THEME A - Fluid Structure Interaction Dam – Reservoir at Seismic loading. Download from Arch http://portal.tugraz.at/portal/page/portal/Files/i2130/Icold_bmws2013/Theme_A/Files_an d_Information_Theme_A.zip, 23.07.2013 [2] FENAS, Walder + Trüeb Engineering AG. (2013). Walder + Trüeb Engineering AG. http://www.waldertrueb.ch/index.php?id=23&L=2%3Fcomply, 23.07.2013 [3] Ghanaat, Y. (1993). Theoretical Manual for Analysis of Arch Dams. US Army Corps of Engineers. http://www.dtic.mil/dtic/tr/fulltext/u2/a269682.pdf, 24.07.2013 [4] Kuo, J. (1982). Fluid-structure interactions: Added mass computations for incompressible fluid. Earthquake Engineering Research Center. Berkeley: University of California.
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Fluid Structure Interaction Arch Dam-Reservoir at Seismic Loading S. Shahriari1 1
Institute of Hydraulic Engineering and Water Resources Management, Stremayrgasse 10/2, A-8010 Graz, AUSTRIA E-mail: [email protected]
Abstract In the present paper, a direct time domain procedure is used for dynamic linear analysis of the coupled arch dam-reservoir-foundation system. The hydrodynamic force on the upstream face of the dam is modeled by added mass method and compressible fluid elements for comparison. The concrete and massless foundation rock was assumed to be linear elastic. Connection between dam and foundation is modeled by coupling all DOF of the corresponding nodes. Also, viscous damping was applied to the materials by using RayleighDamping. Numerical results showed that the different modeling techniques of the interaction lead to different response of the system and stress distribution in the dam’s body. Modeling the interaction by added mass method, increases the period of the system as well as overestimating or underestimating the stresses compare to the model with fluid elements.
Conclusion The following conclusions are drawn based on the numerical experiments conducted herein:
Choosing appropriate damping parameters is a very important step in the dynamic analysis of the dam-water-foundation system. Underestimating the system’s damping by choosing higher than 7th angular frequency as the second frequency for calculation of Rayleigh-Damping parameters can lead to overestimating the stresses in the model.
Modal responses of the dam-water-foundation system are calculated using the finite element software, ANSYS. The results showed that the eigenfrequencies from the model with added mass approach were lower than those calculated from the model which utilized acoustic element to model the reservoir.
Direct time integration procedure is used for dynamic analysis of the system and deformations and stresses are calculated for static and dynamic load cases. Maximum deformation due to static loading was 7.4 cm at the crest level followed by the maximum deformation of 13.6 cm after dynamic analysis. The tensions in arch dams are not desirable; therefore, the tension stresses were studied. Evaluation of hoop stresses indicated that there is no significant tension developed in the main section with maximum value of 0.6 MPa. Furthermore, the maximum tension stress value in the U/S face of the right and left sections is higher than the main section(S=3.6 MPa). Vertical stresses evaluation showed that the maximum tension was occurred at the base of the U/S face of the main section and found to be 6.69 MPa. Also, the maximum compression stress is developed (S=12.5 MPa) at 180 m from the foundation level of the U/S face of the main section.
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The Seismic behaviour of an Arch Dam-ReservoirFoundation System A. Frigerio1 and G. Mazzà2 1
Ricerca sul Sistema Energetico – RSE S.p.A., via R. Rubattino 54, Milan, ITALY E-mail: [email protected]
Abstract The paper summarizes the main features and the related results of the linear static, modal and seismic analyses carried out on the concrete arch dam proposed as Theme A of the 12 th ICOLD International Benchmark Workshop. The analyses have been performed by means of the COMSOL Multiphysics software, making reference to the coarse finite elements mesh provided by the Formulators, but some slightly changes. In the modal and dynamic analysis, the water-dam-foundation interaction has been taken into account. The impounded water has been modeled by means of acoustic finite elements, assuming appropriate boundary conditions to simulate the wave reflecting conditions between the fluid and the foundation, as well as the fluid and the upstream face of the dam, and the non-reflecting condition at the end of the reservoir channel. Results have been provided according to the requests of the Formulators in terms of radial displacements, hoop and vertical stresses on three vertical sections of the dam.
Introduction Seismic analyses have been proposed and discussed during some of the first Benchmark Workshops organized by the Committee on Computational Aspects of Analysis and Design of Dams [1,2]. Hence, it appears appropriate the proposal of the present seismic analysis of a large arch dam, [3], in order to verify the possible improvements both on the modeling and on the computational aspects. In the present paper, all the analyses have been carried out with COMSOL Multiphysics, a software that allows modeling and simulating any physics-based system. In particular, the COMSOL Acoustic, [4], and Structural Mechanics Modules, [5], have been used. In the following paragraphs the geometrical and physical model will be described and the results will be provided according to the requests of the Formulators.
Geometrical and physical model The coarse Finite Element mesh provided by the Formulators, [3], has been slightly modified in order to have an exact geometrical match at the interface between the nodes of the fluid and those of the foundation rock. The reservoir and the foundation domains have been extended upstream to avoid possible numerical influences of the non–reflecting water surface at the end of the reservoir on the dynamic response of the dam under seismic loading conditions. Finally, the finite elements of the central vertical sections of the dam have been refined to discretize with a more regular grid the whole mesh of the dam (Figure 1). The displacement field of each finite element of the numerical model has been discretized with quadratic shape functions.
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Figure 1: The coarse mesh used to study the dam-reservoir-foundation system Material properties Linear elastic constitutive models have been assigned to the dam and the foundation rock. The physical-mechanical properties are summarized in Table 1. Table 1: Concrete and rock properties Domain
Material
Dam Foundation
Concrete Rock
Density [kg/m3] 2400 0
Poisson ratio 0.167 0.200
Young modulus [MPa] 27000 25000
The Rayleight damping model has been taken into account to define the dam behavior during seismic loading conditions. Assuming a 5% structural damping ratio, the mass and stiffness damping parameters of the Rayleight formulation are as follows: = 0.94; = 2.65E-03 The reservoir has been discretized by means of acoustic finite elements whose properties are reported in Table 2. Table 2: Water properties Domain Reservoir
Material Water
Density [kg/m3] 1000
Speed of sound in water [m/s] 1500
Bulk modulus [MPa] 2200
Loading and boundary conditions According to the formulation of the Theme A, the following loadings sequence has been considered: 168
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Dead loads Hydrostatic water pressure with the maximum water level equal to the dam crest height (i.e. 715 m a.s.l.) Seismic loads, provided by the Formulators in terms of accelerations along the three Cartesian directions.
Different boundary conditions have been used for the foundation domain depending on the type of the analysis performed. In case of static and modal analyses, symmetric conditions have been applied to the lateral rock walls whereas fixed constraints have been assigned to the basement rock. In seismic analysis, the accelerations - or the equivalent related displacement varying in time - have been applied to the lateral and bottom surfaces of the foundation. In modal and seismic analyses, the fluid domain has been modeled by means of acoustic elements, assigning the following boundary conditions:
Dam-reservoir interface (refer to the COMSOL “node” acoustic-structure boundary, [4]):
(
)
(1)
(2)
where n is the normal to the interface, the water density, p the fluid “acoustic” pressure, qd the dipole source (equal to zero in the present case), utt the acceleration field of the structural domain at the fluid interface and the stresses tensor.
Foundation-reservoir interface (refer to the COMSOL “node” impedance, [4]):
(
169
)
(3)
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where n is the normal to the reservoir-foundation interface, the water density, p the fluid “acoustic” pressure, qd the dipole source (equal to zero in the present case), Zi the acoustic input impedance assumed equal to: (4)
/q
being q a damping coefficient that characterizes the effects of absorption of the hydrodynamic pressure waves at the boundary, according to the following equation [6]: (5) is the wave reflection coefficient that accounts for the behavior of the absorption of hydrodynamic pressure waves at the boundary, whereas c is the speed of sound in water. According to some literature case studies, has been considered equal to 0.75, [7].
Upstream-reservoir surface (refer to the COMSOL “node” plane wave radiation, [4]):
(
)
(
)
(6)
where n is the normal to the reservoir-foundation interface, the water density, p the fluid “acoustic” pressure, qd the dipole source (equal to zero in the present case), c is the speed of sound in water and Qi the monopole source (equal to zero in the present case).
Free surface (refer to the COMSOL “node” sound soft boundary, [4]):
(7)
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Numerical analyses In the following paragraphs the main features of each numerical analysis will be provided, specifying the assumed hypotheses. Static analysis A linear static analysis has been carried out in order to apply the dead load and the hydrostatic water pressure, thus the water domain has not been included in the geometrical model. The applied boundary conditions are reported in the previous paragraph. Modal analysis The first 10 eigenfrequencies of the dam have been computed taking into account the damreservoir interaction. The applied boundary conditions are reported in the previous paragraph. Seismic analysis Seismic analyses in time domain are generally carried out considering a massless foundation and applying a spatially-uniform ground motion directly at the basement rock. In the present paper, at first an ordinary differential equations problem has been solved, making reference to the foundation domain only, in order to compute the displacements associated to the earthquake motions (Figure 2). In the subsequent dynamic analysis, these displacements have been applied to the bottom and lateral rock walls to calculate the dynamic response of the dam-reservoir-foundation system. This procedure has been chosen because it allows computing easily the relative displacements of the system; anyway, as an alternative, the same results could be attained applying directly the accelerations to the foundation boundaries. The boundary conditions taken into account during the seismic analysis are reported in the previous paragraph.
Figure 2: Acceleration time history (amax 0.1g)
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Results With reference to the results of modal analysis, the first 10 eigenfrequencies and the related modal shapes are reported in Table 3. Table 3: Eigenfrequencies and modal shapes 1
1.54247 [Hz]
6
2.51800 [Hz]
2
1.55194 [Hz]
7
2.83905 [Hz]
3
2.09799 [Hz]
8
2.96151 [Hz]
4
2.22299 [Hz]
9
3.19079 [Hz]
5
2.33002 [Hz]
10
3.37707 [Hz]
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According to the requests specified by the Formulators, the results of the static and seismic analyses are provided in terms of radial displacements [m], hoop and vertical stresses [MPa] on the upstream and downstream faces of three vertical sections of the dam. Table 4: Upstream face – Central section (US-C) – Radial displacements, hoop and vertical stresses
Table 5: Downstream face – Central section (DS-C) – Radial displacements, hoop and vertical stresses
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Table 6: Upstream face – Right section (US-R) – Radial displacements, hoop and vertical stresses
Table 7: Downstream face – Right section (DS-R) – Radial displacements, hoop and vertical stresses
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Table 8: Upstream face – Left section (US-L) – Radial displacements, hoop and vertical stresses
Table 9: Downstream face – Left section (DS-L) – Radial displacements, hoop and vertical stresses
Conclusion
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The paper presents the dynamic response of the dam-reservoir-foundation system according to the basic requests of the Formulators. Only the coarse mesh has been considered as some preliminary analyses have demonstrated that, compared to a high computational effort, results showed only slight differences. A second aspect to be emphasized refers to the boundary conditions assumed for the fluid domain at the rock interface: if a fully reflecting condition is assumed, the stresses computed on the dam are considerably higher than those attained in the case discussed in this paper where a partial absorption conditions has been assumed. This means that experimental investigations should be undertaken in order to define more realistic boundary conditions. A suggestion for the Formulators, in the synthesis phase, refers to the opportunity to make some comparisons between the main outcomes of Theme A and those of the previous Benchmark Workshops, cited above.
Acknowledgements This work has been financed by the Research Fund for the Italian Electrical System under the Contract Agreement between RSE S.p.A. and the Ministry of Economic Development General Directorate for Nuclear Energy, Renewable Energy and Energy Efficiency in compliance with the Decree of March 8, 2006.
References [1] Theme A – Seismic analysis of the Talvacchia dam. (1992). 2nd ICOLD International Benchmark Workshop on Numerical Analysis of Dams. Committee on Computational Aspects of Dam Analysis and Design. Bergamo, Italy. [2] Theme A1 – Earthquake of an arch dam including the nonlinear effects of contraction joint opening. (1996). 4th ICOLD International Benchmark Workshop on Numerical Analysis of Dams. Committee on Computational Aspects of Dam Analysis and Design. Madrid, Spain. [3] Zenz G., Goldgruber M. (2013). Theme A formulation. Fluid Structure Interaction. Arch Dam – Reservoir at Seismic loading. 12th ICOLD Benchmark Workshop on Numerical Analysis of Dams, Graz, Austria. [4] COMSOL Multiphysics. (2013). Acoustic Module – User’s Guide. Version 4.3b [5] COMSOL Multiphysics. (2013). Structural Mechanics Module – User’s Guide. Version 4.3b [6] Ka-Lun Fok, Anil K. Chopra. (1986). Earthquake analysis of arch dams including damwater interaction, reservoir boundary absorption and foundation flexibility. Earthquake engineering and structural dynamics, Vol.1, pp 155-184. [7] Ghanaat Y., Redpath B.B. (1995). Measurement of reservoir-bottom reflection coefficient at seven concrete dam sites. QUEST Structure Report No. QS95-01.
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12th International Benchmark Workshop on Numerical Analysis of Dams Theme A: Fluid Structure Interaction Arch Dam – Reservoir at Seismic Loading A. Diallo1 and E. Robbe1, 1
EDF-CIH, 15 Avenue Lac du Bourget Savoie Technolac, 73370 Le Bourget du Lac, FRANCE E-mail: [email protected], [email protected]
Abstract Dynamic response of arch dams to earthquake ground motion is significantly affected by the interaction between the dam and the impounded water. There are several approaches to take into account this dynamic dam-water interaction. The purpose of this benchmark is to compare different modelling techniques with different levels of precision and therefore with different computing efforts. This paper presents the results of the 3 approaches investigated and the fundamental hypotheses adopted for each them. The first approach is a generalized Westergaard added mass, the second approach is an incompressible finite-element added mass and the third approach is based on a sub structuring method where the fluid is compressible. All investigations are carried out for an artificially generated symmetrical arch dam and simplified loading and boundary conditions. In general, Westergaard added mass yields higher compressive and tensile stresses, as well as higher radial displacement. The sub structuring approach, where the compressibility of the water and the impedance of the foundation are taken into account, yields lower stresses.
Introduction The objective of this paper is to present the seismic analysis of a 220 m high double curvature arch dam undertaken as part of the 12th benchmark study. The results of the analysis are presented for the 3 approaches used to take into account dam-water interaction in accordance with the general assumptions made by the benchmark organizing committee. The analyses of the results focus on the impact of the hydrodynamic approach used on the computed stresses and displacements of the dam. The following approaches are presented: Generalized Westergaard added mass Incompressible finite element added mass Sub structuring method where water compressibility is taken into account. For each approach, the fundamental hypotheses are presented and the physical justification is given.
Description of arch dam analysis - modelling methods The analyses were performed with Code_Aster. A computer program developed by EDF; it offers a full range of multiphysical analysis and modelling methods.
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Materiel parameters Because a modal superposition method is used for the three approaches, the Rayleigh damping cannot be used, rather, a value of modal damping is specified for each mode. This value is fixed at 5% which is the value usually considered in real projects. The mass density of rock is not specified by the benchmark organizing committee because the traditional method assumes that the foundation is massless, however the sub structuring method investigated takes into account the impedance matrix of the foundation rock. It is therefore necessary to specify the mass density of rock; a value of 2400 kg/m3 is considered for the rock in this analysis. The other material properties are considered in accordance with the general assumptions made by the benchmark organizing committee. Although in practice the dynamic modulus of elasticity is usually considered 25 percent greater than static modulus of elasticity, in this analysis the supplied static values are respected. Static analysis The application of the dead load should consider the manner in which the dam was constructed. Arch dams are often constructed as independent cantilever blocks separated by vertical joints. Since these joints are not capable of transferring dead load horizontally until they are grouted, dead loads should be applied to individual cantilever to simulate this condition. This may be accomplished by performing dead load analysis in two steps. First, dead loads are applied to even cantilevers (Set-1) and the stresses are extracted. In second analysis, the dead loads are applied to odd cantilevers (Set-2) separately and the stresses are extracted. The addition of the stresses obtained from the two steps is considered as the initial stress state of the dam and the displacement of the dam due to the dead load is not considered. Water loads due to the hydrostatic pressures of the normal water level are external forces acting on the u/s face of the dam. The hydrostatic pressures are applied to the monolithic arch structure after the construction joints are grouted. Dynamic modelling The methods used in this analysis are based on the modal superposition method. The modal analyses performed for the empty dam show that 90 percent of the mass of the dam is excited by a frequency range between 1.93 and 10 Hz. The maximum of the seismic is also in the same range (1-10 Hz).A significant amplification can therefore be expected. The different approaches used to take into account the fluid-structure interaction are: a) Generalized Westergaard Added-mass The added-mass representation of dam-water interaction during earthquake ground shaking was first introduced by Westergaard [1]. In his analysis of a rigid 2D gravity dam with a vertical upstream face, Westergaard showed that the hydrodynamic pressures exerted on the face of the dam due to the earthquake ground motion is equivalent to the inertia forces of a body of water attached to the dam and moving back and forth with the dam while the rest of reservoir water remains inactive. A general form of the Westergaard added-mass concept which accounts for the 3D geometry [2] can be applied to the earthquake analysis of arch dams. The general formulation is based on the same parabolic pressure distribution with depth used by Westergaard, except that it makes use of the fact that the normal hydrodynamic pressure at any point on the curved surface of the dam is proportional to the total normal acceleration, ̈ : (1) ̈
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(2)
√
Density of water; Westergaard pressure coefficient; H: Water depth; Z: Level of the point on the curved surface of the dam. The normal pressure at each point is then converted to an equivalent normal hydrodynamic force by multiplying by the tributary area associated with that point. b) Incompressible Finite-Element Added Mass – Potential Approach The added-mass representation of the impounded water can be obtained more accurately by a finite-element solution of the pressure wave equation, which fully accounts for the complex geometry of the dam and the reservoir. The impounded water represented by the wave equation is discretized using a finite element mesh of incompressible liquid elements. The solution is obtained by numerical procedures with the following boundary conditions: (3) The hydrodynamic pressures at the water – free surface are assumed to be zero, that is, the effects of surface waves are neglected. The reservoir bottom and sides, as well as a vertical plane at the upstream end of the reservoir model, are assumed to be rigid. For rigid boundaries the normal pressure gradients or the total normal accelerations are zero. The normal pressure gradients at the dam-water interface are proportional to the total normal accelerations of the fluid. The computed pressures for the nodal points on the upstream face of the dam are then converted into equivalent nodal forces, from which an added-mass matrix representing the inertial effects of the incompressible water is obtained.
c) Compressible Water with Absorptive Reservoir Boundary- Substructuring Method The substructuring method consists of dividing the complete system into three substructures: the structure, the water, and the foundation, each of which can be partially analysed independently of the others. The structure is represented by a 3D finite element, which permits modelling of a general geometry and linear elastic material properties. The water domain and the foundation region are represented by boundary elements. The added mass representation of the impounded water described above ignores the effect of water compressibility and reservoir boundary absorption. However, the water compressibility and reservoir boundary absorption can significantly affect the hydrodynamic pressure and hence response of arch dams to earthquakes [3]. Interaction of the dam with the foundation rock leads to an increase in vibration periods, primarily due to the flexibility of the foundation rock. Dam-foundation interaction also decreases the dam response if damping arising from material damping in the foundation rock and radiation damping associated with wave propagation away from the dam are considered in the analysis. Procedures for earthquake response analysis of arch dams including dam-water interaction, water compressibility, reservoir boundary effects and dam-foundation interaction are developed by EDF [4]. In this procedure, the radiative damping and the hysteretic damping of the foundation are also considered. To represent the infinity domain, Green’s functions are used in the fluid domain and in the foundation domain. In the fluid domain, the Helmholtz equation is discretized using boundary elements. 179
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(4) The solution is obtained by numerical procedures with the following boundary conditions: The hydrodynamic pressures at the water – free surface are assumed to be zero, that is, the effects of surface waves are neglected.
The normal pressure gradients at the dam-water interface are proportional to the total normal accelerations of the fluid.
The normal pressure gradients at the foundation-water interface (reservoir boundary) are proportional to the total normal accelerations of the fluid.
The hydrodynamic pressure wave impinging on the reservoir boundary is partly reflected into the water, and partly refracted (absorbed) into the boundary materials. The partial absorption at the reservoir boundary is approximately represented by a reflection coefficient known as “α”, which is the ratio of reflected to incident wave amplitude.
In the foundation domain, the Navier equation is also discretized using boundary elements. To take into account the infinity domain, Green functions are used. (5) The solution is obtained by numerical procedures with the following assumptions:
The rock foundation is assumed homogeneous, isotropic and linear elastic.
A value of 5% is considered for the modal damping.
Results and comparison of the different approaches Modal Analysis The interaction of an arch dam with the impounded water leads to a decrease in the dam vibration frequencies. This is because the dam cannot move without displacing the water in contact with it. The fact that water moves with the dam increases the total mass that is in motion. This added mass decreases the natural frequencies of the dam, which in turn affects the response spectrum ordinate and hence the effective earthquake inertia forces. The flexibility of the foundation rock also decreases vibration frequencies of the dam. Table 1 gives the fundamental modes obtained by the 3 approaches investigated in this analysis. The results show that the Westergaard method gives the largest added-mass value, as evidenced by its decreasing the fundamental frequency the most. However, this does not automatically means that Westergaard approach gives the largest stresses, because the response of the dam also depends on the characteristics of the earthquake ground motion.
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Table 1: fundamental period of the dam Empty reservoir
Fundamental frequency
1.93 Hz
Full reservoir Westergaard Incompressible fluid elements
1.29 Hz
1.57 Hz
Compressible fluid element with 50% wave absorption
1.49 Hz
Stresses and displacement Under static loads, the maximum radial displacement reaches 8.32 cm at the top of the central cantilever. This displacement is only due to hydrostatic pressure acting on the u/s face of the dam. The relatively symmetrical deformation of the arch dam leads to the compression of the lower portion of the d/s face of the dam and to tractions in the abutments on the u/s face of the dam. The maximum static principal stress (compression) reaches 9.32 MPa at lower portion in the abutments on the d/s face of the dam. The minimum static principal stress (tensile) also reaches 4.40 MPa at the lower portion of the u/s of the dam. This tensile stress (4.40 MPa) could lead to the opening of a joint at the contact between dam and the foundation rock. However the contraction joint opening/closing is not modelled. Figures 1 and 2 give the principal stress contours under static loads.
Figure 1: Principal stresses (compression) under static loads
Figure 2: Principal stresses (tensile) under static loads During the ground motion, the maximum displacement of dynamic vibrations is about 6 cm with the respect to the initial displaced shape of the dam. The maximum displacements represented in fig 3 show that the Westergaard method yields higher radial displacements at the crest of the dam. The substructuring method where we take 181
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into account the radiative damping and hysteric damping in the foundation as well wave absorption at the reservoir boundary, yields lower displacements (20% of reduction). 250
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200 150 100 50 0 0.00
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0.16
displacement (m) min_westergaard static max_substructuring
min_potentiel_approach max_westergaard
min_substructuring max_potentiel_approach
Figure 3: Maximum displacement of the central cantilever at the d/s face of the dam In general, the Westergaard method yields higher compressive and tensile stresses. Indeed, the seismic tensile vertical stresses at the u/s face of the main cantilever vary from 2.4 MPa for the Westergaard method to 0.2 MPa for the substructuring method at a point at ¾ of the dam height (cf fig4). 250
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Figure 4: Vertical stresses at u/s face during the ground motion
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Figure 5 gives maximum hoop stresses at the d/s face of the central cantilever during the earthquake. It shows a tensile stress varying from 1.6 MPa for Sub structuring method to 2.2 MPa for Westergaard and potential approach at the top part of the cantilever; this could lead to the opening of vertical joints. 250
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Figure 5: Hoop stresses at d/s face during the ground motion In general, the ground motion increases the maximum principal stress observed at the lower portion of the abutments on the d/s face of the dam as well the minimum principal stress observed at the lower portion of u/s face of the dam. Thus, the maximum compressive stresses during the ground motion varying from 11.26 MPa for sub structuring method to 12.77 MPa for added mass approach on the d/s face of dam (cf fig6 right_section). At ¾ dam height on the u/s face, the maximum compressive stresses also varying slightly, from 9.82 MPa for sub- structuring approach to 10.45 MPa (cf fig 6 main_section).
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Figure 6: Maximum principal stresses at u/s face of central cantilever and d/s face of right section In figure 7, we notice that the substructuring method increases slightly the tensile stress at the contact between dam and the foundation (2.84 MPa under static loads and 3.14 MPa during earthquake). However this amplification is very significant with added-mass methods, thus with Westergaard added mass, the tensile stress reaches 5.96 MPa during the earthquake and with the potential approach it is about 5.24 MPa. These tensile stresses could lead to the opening of the contact between the dam and the foundation.
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240 220
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200 180 160 140 120 100 80 7.00
6.00
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4.00
3.00 2.00 Stresses (MPa)
1.00
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0.00
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Figure 7: Minimum principal stresses (tension) at the u/s face of right section
Influence of reservoir boundary absorption A hydrodynamic pressure wave impinging on the reservoir boundary is partly reflected into the water, and partly refracted (absorbed) into the boundary materials. If the reservoir boundary materials are relatively soft, an important fraction of the reservoir water energy can be absorbed, leading to a major reduction in the dynamic response of the dam. Therefore, the values of the absorption ratio for the design and safety evaluation of dams subjected to earthquake loading should be measured or selected conservatively. The purpose of this section is to show the effect of this absorption on the dynamic response of the dam by studying 3 cases of absorption (0%, 50% and 100% absorption). As expected, wave absorption at the reservoir boundary reduces significantly the dynamic response of the dam. Figure 8 gives the radial spectrum at the crest of the dam and for fundamental frequency (f1=1.49 Hz), we observe that the correspondent pseudo acceleration varies from 1.93g (total absorption) to 4.14 g (without absorption).
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Pseudo-acceleration en (g)
10.00
1.00
0.10
0.01
0.00 0.10
1.00 Without_absorption
frequence in (Hz)
10.00
50%- absorption
100.00 total_absoption
Figure 8: radial spectrum at the crest of the dam for 3 cases of absorption Reservoir boundary absorption also decreases compressive and tensile stresses (cf fig 9), as well as the radial displacement (cf fig 10). In figure 9, we observe that the maximum stress varies from 8.95 MPa (total absorption) to 11.26 MPa (without absorption) at ¾ dam height on the u/s face of the main section. Figure 10 gives the radial displacement of the central cantilever for 3 cases of absorption and we can observe that the crest displacement varies from 11.4 cm (total absorption) to 15.9 cm (without absorption). 250
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Stresses(MPa) min_without_abs min_50%_abs
static max_50%_abs
max_without_abs min_total_abs
Figure 9: Hoop stress of the central cantilever for 3 cases of absorption
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250
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displacement (m) min_without_abs
static
max_without_abs
max_50%abs
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max_total_abs
min_50%abs
Figure 10: Radial displacement of central cantilever for 3 cases of absorption
Conclusion The different approaches to hydrodynamic effect modelling investigated in the present study lead to similar behaviour regarding the structural response of the example arch dam. In general, the Westergaard approach yields higher compressive and tensile stresses, as well as higher radial displacements. The substructuring method where wave absorption is taken into account at the reservoir boundary decreases significantly the dynamic response of the dam due to the increased damping of the coupled system.
Acknowledgements The authors wish to thank Electricité de France (EDF-CIH) who financed this study. Particular thanks to Frederic LAUGIER, head of the Civil Engineering – Structures department for his assistance in preparing the study, and to EDF-R&D for their collaboration.
References [1] Westergaard, H.M (1933), “Water pressure on dam during earthquakes”, transactions American Society of Civil engineers, Vol 98. [2] Kuo, 1982 “Fluid structure interactions: Added Mass Computations for incompressible fluid”. Report No. UCB/EERC-82/09, University of California Earthquake Engineering Research Center, Berkeley. [3] Fok and Choppra 1985 “ Earthquake Analysis and Response of Concrete Dams” Report No UCB/EERC-85/07, University of California Earthquake Engineering Research Center, Berkeley [4] www.code_aster.org
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THEME B
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Theme B Long Term Behaviour of a Rockfill Dam
Formulators: Camilo Marulanda, Ph.D., & Joan Manuel Larrahondo, Ph.D. INGETEC Cra 6 No. 30A-30 Bogotá, Colombia [email protected] & [email protected]
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Introduction Overview: Long Term Rockfill Behhaviour It is widely accepted that rockfill in dam shells exhibit time-dependent, creep-like behaviour ([1]-[3]). Early work by Terzaghi (1960) [2] proposed that such behaviour was due to breakage of rock particles at highly-stressed contacts. Once broken, the rockfill particles would rearrange to attain more stable configurations, thus inducing displacements. Interestingly, most of the experimental evidence available to date still suggests that the main cause of the time-dependent rockfill behaviour is particle breakage and crushing, phenomena enhanced by the presence of water [2]. Furthermore, a number of constitutive models have been proposed to describe such time dependence, including logarithmic, hyperbolic, and visco-elastic relationships between long-term strain and time [1]. Earth-core rockfill dams (ECRD) typically show dramatic deformations upon reservoir first filling [1]. Time-dependent deformation depends on water content of rock particles, such that yield is highest in saturated rock particles, while tends to fade for very dry materials [2]. Since the work by McDowell and Bolton (1998) [2], successive deformation stages are recognized in rockfill-type granular materials subjected to compression stresses as the confining stress is increased. At low stresses, deformation is due to particle rearrangement. From a micromechanical point of view, rockfill deforms mainly because of interparticle sliding and rotation [1]. As the confining stress increases, the granular skeleton becomes gradually blocked, and particle breakage and crushing are triggered. Such mechanism is called “clastic yielding”. Finally, for very high stresses the strain vs. log stress plots have upwarddirected concavity. This is attributed to the comminution limit of small particles. The time-dependent deformation of rockfill can thus be explained by progressive breakage and crushing of stressed particles [1]. Particle breakage takes place as subcritical propagation of preexisting microcracks within loaded rock particles. The velocity of crack propagation is a function of the local stress intensity factor and the prevailing relative humidity. Particle breakage combines with complex localised crushing at interparticle contacts to boost rockfill deformation [1]. Not a large number of databases are available that comprise 25 years or more of instrumentation data from a rockfill dam. The problem proposed in this document consists in reproducing the development of displacements and stresses of a real-case dam during construction, reservoir impoundment, and operation.
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Benchmark Example: Alberto Lleras Dam (Guavio Hydroelectric Project), Colombia Dam Description The case history that is being proposed as benchmark example for the Theme B of the 12th Workshop on Numerical Analysis of Dams is that of the Alberto Lleras Dam, part of the Guavio Hydroelectric Project in Colombia. The Guavio Hydroelectric Project is located in the Cundinamarca region of central Colombia, some 120 km northeast from Bogotá. The project spans between the towns of Ubalá, where the dam is located, and Mambita, where the underground powerhouse is located [4]. The project harnesses the hydropower potential of the lower Guavio River (1100 m nominal head) and comprises the following components designed by INGETEC ([4]-[6]):
The Alberto Lleras Dam: a 245-m high, clay-core rockfill dam, the highest of its kind in South America [3]. The total dam volume is about 17 x 106 m3 which impounds a 14-km long, 950 x 106 m3 reservoir. A 34-m high, 4500 m3/s capacity, ogee-type spillway controlled by radial gates and two 600-m long tunnels A 13.3-km long, 8 m diameter power tunnel and a 505-m high pressure shaft A 234-m long, 17-m wide, 35-m high underground powerhose located 560 m below ground. The total installed generating capacity of the Guavio project is 1600 MW, currently delivered by eight Pelton turbines. A 1160-m long diversion tunnel as well as two other tunnels, 2330 m and 2190 m long, respectively, that divert the Batatas and Chivor Rivers into the Guavio reservoir A 5.2-km long, 8-m diameter tailrace tunnel to return the Guavio river water back to its main course
The Guavio project was built between 1979 and 1995 [2] and first filling took place in February 1992. The average flow of the river at the dam site is 62 m³/s. Figure 1 presents an aerial view of the Guavio Project and the Alberto Lleras Dam. Available instrumentation data span from 1987 through 2013. Specifically, measurements include data from:
78 level survey points on the dam crest and faces 8 Vertical Movement Recorders (RMV) 24 Pneumatic Settlement Sensors (SNA) 40 pressure cells (CP) 26 pneumatic piezometers (PZ)
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Figure 1: General view of the Guavio Hydroelectric Project and the Alberto Lleras Dam [4] The objective for the participants of Theme B is to reproduce, via a numerical model, the displacements observed on the maximum cross section of the Alberto Lleras Dam. The minimum requirements that shall be incorporated in their numerical model are a threedimensional model using the provided simplified topography and the use of the known dam zoning and the construction sequence. Geology Overview The greater Guavio project area comprises Palaeozoic rocks overlain by Mesozoic rocks [5]. The dam site is underlain by Palaeozoic quartzite, argillite, and limestone. At the dam site, the river features a 600-m deep, narrow canyon. The upper half of the canyon at the dam site (left abutment) exhibits Cretaceous rocks. The Palaeozoic formations strike diagonally with respect to the river canyon and dip upstream. Most of these rocks below 30 m of depth are fresh, hard, and depict low permeability [5]. The limestone shows frequent karstic features (caverns) which were tackled via an extensive grouting programme. No geological faults exist at the dam foundation area [5]. The steep slopes of the canyon rendered some relief joints parallel to the river flow. No particular treatment was necessary for these joints.
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Input Data All detailed input data information is provided in this manuscript’s attachments. A description of each database follows. Dam Geometry Attachment 1 contains all drawings related to dam location geometry. Specifically, the following data files are provided:
Maximum longitudinal section and description of dam zones (as-built; Oct. 1989; DWG and PDF files) Plan view and topography (as built; 50-m elevation contours; Nov. 1989; DWG and PDF files) Dam zoning description table (XLS file), including placed volumes and as-built compaction specifications
Construction Sequence Construction of the dam started on June 1983 with foundation excavation work [7]. Fill placement operations initiated on September 1984 and ran through August 1989. In general, during the six rainy months of the year (May-October) rockfill was placed and compacted in the outer portions of the shell. During the remaining six drier months (November-April), fill placement was concentrated in the central portion of the dam, namely the core and confining rockfill [7]. Attachment 2 contains all drawings and data related to construction sequence. The following data files are provided:
Construction sequence plan (maximum longitudinal section; as-built; Nov. 1989) Fill placement sequence summary table (XLS file)
Material Properties The outer portions of the shell rockfill comprised weathered fragments of quartzite, argillite, and limestone [7]. On the other hand, the inner part of the dam below the core consisted in sound, fresh, and clean rockfill. The dam core consists mainly in shale fragments in a silty clay matrix [7] obtained from a nearby quarry named San Pedro. The core material features medium plasticity and fines content >30%. Attachment 3 contains all drawings and data related to material characterization. Specifically, the following data files are provided:
DWG, PDF, and XLS files containing rockfill properties varying with dam depth DWG, PDF, and XLS files containing earth-core properties varying with dam depth One XLS file containing as-built grain size distribution bands for each dam zone One XLS file containing compressibility curves (pre-construction) for the earth core and shell rockfill materials
For illustration purposes, material parameter data that were employed during the geotechnical design of the dam are provided next:
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Earth core and shell rockfill compressibility: an experimental programme was undertaken prior to construction to obtain design compressibility parameters from both the earth core material and the rockfill. 30-cm thick samples of dam-core material were subjected to one-dimensional consolidation tests in a 30-cm diameter oedometer. The earth-core materials featured maximum grain size of 6.5 cm and natural water content during testing. Samples were reconstituted via compaction of five layers using a 30-lb hammer falling freely from a height of 24 in. Between 160 and 200 blows per layer were applied. The rockfill material compressibility was also studied using the same large oedometer as that of the core materials. However, rockfill samples were tested dry and using the same grain size distribution as that of the construction specification (with maximum grain size of 6.5 cm). Samples were reconstituted by compaction of four layers with a vibrating table during 1.5 min each layer. During the loading process, the samples were intentionally inundated at the 16 kg/cm2 loading step. The following table presents the experimental programme initial conditions and results in terms of compressibility coefficient. Attachment 3 also contains the compressibility curves obtained from these tests. Results Max Ave Min Max Ave Min
Water content of the < # 4 sieve fraction (%) 28 24 21 -
Dry density (Ton/m3) 1.84 1.77 1.71 2.25 2.17 2.03
av (10 cm2/kg) 4.7 3.6 2.6 2.1 1.6 0.8 -3
Plasticity of the earth core material (measured values during the dam geotechnical design phase)
Liquid limit (%) Plastic limit (%) Plasticity index (%)
39 26 13
Instrumentation data During construction and particularly after reaching 2/3 of the dam height, large displacements took place within the dam core [7]. As a result, most of the cable of the pneumatic and electric instruments was destroyed. Overall, about 50% of the original instruments installed were lost due to construction operations. Available instrumentation data for this workshop span from 1987 through 2013. Attachment 4 contains all drawings and data related to instrumentation data. Specifically, the following files are provided:
Five folders with XLS data files Two layouts of installed instrumentation (as-builts; plan view and maximum longitudinal section; DWG and PDF; July 1990)
A brief description of available instrumentation data follows. 196
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Movements Surface movements: Data from a total of 63 level survey instruments, labelled 13 through 75 are provided. Survey elevation data span from 1989 through 2013.
Vertical Movement Recorders (RMV) or “crossarm gages”: nine folders are provided containing monthly RMV data, spanning from 1989 through 1995. Additionally, for the 1996-2008 period, a XLS file is provided with data collected in eight of the nine available RMVs. RMV instruments or crossarm gages consist of a series of telescopic pipe sections with alternate sections anchored to the embankment fill by horizontal channel crossarms located at certain depth intervals, typically 3 m ([8] See Figure 2). The crossarms ensure that the coupled pipes move together as compression of the fill progresses. Measuring points are located at the lower end of each inner pipe; depths to such measuring points are usually sounded by a probe with spring-loaded sensing pawls, lowered on a steel tape. To produce a measurement, the probe is lowered just below a given inner pipe and then raised until the paws latch against the lower end.
Pneumatic Settlement Sensors (SNA): two folders; instruments 28-41 with monthly data from 1987 to 1996; instruments 49-58 with monthly data from 1988 to 1995.
Figure 2: RMV (crossarm gage) schematic [8]. Left: telescopic pipe array. Right: measurement probe Pressure Cells (CP) Three folders are provided. All total pressure cell data is reported in kg/cm 2. Monthly data are provided as follows:
CP 1-24: data from 1986 to 1998; CP 1, 3, 5, 6, and 7 are at 1405 masl elevation; CP 8 through 24 are at 1463 masl elevation CP 28-37: 1987 to 1998 at 1523 masl elevation
CP 41-46: data from 1988 to 1997 at 1589 masl elevation
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Pneumatic Piezometers (PZ) One XLS file is provided that contains data collected from the dam foundation piezometers (PZ 82, 83, and 84). Data were collected monthly spanning from 1988 through 2004. 15 XLS data files are provided with data collected from the dam core piezometers (PZ 45 through 49) at 1523 masl elevation. Data were collected monthly spanning from 1987 through 2001.
Requested Results All workshop participants are kindly requested to provide a paper, 15 pages maximum, in which all assumptions are clearly stated, particularly regarding initial and boundary conditions. To facilitate results rating, the participants must also submit their key results under a prescribed format whereby an Excel file under the name ThB_ResXXXX.xls shall be created (XXXX represents a name or acronym of the participant’s organisation). Fill Displacements Displacement contours for three longitudinal sections and one cross section along the dam crest shall be presented. Results at the end of construction, at the condition prior to impoundment, and after reservoir impoundment shall be included. Stresses within the Dam Calculated principal and vertical stresses obtained from the analysis on five locations within the fill are requested. The results shall be presented both at the end of construction and after partial and full impoundment. Creep and Arching Effects A discussion supported with simulation results is requested regarding long-term displacement behaviour including possible arching effects within the dam body. Groundwater Flow The participants are requested to submit the resulting groundwater flow net under 1620 masl reservoir level water level conditions. Acknowledgements The formulators are grateful to EMGESA Colombia for allowing the use of the instrumentation data for this workshop
References [1] Oldecop, L. A. and Alonso, E. E. (2007). Theoretical Investigation of the TimeDependent Behaviour of Rockfill. Géotechnique, V. 57, No. 3, pp. 289–301 [2] Oldecop, L. A. and Alonso, E. E. (2001). A Model for Rockill Compressibility. Géotechnique, V. 51, No. 2, pp. 127-139 [3] Justo, J. L., Durand, P. y Justo, E. (2003). Un Modelo Tridimensional para el Estudio de la Fluencia en Presas de Materiales Sueltos. Rev. Int. Mét. Num. Cálc. Dis. Ing. V. 19, No. 3, pp. 313-330 (in Spanish) [4] INGETEC S.A. (2013). Hydropower Projects – Guavio http://www.ingetec.com.co/experiencia-ingles/textos-proyectos-ingles/proyectohidroelectricos-ingles/guavio-ingles.htm (Accessed 2013)
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[5] Marulanda, A. and Amaya, F. (1989). The Design and Construction of Colombia’s Guavio Dam. Water Power and Dam Construction, December 1989. pp. 41-60 [6] INGETEC S.A. (2013). Hydroelectric Developments. http://www.ingetec.com.co/brochures-ingles/BROCHURE-HIDROELECTRICASINGLES.pdf (Accessed 2013) [7] Amaya, F. and Marulanda, A. (1993). Behavior of Guavio Dam during Construction and First Filling. Proceedings of the International Symposium on High Earth-Rockfill Dams. ICOLD-CSHEE. Beijing, China. [8] Dunnicliff, J. (1988). Geotechnical Instrumentation for Monitoring and Performance. John Wiley & Sons. 577 p.
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PAPERS THEME B
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Solution to Theme B Long Term Behaviour of a Rockfill Dam F. Ezbakhe1, I. Escuder-Bueno1 1
Universidad Politécnica de Valencia, Spain E-mail: [email protected]
Preface The authors acknowledge that the solution herein presented is very simplified and should not be taken as a professional reference by any means. However, with the purpose of enabling the formulator to estimate the order of magnitude of the lack of accuracy and/or physical inconsistencies by using methods as simplified and empirical as those herein presented, the paper may be of certain interest.
Introduction Theme B of the 12th Benchmark Workshop on Numerical Analysis of Dams (ICOLD) consists in the evaluation of the long term behavior of a rockfill dam, analyzing the case of the Alberto Lleras Dam, part of the Guavio Hydroelectric Project in Colombia. The objective is to reproduce, via a numerical model, the behavior of the dam during the construction, impoundment and exploitation, providing the evolution of displacements and stresses within the dam as results. For the solution presented in this paper, the problem has been solved by FLAC3D (Itasca), a finite differences program developed by Itasca Consulting Group, Minnesota. This software uses the finite differences method to analyze the mechanical behavior of a continuous medium in 3D until it reaches an equilibrium state.
Figure 1: View of the Alberto Lleras Dam 203
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Statement of the problem Alberto Lleras Dam is a clay-core rockfill dam which main features are given in the next table: Table 1: Main features of the Alberto Lleras Dam Height Width Dam volume Reservoir volume
245m 950m 7hm3 950hm3
Construction of the dam lasted five years, from September 1984 to August 1989. In general, during the six rainy months of the year (May-October) rockfill was placed and compacted in the outer portions of the shell. During the remaining six drier months (November-April), fill placement was concentrated in the central portion of the dam. The impoundment started on February 1992 and run through September 1992. The dam consists in three rockfill zones, with similar mechanical parameters but different granulometric composition. It includes also two transition zones and a clay core, featuring a medium plasticity clay matrix.
Figure 2: View of the Alberto Lleras Dam The dam site is underlain by quartzite, argilite and limestone, with no geological faults. The formations strike diagonally with respect to the river canyon and are fresh, hard and depict low permeability.
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Features of the implemented model Geometrical model From a simplification of the real topography and geometry of the problem, a 3D geometrical model of the dam was implemented, formed by 7286 nodes and 5800 solid elements. Next figures show the FLAC 3D geometrical model of the dam.
Figure 3: FLAC 3D geometrical model 205
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Mechanical model The mechanical model selected for the solution herein presented is linear-elastic. This model was chosen for its simplicity and wide applicability solving soil and rock mechanics problems. In this model, strain increments generate stress increments according to the linear and reversible law of Hooke: (1) where the Einstein summation convention applies, δ_ij is the Kroenecker delta symbol, and α_2 is a material constant related to the bulk modulus, K, and shear modulus, G, as: (2) New stress values are then obtained from the relation: (3) Next table provides the values that have been adopted for the mechanical model: Table 2: Mechanical parameters
Core Rockfill Foundation
Density (kg/m3) 2010 2750 2500
Bulk modulus (Pa)
Shear modulus (Pa)
7.5·106 8.5·107 4·1010
5.4·106 4.7·107 3·1010
The boundary conditions applied are: - Plane x=-300 (lateral surface). Movements constrained on the x-axis - Plane x=400 (lateral surface). Movements constrained on the x-axis - Plane y=-300 (front surface). Movements constrained on the y-axis - Plane y=1300 (back surface). Movements constrained on the y-axis - Plane z=-300 (underside). Movements constrained on all axis Constructive behaviour The construction process may affect the stress state, particularly if excess pore pressures develop in the soils and do not dissipate completely during the construction stages. In addition, staged modeling of the embankment lift construction also provides a better representation of the stresses in the embankment. In this case, due to the difficulty involved in modeling the actual construction sequence, it has been simplified considering that it takes place in uniform layers of 10m high. Thus, the numerical process is to activate each layer, assigning materials and properties, and calculating the model to reach equilibrium. Post-constructive behaviour In Geotechnical Engineering, time-dependent settlement is associated to the process of consolidation. This behavior would be therefore determined by the rate at which water is able to flow through the ground pores under a certain hydraulic gradient. Boughton (1970) found that this settlement also occurred during the construction process. For calculation purposes, Boughton got a good fit for the form of post-construction deformations 206
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by applying the entire weight of the structure at once. Then, adopting a scale factor, he got a good fit for the magnitude of the deformations. In this case, the scale factor used is 1.05 for dry behavior (from the completion of construction to the start of filling) and 1.11 for the behavior of the wet rockfill (with filled reservoir).
Results Settlements Settlements of the dam body are obtained at three different times: - August 1989 completed the construction of the dam - January 1992 immediately before the first filling - September 1992 1630 masl reservoir Settlements are also obtained in nine point of the dam: Table 3: Settlements in the dam
M-30 M-36 M-48 M-53 M-64 M-66 M-72 SNA-30 SNA-38
aug-89 Settlement Elevation (m) (masl) 0.00 1641.081 0.00 1641.020 0.00 1641.751 0.00 1640.420 1.54 1604.027 1.60 1570.985 1.46 1515.672 3.60 1517.435 3.42 1518.242
jan-92 Settlement Elevation (m) (masl) 0.810 1640.271 0.875 1640.145 0.363 1641.388 0.290 1640.130 1.974 1603.594 1.827 1570.758 1.617 1515.513 4.037 1516.995 3.872 1517.790
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sep-92 Settlement Elevation (m) (masl) 0.971 1640.110 1.043 1639.977 0.470 1641.281 0.368 1640.052 2.070 1603.498 2.032 1570.553 1.715 1515.415 4.593 1516.439 3.991 1517.671
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Figure 4: Settlements in August 1989
Figure 5: Settlements in January 1992
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Figure 6: Settlements in September 1992 Stresses Next figure represents the stresses of the dam after dam construction:
Figure 7: Principal stresses in August 1989 209
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Principal stresses (s1) and vertical stresses (sv) are also obteined in five different points: Table 4: Stresses in the dam @ EOC
Reservoir at EL 1575
Reservoir at EL 1630
ID
s1 [Kpa]
sv [Kpa]
s1 [Kpa]
sv [Kpa]
s1 [Kpa]
sv [Kpa]
CP-11
1594.3
1556.5
1619.1
1599.9
1752.3
1725.6
CP-6
1345.2
1281.2
3015.2
1964.7
3237.5
2482.1
CP-19
1909.4
1554.0
2240.3
1914.9
2806.5
2019.0
CP-22
2871.0
1459.6
2884.9
1947.6
2916.8
1992.0
CP-3
2769.7
2653.6
3438.1
2949.6
3475.0
2969.2
Conclusion In order to study the long dam’s long term behavior, a linear elastic analysis has been realized. This means that a linear analysis is performed for each time step and when the equilibrium is reached the calculus moves to next instant of time. The implemented model considers the dam body as a structure without interfaces and permits the evolution of the displacements and stresses over time. The simulation of this evolution can resembles to some extent to the data collected by auscultation, with a margin of error due to the different simplifications made in the model.
References [1] Escuder, I. “Estudio del comportamiento tenso-deformacional de pedraplenes inundables mediante simulaciones numéricas formuladas en diferencias finitas y calibradas con lecturas de instrumentación”. Doctoral Thesis [2] Itasca Consulting Group, Inc., “Fast Lagrangian Analysis of Continua in 3 Dimensions version 3.1: Theory and Background, Optional features”, Minneapolis, Minnesota, 2006 [3] ICOLD, “Guidelines for use of numerical models in dam engineering”, Bulletin 155
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THEME C
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Theme C
Computational Challenges in Consequence Estimation for Risk Assessment: Numerical Modelling, Uncertainty Quantification, and Communication of Results
Part 1 – Hydraulic Modelling and Simulation Sponsoring Organizations: U.S. Army Corps of Engineers (USA) U.S. Department of Homeland Security (USA) Formulators: Yazmin Seda-Sanabria (Formulation Team Co-Chair) National Program Manager, Critical Infrastructure Protection & Resilience Program, U.S. Army Corps of Engineers, Headquarters, 441 G Street NW (ATTN: CECW-HS), Washington, DC 20314 (USA), Email: [email protected] Enrique E. Matheu (Formulation Team Co-Chair) Chief, Critical Lifelines Branch, Sector Outreach and Programs Division, Office of Infrastructure Protection, U.S. Department of Homeland Security, 245 Murray Lane Arlington, VA 20598-0608 (USA), Email: [email protected] Timothy N. McPherson (Formulation Team Technical Lead) R&D Group Leader, Energy and Infrastructure Analysis, Los Alamos National Laboratory, Los Alamos, New Mexico 87544 (USA), Email: [email protected] Mustafa Altinakar Director and Research Professor, National Center for Computational Hydroscience and Engineering, The University of Mississippi, Brevard Hall Room 327, P.O. Box 1848, University, MS 38677-1848 (USA), Email: [email protected] Mark Jourdan Research Hydraulic Engineer, Coastal and Hydraulics Laboratory, U.S. Army Engineer Research and Development, 3909 Halls Ferry Road, Vicksburg, Mississippi 39180 (USA), Email: [email protected] Michael K. Sharp Technical Director, Geotechnical and Structures Laboratory, U.S. Army Engineer Research and Development, 3909 Halls Ferry Road, Vicksburg, Mississippi 39180 (USA), Email: [email protected] For technical information, please contact Dr. Tim McPherson at [email protected]
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Introduction In the last decade, computational capacity has grown dramatically such that multiprocessor computing techniques are now widely available. This increase in resource availability has allowed the development of a vast array of new models for flooding and consequence assessment. Many of these models are computing flood wave propagation at extremely high temporal and spatial resolutions. When these models are coupled to equally complex models of population mobility, infrastructure impact and economic consequence, simulation frameworks are created that can support a paradigm shift from standard approaches to dam risk analysis. Although computational advances have increased the availability and applicability of novel tools, there is a deficiency in benchmarks on the use of those tools in risk assessments. An obvious application of increased availability and efficiency of computational resources is to conduct probabilistic risk assessments using Monte Carlo techniques, but the application of such approaches entails a wide range of assumptions and technical decisions regarding the management of uncertainty such as variable uncertainty, parameter uncertainty, uncertainty in probabilistic sub-model, measurement error, computational errors, and numerical approximation to name a few. Universities, engineering companies and regulatory bodies are invited to contribute to the benchmark and take part in the discussion of results gained. This document is part 1 in a 2-part series for Theme C. Part 1 pertains to the hydraulic modelling and simulation of the dam breach and subsequent flood wave and provides details regarding the available data, dam geometry and failure, and expected modelling and simulation solution requirements. Part 2 primarily focuses on consequence estimation using the modelling and simulation results from Part 1. Benchmark Focus The numerical problem proposed for the workshop consists of estimating the consequences of failure of a dam near populated areas with complex demographics, infrastructure and economic activity. The dam in question will be near the city of Hydropolis, a virtual testbed for flood risk analysis to be built in preparation for the benchmark study. Theme C participants are free to select the type and sophistication of the simulation engines used to solve the problem, including 1-d, 2-d and 3-d flood simulation tools, Population at Risk (PAR) and Loss of Life (LOL) estimation techniques, and infrastructure and consequence assessment models.
Flood Modelling and Simulation The following sections are intended to provide information regarding the data provided for the dam failure modelling and simulation benchmark. Specifically this information includes the topographic data and dam geometric and construction information. It is not the intent of this benchmark to set requirements as to which modelling and simulation environment should be used. Therefore, the descriptions and data provided are intended to be useful to a wide range of modelling and simulation environments at many levels of fidelity. Dam Information A hypothetical embankment dam was constructed in a mountainous region. The high- hazard dam sits directly above a lightly populated area and 3.5 kilometres away from an urban environment. The front and rear views of this dam are shown in Figure 2-1 and 2-2, respectively.
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The primary function of the dam is flood control for heavy snowmelt and strong monsoonal weather patterns. In addition, the reservoir provides some water supply and recreational activities to nearby communities. The following sections provide more detail regarding the geometry and the construction of the hypothetical dam.
Figure 1: Front view of dam and surrounding topography
Figure 2: Rear view of dam, reservoir, and surrounding topography
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Dam Geometry The hypothetical embankment dam is considered high head (61 m) with moderate storage (38 million cubic meters). An overview of the dam geometry is provided in Table 5 and a crosssectional profile is shown in Figure 3. Table 5: Dam geometric parameters Parameter Description
Value
Dam Location (x, y)
4499.66, 6681.57
Crest Length (m)
360
Crest Width (m)
24
Crest Elevation (m)
272
River Bed Elevation (m)
211
Upstream Embankment Slope (?H:1V)
3
Downstream Embankment Slope (?H:1V)
3
Figure 3: Cross-sectional view of the hypothetical dam The storage capacity at crest elevation is more than 38 million cubic meters. The stagevolume curve for the reservoir is shown in Table 6. Table 6: Reservoir stage-volume curve Elevation (m)
Surface Area (m²)
Volume (m³)
Elevation (m)
Surface Area (m²)
Volume (m³)
211 213
0 898
0 266
243 245
605,559 682,169
6,214,463 7,500,447
215
7,812
7,432
247
762,461
8,944,612
217
25,502
39,871
249
848,052
10,553,044
219
43,821
109,250
251
916,938
12,322,332
221
62,409
216,078
253
978,640
14,218,482
223
85,038
364,294
255
1,035,581
16,233,223
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225
117,544
565,695
257
1,094,197
18,365,080
227
153,732
836,860
259
1,155,316
20,614,116
229
190,728
1,181,477
261
1,216,975
22,987,820
231
222,068
1,594,314
263
1,278,543
25,483,100
233
252,239
2,067,481
265
1,338,586
28,100,382
235
294,797
2,613,155
267
1,398,359
30,837,356
237
351,808
3,254,198
269
1,467,826
33,701,532
239
461,611
4,084,756
271
1,543,126
36,712,416
241
531,463
5,077,507
272
1,584,052
38,276,344
Dam Construction Information Homogenous Dam The dam is a rolled earth fill structure composed of predominantly sandy clays and clayey sands. Compaction was achieved using 4 passes of a 50-ton pneumatic-tired roller on 0.3meter loose lifts. Strength properties for the dam were developed from results from undrained triaxial strength tests. The drained strength parameters were based on 5% axial strain and were selected to represent initial confining stresses up to 478 kPa. The undrained strengths were interpreted from an approximate evaluation of Su/mc ratios estimated from the reported undrained tests. The undrained strength S u was taken as one-half the maximum deviator stress for axial strains up to 10%. The estimated strengths are summarized in Table 7. Table 7: Selected parameters for dam Parameter Description Strength Parameters: c Effective (drained) cohesion in kPa Effective (drained) friction angle
Value
Su
43.09+ 0.175mc
19.15 kPa 14°
Undrained strength in kPa
Stiffness Parameters: Vs1 Shear wave velocity at v = 1 atm Gmax,1 Maximum shear modulus at v = 1 atm Others: Saturated unit weight sat k Permeability
152.4 m/s 46443 kPa 2002 kg/m³ 1.910-6 cm/s
Foundation Foundation stiffness parameters were based on shear wave velocity values without reduction for strain level or changes in effective stress. A simplified stiffness distribution was used to reflect the gradual increase in stiffness with depth. The selected properties are summarized in Table 8.
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Table 8: Selected parameters for foundation Parameter Description
Value Base of dam to Foundation depth of below 3.6 m 3.6 m
Stiffness Parameters: Vs
Shear wave velocity
167 m/s
185 m/s
G
Shear modulus
6.75105 kPa
8.33 106 kPa
Poisson’s ratio
0.25
0.25
Unit weight
2,242kg/m³
2,402 kg/m³
Permeability
9.510-7 cm/s
9.510-7 cm/s
Others: K
Dam Failure Guidance The dam failure for this benchmark takes place when the pool elevation is at crest elevation. The mode of failure will be assumed as an overtopping failure. While it is open for participants to conduct detailed (physically-based) modelling of the breaching process as part of the benchmark, several breach parameter estimation approaches available in the literature are summarized in the appendix of this document. Data Provided In addition to the information provided in this document, gridded data representing the topography and the land use classification are provided. All gridded datasets conform to the same domain and cellsize. The resolution of data provided is considered a base dataset and may be altered if required by the modelling and simulation environment used by the participant. Digital Elevation Data Two digital elevation models (DEM) are provided to benchmark participants: DEM representing pre-dam construction. DEM representing post-dam construction. Participants in the benchmark may decide which DEM is more appropriate for use based on individual requirements for modelling and simulation. Land Use/Cover Benchmark participants are provided a gridded dataset representing the hypothetical land use/cover for the simulation region. These data follow the classification guidance and values provided in the National Land Cover Dataset (NLCD). For completeness, a description of land use classifications is provided in Table 9.
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Table 9: Land use/cover classifications and descriptions NLCD Class 11 12 21 22 23 24 31 41 42 43 52 71 81 82
Description Open Water Perennial Ice/Snow Developed-Open Space Developed- Low Intensity Developed- Med. Intensity Developed- High Intensity Barren Land Deciduous Forest Evergreen Forest Mixed Forest Shrub/Scrub Grassland/Herbaceous Pasture/Hay Cultivated Cropland
Correlation of these classifications to surface roughness for modelling and simulation should be reported by the participant. Flood Modelling and Simulation Reporting Requirements For consistency between benchmark participants, each participant is requested to generate information described in Table 10 from the results of the flood simulation. These data focus on the hydraulics of the simulation. Table 10: Reporting requirements for flood modelling and simulation Required Data Breach Discharge Cross-Section Discharge
Peak Flood Depths Flood Wave Arrival Time Peak Unit Flow Rate Flooded Area
Data Description This hydrograph should show the discharge rate from pre-failure to empty reservoir, in units of m3/ These shall consist of a complete hydrograph for unsteady simulation environments and peak discharges for steady-state simulation environments Gridded dataset representing the peak flood depth, units of meters Gridded dataset with flood wave arrival time in 5-minute intervals Gridded dataset with a value representing the peak unit flow rate in units of m²/s Summation of the total flooded area, units of m². In addition, participants will provide flooded area categorized by range of flood depths at .5 meter intervals
The locations in which participants should provide hydrographs are shown in Table 11
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Table 11: Cross-section locations Cross Section ID 1 2 3 4 5
X Location (m) 4737.18 5971.79 7486.60 9100.84 10716.96
Y Location (m) 6755.39 7053.10 7582.85 7773.91 7397.67
Solution Metadata As described above, Participants are expected to provide gridded data for some of the reporting requirements. These gridded data files must conform to the metadata of the original data provided in the benchmark. These are summarized in Table 12. Table 12: Gridded data metadata requirements Metadata Parameter Left Extent Bottom Extent Right Extent Top Extent Cellsize Columns Rows
Value 0 0 25831.81905 9930.941439 9.4760892 2726 1048
Acronyms and Abbreviations DEM NLCD kPa
Digital Elevation Model National Land Cover Dataset Kilopascal
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Theme C
Computational Challenges in Consequence Estimation for Risk Assessment: Numerical Modelling, Uncertainty Quantification, and Communication of Results
Part 2 – Consequence Estimation Sponsoring Organizations: U.S. Army Corps of Engineers (USA) U.S. Department of Homeland Security (USA) Formulators: Yazmin Seda-Sanabria (Formulation Team Co-Chair) National Program Manager, Critical Infrastructure Protection & Resilience Program, U.S. Army Corps of Engineers, Headquarters, 441 G Street NW (ATTN: CECW-HS), Washington, DC 20314 (USA), Email: [email protected] Enrique E. Matheu (Formulation Team Co-Chair) Chief, Critical Lifelines Branch, Sector Outreach and Programs Division, Office of Infrastructure Protection, U.S. Department of Homeland Security, 245 Murray Lane Arlington, VA 20598-0608 (USA), Email: [email protected] Timothy N. McPherson (Formulation Team Technical Lead) R&D Group Leader, Energy and Infrastructure Analysis, Los Alamos National Laboratory, Los Alamos, New Mexico 87544 (USA), Email: [email protected] Mustafa Altinakar Director and Research Professor, National Center for Computational Hydroscience and Engineering, The University of Mississippi, Brevard Hall Room 327, P.O. Box 1848, University, MS 38677-1848 (USA), Email: [email protected] Mark Jourdan Research Hydraulic Engineer, Coastal and Hydraulics Laboratory, U.S. Army Engineer Research and Development, 3909 Halls Ferry Road, Vicksburg, Mississippi 39180 (USA), Email: [email protected] Michael K. Sharp Technical Director, Geotechnical and Structures Laboratory, U.S. Army Engineer Research and Development, 3909 Halls Ferry Road, Vicksburg, Mississippi 39180 (USA), Email: [email protected] For technical information, please contact Dr. Tim McPherson at [email protected]
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Introduction In the last decade, computational capacity has grown dramatically such that multiprocessor computing techniques are now widely available. This increase in resource availability has allowed the development of a vast array of new models for flooding and consequence assessment. Many of these models are computing flood wave propagation at extremely high temporal and spatial resolutions. When these models are coupled to equally complex models of population mobility, infrastructure impact and economic consequence, simulation frameworks are created that can support a paradigm shift from standard approaches to dam risk analysis. Although computational advances have increased the availability and applicability of novel tools, there is a deficiency in benchmarks on the use of those tools in risk assessments. An obvious application of increased availability and efficiency of computational resources is to conduct probabilistic risk assessments using Monte Carlo techniques, but the application of such approaches entails a wide range of assumptions and technical decisions regarding the management of uncertainty such as variable uncertainty, parameter uncertainty, uncertainty in probabilistic sub-model, measurement error, computational errors, and numerical approximation to name a few. Universities, engineering companies and regulatory bodies are invited to contribute to the benchmark and take part in the discussion of results gained. This document is part 2 in a 2-part series for Theme C. Part 1 pertains to the hydraulic modelling and simulation of the dam breach and subsequent flood wave and provides details regarding the available data, dam geometry and failure, and expected modelling and simulation solution requirements. Part 2 focuses on the consequences of the flood using the modelling and simulation results from Part 1. Benchmark Focus The numerical problem proposed for the workshop consists of estimating the consequences of failure of a dam near populated areas with complex demographics, infrastructure and economic activity. The dam in question will be near the city of Hydropolis, a virtual testbed for flood risk analysis to be built in preparation for the benchmark study. Theme C participants are free to select the type and sophistication of the simulation engines used to solve the problem, including 1-d, 2-d and 3-d flood simulation tools, Population at Risk (PAR) and Loss of Life (LOL) estimation techniques, and infrastructure and consequence assessment models.
Consequence Estimation A full consequence assessment for dam failure generally includes four main categories- public health and safety, economic impact, psychological impact, and governance/mission impact. At a minimum, the consequence assessment should focus on impacts related to human consequences and direct economic impact. This document outlines the requirements for consequence assessment within Theme C of the benchmark, addressing human consequences (e.g., population at risk and loss of life) and direct economic impact. The intent is not to limit the approaches or techniques used by participants but to guide the assessment into categories for comparison. Additional consequence assessments are also encouraged, but will not be used in the assessment of uncertainty.
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Dam Failure Assumptions Over a series of fall rainfall events, the reservoir slowly approaches crest elevation. Due to the frequency of fall rainfall events, there are few people recreating at or near the facility. At 11:00 pm on Saturday evening, the dam fails, releasing a torrent of water through the canyon and towards Hydropolis. The failure was sudden and unexpected, and for those in the canyon witnessing the failure, there are no mobile phone services. The reservoir and dam does not have a completed emergency action plan, thus the potential impact related to failure is unknown to residents in Hydropolis. Data Provided Participants are requested to evaluate the downstream consequences as it relates to population at risk, loss of life and direct economic impact. Participants in Benchmark Theme C are provided additional data to assist in consequence assessment. This data includes Hydropolis census information in shapefile format and parcel data in shapefile format. Census Data Participants are provided with a shapefile representative of census data for the city of Hydropolis. This data includes information regarding the population and the economic activity. Table 5 and Table 2 provide attribution information for these data. Table 1: Population data field names and descriptions Field Names Total Male Munder5 M5to9 M10to14 M15to17 M18to19 M20 M21 M22to24 M25to29 M30to34 M35to39 M40to44 M45to49 M50to54 M55to59 M60to61 M62to64 M65to66 M67to69 M70to74 M75to79 M80to84 M85over
Description Total Male and Female Total Male Male: Under 5 years Male: 5 to 9 years Male: 10 to 14 years Male: 15 to 17 years Male: 18 and 19 years Male: 20 years Male: 21 years Male: 22 to 24 years Male: 25 to 29 years Male: 30 to 34 years Male: 35 to 39 years Male: 40 to 44 years Male: 45 to 49 years Male: 50 to 54 years Male: 55 to 59 years Male: 60 and 61 years Male: 62 to 64 years Male: 65 and 66 years Male: 67 to 69 years Male: 70 to 74 years Male: 75 to 79 years Male: 80 to 84 years Male: 85 years and over
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Female Funder5 F5to9 F10to14 F15to17 F18to19 F20 F21 F22to24 F25to29 F30to34 F35to39 F40to44 F45to49 F50to54 F55to59 F60to61 F62to64 F65to66 F67to69 F70to74 F75to79 F80to84 F85over
Total Female Female: Under 5 years Female: 5 to 9 years Female: 10 to 14 years Female: 15 to 17 years Female: 18 and 19 years Female: 20 years Female: 21 years Female: 22 to 24 years Female: 25 to 29 years Female: 30 to 34 years Female: 35 to 39 years Female: 40 to 44 years Female: 45 to 49 years Female: 50 to 54 years Female: 55 to 59 years Female: 60 and 61 years Female: 62 to 64 years Female: 65 and 66 years Female: 67 to 69 years Female: 70 to 74 years Female: 75 to 79 years Female: 80 to 84 years Female: 85 years and over Table 2: Economic table field names and descriptions
Field Names jobs11 jobs21 jobs22 jobs23 jobs3133 jobs42 jobs4445 jobs4849 jobs51 jobs52 jobs53 jobs54 jobs55 jobs56 jobs61 jobs62 jobs71
Description Agriculture, Forestry, Fishing and Hunting Mining, Quarrying, and Oil and Gas Extraction Utilities Construction Manufacturing Wholesale Trade Retail Trade Transportation and Warehousing Information Finance and Insurance Real Estate and Rental and Leasing Professional, Scientific, and Technical Services Management of Companies and Enterprises Administrative and Support and Waste Management and Remediation Services Educational Services Health Care and Social Assistance Arts, Entertainment, and Recreation 224
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jobs72 jobs81 jobs92
Accommodation and Food Services Other Services (except Public Administration) Public Administration
Parcel Data Benchmark participants are provided parcel data in shapefile format. This includes information regarding the zoning (e.g., residential, commercial, etc.) and the structure (e.g., stories, basement, quality). Table 3 provides a description of the zone classification provided in this data. Table 3: Parcel descriptions for zone classifications Code A-1 A-2 AP C-1 C-2 C-3 IP M-1 O-1 P P-R R-1 R-1 R-2 R-3 RA-1 RA-2 R-D R-LT RO-1 RO-20 R-T SD-LC-1 SD-RO SU-1 SU-2 SU-3 UCO WO
Description Rural Agricultural—1-Acre Minimum Zone Rural Agricultural—2-Acre Minimum Zone Airport Protection Overlay Zone Neighborhood Commercial Zone Community Commercial Zone Heavy Commercial Zone Industrial Park Zone Light Manufacturing Zone Office & Institutional Zone Parking Zone Reserve Parking Zone Residential Zone: Houses Single Family Residential Zone Residential Zone: Houses, Townhomes & Medium Density Apartments Residential Zone: Houses, Townhomes & High Density Apartments Residential and Agricultural Zone, Semi-Urban Area Residential and Agricultural Zone Residential and Related Uses Zone, Developing Area Residential Zone: Houses & Limited Townhomes Rural and Open Zone Rural and Open Agricultural Zone Residential Zone: Houses & Townhomes Limited Neighborhood Commercial Zone Residential/Office Zone Special Use Zone Special Neighborhood Zone, Redeveloping Area Special Center Zone Urban Conservation Overlay Zone Wall Overlay Zone
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Consequence Assessment Reporting Requirements All participants should summarize methods, techniques, assumptions, and key results in a paper not to exceed 15 pages. In addition, each participant is requested to generate the information described in Table 4 using results of the consequence assessment. These data focus on the consequences of failure. Table 4: Reporting requirements for consequence assessment Required Data Population at Risk
Loss of Life
Flood Severity Grid Direct Economic Impact
Data Description Participants should use the provided consequence results spreadsheet to populate the requested information. This information includes age demographic break‐downs by time to flooding and depth of flooding. A gridded dataset indicating the spatial variability of population at risk (regardless of age) is also requested. Participants should use the provided consequence results spreadsheet to populate the requested information. The loss of life estimation includes break‐downs by time from breach failure. A gridded dataset indicating the spatial variability of loss of life is requested. Gridded dataset representing the peak flood severity, classified as low, medium and high (e.g., 1, 2, 3) severity. Participants are expected to describe the assumptions used in categorizing flood severity in the submitted paper. Participants should use the provided consequence results spreadsheet to populate the requested information. This information includes break‐ downs by time from dam failure. A gridded dataset indicating the spatial variability of direct economic impact is also requested.
Consequence Metadata As described above, participants are expected to provide gridded data for some of the reporting requirements. These gridded data files must conform to the metadata of the original data provided in the benchmark. These are summarized in Table 5. Table 5: Gridded data metadata requirements Metadata Parameter Left Extent Bottom Extent Right Extent Top Extent Cellsize Columns Rows
Value 0 0 25831.81905 9930.941439 9.4760892 2726 1048
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RESULT COMPARISON OF THE PARTICIPANTS THEME C
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Results Comparison Theme C Computational Challenges in Consequence Estimation for Risk Assessment Introduction Theme C of the 12th International Benchmark Workshop on the Numerical Analysis of Large Dams involved 8 participants who submitted at least partial solutions of the Hydropolis Dam Break Case Study. This synthesis is not a complete representation of each solution. In many cases, participants conducted their own sensitivity/uncertainty analyses. For brevity, this synthesis document considered only one set of solutions from each participant. Full description of solutions can be found in the papers authored by each participant team. This synthesis of results is based on statistical methods to facilitate comparison to the greatest extent possible. The Kappa statistic is used to quantify the similarities among the participants’ study results. Kappa corresponds to a numerical rating of the degree of agreement between two raters, which in this case are the outputs (e.g., gridded data peak flood depths) of two participant models. Kappa is calculated in Equation 1. (1) where P(a) is the relative observed agreement between the two models and P(e) is the probability that the agreement is due to chance. Kappa ranges from -1 to 1. If the two models are in perfect agreement then Kappa equals 1, whereas if there is no agreement among the models except what would be expected by chance. A Kappa value less than 0 indicates more disagreement than what one can expect by chance. A commonly cited scale for interpreting Kappa values is given by Landis and Koch1 and reported in the following Table 1. The Kappa statistic quantifies the aggregate agreement between two different spatial grids, but does not provide any spatial information about where the two models differ more or less. Table 1: Kappa value interpretation as proposed by Landis Kappa
Agreement Interpretation
<0 0.01-0.20
Less than chance agreement Slight agreement
Agreement Index (used in Results Tables) Chance Slight
0.21-0.40
Fair agreement
Fair
0.41-0.60 0.61-0.80
Moderate agreement Substantial agreement
Moderate Substantial
0.81-0.99
Almost perfect agreement
Almost perfect
1
Landis J.R., and Koch G.G., The measurement of observer agreement for categorical data. 1977 Biometrics 33, 159-174 229
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Overview of Participation The benchmark case includes a diverse set of analyses and several participants only submitted results against a subset of the complete problem. Table 2 lists each of the lead authors, an ID assigned to the participant team for solution comparison, and an indication of submission completion for topics of comparison. Table 2: Summary of Participant Submissions Authors
ID
Hydraulic Solution
Davison et al. Bent et al. Mancusi et al. Williams & Buchanan Thames & Kalyanapu Altinakar et al. Saberi et al. McVan et al.
HRW LANL MANC MMC TK MUST SAB MCV
Yes Yes Yes Yes Yes Yes Yes Yes
Population at Risk Solution Yes Yes Yes Yes Yes Yes No Yes
Loss of Life Solution Yes Yes Yes Yes Yes Yes No No
Economic Solution Yes Yes Yes Yes Yes No No No
Discussion Breach formulation Participants used a range of models, including physics-based breach models using dam material information and regression equations based on previous dam failures. Figure 1 shows the breach hydrographs produced by each team. It is clear that choice of model, method, and parameters can significantly affect the timing and the magnitude of the peak discharge. In general, models using regression equations had a much earlier peak than physics-based models. The SAB and HRW hydrographs were shifted in time to better illustrate the data for comparison. The peak discharge ranges in time from under 0.5 hours to more than 18 hours and the magnitude of peak discharge ranges from near 10,000 cubic meters per second (cms) to more than 40,000 cms. SAB, MMC, LANL, MANK, and TK all have peak discharges above 20,000 cms, while MCV, MUST, and HRW have peaks below 20,000 cms. ~4 hr shift for comparison
~16 hr shift for comparison
Figure 1: discharge hydrographs 230
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Hydrodynamics Each participant used different models, which solve the same equations using different numerical schemes. The most notable difference between models appeared to be the difference in mesh or grid type used. Models using structured grids tended to have more similar results relative to peak depths than models using irregular meshes. This may be a result of interpolation between the two types of meshes for comparison. A wide variety of techniques were used by the participant teams to produce the necessary flood output data for consequence analyses. Each participant submission details the methods used. These differences can be a strong explanatory variable in the comparison of consequence results. For example, dam failure hydrodynamic simulation results are dependent on the input datasets (e.g., breach discharge, topography, roughness) and modeling\method approach. Participants used different approaches to estimate roughness coefficients. Each participant was required to interpret the roughness coefficient associated with an NLCD land use type. Because there isn’t a single accepted dataset, there were a range of coefficients used by participants. Cross Sections The participants were provided a single point location for each section, and the participants estimated the line upon which flow should be summed. Figure 2 shows the hydrographs submitted by each of the participants for all cross sections. SAB and HRW have the same time shifts as previously noted.
Figure 2: Hydrographs for all cross sections submitted by benchmark participants 231
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Peak Flood Depths The peak flood depth was defined by all participants as the peak depth at every cell regardless of the time at which it occurred. The maximum flooded area and flooded area categorized in half-meter increments of peak depths are summarized in Figure 3 and Figure 4. The total flooded area ranges from near 47 km² to just under 30 km². The Kappa statistic was applied to the gridded peak flow depths. A Kappa value was calculated for each unique combination of two models. The results of the peak flood depth Kappa analysis are summarized in Table 3. There is generally strong agreement between the models. The chance and moderate differences between MCV and SAB and other models is likely due to interpolation of the irregular mesh output from those submissions to a new mesh for comparison with other models.
Figure 3: Summary of total flooded area
Figure 4: Flood area summarized by half-meter increments of peak depth
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Table 3: Kappa statistic (model similarity) for peak flood depth HRW HRW
LANL
MANC
MMC
TK
MUST
MCV
SAB
Substantial
Substantial
Substantial
Substantial
Substantial
Moderate
Chance
Substantial
Substantial
Substantial
Substantial
Moderate
Chance
Substantial
Substantial
Substantial
Moderate
Chance
Substantial
Substantial
Moderate
Chance
Substantial
Moderate
Chance
Moderate
Chance
LANL
Substantial
MANC
Substantial
Substantial
MMC
Substantial
Substantial
Substantial
Substantial
Substantial
Substantial
Substantial
Substantial
Substantial
Substantial
Substantial
Substantial
Moderate
Moderate
Moderate
Moderate
Moderate
Moderate
Chance
Chance
Chance
Chance
Chance
Chance
TK MUST Chance
MCV Chance
SAB
Flood Wave Arrival Time There were differences among participants in defining thresholds for determining when the flood wave arrived. For example, LANL and MUST defined the flood wave arrival as the time at which a grid cell becomes wet without implying a threshold. MMC defined the flood wave arrival as the time in which the depth of water in a grid cell reaches 0.6 meters. Agreement between participants was assessed using the Kappa Statistic and are shown in Table 4. The results indicate that the level of agreement between any two models varies more for the flood wave arrival time than for the peak flood depths. It is noted again that HRW used a breach hydrograph with a much longer lag time to peak discharge than other participants. Table 4: Kappa statistic (model agreement) for flood wave arrival time HRW HRW
LANL
MANC
MMC
TK
MUST
MCV
SAB
Chance
Chance
Chance
Chance
Chance
Chance
Chance
Slight
Substantial
Slight
Fair
Substantial
Moderate
Slight
Chance
Fair
Slight
Fair
Slight
Moderate
Substantial
Substantial
Slight
Slight
Slight
Fair
Substantial
LANL
Chance
MANC
Chance
Slight
MMC
Chance
Substantial
Slight
TK
Chance
Slight
Chance
Slight
MUST
Chance
Fair
Fair
Moderate
Slight
MCV
Chance
Substantial
Slight
Substantial
Slight
Fair
SAB
Chance
Moderate
Fair
Substantial
Slight
Substantial
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Peak Unit Flow Rate Table 5 shows the level of agreement between participants for Peak Unit Flow Rate as determined by the Kappa analysis. The results show less agreement between two models for the peak unit flow rate than for peak flood depth and flood wave arrival. Table 5: Kappa statistic (model agreement) for peak unit flow rate HRW HRW LANL MANC MMC TK MUST MCV SAB
Fair Moderate Slight Slight Substantial Fair Fair
LANL
MANC
MMC
TK
MUST
MCV
SAB
Fair
Moderate Fair
Slight Substantial Fair
Slight Chance Slight Chance
Substantial Fair Moderate Fair Slight
Fair Slight Slight Slight Slight Fair
Fair Slight Slight Slight Chance Fair Slight
Fair Substantial Chance Fair Slight Slight
Fair Slight Moderate Slight Slight
Chance Fair Slight Slight
Slight Slight Chance
Fair Fair
Slight
Consequences Population The spatial distribution of population within the flooded area varied among the participants. MCV, MANC, and MUST uniformly distributed the population available in the census blocks to properly sum the affected population within partially flooded census blocks. HRW, TK, and MMC redistributed the population to the parcel data provided. MMC additionally accounted for residential population and workforce population within the parcels. Finally, LANL used imperviousness defined by developed areas in the NLCD to distribute the population from each census block. Population at Risk Population results include total PAR and PAR by age (e.g., under 14 and over 65 years). These are summarized in Figure 5 and Figure 6.
Figure 5: Summary of total, under 14-yr, and over 65-yr PAR
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Despite the wide range of peak discharges and subsequent flooded areas, the total sum of PAR is fairly consistent among participants. In addition, the participants consistently indicate that the majority of the PAR is located in flood water less than 2 meters deep. While there is consistency in total population by depth, PAR is less consistent with respect to flood arrival time. This indicates that PAR is perhaps sensitive to breach formulation. For example, HRW had nearly a 16-hour lag time for peak discharge. Therefore, the PAR in their solution is not generally at risk until more than 3 hours. This information is likely to have an impact on loss of life models that take into account warning time and evacuation plans.
Figure 6: PAR by peak depth Flood Severity Figure 1-7 shows the standard deviation among the different flood severity calculations.
Figure 7: Flood severity standard deviation There are significant differences on how the flood severity is defined across the participants. The methods used by TK and MMC were not described. HRW defined flood severity based on effects to population who are exposed (i.e., not within buildings) to the flood waters. LANL, MUST, and MANC used the U.S. Department of Homeland Security’s Dam Sector
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consequence estimation guidelines.2 These classifications are based more on structural stability and population within buildings than people directly exposed to the flood. There is complete agreement between the models on the severity assigned within the canyon (standard deviation of 0). Differences in severity classification are noted within the floodplain area, which the area where population in this case study is located. Loss of Life The total loss of life is shown in Figure 8. Although the method in which the loss of life was estimated varied between participants, the majority of estimates for loss of life was approximately 2,000 people. HRW’s loss of life estimate was the same order of magnitude as the other teams, but nearly twice as great. An explanatory factor in this difference could be the different technique used to calculate flood severity.
Figure 8: Total loss of life Economics There were differences in definition of direct economic impact. For example, LANL defined the direct economic impact as lost jobs, wages, and business outputs. LANL was the only participant to use business gross domestic product (GDP) per employee, but they did not include structural damage, which they defined as an insurable loss and not a direct economic impact. As such, LANL economic results are reported on a $/day basis, but did not estimate the duration of the impact. MAN, MMC, TK, and HRW all evaluated structural damages using the provided parcel information. The key challenge using the parcel data was assigning values to the occupancy classes. Because this information was not provided by the formulation team, each participant had to assume this information. The data used by participants ranged from available data in models, economic subject matter experts, and values reported in the literature. Figure 9 shows the total economic impact. Because of the difference in methods (e.g., GDP vs. insurable losses), and the assumption regarding asset value, the economic impact values range significantly. LANL reported $665,000/day, but the duration of the event is unknown. At the high end, MANC reported $2.6 Billion based on insurable losses.
2
Department of Homeland Security, 2011. “Estimating loss of life for dam failure scenarios.”
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Figure 9: Summary economic impact submitted by the participants
Conclusion In general, the results from each participant team are similar in terms of hydrodynamics. However, flood wave arrival times were different between teams. This is attributable to differences in the calculation of the breach hydrograph and differences in regression and physics-based formulations in addition to definition of thresholds used to quantify the arrival of the flood. The largest differences in peak flood depths were likely due to the requirement for the teams using irregular meshes to report output in a structured grid format. PAR estimates were also similar across teams. Differences are primarily due to different methods used by participants to spatially distribute population. The methods included uniform distribution of the population within the census block, uniform distribution of the population among parcels within the census blocks, and uniform distribution of the population among develop land within the census block. The most significant differences in the population estimates appeared to be the time after failure at which population is at risk. This has significance relative to flood wave warning time and potential evacuations. Finally, there were some significant differences in the economic impact. First, there were differences in interpretation of direct economic impact. This was either interpreted as business losses based on GDP or structural damage/insurable losses using parcel data. In addition, within the insurable loss estimation, there were significant differences in assessing the value of structures within parcels.
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PAPERS THEME C
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A Benchmark study on dam breach and consequence estimation using EMBREA and Life Safety Model M. Davison1, M. Hassan1, O. Gimeno1, M. van Damme1 and C. Goff1 1
HR Wallingford, Howbery Park, Wallingford, Oxfordshire, OX10 8BA, United Kingdom E-mail: [email protected]
Abstract This paper presents the modelled consequences of a hypothetical dam breach as laid out in Theme C of the 12th international Benchmark Workshop on Numerical Analysis of Dams. The EMBREA model was used to model the failure of the dam due to headcut erosion and to derive a breach hydrograph which was then used in InfoWorks ICM to model the 2D flood spreading. The results of the flood model have been used to calculate how severe the flood would be in terms of the total population at risk, loss of life using the Life Safety Model and economic impact. The paper shows that the consequences of the hypothetical flood will be severe in terms of casualties and economic damage. Introduction This paper presents the modelling of a potential breach of a hypothetical dam and the estimation of the consequences in the populated areas and economic activities downstream, within Theme C of the 12th International Benchmark Workshop on Numerical Analysis of Dams [1]. The paper is structured in three sections. Section 1 describes the dam breach model, which provides indicative predictions of a number of breach scenarios; Section 2 focuses on the flood wave propagation and Section 3 presents the human consequences and direct economic impact of the dam failure.
Dam Breach Modelling Choice of Breach Model During the last 15 years, HR Wallingford has undertaken the development of a new breach model, called EMBREA (previously known as HR BREACH). The IMPACT project [2] in 2005, Peeters et al. [3] in 2011, and Morris et al [4] in 2012 have demonstrated that EMBREA performs the best among the available breach models. On that basis, EMBREA was used to undertake the modeling for this benchmark work. Selection of Failure Modes EMBREA can simulate breach failure by overtopping and piping. For this study overtopping is considered to be the main dam failure mode, but piping was also investigated in one of the scenarios. Overtopping failure results in breach formation either due to headcut erosion or surface erosion. With headcut formation the erosion of material forms steps in the downstream face of the embankment which progressively grow in size and cut through the embankment. When the retreating headcuts reach the upstream side of the crest, the hydraulic control rapidly reduces in height and rapid failure ensues. With surface erosion, material is eroded from the embankment face and the crest area more smoothly leading to a reduced cross sectional embankment profile.
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The dominant process depends mainly on the material used in the dam construction. The hypothetical dam in this instance is a rolled earth fill structure composed of predominantly sandy clays and clayey sands with a relatively low permeability (1.9x10-6 cm/s). Based on the type of soil and the permeability provided, and using Shevnin et al. [5], its clay content is estimated to be more than 25%. In addition to the clay content, the matric suction in unsaturated soils adds to the cohesive properties of soils. The low permeability of the embankment soil indicates the high likelihood of the presence of an added cohesion due to matric suction throughout the erosion process. The overall cohesive properties make headcut erosion the most likely erosion mode during overtopping failure. The headcut erosion process is modeled in EMBREA using an erodibility coefficient (Kd), and a headcut migration coefficient (C). These parameters are usually measured on site or in a soil laboratory, but as the dam on this study is hypothetical, they were estimated based upon the soil clay content and compaction effort. To assess their uncertainty, the Monte Carlo analysis feature in EMBREA was used as described in below. A number of modeling runs were undertaken to establish the potential worst case scenarios for the hypothetical dam failure. Within these runs, consideration was given to the uncertainty and sensitivity of results to various aspects such as material erodibility, failure mode and erosion processes, as follows: Runs 1, 2 and 3. In these runs EMBREA was used to perform a Monte Carlo analysis to the overtopping failure with headcut (1000 runs with a triangular probability distribution for Kd and corresponding C values as given in Table 4). The output of the Monte Carlo run analysis was a frequency distribution of the peak breach outflow. The median value of this distribution was taken as the base run (Run 1). The corresponding values of the 75 % (Run 2) and 25 % (Run 3) exceedance probabilities were taken as the lower and upper limits respectively. Run 4. In this run, the erosion process was changed to overtopping through surface erosion rather than headcut to assess the impact of a different erosion process on the breach hydrograph. Other inputs were identical to Run 1. Run 5: In this run, a good grass protection is assumed to be present on the downstream face of the embankment to assess its impact on delaying the breach initiation and hence the wave arrival time at downstream locations. Other inputs were identical to Run 1. Run 6: In this run, the failure mode was changed to piping rather than overtopping to assess the impact of a different failure mode on the breach hydrograph. It should be noted that surface erosion is used in the model once the top of the pipe becomes unstable and fails (i.e. failure mode changes from piping to overtopping). Other inputs were identical to Run 1. Model set up This section provides a description of the model set up, including modeling boundary conditions, initial conditions, dam geometry and soil properties. The selection of such data was mainly based upon the data provided to participants in [1]. Due to the size of the reservoir, any inflow into the reservoir during the breach event has a negligible effect on the reservoir water levels, and has therefore not been included. Based on investigation of the immediate reach downstream of the dam, a low tailwater level was assumed meaning that the breach flow was not ‘drowned’. The dam geometry data is given in Table 1. Table 2 shows
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the initial conditions used in each series of runs. The dam failure for this benchmark takes place when the pool elevation is at crest elevation. Table 3 shows soil properties used in each series of runs and Table 4 shows Kd and C values for each run. The Kd values and C values were obtained from the qualitative relationships for different soil types provided in the EMBREA user manual version 1.3. Table 1: Dam geometry Crest level 272 mAD
Foundation level 211 mAD
Crest Width
Crest length
24m
360m
Upstream slope 1:3
Downstream slope 1:3
Initial breach depth (m) 0.25 NA
Initial breach width (m) 0.5 NA
Table 2: Initial conditions Run No. 1-5 6
Pipe diameter (m) NA 0.03
Pipe Level (mAD) NA 212
Initial water level (mAD) 272 272
Table 3: Soil properties Porosity Unit Wt (KN/m3) Manning’s n
0.40 19.64 0.025
Friction Angle (degrees) Cohesion (KN/m2) Plasticity Index
14 19.15 10
Table 4: Kd and C values Run No. Monte Carlo 1, 5 and 6 2 3 4
Erodibility coefficient, Kd (cm3/N-s) Triangular probability distribution with a lower value of 1, mid value of 5 and an upper value of 10 5.5 4.3 6.8 5.5
Headcut migration coefficient, C (s-2/3) Values calculated from Kd values based on EMBREA user manual version 1.3 0.0032 0.0026 0.0040 NA
Modelling results, observations and conclusions Figure 1 shows a comparison of the breach outflow results for Runs 1 to 6. Runs 1, 2 and 3 hydrographs show that Kd and C values have a significant effect on the breach peak outflow and initiation time in the case of overtopping failure due to headcut formation. Increasing them by a factor of 1.3 (Run 1) and 1.6 (Run 3) has increased the breach peak outflow by about 40 and 100% compared to run 2 respectively. Run 4 shows that changing the type of erosion process results in a significant reduction in the peak breach outflow and initiation time. The peak outflow was reduced by about 78% compared to Run 1 with less than one hour of initiation time. Run 5 shows that for the case considered, a grass protection increases the initiation time of the breach with a negligible change in the peak breach outflow compared to Run 1. Run 6 shows that changing of the failure mode on breach formation to piping
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significantly reduces the peak breach outflow and initiation time. The peak outflow was reduced by about 30 % compared to Run 1, with the shortest initiation time of all the runs.
Figure 1: Breach outflow hydrograph for Runs 1 to 6
Flood Modelling Data provided The data provided for the flood model to participants was: digital elevation model (DEM) representing pre-dam and post-dam construction gridded dataset of land use/cover The horizontal resolution of both datasets is about 9.5m. The land use/cover data follows the classification guidance and values provided in the National Land Cover Dataset (NLCD), 2006. Roughness estimation Surface roughness values for the flood simulation were estimated for each land use/cover class using the Conveyance and Afflux Estimation System (CES/AES) software [6]. The tool provides a database of roughness values extracted from various sources in the literature and given as mid, upper and lower estimates covering the range of roughness values expected within each system. This provides some measure of the uncertainty associated with the estimation of roughness by deriving an upper and a lower roughness credible scenario. The values assigned to each land use/cover class are shown in Table 5, as well as the total areas covered in the test bed region. For the developed areas, the low and high roughness values correspond to the lowest and highest values derived from the possible share between pervious/impervious areas given in the descriptions.
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Table 5: Land use/cover estimated roughness Class
ID Description
Water
11 Open Water Developed, 21 Open Space Developed, 22 Low Intensity Developed, 23 Medium Intensity Developed 24 High Intensity Barren Land 31 (Rock/Sand/Clay) 41 Deciduous Forest 42 Evergreen Forest 43 Mixed Forest 52 Shrub/Scrub Grassland/ 71 Herbaceous
Developed
Barren Forest Herbaceous Planted/ Cultivated Wetlands
Pasture/Hay 81 90 Woody Wetlands
Winter Summer Mid Low High Mid Low High 16.9% 0.010 0.010 0.010 0.010 0.010 0.010 Area
5.2%
0.029 0.025 0.034 0.029 0.025 0.034
10.7% 0.030 0.026 0.035 0.030 0.026 0.035 5.1%
0.031 0.026 0.036 0.031 0.026 0.036
0.5%
0.032 0.025 0.036 0.032 0.025 0.036
0.2% 28.4% 7.9% 0.1% 20.9%
0.022 0.073 0.251 0.142 0.073
3.5%
0.046 0.027 0.083 0.046 0.027 0.083
0.4% 0.2%
0.046 0.027 0.083 0.046 0.027 0.083 0.054 0.039 0.064 0.054 0.039 0.064
0.017 0.049 0.151 0.091 0.048
0.028 0.112 0.341 0.205 0.112
0.022 0.102 0.251 0.142 0.102
0.017 0.072 0.151 0.091 0.072
0.028 0.162 0.341 0.205 0.162
Flood model The 2D Infoworks ICM software was used to simulate spreading of the flood given by the breach hydrograph into the valley downstream of the dam. Three roughness scenarios were considered that correspond with the mid, low and high roughness values for each land use class. For each roughness scenario, the model was run for each of the six dam breach hydrographs from the breach model. The hydrographs were used as an inflow boundary condition for the flood model. This was represented using the breach width and depth corresponding to the maximum breach outflow in combination with the post-dam construction digital elevation model. Results The results of the flood model presented in this section correspond to the overtopping failure with headcut failure mechanism only, as this is the most likely failure mode for the dam. The results for other failure modes can be provided upon request. Figure 3 shows the attenuation of the breach hydrograph of the median scenario (Run 1) between the five cross sections on Figure 2. As shown, the peak flow is reduced by 10% when the flood arrives at cross section 5, which takes about 14 min from the moment of the peak breach discharge. In the scenario with 75% exceedance probability (Run 2), the peak flow attenuation is of a similar amount (9%), but the wave takes 10 min to reach cross section 5. Finally, for the scenario with 25% exceedance probability (Run 3), the peak flow hardly decreases (see Table 6). The flood extents for the three scenarios are very similar, with the flood spreading to the valley on the north between sections 3 and 4 for the median and 75% scenarios (see Figure 2). 245
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The peak flood depths for the 25%, the Median, and 75% exceedance probability scenarios are given in Table 7. Flood depths are mostly below 2m for all three scenarios, and below 4m for more than 90% of the flooded area. The spatial variability of the peak flood depths is shown in Figure 4 at 0.5m intervals for the median scenario. The higher flood depths in the city occur between cross sections 3 and 5 and also along the south end of the flood extent downstream section 5. That is also the main flowpath through the plain, with unit flow rates mainly between 4 and 10 m2/s (Figure 5), increasing to 30m2/s near section 5. The area of the city with higher unit flow rates –up to 80m2/s- is between cross sections 4 and 5. The arrival time of the first inundation relative to the time at which a breach flow initiates is shown in Figure 6. The flood reaches the populated area after 1.5h to 2h, but the majority of the city starts flooding after 2h. That is the arrival time of the small flow at the beginning of the dam breach initiation. However, the peak flow travel time in Figure 7 shows that the flood wave peak travels from the dam to the city in less than 15min, reaching the further edge in less than 90min, and hence the peak flow travel time is the travel time to consider.
Figure 2: Cross-section locations and flood extent
Figure 3: Cross section discharges median run
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Table 6: Cross section discharges Breach Cross Section 5 3 Scenario Peak flow (m /s) Time Peak flow (m3/s) Time Median 13,956 18:51 12,540 (-10.1%) 19:05 (14min) 75% Exc. Prob. 19,257 15:05 17,537 (-8.9%) 15:15 (10min) 25% Exc. Prob. 9,908 23:51 9,877 (-0.3%) 24:05 (14min)
Figure 4: Peak flood depths (m) for the median scenario Table 7: Flooded area (m2) Peak Flood Depth Range (m) 0 - 0.5 0.5 - 1 1 - 1.5 1.5 - 2 2 - 2.5 2.5 - 3 3 - 3.5 3.5 - 4 4 - 4.5 4.5 - 5 5 - 5.5 5.5 - 6 6 - 6.5 6.5 - 7 7 - 7.5 7.5 - 8 >8 TOTAL
Flooded area (m²)
25% exceedance probability 7,684,136 9,977,083 5,110,216 3,012,575 2,115,959 1,231,376 515,700 355,773 186,327 136,850 143,135 95,004 61,959 57,559 61,421 58,368 522,524 31,325,966
Median 3,842,921 8,084,897 6,912,247 3,611,785 2,523,993 2,305,789 1,875,305 1,185,490 683,978 463,618 354,426 222,336 151,037 131,372 113,772 86,025 900,657 33,449,648 247
75% exceedance probability 3,776,472 8,525,347 7,732,446 4,025,567 2,710,321 1,967,706 1,380,169 696,280 424,916 386,304 256,907 162,801 150,050 140,801 82,433 71,119 973,392 33,463,027
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Figure 5: Peak unit flow rate in m2/s for the median scenario
Figure 6: Inundation arrival time from initiation time for the median scenario
Figure 7: Peak flow travel time
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Consequence Estimation Flood severity Flood Severity has been classified according to the Defra methodology presented in technical reports [7, 8]. The method describes the flood hazard as a function of velocity, depth and the presence of debris as (1) Where, HR is the (flood) Hazard Rating, d is the Depth of flooding (m), v is the velocity of the floodwater in (m/s), and DF is the debris factor (assumed to be 0.5 for depths <0.25m, and 1.0 for depths >0.25m). Four flood hazard classes were established based on the thresholds of flood hazard rates (HR) that cause people to lose stability. Thresholds are derived from the experiments on people with different height multiplied by mass values. For this study, a 3 class categorization (low, medium and high) was required. Therefore the thresholds have been redefined as in Table 8. Figure 8 shows that the flood severity is mostly high in the flooded area. Table 8: Flood Severity classification ID
Class
1
Low
Flood Hazard Rate (HR) < 0.75
2
Medium
0.75 - 1.25
3
Significant >1.25 to high
Description Caution. Flood zone with shallow flowing water or deep standing water” Dangerous for some (i.e. children, the elderly and the infirm). Danger: Flood zone with deep or fast flowing water Dangerous for most people to all people. Danger to extreme danger: flood zone with deep fast flowing water
Figure 8: Flood severity
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People at risk Given the reporting requirements of the people at risk by time intervals and flood depths, the calculation of the number of people at risk was based on the flood extent from the hydraulic model and the population from the census data and the buildings. The average number of people in a building was calculated by dividing the total population by the number of buildings for each census zone. To calculate the number of people at risk from flooding the number of buildings in the maximum flood extent was determined from GIS and the population summed. Table 9 shows the total number of people at risk per age group and flood depth. Most of the people are at areas with flood depth below 1.5m. Table 9: Number of people at risk per age group and flood depth Peak Flood Depth Range (m) 0 - 0.5 0.5 - 1 1 - 1.5 1.5 - 2 2 - 2.5 2.5 - 3 3 - 3.5 3.5 - 4 4 - 4.5 4.5 - 5 5 - 5.5 5.5 - 6 6 - 6.5 6.5 - 7 7 - 7.5 7.5 - 8 >8 Total
Total Population At Risk 4,608 10,189 5,336 1,209 710 621 497 369 283 252 211 182 123 91 52 43 128 24,904
12-yr and Under Population at Risk 693 1,528 763 146 113 118 106 93 71 63 55 47 33 23 13 11 37 3,913
65-yr and Over Population at Risk 841 1,878 1,139 326 163 111 72 28 21 21 17 13 7 8 4 3 6 4,658
Loss of life As with the number of people at risk from flooding, the calculation of loss of life uses the same population distribution. There are a number of methods to calculate loss of life. Brown and Graham [9] is a very simple method where the population at risk is multiplied by a factor depending on the length of time between the breach and flood warning. DeKay and McClelland [10] use a simple method that uses different equations for high and low force flooding. Graham [11] expands on the simple methods of Brown and Graham [9] and DeKay and McClelland [10] using the concept of flood severity, flood warning and the understanding of the exposed population to vary the fatality rate. The UK Flood Risk to People methodology [7] calculates the loss of life based on the hazard rating, the type of people exposed and the sensitivity of the area. This is similar to the method of Graham [11] used in the US, although it is more advanced because it uses the magnitude of the flood expressed by the spatially variable hazard rating to define fatality rate rather than a global fatality rate. The sensitivity of the area is defined by the time of travel of the flood wave, type of housing, and the presence of flood warning and evacuation plans. This calculation is most appropriate for large areas for 250
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example census zones. Finally, the Life Safety Model [12] is an agent based model that determines the fate of each person that is exposed to the flood in a dynamic way through time and space. The model is designed for improving evacuation plans. The loss of life function in the model is based on the water depth and velocity. This function can be used in a static way to calculate loss of life from the maximum DV (Depth x Velocity). After consideration of all the above methodologies, the approach taken was to primarily use the loss of life function from the Life Safety Model (LSM) to provide an estimate of loss of life based on people distributed in buildings. This has been chosen ahead of the UK Risk to People (UK R2P) method because of the requirement to produce gridded output at a high resolution. The Theme C2 information [13] states that the dam fails at 11pm on Saturday night with no witnesses that are able to raise an alarm, and that there is no evacuation plan in place. It has therefore been assumed that the total population are in their homes and receive no warning for the loss of life calculations. The loss of life estimated for this pilot from the Life Safety Model approach is 4,966. That is based on conservative assumptions that a) people are on the ground, and b) take no measures to avoid the flood. The spatial variability of fatalities in Figure 9 show that the area between cross sections 3 and 5 is the most affected. All fatalities are after 3h of the breach initiation. An alternative scenario where people shelter in buildings has been performed with the Life Safety Model where loss of life occurs if the building is submerged or collapsed. Assuming that people shelter in buildings the number of fatalities is estimated to be 3,075. A more realistic estimate can be produced assuming that people take refuge in multi-storey buildings or respond to a warning and evacuate the area, for which the dynamic Life Safety Model would be appropriate. A comparison of the methods is shown in Table 10. The UK R2P method was used as a secondary check of LLOL. It is an area wide calculation which uses a simple vulnerability measure multiplied by the Hazard Rating to define the number of people at risk. The number of injuries is then estimated as 2 times the number of old and sick people as a proportion of the number at risk. The fatality rate is 2 times the Hazard Rating, and the number of fatalities is the number of injured multiplied by the fatality rate. This produces a significantly lower estimate of loss of life than the LSM because it assumes that young and healthy people can escape the flood. In this case the lack of flood warning means that the whole population should be considered at risk. The calculation has been modified so that the whole population is exposed producing an estimate of loss of life of 4,134. Table 10: Loss of life Estimation method Loss of life
Life Safety Model approach 1 4,966
Life Safety Model approach 2 (people shelter in buildings) 3,075
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UK Risk to people methodology 1,150
Modified UK Risk to people methodology 4,134
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Figure 9: Loss of life from Life Safety Model, cell size ~9.5m x 9.5m Economic damage Depth damage curves have been taken from the MCM [14] 2010 update for residential and non-residential properties that are similar to the description given in the building data provided. These curves were developed for UK properties. The damages have been converted to US$ using an exchange rate of 1.51, but no other adjustments have been made to account for differences in US and UK properties. Alternatively there is a US method that relates the damage as a percentage of the building value for given water depths [15], but this method could not be used because building values were not provided in the building data. The total economic damage for the base dam breach has been estimated as US$1,237,582,738. The spatial variability of the economic impact is shown in Figure 10, the commercial buildings having the higher damage per grid cell (~89.8m2).
Figure 10: Direct economic impact Observations It would have been preferable to distribute the number of people across the residential buildings based on knowledge of household groupings. However, the census data provided was not detailed enough. In addition the building data provided contained the overall footprint rather than the number of individual properties (in the case of flats). This meant that some census zones had large population with little or no residential buildings. For example census zone 30 has 3 large properties classified as commercial but a population of 565. To reduce 252
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uncertainty in the distribution of people within a census area, more detail is required on the demographics and number and type of buildings. The LSM can also model the behaviour of people in a flood situation as they seek to avoid the flood, thereby reducing the predicted loss of life.
Conclusions The paper presents the consequences of the breach of the hypothetical dam as described in Theme C of the 12th International Benchmark Workshop on Numerical Analysis of Dams. The dam breach analysis undertaken using EMBREA highlights the importance of choosing the correct erosion method and the impact of choosing the correct erodibility of the soil. Based on the data provided, it was estimated that the embankment fails according to headcut erosion leading to a steep hydrograph with a high peak discharge, the consequences of which are more severe than when an embankment fails due to surface erosion or piping. The effects of the different roughness scenarios considered on the flood model was negligible compared to the impact of the different hydrographs used as input, emphasizing again the need for a good dam breach model. The flood model results provide useful information for warning and evacuation planning. The model indicates that the first flow from the dam breach reaches the populated area 1.5h to 2h after the breach initiates, but the peak flow travels from the dam to the city in less than 15min, reaching the further edge in less than 90min. The total direct economic impact is estimated at more than US$1.24b. The loss of life calculations performed with the static Life Safety Model show a loss of life of 4,966, which represents nearly the 20% of the total people at risk.
References [1] 12th International Benchmark workshop on numerical analysis of dams, "THEME C: Computational Challenges in Consequence Estimation for Risk Assessment: Numerical modelling, Uncertainty Quantification, and Communication of Results. Part 1- Hydraulic Modelling and Simulation," ed: International Commission on Large Dams (ICOLD), 2013. [2] M. W. Morris, M. A. A. M. Hassan, and K. A. Vaskinn, "Conclusions and recommendations from the IMPACT Project: WP2 Breach Formation," 2004. [3] P. Peeters, T. Van Hoestenberghe, L. Vincke, and P. Visser, "SWOT analysis of breach models for common dike failure mechanisms," presented at the 34th IAHR World Congress - Balance and Uncertainty, 33rd Hydrology & Water Resources Symposium, 10th Hydraulics Conference, Brisbane Australia, 2011. [4] M. W. Morris, M.A.A.M. Hassan, T.L. Wahl , R.D. Tejral, G.J. Hanson, and D.M. [5] Temple, "Evaluation and development of physically-based embankment breach models," presented at the 2nd European Conference on Flood Risk Management, Rotterdam, The Netherlands, 2012. [6] V. Shevnin, O. Delgado-Rodríguez, A. Mousatov, and A. Ryjov, "Estimation of hydraulic conductivity on clay content in soil determined from resistivity data," Geofisica Internacional, vol. 45, 2006. [7] Defra and Environment Agency, "Reducing uncertainty in river flood conveyance, phase 2, Conveyance manual," Environment Agency, report2004. [8] Defra and Environment Agency, "R&D outputs: Flood Risk to People Phase 2.
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[9] Guidance Document," 2006. [10] Defra and Environment Agency, "Framework and guidance for Assessing and Managing Flood Risk Development, Flood Risk Assessment Guidance for New Development," Technical report2005. [11] C. A. Brown and W. J. Graham, "Assessing the Threat to Life from Dam Failure," [12] Water Resources Bulletin, vol. 24, pp. 1303-1309, 1988. [13] M. L. DeKay and G. H. McClelland, "Predicting Loss of Life in Cases of Dam Failure and Flash Flood," Risk Analysis, vol. 13, pp. 193-205, 1993. [14] W. J. Graham, "A Procedure for Estimating Loss of Life Caused by Dam Failure," ed. [15] Denver, Colorado: U.S. Department of Interior, Bureau of Reclamation, Dam Safety [16] Office, 1999. [17] BC Hydro, "Dam Safety, BC Hydro Life Safety Model: Formal model description," ed: BC Hydro,, 2004. [18] I. B. w. o. n. a. o. dams, "THEME C: Computational Challenges in Consequence Estimation for Risk Assessment: Numerical modelling, Uncertainty Quantification, and Communication of Results. Part 2- Consequence Estimation," International Commission on Large Dams (ICOLD) ed, 2013. [19] E. Penning-Rowsell, C. Johnson, S. Tunstall, S. Tapsell, J. Morris, J. Chatterton, et al., "The benefits of Flood and Coastal Risk Management: A manual of assessment techniques," ed. Middlesex: Middlesex University Press., 2005. [20] C. Scawthorn, P. Flores, N. Blais, H. Seligson, E. Tate, S. Chang, et al., "HAZUS-MH flood loss estimation methodology. II. Damage and loss assessment," Natural Hazards Review, vol. 7, pp. 72-81, 2006.
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Uncertainty in Two-Dimensional Dam-Break Flood Modeling and Consequence Analysis M.S. Altinakar1, M.Z. McGrath1, V.P. Ramalingam1, D. Shen1, Y. Seda-Sanabria2 and E.E. Matheu3 1
National Center for Computational Hydroscience and Engineering, The University of Mississippi, Brevard Hall Room 327, P.O. Box 1848, University, MS 38677-1848, USA 2 Critical Infrastructure Protection and Resilience Program, Office of Homeland Security, U.S. Army Corps of Engineers, Headquarters, Washington, DC 20314, USA 3 Critical Lifelines Branch, Sector Outreach and Programs Division, Office of Infrastructure Protection, National Protection and Programs Directorate, Washington, DC 20528, USA E-mail: [email protected]
Abstract This paper presents the preliminary results of an extensive study to evaluate and quantify uncertainty in two-dimensional numerical dam-break flood modeling and consequence analysis based on a benchmark test case. The benchmark problem consists of a hypothetical 61m-high embankment dam in a mountainous region with lightly populated urban areas located downstream. Although the benchmark problem was intended mainly for estimating the uncertainty in the evaluation of the consequences of failure of a dam near populated areas with complex demographics, infrastructure and economic activity, the present study also investigated the uncertainty in two-dimensional numerical modeling based on three control variables. DSS-WISE™ numerical model was used to calculate dam-break flood and potential loss-of-life for a total of 120 cases, which represented 40 random pairs of breach width and breach formation time, each computed with three different sets of Manning’s coefficients defined based on land use/cover type. Computed results include raster maps of maximum flood depth, maximum flood discharge per unit width and flood arrival time as well as hydrographs at 7 cross sections. Analysis of population at risk (PAR) impacted by the flood and loss-of-life were also computed for all 120 simulations. The paper presents some preliminary results based on the statistical analysis of results.
Introduction Test case consists of synthetic data specially designed for the benchmark study. A hypothetical 61m-high embankment dam located in a mountainous area impounds a reservoir having a normal storage of 38 million cubic meters. A lightly populated urban area, called Hydropolis, is located about 3.5 km downstream from the dam. The benchmark test case data for hydrodynamic simulation consists of (1) digital elevation model (DEM) of the area of interest with and without the dam; (2) dam geometric characteristics; (3) reservoir stagevolume and stage surface area curves; (4) dam and foundation material; (5) dam failure scenario and conditions ate the time of failure; and (6) gridded data of land use/cover. In addition the following data is provided for consequence estimation: (1) shapefile of the census data for Hydropolis with breakdown into various age and gender classes; (2) shapefile for twenty economic activity classes; (3) shapefile for parcel data with 28 zone classes (residential, commercial, etc.). The complete dataset and supporting documents have been published on the conference website under Theme-C and will not be repeated here. It is
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assumed that the dam fails without warning when the water surface in the reservoir is at the crest elevation and the population downstream cannot be warned.
Two-Dimensional Dam-Break Flood Simulation Model Two dimensional simulations of dam-break flood were performed using the DSS-WISE™ software developed by the National Center for Computational Hydroscience and Engineering (NCCHE), The University of Mississippi. DSS-WISE™ is an integrated platform, which couples a state-of-the-art two dimensional numerical model with GIS-based pre- and postprocessors, that are developed as an extension to ArcGIS® software developed and commercialized by ESRI. Its numerical solver – CCHE2D-FLOOD – solves the conservative form of the two-dimensional shallow water equations, given as [
]
[
(1)
]
in which, is the vector of conserved variables, directions, and . These are defined as
[
]
[
]
[
and
are flux vectors in
] [
(
)
(
)]
and
(2)
where, is the flow depth, and components of unit discharge in and directions, the bed elevation, the net source/sink mass flux per cell area per unit time, and the gravitational acceleration. The slopes of energy grade line are given by Manning’s equation √
√
(3)
with as the Manning’s coefficient of roughness. Note that and correspond to the components of the unit discharge in and directions, respectively, i.e., and . Finite volume discretization of Equation (1) over a regular Cartesian mesh provides an explicit equation for advancing the values of the conserved variables in time: (
)(
)
(
)(
)
4
In Equation (4), and define the cell size in and directions, and the time step size. The intercell fluxes are computed using the first order Harten-Lax-van Leer-Contact (HLLC) approximate Riemann solver by Toro et al. (1994), which was implemented following the methodology described by Kim et al. (2007). The details can be found in Altinakar et al. (2013a), Altinakar (2012) and Altinakar and McGrath (2012). The DSS-WISE software and its solver CCHE2D-FLOOD have been fully verified using analytical solutions and validated using laboratory and field data. Blind tests have also been undertaken in collaboration with U.S. Army Corps of Engineers (Altinakar et al., 2012).
Data for Benchmark Test Case Figure 1 shows the digital elevation model without dam. Open water surface to the East of the area of interest was converted into “NoData”. The numerical model treats NoData cells as an
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internal free outflow boundary. This prevents flood propagating over the open water surface, which would otherwise be perceived as bed topography. The dam is represented in the DEM as an idealized structure representing a wall of constant thickness. The embankment dam has a crest width of 24m and the upstream and downstream slopes of 3H:1V. Considering that the dam has a height of 61m, the width of the idealized wall was chosen as 207m, which corresponds to the average width of the embankment dam. The overtopping failure was assumed to take place when the reservoir reaches crest elevation. Thus, the initial condition models the reservoir as a stagnant water body with surface elevation equal to crest elevation. Cross Section 6
Elevation (m) 1823.5
Breach Cross Section
67.2
Idealized Dam
Open water area removed by changing elevations to No Data
Figure 1: DEM without the dam (pre-construction) with open water area removed.
Selection of Test Cases Three parameters were chosen to investigate the uncertainty of unsteady hydrodynamic simulations: (1) bottom breach width, ; (2) breach formation time, ; and (3) the set of Manning’s coefficient of roughness, . Prediction of Breach Parameters Empirical equations have been developed to estimate breach parameters based on the analysis of the data from historic dam failures (Wahl, 2004). The present study uses the empirical equations given by Froehlich (1995) to compute the breach width and breach formation time (5) (6) where is the average breach width (m), i.e. the breach width at half of the breach height (m), is a constant equal to 1.4 for overtopping failures, and is the reservoir volume at the time of failure (m3). Equations (5) and (6) were obtained based on data from past failures of zoned earthen, earthen with a clay core, and rockfill embankment dams. The dam heights ranged from 3.66m to 92.6m (with 90%<30m and 76%<15m), the volume of water in the reservoir ranged from 0.0130×106 m3 to 660.0×106 m3 (with 87%<25.0×106 m3 and 76%<15.0×106 m3) and the lake surface areas ranged from 0.045km2 to 2,165km2. The data for the benchmark test case (61m, 38×106 m3, and 1.584 km2), which is well within the range of cases used to establish Equations (5) and (6). Thus, the average breach width and breach formation time are calculated as and ̅ , respectively. Considering the
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side slopes of 1.4H:1V [6], the bottom with and top width of the trapezoidal breach cross and , respectively. section are Uncertainty of prediction of breach parameters for embankment dams was investigated by Wahl (2004) [12]. It was shown that the prediction errors using empirical equations, including Froehlich (1995) [6], exhibit a lognormal probability distribution (base 10) defined by a mean value, , and a standard deviation, . Table 1 summarizes the uncertainty estimates of the breach parameters predicted using Froehlich (1995). Using the values given in Table 1, the 95% confidence interval for the breach bottom width extends from 25m to 146m whereas the 95% confidence interval for breach formation time extends from 0.3hr to 4.9hr. Table 1: Uncertainty Estimates for Froehlich (1995) Equations (taken from Wahl, 2004). Number of Cases (log Before outlier After outlier cycles) exclusion exclusion Breach width Eq. (5) 77 75 +0.01 Breach formation time Eq. (6) 34 33 -0.22 *95% confidence interval is given around a hypothetical predicted value of 1.0 Parameter
Equation
(log cycles) ±0.195 ±0.32
95% Confidence interval* 0.40 – 2.4 0.38 – 7.3
Manning’s Coefficients Manning’s coefficients were assigned based on the given land use/cover data with 16 size classes that follow the classification guidance and values provided in the U.S. National Land Cover Dataset (NLCD). The land use/cover classes for the area of interests are shown in Figure 2. Table 2 shows the three sets of Manning’s coefficient used in the present study. It should be noted that the area of interest does not have classes 12 and 95. Figure 3 provides a visual comparison of the three sets of Manning’s coefficients for each land use/cover class.
Figure 2: Map of land use/cover classifications. Latin Hypercube Sampling for Breach Parameters Latin Hypercube Sampling (LHS) method was used to select 40 random pairs of bottom breach width, and breach formation time, . LHS is a stratified-random method which provides an efficient way of sampling variables from their distributions (Iman et al. 1981). The cumulative distribution (normal distribution of logarithms) of both variables were divided into 40 intervals of equal probability. For each variable, a value is randomly selected from every interval. The selected values are paired randomly to achieve an optimal filling of the parameter-uncertainty space defined by the 40×40 matrix of and . Design of the LHS was accomplished using the “lhsdesign” function in Matlab® software using an option that 258
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maximizes the minimum distance between the pairings. Figure 4a shows the LHS design used in the present study. Since and have lognormal distributions (base 10), the probability values, , used in LHS were converted into real values using the following expressions: Table 2: Land Use/Cover Classes and Three Sets of Corresponding Manning’s Coefficients. ( ) DSSFEMA NOAA Min. WISE(1) HAZUS(2) C-CAP(3)) Value 11 Open Water 0.0330 0.0010 0.0250 0.0010 12 Perennial Ice/Snow 0.0100 0.0100 0.0100 0.0100 21 Developed-Open Space 0.0404 0.0200 0.0350 0.0200 22 Developed- Low Intensity 0.0678 0.0500 0.1200 0.0500 23 Developed- Med. Intensity 0.0678 0.1000 0.1200 0.0678 24 Developed- High Intensity 0.0404 0.1500 0.1200 0.0404 31 Barren Land 0.0113 0.0900 0.0300 0.0113 41 Deciduous Forest 0.1000 0.1000 0.1600 0.1000 42 Evergreen Forest 0.1000 0.1100 0.1800 0.1000 43 Mixed Forest 0.1200 0.1000 0.1700 0.1000 52 Shrub/Scrub 0.0400 0.0500 0.0800 0.0400 71 Grassland/Herbaceous 0.0400 0.0340 0.0350 0.0340 81 Pasture/Hay 0.0350 0.0330 0.0500 0.0330 82 Cultivated Cropland 0.0700 0.0370 0.1000 0.0370 90 Woody Wetlands 0.1500 0.1000 0.1500 0.1000 95 Herbaceous Wetlands 0.1825 0.0450 0.0500 0.0450 (1) Altinakar et al. (2013b); (2)Luettich and Westerink (2009); (3)ARCADIS (2011) NLCD Class
Description
Mannin's Coefficient, n (m-1/3s)
DSS-WISE
FEMA HAZUS
Average Value 0.0197 0.0100 0.0318 0.0793 0.0959 0.1035 0.0438 0.1200 0.1300 0.1300 0.0567 0.0363 0.0393 0.0690 0.1333 0.0925
Max. Value 0.0330 0.0100 0.0404 0.1200 0.1200 0.1500 0.0900 0.1600 0.1800 0.1700 0.0800 0.0400 0.0500 0.1000 0.1500 0.1825
NOAA C-CAP
0.2000 0.1800 0.1600 0.1400 0.1200 0.1000 0.0800 0.0600 0.0400 0.0200 0.0000
11 12 21 22 23 24 31 41 42 43 52 71 81 82 90 95
NLCD Class
Figure 3: Comparison of three sets of Manning’s coefficients assigned to NLCD classes. ̅̅̅̅
[
]
;
̅
[
]
(7)
In the above expression, stands for the inverse of the cumulative distribution function of standard normal distribution, with mean equal to zero and standard deviation equal to 1, at the probability value of . For each variable, i.e. or , its corresponding and values were used. The selected 40 pairs are provided in Table 3. In Figure 4a, the numbers for the data points correspond to combination numbers in Table 3. Figure 4b and Figure 4c show that the randomly selected of and values display a lognormal probability distribution, as expected. Assumptions Regarding the Progression of the Breach Geometry 259
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Histogram of Generated Wb Data and Lognormal PDF Histogram of Data and Lognormal PDF
Histogram of Generated W Data and Lognormal PDF Histogram of Data band Lognormal PDF
14
16
12
14
Number Observations of Observations Numberof
Number Numberof of Observations Observations
Froehlich (1995) expression, used for estimating breach characteristics, assumes that the final breach cross section for overtopping failures is a trapezoidal with side slopes of 1.4H:1V. Figure 5a shows the mean, minimum and maximum breach sizes selected by LHS. The elevations of the cells under the footprint of the idealized dam were initially raised to the level of the crest. As soon as the simulation starts, the breach occurs by gradually bringing down these cells to the original valley elevation during the breach formation time, , as shown in Figure 5b. The cell elevations are not permitted to go below the original bed elevation.
10
8
6
4
2
12 10 8 6 4 2
0 0
50
100 150 Breach Bottom Width (m)
200
0
250
0
2
Breach Bottom Width, Wb, (m)
(a)
4 6 8 Breach Formation Time (hr)
10
12
Breach Formation Time, tf, (hr)
(b)
(c)
Figure 4: (a) LHS design obtained by maximizing minimum distance between data pairs; (b) Histogram of randomly selected breach widths and the fitted lognormal distribution; (c) Histogram of randomly selected breach formation times and the fitted lognormal distribution. Table 3: List of 40 pairs of breach width and breach formation time selected using LHS. Pair No 1 2 3 4 5 6 7 8 9 10
Pair No 11 12 13 14 15 16 17 18 19 20
(hr) 2.2 1.0 0.6 5.2 2.0 0.6 1.8 0.4 3.4 0.8
(m) 38.3 130.3 116.9 68.8 58.6 83.6 80.6 37.2 66.7 177.3
Pair No 21 22 23 24 25 26 27 28 29 30
(hr) 1.4 2.1 1.6 0.6 1.7 1.3 0.5 0.2 2.5 1.7
300
300
280
290
(m) 70.6 46.2 60.7 88.0 113.1 98.5 44.7 27.4 65.7 48.9
220 200 100
150
200
250
300 350 Chainage (m)
Cross section
Dam Crest
Minimum Breach
Maximum Breach
400
450
500
(m) 88.4 53.2 49.6 78.4 92.9 63.6 31.8 73.8 47.5 36.0
(hr) 0.3 1.4 0.7 2.6 0.5 0.8 1.1 0.9 4.9 0.7
t=0
270
240
Pair No 31 32 33 34 35 36 37 38 39 40
(hr) 0.4 2.8 0.9 1.0 1.2 3.0 1.1 0.7 1.5 1.3
280
260 Elevation (m)
Elevation (m)
(m) 41.9 51.8 103.5 55.1 57.0 59.0 30.6 20.0 74.4 40.1
t = tf / 4
260 250 240
t = 2tf / 4
230
t = 3tf / 4
220
t = tf
210
200
Mean Breach
100
150
200
250
300
350
400
450
500
Chainage (m)
(a)
(b)
Figure 5: (a) Breach geometry for mean, minimum and maximum breach bottom widths; and (b) Linear progression of the breach geometry assumed in the numerical model.
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Results of Hydrodynamic Simulations A total of 120 numerical simulations were performed by combining 40 randomly chosen pairs of breach width and breach formation time with three sets of Manning’s coefficients. Each simulation provided three raster files (maximum flow depth, , flood arrival time, , and maximum specific discharge, ), and discharge hydrographs at seven cross sections (dam crest, five pre-defined cross sections along the main downstream flow, and one cross section to capture the flows towards north) recorded with a time interval of two minutes.
Average Hmax (m)
St. Dev. of Hmax (m)
Minimum Value of Hmax (m)
Maximum Value of Hmax (m)
Figure 6: Statistics of maximum flood depth,
meter 0.25 0.50 0.75 1.0 2.0 3.0 4.0 5.0 8.0 12.0 17.0 22.0 27.0 32.0 37.0 42.0 47.0 52.0 57.0 62.0
(m), based on 120 simulations.
Maps of Maximum Depth, Flood Arrival Time and Maximum Discharge per Unit Width The set of 120 raster files were used to calculate maps of ensemble average, standard deviation, and minimum and maximum values of the maximum flood depth (Figure 6), the flood arrival time (Figure 7), which is defined as the time at which the dry computational cell becomes wet, regardless of the depth, and the maximum discharge (Figure 8). In these maps, cells with a null value are not displayed. As expected, the highest flow depths and unit discharges occur in the narrow valley immediately downstream of the dam. Comparison of minimum and maximum value maps of these three variables indicate that large differences exist between individual runs. The largest standard deviations of maximum flood depth and maximum unit discharge occur in the narrow valley downstream of the dam whereas the largest standard deviation values for the arrival time are observed farthest from the dam. The area on the northern part is inundated only for certain simulations. The longest time arrival and largest standard deviations of arrival time are observed in this area (Figure 7). An understanding of the sensitivity of results to different parameters can be gained by analyzing and comparing results files of individual simulations. Figure 9 shows a comparison of the flood maps of selected pairs of breach width and breach formation time with all three sets of Manning’s coefficients. The upper right hand corner shows a plot of all 40 random pairs selected with LHS in terms of real values of and . This should be compared with the plot in probability space shown in Figure 4a. As shown, there are no simulations in the upper right corner of the plot corresponding to large breach widths with long formation times. The inspection of Figure 9 shows that the area on the north part of the DEM is inundated only for certain combinations of parameters. In space, the simulations with inundation of 261
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north part correspond to small breach formation times and large breach widths. In addition, higher Manning’s values favor inundation of the north part of the DEM. The main land use/cover types in the narrow valley are pasture/hay, woody wetlands, and evergreen forest. The n-values from NOAA C-CAP for these land use/cover types are the highest resulting in higher flow depths and, thereby spilling of the flood into northern region of the DEM for pair numbers 3, 6. 8, 18, and 31. hours
Average tarr (hr)
St. Dev. of tarr (hr)
Minimum Value of tarr (hr)
Maximum Value of tarr (hr)
Figure 7: Statistics of flood arrival time,
0.25 0.50 0.75 1 2 3 4 5 6 7 8 9 10 11 12 16 20 24 30 34
(hr), based on 120 simulations. m2/s
Average qmax (m)
St. Dev. of qmax (m)
Minimum Value of qmax (m)
Maximum Value of qmax (m)
Figure 8: Statistics of maximum discharge per unit width, simulations.
3 4 4.65 5 6 8 10 14.86 25 50 100 150 200 250 300 350 400 450 500 562
(m2/s), based on 120
Hydrographs Discharge hydrographs were recorded at seven cross sections. These include the breach cross section and five cross sections specified in the benchmark test case formulation. An additional cross section was also introduced in order to measure the flood discharge propagating to the north of the DEM under certain combinations of control parameters (see Figure 9). Due to the
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lack of space, only limited number of results will are presented in this paper. Additional details will be provided in subsequent publications. Figure 10 shows the discharge hydrographs at the breach cross section (left plot) and the relationship between peak discharge and the control parameters (middle plot) and (right plot). Runs with different n-values are distinguished with different symbols/colors. Since at the breach cross section the influence of the Manning’s coefficient is not fully felt, for a given and pair, the hydrographs obtained with different n-values coincide fairly well. The peak discharge does not seem to be highly correlated with the breach width (middle plot) but rather with the breach formation time (right plot). The peak discharge is inversely proportional with the breach formation time and all the data collapses onto a single curve. A similar plot is shown for cross section 5 in Figure 11. The peak discharges of hydrographs are attenuated. For a given and pair, the hydrographs computed with different sets of nvalues have different arrival times. The peak discharges are not correlated with the breach width (middle plot) but they still show a relatively good correlation with the breach formation time (right plot) despite the fact that the differences in the peak discharge for simulations with different n-values are larger for and pairs. DSS-WISE™ n-values
FEMA HAZUS n-values
NOAA C-CAP n-values
Figure 9: Comparison of flood depth maps for selected simulations.
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4 3.5 3 2.5 2 1.5 1 0.5 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Breach Cross Section
4
Max. cross section discharge (m3/s)
Cross section discharge (m3/s)
x 10
1.8
time (s)
2
4.5
x 10
4 3.5 3 2.5 2 1.5 1 0.5 0 20
40
60
80
100
120
140
Breach Cross Section
4
160
180
Max. cross section discharge (m3/s)
Breach Cross Section
4
4.5
4.5
x 10
4 3.5 3 2.5 2 1.5 1 0.5 0
0
1
Breach Width(m)
4
x 10
2
3
4
5
6
Breach Formation Time (s)
Figure 10: Discharge hydrographs and the peak discharge at the breach cross section (DSSWISE: black squares, FEMA HAZUS: red “plus” signs, and NOAA C-CAP: blue crosses). 3
2.5
2
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
1.2
time (s)
1.4
Cross Section 5
4
1.6
1.8
2
Max. cross section discharge (m3/s)
Cross section discharge (m3/s)
x 10
4
x 10
3.5
x 10
3
2.5
2
1.5
1
0.5
0 20
40
60
80
100
120
Breach Width(m)
Cross Section 5
4
140
160
180
Max. cross section discharge (m3/s)
Cross Section 5
4
3.5
3.5
x 10
3
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
6
Breach Formation Time (s)
Figure 11: Discharge hydrographs and the peak discharge at cross section 5 (DSS-WISE: black squares, FEMA HAZUS: red “plus” signs, and NOAA C-CAP: blue crosses).
Consequence Analysis In the present study the consequence analyses were restricted to the analysis of population at risk (by age group) to be affected by flood based on flood arrival time and peak discharges per unit width. Direct or indirect economic analyses were not performed. Population by Census Block and PAR Distribution Census block polygons for Hydropolis, colored by total population, are shown in Figure 12. Other polygon attributes include population by age and gender group and jobs in 20 different classes. The loss of life analysis in the present study was based only on total population. The map of population at risk per computational cell shown in Figure 13 was computed by assuming a uniform distribution of population in each census block. As it can be seen, there are two major urban areas (see also Figure 2). The urban area on the west is located in the narrow valley downstream of the dam and directly on the path of the flow. The urban area on the north east corner of the map is farther from the dam and located on a relatively flat area.
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Total Population 1 - 42 43 - 81 82 - 152 153 - 278 279 - 481 482 - 943 944 - 1987 1988 - 3338
Figure 12: Total population by census block (only census blocks with population are shown). PAR Analysis The number of PAR by age group for each one of the 120 simulations were counted for 7 flood arrival-time intervals and for 17 depth intervals and recorded in tables. shows the average value, standard deviation, minimum and maximum values of PAR numbers by arrival time interval. Some of the information listed in is summarized in Figure 14. The plot on the right hand side of Figure 14 shows that the range between cumulative values of minimum and maximum PAR can be significant. Final minimum and maximum cumulative PAR values correspond to 4.4% of 268% of the final average cumulative PAR.
PAR / 100m2 82
1
Figure 13: PAR distribution per computational cell (10m×10m).
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Table 4: Statistics of PAR numbers by age group and flood arrival time interval.
PAR ≥ 65
14
PAR ≤ 14
Total PAR
Age
Value Average St. Dev. Min. Max. Average St. Dev. Min. Max. Average St. Dev. Min. Max. Average St. Dev. Min. Max.
0-15min 0.186 0.548 0.043 5.110 0.035 0.147 0.006 1.286 0.129 0.031 3.507 0.021 0.030 0.007 0.317
15-30min 12.782 14.597 0.019 52.346 3.296 3.802 0.002 13.409 8.623 0.014 35.341 0.862 0.986 0.004 3.597
30-60min 38.370 28.768 8.566 144.341 7.584 3.101 2.014 19.903 25.142 5.926 93.943 5.644 7.590 0.625 30.495
60-90min 75.021 47.800 1.110 148.087 10.381 6.789 0.286 21.096 48.743 0.746 96.662 15.898 10.011 0.078 30.329
120-180min 54.556 39.032 1.210 119.794 7.987 6.049 0.157 18.265 35.977 0.853 77.542 10.592 7.275 0.200 23.987
>180min 14.201 26.624 0.087 132.010 2.037 4.041 0.010 20.306 9.376 0.061 87.380 2.789 4.983 0.016 24.324
Cumulative Total PAR as a Function of Time
80
800
70
700
60
600
Number of PAR
Number of PAR
Average PAR
90-120min 65.333 25.632 0.532 95.128 9.437 4.100 0.140 14.424 42.874 0.346 59.286 13.022 4.963 0.046 21.418
50 40 30 20
500 400 300 200
10
100
0 0-15min
15-30min PAR ≤ 14
30-60min
60-90min
14 < PAR < 65
90-120min 120-180min
PAR ≥ 65
0
>180min
0-15min
15-30min
Cumul. Total PAR
Total PAR
30-60min
60-90min
Min Cumul. Total PAR
90-120min
120-180min
>180min
Max Cumul. Total PAR
Figure 14: (Left) Average PAR by age group and by arrival time; (Right) Cumulative values of average, minimum and maximum total PAR impacted by the flood as a function of time. and Table 6 show the average value, standard deviation, minimum and maximum values of PAR numbers by flood depth interval in two separate tables. Some of the PAR information listed in Table 5 and Table 6 is summarized in Figure 14. The plot on the right hand side of Figure 14 shows that the range between cumulative values of minimum and maximum PAR can be significant. This figure shows that majority of PAR impacted by the flood (93.5%) will be subjected to a maximum flood depth of 3.5m. The range between minimum and maximum PAR values is significant. The largest differences are observed for the depth interval 0.00.5m. Minimum (98.6% PAR in less than 3.5m depth) and maximum (87.8% PAR in less than 3.5m depth) PAR both show the same tendency as the average PAR. The average, minimum and maximum values of the inundated area listed in Table 5 and Table 6 are plotted in Figure 16 as a function of flow depth. As it can be seen most of the inundated area (95.1%) will be subjected to water depths less than or equal to 5m. The largest differences between minimum and maximum values occur for the depth interval 0.0-0.5m.
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Table 5: Average and standard values of PAR numbers by age group and depth interval. Depth (m) 0.0m-0.5m 0.5m-1.0m 1.0m-1.5m 1.5m-2.0m 2.0m-2.5m 2.5m-3.0m 3.0m-3.5m 3.5m-4.0m 4.0m-4.5m 4.5m-5.0m 5.0m-5.5m 5.5m-6.0m 6.0m-6.5m 6.5m-7.0m 7.0m-7.5m 7.5m-8.0m >8.0m
Total PAR Min Max 55.53 19.25 73.38 15.41 67.36 23.75 26.36 12.31 10.30 4.18 4.65 0.87 3.23 0.47 2.57 0.67 2.24 0.83 2.18 0.85 2.02 0.90 1.81 0.95 1.63 1.08 1.41 1.09 1.17 1.06 1.03 1.13 3.55 4.92
PAR ≤ 14 Min Max 8.21 2.91 11.10 2.16 9.97 3.68 3.63 1.83 1.37 0.43 0.81 0.15 0.67 0.19 0.62 0.20 0.58 0.21 0.56 0.21 0.52 0.23 0.46 0.24 0.41 0.27 0.36 0.27 0.30 0.27 0.26 0.28 0.92 1.21
14 < PAR < 65 Min Max 37.03 48.74 44.20 16.67 6.23 2.83 2.02 1.67 1.50 1.46 1.35 1.22 1.10 0.95 0.79 0.70 2.40 -
PAR ≥ 65 Min Max 10.29 3.48 13.54 3.29 13.19 4.05 6.06 2.33 2.70 1.27 1.01 0.41 0.54 0.24 0.28 0.14 0.16 0.06 0.16 0.07 0.15 0.07 0.13 0.08 0.12 0.09 0.10 0.09 0.08 0.08 0.08 0.09 0.23 0.36
Inundated Area (m2) Min Max 7,333,416 3,206,673 9,018,780 1,693,804 8,709,363 1,270,547 5,856,884 1,471,850 3,358,918 1,008,449 2,018,934 709,981 1,453,657 533,611 1,337,561 659,157 674,860 362,878 325,181 106,113 260,150 95,178 218,419 85,614 185,084 79,717 161,414 73,249 137,548 63,610 119,563 57,089 2,226,929 550,383
Table 6: Minimum and maximum values of PAR numbers by age group and depth interval. Depth (m) 0.0m-0.5m 0.5m-1.0m 1.0m-1.5m 1.5m-2.0m 2.0m-2.5m 2.5m-3.0m 3.0m-3.5m 3.5m-4.0m 4.0m-4.5m 4.5m-5.0m 5.0m-5.5m 5.5m-6.0m 6.0m-6.5m 6.5m-7.0m 7.0m-7.5m 7.5m-8.0m >8.0m
Total PAR Min Max
38.76 49.03 12.73 7.50 3.93 2.59 1.25 0.67 0.29 0.28 0.23 0.17 0.02 0.00 0.00 0.00 0.02
134.98 102.42 92.40 45.04 19.90 7.23 4.61 3.63 4.16 4.03 3.65 3.69 4.07 4.05 4.10 4.26 20.47
PAR ≤ 14 Min Max
5.33 7.00 2.05 1.32 0.80 0.54 0.30 0.16 0.07 0.08 0.07 0.06 0.01 0.00 0.00 0.00 0.00
20.13 15.17 14.09 6.95 2.95 1.09 1.15 0.93 1.02 1.00 0.93 0.91 1.01 1.00 1.02 1.05 5.10
14 < PAR < 65 Min Max
25.84 33.25 7.80 4.98 2.82 1.84 0.85 0.46 0.20 0.19 0.16 0.11 0.01 0.00 0.00 0.00 0.01
89.84 67.26 61.01 29.01 12.39 4.38 2.59 2.08 2.78 2.70 2.42 2.45 2.71 2.70 2.72 2.84 13.92
PAR ≥ 65 Min Max
7.59 8.78 2.88 1.20 0.30 0.21 0.10 0.05 0.02 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00
25.01 19.99 17.30 9.09 4.56 1.76 0.87 0.62 0.36 0.33 0.31 0.33 0.35 0.35 0.35 0.36 1.44
Inundated Area (m2) Min Max
3,502,773 6,067,713 4,253,739 2,131,314 740,011 422,222 447,365 171,062 95,543 78,662 73,453 74,621 66,539 59,984 44,808 39,151 1,409,981
17,674,779 11,718,682 10,358,269 7,690,691 5,280,020 3,035,832 2,560,271 2,481,161 1,333,115 521,537 442,965 418,002 330,630 298,393 276,752 277,381 3,827,117
Loss of Life Analysis The loss of life is computed using the U.S. Bureau of Reclamation DSO-99-06 Procedure (Graham, 1999). In this method, the fatality rate of PAR in a given area, which in the case of a two dimensional numerical simulation can be taken as the computational cell, is computed from Table 6 based on three parameters: flood severity, warning time, and flood severity understanding. Warning time can be computed as the time difference between the time the warning issued and the arrival time of the flood. Flood severity understanding is a parameters that must be appreciated based on the characteristics of the population in the inundation area. Since in the benchmark test case it is assumed that the population cannot be warned, the flood severity understanding is no longer applicable, and Table 6 reduces to the three highlighted lines. There are no clear guidelines for estimating flood severity. In the present study, the flood severity was computed based on maximum specific discharge, , which is equal to the product of depth and flow speed. The limits of high, medium, and low flood severity
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shown in Table 7 are taken from guidelines published by the U.S. Department of Homeland Security (DHS, 2011). Average PAR
Minimum, Avarage and Maximum Total PAR
80 60
Number of PAR
Number of PAR
70 50
40 30 20 10 0
PAR ≤ 14
14 < PAR < 65
PAR ≥ 65
160 140 120 100 80 60 40 20 0
Mean Total PAR
Mean Total PAR
Minimum Total PAR
Maximum Total PAR
Figure 15: (Left) Average PAR by age group and by flood depth; (Right) Average, minimum and maximum total PAR impacted by the flood as a function of depth.
Inundated Area (m2)
Minimum, Avarage and Maximum Values of Inundated Area 20,000,000 18,000,000 16,000,000 14,000,000 12,000,000 10,000,000 8,000,000 6,000,000 4,000,000 2,000,000 0
Avg. Inundation Area
Min. Inundation Area
Max. Inundation Area
Figure 16: Average, minimum and maximum values of inundated area as a function of depth. Based on the flood severity criteria given in Table 8 and the fatality rates in Table 7, the lossof-life (LOL) maps for each simulation was established using the corresponding raster. The average value, standard deviation, and minimum and maximum values of loss of life for each computational cell were then computed from 120 loss-of-life maps. Figure 17 shows the average value, standard deviation, minimum and maximum values of estimated loss of life (LOL) based on 120 simulations. As expected, the largest loss of life numbers are observed in the western urban area located in the narrow valley. This area is closer to the dam and directly on the path of the dam-break flood. The values are much larger. The urban area on the north eastern corner of the map has relatively small values of LOL due to lower values of . It is important to note that the areas with no loss of life and with zero standard deviation are not displayed. Looking at the standard deviation map, one can observe that there are areas with zero standard deviation. These areas can be regarded as areas where the LOL has highest likelihood.
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Table 7: Fatality rates for loss-of-life estimation from dam failure (Graham, 1999). Flood Severity
Warning Time (Minutes) 15 to 60
HIGH
More than 60 15 to 60
MEDIUM
More than 60 15 to 60
LOW
More than 60
Flood Severity Understanding Not applicable Vague Precise Vague Precise Not applicable Vague Precise Vague Precise Not applicable Vague Precise Vague Precise
Fatality Rate (Fraction of PAR projected to die) Suggested Value Suggested Range 0.75 0.30 – 1.00 Apply the values shown above to the PAR values who remain in the dam failure floodplain after warnings are issued. No guidance is provided on how many people will remain in the flood plain. 0.15 0.03 – 0.35 0.04 0.01 – 0.08 0.02 0.005 – 0.04 0.03 0.005 – 0.06 0.01 0.002 – 0.02 0.01 0.0 – 0.02 0.007 0.0 – 0.015 0.002 0.0 – 0.004 0.0003 0.0 – 0.0006 0.0002 0.0 – 0.0004
Table 8: Flood severity criteria used for loss of life analysis. Flood severity LOW MEDIUM HIGH
Criteria in SI Units
Criteria in US Customary Units
# of Deaths
Minimum Value of LOL
Maximum Value of LOL
1 - 42 43 - 81 82 - 152 153 - 278 279 - 481 482 - 943 944 - 1987 1988 - 3338
St. Dev. of LOL
PAR
Average LOL
0.00 - 0.02 0.02 - 0.04 0.04 - 0.06 0.06 - 0.08 0.08 - 0.10 0.10 - 0.12 0.12 - 0.14 0.14 - 0.16 0.16 - 0.18 0.18 - 0.20 0.20 - 0.22 0.22 - 0.24 0.24 - 0.26 0.26 - 0.28 0.28 - 0.30 0.30 - 0.32 0.32 - 0.34 0.34 - 0.36 0.36 - 0.38
Figure 17: Average value, standard deviation, minimum and maximum values of estimated loss of life (LOL) based on 120 simulations.
Conclusion The uncertainty in 2D numerical modeling of dam-break flood and the resulting loss of life were studied using 120 simulations representing 40 random pairs of breach width and breach formation time calculated with three different sets of Manning’s coefficients. The results show that large variations in the extent of the inundated area, water depths, and loss-of-life occur based on the particular combination of control parameters. The paper presents the average value and standard deviation of selected computed results as well as their upper and lower bounds. Peak discharge at the breach cross section and the cross sections downstream seem to be highly correlated with the breach formation time rather than the breach width. 269
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Only part of the results could be presented. Additional findings will be presented in subsequent publications.
Acknowledgements The research for the present benchmark study and the paper was sponsored by the U.S. Army Corps of Engineers, Headquarters Office of Homeland Security. The development of the DSS-WISE™ software was funded by the Department of Homeland Security-sponsored Southeast Region Research Initiative (SERRI) managed by the Department of Energy’s Oak Ridge National Laboratory.
References [1] Altinakar, M.S., McGrath, M.Z., Yucel, O., and Ger, M. (2013a). Simulation of Supercritical Flow in an Open Channel Bend Using Cut-Cell Boundary Method. 6th Int. Perspective on Water Res. and Env. (IPWE 2013), January 7-9, 2013, Izmir Turkey. [2] Altinakar, M.S., McGrath, M.Z., Ramalingam, V.P., Strati, O. (2013b). Dam Break Flood Simulation – Doing It Faster and Simpler: DSS-WISE™, DSS-WISE™ Lite, and DSAT. 1-day Short Course, ASDSO West Regional Conf., May 8, 2013, Seattle, Washington. [3] Altinakar, M.S. (2012). Advances in Numerical Modeling and Simulation of Floods, Inundation Mapping and Consequence Analysis. Keynote Lecture presented at the 10th International Congress on Advances in Civil Engineering (ACE 2012), October 17-19, 2012, Middle East Technical University (METU), Ankara, Turkey. [4] Altinakar, M.S., McGrath, M.Z., Matheu, E.E., Ramalingam, V.P., Seda-Sanabria, Y. Breitkreutz, W., Oktay, S., Zou, J.Z., Yezierski, M. (2012): Validation of Automated Dam-Break Flood Simulation and Assessment of Computational Performance, Proc. Dam Safety 2012, September 16-20, Denver, Colorado. [5] Altinakar, M.S. and McGrath, M.Z. (2012). Parallelized Two-Dimensional Dam-Break Flood Analysis with Dynamic Data Structures. ASCE-EWRI, 2012 World Environmental & Water Resources Congress, May 20-24, 2012, Albuquerque, New Mexico. [6] ARCADIS (2011). ADCIRC Based Storm Surge Analysis of Sea Level Rise in the Galveston Bay and Jefferson County Area in Texas. Report prepared for The Nature Conservancy, Nov 28, 2011. [7] DHS (2011). Estimating Loss of Life for Dam Failure Scenarios. U.S. Department of Homeland Security, September 2011. Washington DC. [8] Froehlich, D.C. (1995a). Embankment Dam Breach Parameters Revisited. Water Res. Engineering, Proc. 1995 ASCE Conf. on Water Res. Eng., New York, pp. 887-891. [9] Graham, Wayne J. (1999). A Procedure for Estimating Loss of Life Caused by Dam Failure. DSO-99-06, U.S. Bureau of Reclamation, Denver, Colorado, September 1999. [10] Iman, R.L.; Helton, J.C.; and Campbell, J.E. (1981). An Approach to Sensitivity Analysis of Computer Models, Part 1. Introduction, Input Variable Selection and Preliminary Variable Assessment. Journal of Quality Technology, Vol. 13, No. 3, pp. 174–183. [11] Kim, D.-H., Cho, Y.-S., and Yi, Y.-K. (2007). Propagation and Run-up of Nearshore Tsunamis with HLLC Approximate Riemann Solver. Ocean Engineering, Volume 34, Issues 8-9, June 2007, pp. 1164-1173. [12] Luettich, R. and Westerink, J. Storm Surge Inundation Modeling: State of Science. Hurricane Surge Workshop, February 11, 2009, St. Petesburg, Florida [13] Toro, E. F., Spruce, M., Speares, W. (1994). Restoration of the Contact Surface in the HLL-Riemann Solver. Shock Waves, Vol. 4, No. 1, pp. 25-34. [14] Wahl, T.L. (2004). Uncertainty of Predictions of Embankment Dam Breach Parameters. Journal of Hydraulic Engineering, Vol. 130, No. 5, pp. 389-397. 270
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Risk assessment for hypothetical dam break A method for the rapid and consistent evaluation L.Mancusi1, L.Giosa2, A.Cantisani2, A.Sole2 and R.Albano2 1
Sustainable Development and Energy Resources Department, Research on Energy Systems (RSE) Spa, via R. Rubattino 54, 20134 Milano, Italy 2 School of Engineering, University of Basilicata, viale dell’Ateneo Lucano 10, 85100 Potenza, Italy E-mail: [email protected]
Abstract This paper details the technical contribution to the theme of flood risk analysis as consequence of a dam failure. According to the numerical problem proposed for the workshop, the analysis consists of the evaluation of the dam break and its consequences. The simulation includes two scenarios of dam breach: the scenario 1 that represents the case of an easy erodible dam and the scenario 2 the case of an erosion resistant dam. For each scenario, a dam failure discharge hydrograph was calculated and the subsequent flood wave and consequences have been evaluated. The methodology adopted involves, for a first part, the use of standard models for the hydraulic modelling of the dam breach and flood wave propagation. For the second part, a set of GIS scripts was written, tested and developed using the python scripting language to obtain a rapid appraisal of consequences for the population and to assess the direct economic damages for residential, commercial, and industrial buildings. Since the latter elaboration depends greatly on the type and the level of detail of available data, in this study we have been used data as generic as possible and GIS scripts that allow, for a great variety of cases, a rapid initial assessment.
Introduction Modern society considers it essential to increase the safety of the infrastructure. Risk analysis is a helpful tool for the evaluation and management of risks which can affect people, environment and human development. The purpose of this paper was to demonstrate the application of an example of quantitative risk assessment technique that consists of estimating the consequences of failure of a dam near populated areas with complex demographics, infrastructure and economic activity. The first chapter concerns the hydraulic modelling and simulation of the dam breach, the second the subsequent flood wave propagation and the last focuses on consequence estimation.
Dam failure This section includes the description of modelling of the breaching process and the subsequent discharge hydrograph for the hypothetical overtopping of the dam. Two methods have been adopted and the results are named respectively scenario 1 and 2. Scenario 1 In this case were used, a statistical method available in the literature for the dam failure peakdischarge estimation of and a physically based mathematical model for calculating the total discharge hydrograph. 271
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P. Molinaro [1] utilised 31 data sets (predominantly earthfill and some rockfill) extracted from the report of J. E. Costa [2] to develop a relationship from the peak-discharge, the height of dam and the reservoir volume at time of failure. Qmax gH
5
2
V 0.116 3 H
0.221
(1)
Qmax = peak-discharge (m3/s) g = gravity of Earth that has an approximate value of 9.81 m/s2 H = height of the dam (m) V = volume of water at breach time (m3) Applying the formula (1) for this case (H=61 m and V=30.3*106 m3) we obtain: Qmax =32800 m3/s Where:
The latter value was used to calibrate parameters of the mathematical model. The mathematical model was developed by Molinaro [3] and simulates the breach development process through an earthen dam due to overtopping. The model is developed by coupling the conservation of mass of the reservoir inflow, spillway outflow, and breach outflow with the sediment transport capacity of the quasi-steady uniform flow along an erosion-formed breach channel. The rate at which the breach is eroded is evaluated using the Engelund and Hansen [4] sediment transport relation. The dam is modeled as an isosceles triangle formed by a noncohesive material of uniform diameter D. The storage characteristics of the reservoir are described by specifying a table of volume vs. water elevation. The overtopping failure simulation starts by assigning a small initial breach whose bottom elevation must be below the reservoir water level. The first stages of erosion are along the downstream face of the dam while the breach bottom erodes vertically downward. An erosion triangular channel is gradually cut into the downstream face of the dam. The sides of the breach channel has a constant angle (α) with the vertical which is a function of the internal friction (φ) of dam’s material. The flow into the channel is determined by the broad-crested triangular weir relationship. The breach bottom is allowed to progress downward until it reaches the bottom elevation of the dam, subsequently the channel becomes trapezoidal with the sides that maintain the same slope α of the previous triangle. The following figure Figure 1 shows the sequence of the simulation of the breach formation.
Figure 1: Breach formation sequence
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The most important parameter for model calibration is the characteristic diameter D. In this case, taking into account the result of the equation 1, we have adopted the value D = 0.01 m. The discharge hydrograph obtained is shown in the Figure 3. This hydrograph was used for the first propagation of a flood wave referred to the next chapter. It is characterized by Qmax =28936 m3/s and a breach formation time of 2040 sec (0.57 hrs). Scenario 2 In addition to the above, a second method was applied for the evaluation of failure discharge hydrograph. Even in this case a regression equations was used for the estimation of dam breach parameters, and then a mathematical model was applied. Table 1 summarizes the resulting breach parameters (Wb: bottom width of the breach and tf: breach formation time) computed by several approaches available in the literature. Table 1: Breach Parameters Method MacDonald and Langridge - Monopolis (1984) Floehlich (1995a) Floehlich (2008) Von Thun and Gillette (1990)
Wb (m) 167 147 110 207
tf (hrs) 1.99 0.66 0.57 1.47
In the last column you can see that the breach formation time ranges from 0.57 to 2 hours. As in scenario 1 the resulting time corresponds with the minimum, then for this scenario the case of the maximum value of tf=2 hours was investigated. For performing the dam breach outflow hydrograph computation, HEC-RAS model was adopted. The implementation of these breach parameters in the HEC-RAS modelling system is depicted on Figure 2. The resulting discharge hydrograph compared to the previous scenario is depicted in Figure 3. As can be seen in the latter figure, the second hydrograph, having the same volume, is characterized by a peak value cut in half compared to the first, but a duration in time approximately double. For further analysis, we can consider the two hydrographs: the first as representing of a easy erodible dam and the second an erosion resistant dam.
Figure 2: HEC-RAS Dam Breach Model of scenario 2
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Dam Failure discharge hydrographs for scenario 1 and 2 35000
Discharge for scenario 1 30000
Discharge for scenario 2
Q(m3/s)
25000 20000 15000 10000 5000 0 0
20
40
60
80
100
120
Time (min)
Figure 3: Discharge hydrograph of scenario 2 compared with scenario 1
Flood simulation Hydraulic modelling has carried out using the MIKE 21 software by the DHI Water Environment Health to simulate flood wave propagation in the river and to describe the inundation on the floodplain. This software solves the shallow water equations by means of a finite difference scheme. For each scenario, the simulation has been constructed using, as upstream boundary condition, the corresponding discharge hydrograph for the hypothetical overtopping of the dam. According to Bunya et al. [5], the Manning n coefficient is spatially assigned associating the value of n with the land cover definition of 2001 from the USGS National Land Cover Data (NLCD) (Table 2), These values are selected or interpolated from standard hydraulic literature. Table 2: Manning n value for 2001 NLCD classification Lu Code 11 12 21 22 23 24 31 41
Description Open Water Perennial Ice/Snow Developed-Open Space Developed-Low Intensity Developed-Med Intensity Developed-Hight Intensity Barren Land Deciduous Forest
n Mann. (s/m1/3) 0.020 0.022 0.050 0.120 0.120 0.121 0.040 0.160
Lu Code 42 43 52 71 81 82 90 95
Description Evergreen Forest Mixed Forest Shrub/Scrub Grassland/Herbaceous Pasture/Hay Cultivated Cropland Woody Wetlands Herbaceous Wetlands
n Mann. (s/m1/3) 0.180 0.170 0.070 0.035 0.033 0.040 0.140 0.035
The results of flood modelling consist of values, for each grid cell in the study area, depth (m) and the two components of the vector unit flow rate (m2/s) for 15 minute intervals and the envelope of their maximum. Using a GIS scripts, hydrographs flow at different cross sections were extracted. Some of these are shown in Figure 4.
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Comparison of hydrographs of the two scenarios at different cross sections 35000
Scenario1 - Cross-sec.1 30000
Scenario1 - Cross-sec.5 Scenario2 - Cross-sec.1
25000 Q (m3/s)
Scenario2 - Cross-sec.5 20000 15000 10000 5000 0 0
20
40
60
80
100
120
140
160
Time (min)
Figure 4: Discharge hydrographs of the two scenarios at different cross sections Comparing the hydrographs of two scenarios in different cross sections we may note that: while having at the initial cross-section a large difference in peak flow rate, however, during the process of propagation downstream, an attenuation of the difference occurs.
Impacts assessment The potential risks associated with the failure or disruption of dams could be considerable and potentially result in significant destruction, including loss of life, massive property damage, and severe long-term consequences. The following sections contain the analysis of two categories: public safety and direct economic impact. Population at Risk and Loss of Life estimation The analyzes described in this section have been carried out by adopting the published guidelines of the report [6] that provides guidelines and recommendations for estimating loss of life resulting from dam failure or disruption. The results of flood modelling and the data from the population census are used. Geographic analyzes were carried out using Map Algebra techniques implemented in a set of scripts written, tested and developed using the python scripting language and the Open Sources GDAL libraries and NumPy Python module. To combine multiple maps in Map Algebra all data have been converted into grid format. The outputs of the hydrodynamic model have been processed to derive the information required for the analysis. Using a GIS scripts, a Flood Wave Arrival Time grid was obtained, in addition the two components of the vector unit flow rate are combined to obtain the maximum Peak Unit Flow Rate values (m2/s). These values, called parameter DV, are representative of the general level of destructiveness that would be caused by the flooding. The DV values are then categorized, as suggested in the Figure 5 extracted from guidelines, into ranges of values which define low, medium, and high severity zones.
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Figure 5: Flood severity rating criteria reported in the guidelines The vector polygons of the population census block were converted into grid format: the hypothesis assumed for the different values of the fields is that their distribution within the polygon is homogeneous. By overlaying grid maps of flood with the grid of the population is achieved as a result the map of Population at Risk (PAR). The estimate of loss of life is finally obtained by multiplying the PAR with the Fatality Rate (Fraction of people at risk projected to die). The latter was obtained by using the values of the Table 3 (from tab. 4 of guidelines) as a function of warning time and flood severity. Table 4 shows the results for the two scenarios for the hypothesis of event occurred at night (understanding=vague). In the case of the first scenario, there is a population at risk greater than 11 percent, while the estimate of the largest loss of life is almost 30 percent. The last result is mainly caused by the shorter warning time of the first scenario. It should be noted that in each case that the differences in terms of the consequences are less than the differences of the peak discharge of the two scenarios at the breach of the dam, . This result is due to the fact that the volume released from the dam is still the same.
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Table 3: Recommended Fatality Rates for Estimating Loss of Life as reported in the guidelines Flood Severity HIGHT MEDIUM
Warning Time (min) Not applicable No warning 15 to 60 More then 60 No warning
LOW
15 to 60 More then 60
Understanding Not applicable Not applicable Vague Precise Vague Precise Not applicable Vague Precise Vague Precise
Fatality Rate 0.75 0.15 0.04 0.02 0.03 0.01 0.01 0.007 0.002 0.0003 0.0002
Table 4: Population at Risk and Loss of Life estimation Time Interval (min) 0-15 15-30 30-60 60-90 90-120 120-180 >180 Total Peak Flood Depth Range (m) 0.0-0.5 0.5-1.0 1.0-1.5 1.5-2.0 2.0-2.5 2.5-3.0 3.0-3.5 3.5-4.0 4.0-4.5 4.5-5.0
14-yr and Under 65-yr and Over Total Population At Risk Population at Risk Population at Loss of Life Risk Scen. 1 Scen. 2 Scen. 1 Scen. 2 Scen. 1 Scen. 2 Scen. 1 Scen. 2 0 3 0 0 0 0 0 2 2 4 0 0 0 1 2 2 4 529 2 794 1 169 732 308 196 1 783 1 262 5 111 641 721 110 1 057 105 2 0 10 411 8 529 1 491 1 084 2 163 1 996 0 0 8 935 12 346 1 350 1 899 1 662 2 290 0 0 688 1 911 85 265 121 360 0 0 29 676 26 228 4 816 4 090 5 311 4 948 1 787 1 266 Flooded Area (m2) Scen. 1 8 308 132 6 025 869 7 854 660 5 612 806 3 711 550 2 547 251 1 746 448 1 279 417 546 680 391 332
Total Population At Risk
14-yr and Under 65-yr and Over Population at Population at Risk Risk
Scen. 2 Scen. 1 Scen. 2 Scen. 1 Scen. 2 Scen. 1 Scen. 2 4 088 784 5 854 4 825 915 706 1 017 937 7 190 976 5 683 6 770 895 1 021 987 1 237 7 858 701 8 937 7 769 1 373 1 164 1 664 1 453 5 141 466 3 768 2 879 543 391 781 676 2 970 641 1 562 1 038 197 118 404 304 1 901 885 510 380 72 60 137 98 1 453 532 274 292 43 49 68 70 711 995 209 220 37 53 49 28 392 859 125 228 35 62 8 17 310 246 153 237 43 65 11 17 277
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5.0-5.5 5.5-6.0 6.0-6.5 6.5-7.0 7.0-7.5 7.5-8.0 >8 Total
380 736 209 944 242 630 250 352 201 323 294 532 192 882 221 527 176 180 128 947 186 417 101 919 1 880 693 850 640 41 285 006 34 078 946
211 190 194 162 176 254 1 414 29 676
216 271 346 337 154 84 182 26 228
57 51 53 44 44 61 353 4 816
54 67 90 83 37 18 52 4 090
15 15 14 9 11 16 104 5 311
13 16 25 30 11 7 9 4 948
Direct Economic Impact estimation Methods and values of the parameters used in this section are drawn mostly from the report [7]. They concern the assessment of the direct economic damages for residential, commercial, and industrial buildings. The input data consist of map of land use and parcel zone map of the study area. As in the previous paragraph, for the analysis, all the data are preliminarily converted into grid format. The following assessments do not take into account Agricultural, Roads, Infrastructure and Vehicles damages. The assessment however allows the estimation of the damage to buildings and their contents, and when applied to different scenarios allows an effective comparison of the impact. The extent of damage to the buildings and its contents is estimated from the depth of flooding by the application of a depth-damage curve associated with each occupancy type. Depth damage curves demonstrate the relationship between the depth of the flood relative to the first finished floor level of buildings and the damage caused to the structures and contents. Damages are typically expressed as a percentage of depreciated building replacement value. Adopting a non-traditional approach, the adopted method models directly the content damage as a percentage of structure value rather than using a content-to structure value ratio. Not having a map of buildings, the area covered by the buildings has been derived from the land use map according to the hypothesis of Building Coverage shown in the following table. Table 5: Relationship between Land Use and Building Coverage Lu Code 11 12 21 22 23 24 31 41
Description
Building Cover. % Open Water 0% Perennial Ice/Snow 0% Developed-Open Space 10% Developed-Low Intensity 20% Developed-Med Intensity 35% Developed-Hight Intensity 50% Barren Land 0% Deciduous Forest 0%
Lu Code 42 43 52 71 81 82 90 95
Description Evergreen Forest Mixed Forest Shrub/Scrub Grassland/Herbaceous Pasture/Hay Cultivated Cropland Woody Wetlands Herbaceous Wetlands
Building Cover. % 0% 0% 0% 0% 0% 0% 0% 0%
To calculate damages, each structure must be assigned to a structure occupancy type. For each structure occupancy type an estimated replacement value and a structure depth-damage and a content depth-damage relationship must be defined. In our case, replacement values were extracted from the "Table C-3 Estimated Replacement Value", depth-damage relationship from “Table C-1 Depth Damage Curves, Defining Damages as a Percentage of Depreciated Building Value for Depth of Flooding Above Floor 278
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Height” and the height of the floor of buildings from the ground level was taken from "Table C-2 Foundation Heights" of report [7]. The following table contains the list of occupancy type categories adopted. Table 6: Occupancy types Occup. Type RES1
Description
COM
Residential One Story, No Basement Residential Two or more Stories, No Basement Commercial buildings
IND PUB FAR
Industrial buildings Public buildings Homestreads
RES2
Unit Cost ($US/sqm) 1711 3336 1528
Origin data of Depth Damage Curve USACE Generic Depth Damage Curves for residential buildings " USACE depth damage, as used in Ford (2005) [8] " " "
1528 1711 1711
The Figure 6 below shows the graph of the depth-damage curves. Contents Depth-Damage Curves as a Percentage of Depreciated Building Value for Depth of Flooding Above on the Ground Level
90%
90%
80%
80%
70%
70%
60%
60%
Damage (%)
Damage (%)
Structural Depth-Damage Curves as a Percentage of Depreciated Building Value for Depth of Flooding Above on the Ground Level
50% 40%
RES1_S RES2_S COM_S IND_S PUB_S FAR_S
30% 20% 10% 0% -1
0
1
2
3
4
RES1_C RES2_C COM_C IND_C PUB_C FAR_C
50% 40% 30% 20% 10% 0%
5
-1
0
1
2
3
4
water depth (m)
water depth (m)
Figure 6: Depth damage curves To assign at each parcel the occupancy type we chosen the values according to the Table 7. Table 7: Reclassify table: from parcel ZONINGCATE to occupancy type ZONINGCATE COMMERCIAL INDUSTRIAL / WHOLESALE / MANUFACTURING INSTITUTIONAL / GOVERNMENT OFFICE OFFICE OFFICE OPEN SPACE / RECREATION / AGRICULTURAL RESIDENTIAL RESIDENTIAL RESIDENTIAL 279
Stories any any any 1 2 3 any 1 2 3
Occupancy Type COM IND PUB RES1 RES2 RES2 FAR RES1 RES2 RES2
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RESIDENTIAL / AGRICULTURAL UTILITIES / TRANSPORTATION
any any
FAR RES1
The results of applying the method for the two scenarios are listed in the following table. Table 8: Direct Economic Impact Time Interval (min)
Direct Economic Impact ($US)
Scenario 1 Scenario 2 0-15 0 0 15-30 0 0 30-60 499 417 507 333 141 773 60-90 671 323 845 151 037 801 90-120 839 403 518 854 746 410 120-180 577 769 953 822 621 093 >180 12 807 947 64 207 745 Total 2 600 722 770 2 225 754 822 The results show that the total damage, in the case of the first scenario, are greater than 14% and that difference occurs in the first 120 minutes. Also in terms of economic loss the difference between the two scenarios are less than the differences of the peak discharge in the breach of the dam.
Conclusion In this paper we present the results of the analysis of a possible dam failure. The development of a dam break is a complex process involving numerous uncertainties: the methodology adopted in this work is a medium-scale approach type and can be used for the rapid and consistent evaluation of consequences for the population and to assess the direct economic damages for residential, commercial, and industrial buildings. Rapidity is allowed by using aggregate data: maps of land-use, population census and parcel zone. Consistency is required to ensure comparability between evaluations. For that reason the method can be used to prioritize corrective actions to achieve the greatest and quickest possible risk reduction or for identification of the most effective and better-justified measures of risk mitigation. The comparison carried out for the two scenarios is an example of use of the methodology to estimate the sensitivity of results with respect to an uncertain parameter which is the breach formation time.
Acknowledgements This work was carried out also thanks to the Research Fund for the Italian Electrical System under the Contract Agreement between RSE S.p.A.(Research for Energetic System) and the Ministry of Economic Development - General Directorate for Nuclear Energy, Renewable Energy and Energy Efficiency, stipulated on July 29, 2009 in compliance with the Decree of March 19, 2009.
References [1] Molinaro P. (1990). Metodi statistici per la stima della portata di picco defluente in una valle per la rottura di una diga in materiale sciolto. ENEL DSR-CRIS Milano. [2] Costa J. E. (1995). Floods from dam failures. U.S. Geological Survey, Open-File Report 85-560 Denver Colorado. 280
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[3] Molinaro P. (1986). Modello matematico della breccia che si sviluppa in uno sbarramento in materiale sciolto per tracimazione. ENEL DSR-CRIS Milano. [4] Engelund F., and E. Hansen (1972). A Monograph on Sediment Transport in Alluvial Streams, Teknisk Folag, Copenhagen. [5] S. Bunya, J. C. Dietrich, J. J. Westerink, B. A. Ebersole, J. M. Smith, J. H. Atkinson, R. Jensen, D. T. Resio, R. A. Luettich, C. Dawson, V. J. Cardone, A. T. Cox, M. D. Powell, H. J. Westerink, H. J. Roberts. (2010). A High-Resolution Coupled Riverine Flow, Tide, Wind, Wind Wave, and Storm Surge Model for Southern Louisiana and Mississippi. Part I: Model Development and Validation. Monthly Weather Review 138:2, 345-377. [6] Dams Sector. Estimating Loss of Life for Dam Failure Scenarios. U.S. Department of Homeland Security, Washington, D.C. , September 2011. [7] Department of Water Resources Division of Flood Management (2008). Flood Rapid Assessment Model (F-RAM) Development 2008, State of California, The Resources Agency [8] David Ford Consulting Engineers (2005). Urban flood scenario: Sacramento area levee breach scenario, report prepared for California Department of Water Resources, December 5th 2005.
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2-D hydraulic modelling of a dam break scenario O. Saberi1 , C. Dorfmann2 and G.Zenz3 1, 2
PhD student, Institute of Hydraulic Engineering and Water Resources Management, Stremayrgasse 10/2, A-8010 Graz, AUSTRIA 3 Univ. Prof. Institute of Hydraulic Engineering and Water Resources Management, Stremayrgasse 10/2, A-8010 Graz, AUSTRIA E-Mail: [email protected]
Abstract Dam failure is a catastrophic event, and study on the structure of dam is important in the field of water resource engineering. The reason is the risk to life and property below the dam structure. The dam failure occur due to many reasons, some of these reasons are related to structural problems while others are related to the hydraulic conditions. For embankment dams, main reasons are overtopping and piping through the dam body or foundation. However there are some situations that make dam failure happen suddenly, like earthquakes, landslides or wars. This study provides a two-step numerical simulation of dam failure analysis for an embankment dam which its failure happened in overtopping conditions. First step is about simulating breach of the dam and calculating breach hydrograph, maximum discharge of breach, initial time of the breach formation, and time of maximum breach discharge and other parameters of breach. BREACH GUI software was used for this step. The second part simulates flood after dam break. TELEMAC2D software was used for simulating the flood. The results include the travel time (warning time) of the flood wave to various locations in the downstream valley and the representative valley cross-sections depicting flow depth and unit flow rates. Furthermore the results of this paper can provide information to build an inundation map which can help to develop risk management analyzes.
Introduction A hypothetical embankment dam was constructed in a mountainous region. This high hazard dam sits directly above a lightly populated area which is 3.5 kilometers away from an urban environment. The primary function of this dam is flood control for heavy snowmelt and strong monsoonal weather patterns. In addition, the reservoir provides some water supply and recreational activities to nearby communities. The dam failure for this dam takes place when the pool elevation is at crest elevation. The mode of failure is assuming an overtopping failure [6]. Two primary tasks in the analysis of a potential dam failure are the prediction of the reservoir outflow hydrograph and the routing of that hydrograph through the downstream valley to determine the dam failure consequences. When populations are located close to a dam, it is important to accurately predict the breach outflow hydrograph and its timing relative to events in the failure process that could trigger the start of evacuation efforts.
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Breach Parameter Estimation Empirical Method The empirical method are used to predict breach parameters estimation and breach peak discharge and breach hydrograph estimation including geometry of breach, time to reach failure and peak flow discharge, the empirical methods are get from documented failures. The recommended empirical method for predicting dam breach parameters are MacDonald&langridge-Monopolis (1984), Froehlich (2008), Froehlich (1995 a) and Von Thun and Gillite. In this paper we calculate the dam breach parameters with Froehlich (2008) Formula. For more details we refer to the 12th International Benchmark Workshop [6]. Froehlich (2008) is depending only on the height of breach ( , the reservoir volume and the breach side-slope, also we have the failure mode factor ( ) in the Froehlich (2008) for distinguish between overtopping and piping failure. After put mentioned parameters in the Froehlich (2008) formula we have breach hydrograph in the below shape: Table 1: Results of the Froehlich (2008)
Discharge (m³*1000/s)
Froehlich (2008)
⁄ 39500
0.57
110
50000 40000 30000 20000 10000 0 0
0.2
Time (hours)
0.4
0.6
Figure 1: Outflow hydrograph from the Froehlich (2008) BREACH modelling In this project we use the National Weather Service (NWS) software Brach GUI for predicting the breach parameters and the resulting outflow hydrograph of an earthen dam [1]. BREACH model was primarily developed by Fread [9] from the National Weather Services. Since 1988 it has been used broadly. This software use a physically-based mathematical model using the principles of hydraulics, sediment transport, soil mechanics, the geometric and material properties of the dam, and the reservoir properties [1]. Furthermore it uses critical properties which are measurable from dam material descriptions. Therefore it considers a more robust model. However it is the responsibility of the engineer to determine the appropriate combination of values as the measure characteristic are normally changing with in a wide range. Available parameter of breach hydrograph The dam used in this study is a homogenous embankment dam which is a rolled earth fill structure and composed of predominantly sandy clays and clayey sands. It is located in a 284
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mountain region. The width of crest is 24 m, the length of crest is 360 m and the height of the dam is 61m. The total water volume of water behind the dam is 38,276,344 m³. The distance from this dam to the population area is 3.5 km. The dam failure will take place when the water level behind the dam is at crest elevation and the mode of failure will be over topping failure. For more details we refer to the 12th International Benchmark Workshop [6]. As base inflow we assumed a discharge equal to 100 m³/s. In Table 1 the main physical parameters of the dam are listed. Table 2: Strength parameters of soil [6] Effective cohesion (KPa) Effective friction angle Undrained strength (KPa) Shear Wave velocity (m/s) Maximum shear modulus (KPa) Saturated unit weight (kg/m3) Permeability (cm/s)
19.15 14° 43.09+0.175 152.4 46443 2002 1.9*10^-6
Calculating input parameter for Breach software For start a simulation with the BREACH software we need to calculate some parameters as input data. The most important input parameters to be provided in the BREACH software are the mean grain size D50 (mm), the initial elevation of water surface, the elevation of the top of the dam, the elevation of the bottom of the dam, the average plasticity index (PI), the critical shear stress coefficient, the slope of the upstream and downstream faces of the dam (1:3), the base inflow hydrograph, the surface area behind the dam vs. the elevation of the dam, the porosity specifications of the dam material and the breach parameter. Table 1 and the equations 1-4 according to [ ] can help with defining these input parameters: √ ⁄
(1)
where V is the shear wave velocity (m/s), shear modulus (KPa)
is the density of material (kg/m3) and G is the (2)
where is the specific gravity, of water (N/m³)
is the unit weight of soil (N/m³) and
is the unit weight
(3) where
is the specific gravity,
is the unit weight of water (N/m³) and e is the void ratio (4)
where e is the void ratio (%) and n is the porosity (%). With the parameters in Table 1 and the equations 1-4 the porosity n is calculated as 49%. D50 can be considered approximately equal to 0.066 mm [4]. The basis for this approximation comes from soils mechanics fields.
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When the diameter of Clay soil is less than 0.002 mm and the diameter of sand soil is between and 0.05 and 2.00 mm then sandy clay is a soil with 33 percent till 55 percent clay and up to 65 percent sand [5]. The average plasticity index (PI) is calculated according to the USCS plasticity chart and the relationship equations of Skempton and Henkel (1953) [3], Psterman (1959) [3] and Bjerum(1960) [3] which is equal to 42%. Furthermore PI will be used in the following formulation to calculate two more parameters required by BREACH software (CA, CB) [1]: (5) where is the critical shear stress (kPa), PI is the plasticity index (%) and CA, CB are the critical shear stress parameters. Having PI=42% and derived from Table 1, CA and CB are calculated as .
Discharge (m³*1000/s)
Output of the BREACH software Having all the above inputs into the software we can run the BREACH model to calculate the outflow hydrograph given in Figure 1. 45 40 35 30 25 20 15 10 5 0 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
Time (hours)
Figure 2: Outflow hydrograph from the breach software Furthermore the software output provides some other parameters which can be used in risk management studies. These parameters are given in Table 2: Table 3: Results of the BREACH software Maximum discharge of breach (m3/s)
41600
Time for peak out flow (hours)
4.62
Time for starting outflow (hours)
4.44
Final depth of breach (m)
61
Top width of breach at peak breach flow (m)
250
Time to reach final breach bottom elevation (hours)
4.97
Bottom width of breach at peak breach flow (m)
286
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Comparison between result of Froehlich (2008) and BREACH modelling
Discharge (m³*1000/s)
After calculating breach hydrograph from Froehlich (2008) and breach modeling in follow picture, we find out all the dam breach parameters including the failure time, the peak discharge outflow and the average of breach width of Froehlich less than BREACH software. 45000 40000 35000 30000 25000 20000 15000 10000 5000 0 4.463
BREACH software Froehlich (2008)
4.963
5.463
Time (hours)
Figure 3: merged discharge of cross sections (m3)/s 2-D hydrodynamic flood with Telemac-2D The flood from dam break can go to downstream valley and in this part we are going to determine dam failure consequences and calculating the water surface, the water depths, the unit flow rates as well as the flow rates in various cross sections. The TELEMAC-2D modelling software is used for this purpose. TELEMAC-2D is a numerical simulation software and belongs as module to the open source Telemac-Mascaret suite for the simulations of hydrodynamic flow, contaminant and sediment transport. It solves the twodimensional depth-averaged Saint Venant equations for free surface flow. A detailed description of the Telemac System is given in Hervouet’s book (2007)[7]. The main results of Telemac-2D are the water depths and the depth- averaged velocity components. The original geometry provided by the ICOLD benchmark specifications was modified by deleting a priori non flooded mesh areas in order to save computation time. Table 3 shows the resulting mesh properties: Table 4: Mesh properties Number of element: 213750 Interior elements: 210652 Edge elements: 3038 Number of nodes: 108433 Edge nodes: 3114 Interior nodes: 105319 The upstream boundary condition is located at the dam and the hydraulic boundary condition is the unsteady flow hydrograph from the BREACH software, see Figure 1. The downstream boundary condition is a free outflow.
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Results overview The results of TELEMAC-2D are stored in the binary SEARAFIN format which can be analyzed and visualized by using the free software Fudaa or BlueKenue [8]. In this project BLUE KENUE software was used. In the following the results are shown graphically for breach discharge, cross section discharge, peak flood depth, flood wave arrival time, peak unit flow rate and flooded area. Furthermore interpretation on each subject is given along with the output figure. Breach Discharge This result was calculated in the first part with BREACH GUI software (shown in Figure 1). Also from the calculated breach hydrograph we find out: With the inflow of 100 m³/s into the dam reservoir it takes 4.36 hours for the dam breach initiation. The maximum discharge takes place after 648 s after the breach initiation. The duration of flood is 2844 s. Cross-Section Discharge For the calculation of the cross section discharge the following steps are performed: First the unit flow rates are calculated (m²/s) Second the unit flow rates are integrated along the cross section width Figure 2 shows the discharges in various cross sections and the reduction of the peak flows going downstream from cross section 1 to 5. Discharge of cross sections (m3/s) 45000 40000 35000
Discharge (m3/s)
30000 cross section 1
25000
cross section 2 20000
cross section 3
15000
cross section 4 cross section 5
10000 5000 0 00:00:00 00:28:48 00:57:36 01:26:24 01:55:12 02:24:00 02:52:48 time (HR)
Figure 4: merged discharge of cross sections (m3)/s
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Peak flood depths In BLUE KENUE software we can also calculate the maximum flood depths in the project area. Figure 3 shows the results for the peak flood depths where the blue color indicates the safe area with maximum water depths less than 0.5 m and the red color indicates dangerous areas with minimum water depths of 4.5 m.
Figure 5: Peak flood depths (m) Flood wave arrival time and water depths In this part we calculated the arrival time of flood. This will help us to find out how long it takes for the flood to reach each part of downstream area. BlueKenue software visualizes the water depths separately for each chosen time interval. It means BlueKenue gives us one gridded dataset for each time intervall. Figures 4, 5 and 6 shows the water depths or flood wave arrival times for certain time intervals.
Figure 6: Water depths after 30 minutes
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Figure 7: Water depths after 2 hours and 30 minutes As we can see in Figure 4, 5 and 6 these visualizations are useful to analyze the flood wave arrival time and the locations of impact. Peak unit flow rate The unit flow rate is given in equation 6: (6)
√
where q is the unit flow rate (m²/s), U and V are the water velocity components in x and y direction (m/s), and H is water depth (m). The peak unit flow rate in every mesh node can be extracted from the time varying unit flow rate and is shown in Figure 7.
Figure 8: Peak unit flow rate (m²/s) Flooded area 290
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For calculating the flooded area we can define the total wet area by defining iso-lines. This option categorizes the flooded area for arbitrary water levels. The total flooded area is given in Table 4 and the flooded areas for different water depths with interval of 0.5 m is given in Table 5. Table 5: Total flooded area Total flooded area (m²)
47626580
Table 6: Flooded areas, intervals 0.5 m Water heights (m) >8 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5
Flooded area (m²) *10^3 4179 4290 4805 5565 6827 7910 9131 10269 11569 13349 15432 18145 22200 27109 32959 38433 41043
The results of these tables can be used in the risk management study for calculating the damage property and loss of life.
Conclusion The first part of this study describes the BREACH model for predicting the breach hydrograph of a hypothetical embankment dam. The reason for the failure of dam is assumed to be overtopping. The following results can be deduced: The maximum outflow in this project is calculated as 41558 m³/s. The water flow is transferred into the downstream direction of the dam which can cause catastrophic consequences.
The maximum discharge happens near the dam place at cross section number 1. One of the most important parameter in the risk management is the alarm time. In this
study we have calculated the break initiation time as well as the formation time. Furthermore these data can help us to determination the Property damage and loss lives. The above mentioned results used in the successive 2-D depth-averaged calculations with TELEMAC-2D. The following conclusions can be made based on the 2-D simulation results:
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The height of water and the maximum height of water which are important parameters in risk management fields are calculated to define inundation regions. These results also can be used to define warning zones according to the flooded areas.
The wave arrival time is analysed for 15 minutes intervals. This time model shows us how long it takes for the flood to reach each part of the downstream area which is useful for reducing loss of human lives in risk management studies.
Acknowledgements I would like to express my deepest appreciation to all those who provided me the possibility and help me to complete this report.
References [1] Janice Sylvestre, Developer, 2010, BREACH GUI (BREACHJ) Version 1-0-0 08-012010 [2] TELEMAC modelling system, 2D hydrodynamics, TELEMAC-2D software Version 6.0, USER MANUAL [3] A. S. Balasubramaniam, H. Cai, D. Zhu, C. Surarak and E. Y. N. Oh,Griffith School of Engineering Griffith University, Gold Coast, Australia, Settlements of Embankments in Soft Soils [4] AFTA 2005 Conference Proceedings, Placement of Riparian Forest Buffers to Improve Water Quality [5] Braja M.Das, 2008. Advance soil mechanic, third edition. This edition published 2008 [6] By Taylor & Francis 270 Madison Ave, New York, NY 10016, USA [7] 12th International Benchmark Workshop on numerical analysis of dam, 2013, Austria, Graz [8] Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method by Jean-Michel Hervouet(May 29, 2007) [9] CHC - Canadian Hydraulics Centre, National Research Council, 2010. Blue Kenue, Reference Manual, August 2010. [10] The NWS simplified dam-break flood forecasting model by Jonathan N.wetmore and Danny L.Fread (Reviesed 12/18/91) by Danny L.Fread, Janice M.Lewis, and Stephen M.Wiele
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Risk Assessment Analysis of a Hypothetical Dam Breach Using Adaptive Finite Element Methods Adaptive Hydraulic Model, ADH D. McVan1, J. Ellis2, G. Savant3 and M. Jourdan1 1Research Hydraulic Engineer, Engineer Research and Development Center, U.S. Army Corps of Engineers, Vicksburg, MS, 39180. E-mail: [email protected] 2Engineering Technician, Bowhead Science and Technology, LLC, and Onsite Contractor, Engineer Research and Development Center, U.S. Army Corps of Engineers, Vicksburg, MS 39180. E-mail: [email protected] 3
Research Water Resources Engineer, Dynamic Solutions LLC and Onsite Contractor, Engineer Research and Development Center, U.S. Army Corps of Engineers, Vicksburg, MS 39180. E-mail: [email protected]
Abstract A virtual computational test bed was developed by a committee sponsoring a session at the International Commission on Large Dam (ICOLD) 12th International Benchmark Workshop. The purpose of this virtual test bed was to serve as a platform for comparison of dam breach models and methodologies. This paper provides the results of one of the model comparisons. The ADH model, developed by the U.S. Army Engineer Research and Development Center (ERDC) was used to simulate a catastrophic dam failure and gradual dam failure. The ADH Model is described, as well as the domain, setup, and boundary conditions for the specific simulations. Results are provided as gridded and tabular data, as well as hydrographs. The gridded results include peak flood depths, flood wave arrival times, peak unit flow rates and total population at risk. Tabular results include flooded area and population as risk (including age demographics). Hydrographs are presented for both the dam failure discharge and at five down stream cross sections. Loss of life and economic impacts were not calculated.
Introduction Computational capacity, along with multi-processors computing techniques, has grown resulting in development of new models for flooding and consequence/risk assessment. These new models which are computing flood wave propagation at higher spatial resolutions can be coupled with risk assessment models which allow for more detailed analyses of the effects of these flows. This paper represents results of the numerical dam breach done using the virtual testbed developed for the International Commission on Large Dam (ICOLD) 12th International Benchmark Workshop. This virtual testbed was developed as a tool for comparison of different numerical model results and the model capabilities. The testbed includes a hypothetical dam above a hypothetical city named Hydropolis. The focus of this study was to estimate the consequences of failure of the dam. Demographics, infrastructure and dam 293
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characteristics were provided so that comparison of these models could be made which will aid in the development of simulation frameworks that can support dam risk analysis. Two model scenarios were modeled; the first model represents a catastrophic event in which the entire dam is destroyed instantaneously. The second model represents a gradual breach using Froehlich (1995a) equations.
Methodology Model Code Description ADH is a state-of-the-art code developed by the U.S. Army Engineer Research and Development Center (ERDC) to simulate both saturated and unsaturated groundwater, overland flow, three-dimensional Navier-Stokes flow, and two and three-dimensional shallow water equations, including super-critical flows and shock capturing (Berger el.al. 2010). The ADH code is parallelized and is capable of running on high performance computing systems. ADH is also an implicit code meaning its time step size is not limited by the element size as with explicit codes. These features make ADH computationally efficient for large-scale applications. ADH has been utilized to study varied phenomena such as estuarine circulation (McAlpin et al. 2009; Tate et al. 2010; Martin et al. 2010, Martin et al. 2011), riverine flow (Stockstill and Vaughan 2009; Stockstill et al. 2010), and dam breach (Savant et al. 2011). The two-dimensional shallow water (SW2) equations are used for the application presented here. This module solves the conservative form of the SW2 equations, allowing for local and global mass conservation. ADH is a temporally and spatially adaptive code. For temporal adaption, the code utilizes a variable time step such that failure to reach convergence at a given time step size does not end the simulation; rather, the time step size is reduced and the solve is attempted again (Savant et al. 2011). ADH also has the ability to allow continuous wetting and drying such that the flood front is computed and visualized accurately (Berger and Lee 2004; Savant and Berger 2011). For this model simulation, a maximum time step of 10 seconds was specified with smaller time steps being performed automatically as needed. Model Setup All hydrodynamic numerical models require certain basic input data to perform simulations. These data are bathymetric and topographic elevations for defining the model domain, inflow hydrographs or water elevations, as well as other boundary information such as rainfall and ocean water surface elevation (if the area of interest is in a tidal zone) for driving the model, as well as roughness characteristics and the ground surface and other parameterizations depending upon the objective of the modeling application. In the following sections, these domain construction data, driving data, and parameterization needs are addressed. Model Domain The bathymetric and topographic information for this workshop were provided from the ICOLD “Theme C” Formulation Team in the form of a DEM with 9.4 meter resolution. Two DEMs were provided, one representing the domain without the dam, and the other with the dam. The DEM covered an area that was approximately 250 km2. The hypothetical dam was located at coordinate 4499.66 meters and 6681.57 meters and has a crest width of 24 meters and a crest length of 360 meters. The river bed elevation at the base of the dam is 211 meters and the crest elevation is 272 meters with 3H:1V embankment slopes. The maximum storage capacity of the reservoir is approximately 38 million cubic meters.
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The ADH mesh was created using the Surface-water Modeling System (SMS) (Aquaveo 2009) meshing algorithms and consists of 13,790 triangular elements comprised of 7,138 nodes at the triangle corners. Both the catastrophic model and the gradual breach model used the same mesh. The dam was ‘built’ into the mesh at the provided coordinates specifying the width of the elements along the dam’s crest and determining the upstream and downstream length of the embankment. Element areas range from approximately 250 m2 to 25,000 m2, with the finest resolution located at the dam and along the channel and the larger areas located downstream in the lake. After the mesh was constructed, the separate DEMs were interpolated to the mesh. The DEM without the dam was interpolated to the mesh for the catastrophic model and the DEM with the dam was for the gradual breach model. Figure 1 shows the mesh at the dam site, the model domain and bathymetry. Model Boundary Conditions and Parameterization Numerical modeling of any type requires the specification of driving boundary information. For the dam breach model, this driving force is maximum volume of the reservoir at the crest elevation, 272.0 meters. This is achieved by creating initial depth conditions at each node. For the nodes representing the reservoir, the water surface elevation was entered and the initial water depth was determined from the mesh bathymetry. The rest of the mesh was
Figure 1: Model Domain. given a -0.1 meter water depth in order to simulate a dry mesh. In an attempt to prevent backup of the flow which may cause artificial inundation not created by the flooding, an evaporation boundary was applied to the downstream boundary that allows the water to be transported out of the domain. However, after the model was complete, it was observed that after the flood wave hits the boundary, the wave is reflected back into the model domain. This only occurred in the area of the lake and it was determined that it would not affected the area of interest. Bed roughness was an additional user specified parameter that defines the frictional resistance. ICOLD’s “Theme C” Formulation Team provided a gridded dataset representing the land use/land cover of the region. This data was used to identify areas of similarity and applied to the mesh as material types. Figure 2 shows the various material types used in the model. The elements and nodes associated with each material type will have the same material properties specified in the boundary conditions. Literature search describing the varying roughness coefficients for differing floodplains were used to determine the bed roughness for each material type (Arcement, G.J., Schneider, V.R., and Oregon Department 295
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of Transportation, 2005). Table 1 shows the material types used in the model along with its roughness value. Table 1: Bed Roughness Material Type Manning’s n Value Channel 0.018 Forest 0.100 Developed 0.020 Shrub/Cropland 0.050 Shrub/Forest 0.070 Lake 0.018 Reservoir 0.018 Dam Crest 0.018 Upstream Dam Face 0.018 Downstream Dam Face 0.018 Downstream Boundary 0.018 (Arcement, G.J., Schneider, V.R., ) (Oregon Department of Transportation, 2005) In the ADH numerical code, the Manning’s value is converted to an equivalent roughness height that depends on the instantaneous water depth at a particular node. The friction formulation is then derived from the logarithmic velocity profile based on open channel flow. Thus, while the user-specified frictional value (Manning’s number) is constant over the specified material type, the actual applied resistance is spatially varying according to the water depth and velocity magnitude at each node in the mesh. This results in higher frictional values for locations of shallower depths as is experienced in nature and can be extremely important when estimating flood extents.
Figure 2: Material Types For the catastrophic model, the dam does not exist in the mesh therefore the model’s initial conditions at time 0.0 simulated a wall of water at a depth of 61 meters. For the gradual breach, the dam does exist in the mesh. Equations developed by Froehlich (1995a) were used to determine the average breach width and the breach formation time and are shown below. Failure of the dam was assumed to be caused by overtopping resulting in 80-percent of the dam height removed or approximately 48.8 meters. Therefore, the maximum width of the breach was approximately 141 meters and the breach formation time was approximately 48 minutes (2940 seconds). (The height of the breach and the formation 296
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time were rounded to 49 meters and 49 minutes). To achieve the gradual breach, elevations of the nodes representing the dam were gradually lowered at specified time intervals. At the end of the breach, the elevations of the nodes representing the breach were 223 meters. (1) (2) Where:
Bave Ko Vw hb tf
= = = = =
Average Breach Width (m) Constant (1.4 for overtopping failures, 1.0 for piping) Reservoir volume at time of failure (m3) Height of the final breach (m) Breach formation time (hours)
Model Results Model Simulations The ADH model simulations for both the catastrophic and the gradual breach were performed on the SGI Altix Ice machine (Diamond) at the USACE ERDC (ERDC DSRC 2011). This machine uses an SLES (Linux) 11 operating system with a 2.8 GHz Intel Xeon Nehalem-EP Processor. The catastrophic model was run on 16 processors and simulated 6 hours 40 minutes (24,000 seconds) and required approximately 60 minutes to complete (wall time). The gradual breach was run on 12 processors and simulated 12 hours (43,200 seconds) and required approximately 2 hours to complete. Both models had output intervals of 100 seconds. In accordance to ICOLD’s “Theme C” Formulation Team, the information obtained for both the catastrophic event and the gradual breach were discharge hydrographs for the dam failure and along five cross-sections specified by the Formulation Team. The model also provided contours of the peak flood depths, peak unit flow rate, and the flood wave arrival times. Because ADH uses a mesh, the output results are also in the mesh format. In other words, each node in the mesh contains all the information for each time step. To determine the total population at risk including the age demographics as well as the flooded area, the ADH depth file were converted to a raster for each output time interval. The raster was then converted to a shapefile which contained the depth at 1 meter intervals. The Formulation Team specified 0.5 meter intervals, however due to the constraints of this methodology, only integer values could be used. The census block shapefile provided by the Formulation Team was clipped to the depth shapefile and the ratio of the clipped area compared to the entire area determined the total population potentially affected by the flood. Catastrophic Model Results As mentioned previously, the catastrophic model simulated the worst case scenario where the entire dam is removed instantaneously. The maximum discharge at the time of the breach was approximately 84,000 cms. At each cross section, the maximum discharge varied between approximately 61,000 cms and 81,000 cms. Figure 3 shows the discharge hydrographs for each location. The total volume at the reservoir was approximately 38.93 km3 and at each of the cross sections, the volume varied between 39.4 km3 and 40.9 km3. Table 2 shows the maximum discharge with the associated times of occurrence at the dam and for each of the 5 cross-sections along with the total volume for each location.
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Though ADH conserves fluid and constituent mass to machine precisions, for wet-dry problems this conservation is dependent on user specified tolerance parameters. These parameters control how accurately ADH solves the continuity and momentum equations using the Newton-Raphson iterative approach for solving a system of implicit equations. Very small tolerance parameters will drive the solution to conserve mass to machine precision but can significantly increase run time, large tolerances will significantly reduce run time but will not provide mass conservation or the correct results. A balance has to be achieved between the requirements of a particular project or simulation and whether exact mass conservation is required. For the simulations presented in this document the mass is conserved to 3-percent for catastrophic breach. ADH utilizes a diffusive wave type shock capturing scheme in wetdry areas, this scheme, for extreme wet-dry scenarios, can hide fluid mass in dry cells and is an additional, though rare, source of mass errors. All elements/nodes are included in computations at all times and no minimum ‘include in computation’ depth is used by ADH. The error shown in Table 2 shows how well ADH conserved the mass at each cross-section. Table 2: Maximum Discharge for Catastrophic Event
Location At the Dam X-Section 1 X-Section 2 X-Section 3 X-Section 4 X-Section 5
Time of Maximum Discharge (seconds) 1 26 207 293 385 546
Maximum Discharge (cms) 83,828 80,855 66,027 60,738 62,089 67,166
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Total Volume (km3) 38.93 39.37 39.47 39.69 39.65 40.91
Error 1.01 1.01 1.01 1.00 1.03
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Figure 3 Maximum Discharges for Catastrophic Event. The reservoir was completely emptied in approximately 50 minutes (3000 seconds). The flood wave was completely at the lake after 2.7 hours. As one would expect, the greatest depths occur within the channel. Maximum depths in the floodplain range between 2 and 12 meters. The unit discharge in the floodplain is less than 50 m2/s. Figures 4 and 5 show the maximum peak depth for the catastrophic model and the peak unit flow rate, respectively. The flood wave reaches the floodplain about 5 minutes after the dam is breached and reaches the downstream boundary at approximately 1.67 hours (100 minutes). Figure 6 shows the arrival time in 15 minute intervals.
Figure 4 Peak Flood Depth in meters for Catastrophic Event
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Figure 5 Peak Unit Flow Rate (m2/s) for Catastrophic Event
Figure 6 Flood Wave Arrival Times for Catastrophic Event The total flooded area at peak depths, excluding the area of the lake, was approximately 41.1 km2 and affected well over 28,000 people. The total population at risk after 180 minutes is over 26,000. These differences are the results of taking a single snap shot of the population in time. As the flood wave travels downstream, the inundated area covers a finite area. For the next time, the inundated area covers a completely different area. Sometimes these areas overlap which result in duplicating the population at risk and at other times there is a gap resulting in the population at risk not being accounted for. Therefore, when assessing loss of life and economic consequences, it is best to use the peak flood depth values. Figure 7 shows the clipped total population at risk for the peak depths. Table 3 shows the population at risk at 15 minute time intervals and includes age demographics and Table 4 shows the flooded areas at 1 meter intervals along with the total population at risk and the age demographics. For the 12-year and under age demographic, the total population at risk after 3 hours is over 4,300 and about 4,600 for the maximum flooded area. The 65-year and over age demographic has over 4,700 at risk after 3 hours and around 5,200 for the maximum flooded area. The total flooded area with depths greater than 8 meters is over 3.35 km2 which affects a total population at risk of less than 2,000. For the 12-year and under demographics, the total population at risk affected by depths greater than 8 meters is slightly over 400 and over 130 for the 65-year and over demographics.
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Table 3: Total Population at Risk for Catastrophic Event Time Intervals (minutes 15 30 45 60 75 90 105 120 135 150 165 180 Total
Total Population at Risk 3,179 12,516 9,807 629 33 19 13 12 12 11 11 11 26,253
12-yr and Under Population at Risk 815 1,936 1,449 97 7 4 2 2 2 2 2 2 4,320
65-yr and Over Population at Risk 225 2,487 1,910 132 1 1 0 0 0 0 0 0 4,759
Table 4: Flooded Area Population at Risk for Catastrophic Event Peak Flood Depth Range (meters) >/= 1 >/= 2 >/= 3 >/= 4 >/= 5 >/= 6 >/= 7 >/= 8 Total
Flooded Area excluding lake (km2) 41.09 32.37 17.55 9.73 5.94 4.56 3.82 3.35 41.09
Total Population at Risk 28,802 21,273 6,274 3,494 2,886 2,598 2,201 1,710 28,802
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12-yr and Under Population at Risk 4,641 3,429 1,207 835 710 635 535 413 4,641
65-yr and Over Population at Risk 5,208 3,994 1,044 327 220 202 167 136 5,208
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Figure 7 Total Populations at Risk for Catastrophic Event Gradual Breach Model Results Failure of the dam for the gradual breach scenario was assumed to be caused by overtopping resulting in 80-percent (49 meters) of the dam height removed. The maximum width of the breach was 141 meters and the breach formation time was 49 minutes (2940 seconds). Elevations of the nodes representing the dam were gradually lowered by 1 meter at 1 minute intervals. At the end of the breach, the elevations of the nodes representing the breach were 223 meters. The maximum discharge at the dam location was over 17,000 cms and occurred 33.33 minutes (2000 seconds) after the initial time of breach. The maximum discharges for the 5 cross-sections ranged from between 16,400 cms and 20,500 cms. Table 5 shows the maximum discharges at the dam and the 5 cross sections along with the calculated volumes of each hydrograph. Figure 8 shows the discharge hydrographs for each location. Since the lowest elevation of the final breach is 223 meters, the reservoir is assumed to retain a volume of approximately 0.36 km3. The volumes at the cross sections range from 38.5 km2 to 42.2 km2. As previously mentioned, for wet-dry problems, the conservation of mass is dependent on the tolerance parameters specified by the user which controls how accurately ADH solves the continuity and momentum equations. For the gradual breach simulation, the mass was conserved to 8-percent. The error in Table 5 shows how well ADH conserved the mass at each cross section. The maximum peak depths for the gradual breach inundated area are over 20 meters and occur in the channel. The floodplain depths range from 0.5 meters to less than 6 meters, with a few scattered pockets of slightly higher depths. The peak unit flow rates in the floodplain are less than 10 m2/s and in the channel, the peak unit flow rate is over 120 m2/s. The flood wave arrives at the floodplain about 45 minutes after the initial breach and reaches the downstream boundary after a little over 140 minutes. The flood wave was completely in the lake after 3.3 hours. Peak flood depth, peak unit flow rate and the flood arrival times are shown in figures 9 through figure 11, respectively. Table 5: Maximum Discharge for Gradual Breach Event
Location At the Dam X-Section 1 X-Section 2 X-Section 3 X-Section 4 X-Section 5
Time of Maximum Discharge (seconds) 2000 1690 2246 1029 1174 1409
Maximum Discharge (cms) 17,357 16,717 16,839 16,447 20,527 18,185
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Total Volume (km3) 38.93 38.54 39.03 40.19 40.82 42.21
Error 0.99 1.0 1.03 1.05 1.08
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Figure 8 Maximum Discharges for Gradual Breach Event.
Figure 9 Peak Flood Depth in meters for Gradual Breach Event.
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Figure 10 Peak Unit Flow Rate in square (m2/s) for Gradual Breach Event.
Figure 11 Flood Wave Arrival Times for Gradual Breach Event. The total flooded area, excluding the lake, was slightly over 20 km2. The total population at risk from this inundation is over 9,000 people. The highest population at risk occurs within the first 75 minutes of the breach and affects over 15,500 people. As mentioned previously, as the flood wave travels downstream, the single snap shots in time may over lap resulting in the duplication of the number of population at risk. This is the case for the gradual breach which results in the total population at risk at the end of 3 hours being double the total population at risk from the peak flooded area. Again, the more reasonable assessment of the total population at risk should be based on the peak depth flooded area. For the 12-year and under age demographic, the total population at risk after 3 hours is over 3,500 and half that, about 1,600 for the maximum flooded area. The 65-year and over age demographic has over 3,000 at risk after 3 hours and around 1,700 for the maximum flooded area. The maximum inundated area affected by depths greater than 8 meters is 1.99 km2. The total population affected by these depths is less than 300, with the 12-year and under population at risk slightly over 50 and under 30 for the 65-year and over demographics. Table 6: Total Population at Risk for Gradual Breach Event Time Intervals (minutes)
Total Population at Risk
12-yr and Under Population at Risk 304
65-yr and Over Population at Risk
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15 30 45 60 75 90 105 120 135 150 165 180
Total
217 2,871 2,966 4,795 4,812 1,664 800 182 82 32 30 38 18,489
32 738 760 882 728 233 114 18 6 3 3 4 3,523
29 208 223 890 1,024 395 199 62 28 8 8 10 3,084
Table 7: Flooded Area Population at Risk for Gradual Breach Event Peak Flood Depth Range (meters) >/= 1 >/= 2 >/= 3 >/= 4 >/= 5 >/= 6 >/= 7 >/= 8 Total
Flooded Area excluding lake (km2) 20.02 10.00 5.85 3.99 3.05 2.49 2.21 1.99 20.02
Total Population at Risk 9,226 4,107 2,768 2,121 1,302 326 401 272 9,226
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12-yr and Under Population at Risk 1,600 892 699 531 321 142 86 52 1,600
65-yr and Over Population at Risk 1,699 541 220 166 101 48 34 27 1,699
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Figure 12 Total Populations at Risk for Gradual Breach Event
Conclusion The results from the catastrophic breach event showed a total flooded area, excluding the lake, of 41.1 km2 which affects over 28,800 people. The total number of 12-year and under affected by the maximum inundated area is over 4.600 and the total number of 65-year and over is over 4,200. The flooded area with depths greater than 8 meters is 3.3 km 2 and affects about 1,700 individuals, over 400 are 12-year and under and over 130 are 65-year and over. The greatest number of the population at risk occurs within the first hour of the dam breach resulting in over 26,000 being affected. Three hours after the initial dam breach, the majority of the flood wave has reached the lake and no longer affects the city of Hydropolis. The gradual breach scenario assumed breaching was caused by overtopping resulting in 80percent of the dam being removed. The time to reach the maximum breach and the maximum width were determined using equations developed by Froehlich (1995a). The results showed a total flooded area excluding the lake of over 20 km2 which affects over 9,200 people. The number of 12-year and under affected by the maximum flooded area is 1,600 and the number of 65-year and over is 1,699. The flooded area with depths greater than 8 meters is 1.99 km2 affects less than 300 individuals with over 50 being 12-year and under and less than 30 being 65-year and over. The majority of the population at risk occurs within the first 90 minutes of the dam breach; however the total number is skewed because the inundated area for each time interval overlaps. These results represent the impacts of a dam breach, whether that breach is catastrophic or gradual. Though loss of life was not calculated using the described model, the population at risk was significant. Additionally, economic impacts were not considered, as this is not part of the applied model. Even without these values available for comparison, there are many data elements that will be useful in comparing the ADH model to other models presented at this workshop. The use of the virtual test bed should prove to be a valuable tool, both now and in the future, for comparison of such models.
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References [1] Aquaveo (2009). “Surface-water Modeling System Version 11.0”, Aquaveo. http://www.aquaveo.com/pdf/SMS_11.0.pdf. [2] Arcement, G.J., Schneider, V.R. “Guide for Selecting Manning’s Roughness Coefficients for Natural Channels and Flood Plains” U.S. Geological Survey Watersupply Paper 2339. http://www.fhwa.dot.gov/bridge/wsp2339.pdf [3] Berger, R.C. and Lee, L.M. (2004) “Multidimensional Numerical Modeling of Surges Over Initially Dry Land.” Coastal and Hydraulics Engineering Technical report, ERDC/CHL TR-04-10. Vicksburg, MS: U.S. Army Engineering Research and Development Center. [4] Berger, R.C., Tate, J.N., Brown, G.L., and Savant, G. (2010). “Adaptive Hydraulics: User Manual.” Vicksburg, MS: U.S. Army Engineering Research and Development Center. https://adh.usace.army.mil/. [5] Engineer Research and Development Center (ERDC) DoD Supercomputing Resource Center (DSRC), http://www.erdc.hpc.mil/docs/diamondUserGuide.html, accessed December 5, 2011. [6] Martin, S.K., Savant, G., and McVan, D.C. (2010). “Lake Borgne Surge Barrier Study.” Coastal and Hydraulics Engineering Technical Report, ERDC/CHL TR-10-10. Vicksburg, MS: U.S. Army Engineering Research and Development Center. [7] Martin, S.K., Savant, G. and McVan, D. (2011) “Two Dimensional Numerical Model of the Gulf Intracoastal Waterway Near New Orleans: Case Study”, Journal of Waterway, Port, Coastal and Ocean Engineering, doi:10.1061/(ASCE)WW.1943-5460.0000119. [8] McAlpin, T.O., Berger, R.C., and Henville, A.M. (2009). “Bush Canal Floodgate Study.” Coastal and Hydraulics Engineering Technical Report, ERDC/CHL TR-09-09. Vicksburg, MS: U.S. Army Engineering Research and Development Center. [9] Oregon Department of Transportation (2005). “Hydraulics Manual (Part 1); Chapter 8, Appendix A”. Engineering and Asset Management Unit Geo-Environmental Section. ftp://ftp.odot.state.or.us/techserv/GeoEnvironmental/Hydraulics/Hydraulics%20Manual/Chapter_08/Chapter_08_Appendix_A/ Chapter_08_Appendix_A.pdf [10] Savant, G., Berger, R.C., McAlpin, T.O., and Tate, J.N. (2011). “An Efficient Implicit Finite Element Hydrodynamic Model for Dam and Levee Breach.” Journal of Hydraulic Engineering, ASCE. Vol. 137, No. 9, pp: 1005-1018. [11] Savant, G and Berger, R.C. (2011) “Wetting and Drying Characteristics of Adaptive Hydraulics.” USACE, ERDC Technical Note, USACE/ERDC-11-DRAFT, In Progress. [12] Stockstill, R.L. and Vaughan, J.M. (2009). “Numerical Model Study of the Tuscarawas River below Dover Dam, Ohio.” Coastal and Hydraulics Engineering Technical Report, ERDC/CHL TR-09-17. Vicksburg, MS: U.S. Army Engineering Research and Development Center. [13] Stockstill, R.L., Vaughan, J.M., and Martin, S.K. (2010). “Numerical Model of the Hoosic River Flood-Control Channel, Adams, MA.” Coastal and Hydraulics Engineering Technical Report, ERDC/CHL TR-10-01. Vicksburg, MS: U.S. Army Engineering Research and Development Center. [14] Tate, J.N., Lackey, T.C., and McAlpin, T.O. (2010). “Seabrook Fish Larval Transport Study.” Coastal and Hydraulics Engineering Technical Report, ERDC/CHL TR-10-12. Vicksburg, MS: U.S. Army Engineering Research and Development Center.
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Assessment of ICOLD Benchmark Case Study using Flood2D-GPU and HEC-FIA B. A. Thames 1 and A. J. Kalyanapu 2 1
2
US Army Corps of Engineers, 801 Broadway, Nashville, TN 37203, USA Department of Civil & Environmental Engineering, Tennessee Technological University, 1020 Stadium Drive, Prescott Hall 334 Box 5015, Cookeville, TN 38505, USA E-mail: [email protected]
Abstract This paper presents a benchmark case study of a dam failure using Flood2D-Graphics Processing Unit (GPU), a two-dimensional hydraulic modeling software, coupled with the Hydrologic Engineering Center’s Flood Impact Analysis (HEC-FIA) software, a flood consequence model, to evaluate flood risk in a hypothetical test bed environment. The defined objective of this study is to present the findings prescribed by the ICOLD Theme “C” Formulation Team, which include inundation mapping, population at risk, loss of life, and direct economic impacts. The increased computational capability afforded by Flood2D-GPU allows for a much deeper analysis of the hypothetical scenario. Many studies have been conducted to evaluate the uncertainty in the estimation of breach parameters (breach width and time of failure); however, how breach parameter estimation uncertainty affects the uncertainty in consequence estimation has been rarely evaluated. A secondary objective of this study is to determine the affects of breach parameter estimation uncertainty on consequence estimation uncertainty through a comparative analysis of four breach parameter estimation (BPE) methods and a sensitivity analysis of the main breach parameters such as breach depth, breach width, time of failure, and breach side slope. Through these analyses, breach depth, typically assumed using engineering judgment, is found to be the most sensitive breach parameter. Similar breach formation times are computed for three of the BPE methods, Froehlich (1995a), Froehlich (2008), and Von Thun and Gillette, evaluated while varying breach widths are computed; however, the consequence results are similar despite the varied breach width computations highlighting the importance of breach formation time.
Introduction With the increased awareness of the infrastructure deficiencies within the United States and world-wide, dam safety and understanding the impacts of dam failures has become much more emphasized due to the large number of dams world-wide with population centers within the destructive path of dam failure discharges. Studies such as these strive to better define the hydraulics of a dam failure and the resulting flood impacts. Many US federal agencies, such as the US Army Corps of Engineers (USACE), the Department of Homeland Security (DHS), and the Bureau of Reclamation (USBR), are working diligently through research and practical analysis to better understand the mechanics of a dam failure. This case study provides insight into hydraulic and consequence modeling tools available to simulate a dam failure and evaluate possible impacts from flooding.
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Methodology Various modeling software are used to obtain the risk assessment results prescribed by the ICOLD formulation team for Theme C, which consist of an array of hydraulic and consequence-based products. To evaluate the proposed problem, a dam breach discharge hydrograph is first developed using the unsteady flow component of the Hydrologic Engineering Center’s River Analysis System (HEC-RAS) version 4.1. The dam breach discharge hydrograph is a necessary input into Flood2D-GPU, which leverages the computer’s graphics card for much improved computational capabilities of the model input [8]. Output from Flood2D-GPU, which includes inundated areas, depth grids, and flood wave arrival time grids, serve as inputs into HEC-FIA, which provides the consequence assessment for the solution to the problem statement. HEC-FIA is a single event GIS-based software that determines impacts from flooding, such as population at risk (PAR), economic damages, and loss of life (LOL) on a structure-by-structure basis [9]. Ultimately, a dam failure within the proposed hypothetical test bed is evaluated using this combination of modeling software through a comparative analysis of the four suggested BPE techniques and a sensitivity analysis of four breach parameters to evaluate the range of possible PAR, LOL, and economic damage outcomes and the sensitivity of these outcomes to breach parameter estimations. Development of Dam Failure Hydrograph (HEC-RAS) HEC-RAS is utilized to generate dam failure discharge hydrographs for input into Flood2DGPU. HEC-RAS is one-dimensional hydraulic model capable of performing unsteady flow dam failure simulations utilizing user-provided hydraulic inputs (cross-section and inline structure geometry, boundary conditions, Manning’s n-values, etc.) and breach parameters (breach opening dimensions and location, breach weir coefficients, breach formation time, etc.) [13]. The HEC-RAS model developed for this evaluation uses the information provided by the formulation team including a digital elevation model (DEM), a land use layer, and information about the dam embankment and failure scenario. Because the only purpose of the HEC-RAS model is the generation of a dam failure discharge hydrograph, the extent of the model is limited to the reach just adjacent to the proposed dam. The HEC-RAS model geometry is developed by processing the DEM using HEC-GeoRAS, a data pre- and post-processing software developed for HEC-RAS. The HEC-RAS model consists of a storage area to represent the reservoir above the dam, an inline structure representing the dam and the necessary bounding cross sections upstream and downstream of the dam. The stage-volume curve provided by the formulation team is utilized in the HECRAS model to represent the storage characteristics of the reservoir. The cross section geometry is developed directly from the DEM and imported into HEC-RAS using HECGeoRAS. The inline structure geometry is developed using a combination of information provided by the formulation team in the problem statement and the DEM. The most critical input into the HEC-RAS model is the determination of the breach geometry and parameters that include the final breach bottom width (Wb), final breach invert elevation (Elb), breach side slopes (SS), and breach formation time (tf). [16] conducted an uncertainty analysis of embankment dam breach parameters for all of the proposed BPE approaches provided by the formulation team with the exception of Froehlich (2008), which did not exist at the time of publication. This study focuses on four BPE methods: Froehlich 1995a [3], 310
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Froehlich 2008 [5], the MacDonald and Langridge-Monopolis (MLM) [10], and the Von Thun and Gillette (VTG) [15] methods, and the calculated or assumed values are presented in Table 1. [16] shows through an uncertainty analysis, that in predicting breach width, breach failure time, and peak flow through the breach, the Froehlich (1995a) technique produced the lowest mean prediction error and the narrowest width of uncertainty band for all three estimations. The only exception to this is the width of the uncertainty band for the breach width estimation which is only slightly larger than the VTG technique. Based on these results, the Froehlich (1995a) BPE technique emerges as the best compromise; however, the current study compares the four methodologies across the range of sensitivity for all methods to highlight the range of possible outcomes in terms of hydraulic and consequence results. In addition, this evaluation provides a better understanding of how breach parameter sensitivity affects consequence estimation uncertainty through a sensitivity analysis of four breach parameters, SS, Wb, Elb, and tf, for each BPE method. The sensitivity analysis approach involves computing these four breach parameters using each BPE method (Table 1) and adjusting these values to create a sensitivity range. SS, W b, and tf are increased and decreased by 25 and 50%. Because the initial assumption is that the breach height formation will extend to the toe of the dam for all BPE methods, the sensitivity range for Elb is developed by increasing the base assumption of 211m by 7.625m or one eighth of the total breach depth (hb) until the value reaches half of the total breach depth. Using the computed values from Table 1, a base condition breach failure hydrograph is developed for each BPE method using HEC-RAS serving as the basis for the comparative analysis. In addition, breach hydrographs are developed for the range of sensitivity values. Sixty eight simulations are computed to produce breach hydrographs for input into Flood2DGPU. In interest of space, the resulting breach hydrographs for the four base condition simulations are presented in Figure 1. Table 1: Computed Breach Parameter Estimates using the Four BPE Methods Parameter SS Wb hb tf
Froehlich (1995a) 1.4 62 m 61 m 0.7 hrs
Froehlich (2008) 1.0 110 m 61 m 0.6 hrs
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MLM (1984) 0.5 12 m 61 m 2.0 hrs
VTG (1990) 1.0 207 m 61 m 0.9 hrs
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50000 45000 40000 Froehlich 1995a Froehlich 2008 MLM VTG
Discharge (CMS)
35000 30000 25000 20000 15000 10000 5000 0 1
1.5
2
2.5
3
3.5
Time (hrs)
Figure 1: Computed Discharge Breach Failure Hydrographs from unsteady flow HEC-RAS Model for Base Simulations Two-Dimensional Hydraulic Modeling (Flood2D-GPU) In this study, a GPU flood model, named Flood2D-GPU, developed in NVIDIA's CUDA programming environment is used [8]. The modelling framework uses the 2D unsteady numerical flood model that solves the non-linear hyperbolic shallow water equations using a first-order accurate upwind difference scheme. These equations are developed from the Navier-Stokes equations by integrating the horizontal momentum and continuity equations over depth often referred to as the depth averaged or depth integrated shallow water equations (i.e., Saint Venant equations). The non-conservative form of the partial differential equations is [7;8;12]: h uh vh 0 ContinuityEquation(1) h uh vh t x y 0 ContinuityEquation(1) t x y u uuu uuuu vHuu gHH gS 0 Momentum Equation inxx--direction direction (2) u v u g v gS Equation in x - direction (2) g fx 0 gS fxfx 0 Momentum Momentum Equation in (2) t xtt yxx xyy xx v vvv vvv H vv gHH gS 0 Momentum Equation inyy--direction direction (3) u vuu gvv gS Equation in y - direction (3) g fy 0 gS fyfy 0 Momentum Momentum Equation in (3) t x y y t xt yx yy y where, h is the water depth, H is the water surface elevation, u is the velocity in the xdirection, v is the velocity in the y-direction, t is the time, g is the acceleration due to gravity, Sfx is the friction slope in the x-direction, and Sfy is the friction slope in the y-direction. The upwind finite difference numerical scheme is used to discretize governing equations (1-3), as it yields non-oscillatory solutions, through numerical diffusion [2;11]. A staggered grid computational stencil is used to define the computational domain with the water depth (h) in the centre of the cell and u and v velocities on the cell edges. The model requires a digital elevation model (DEM) to represent topography, Manning’s n for surface roughness representation and a flow hydrograph. The CPU environment in this module consists of the 312
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random number generator, and storage of input data and parameters and source code to calculate future time step using Courant condition, and implement hydrograph update. The GPU environment contains the computational engine that solves continuity and momentum equations, and boundary condition implementation. A 2D model is chosen because of its better representation of flood flow (especially in floodplains) simultaneous flood extent delineation and instantaneous flood velocities at all nodes in the computational domain. Flood2D-GPU is validated for accuracy and found to provide significantly reduced computational time (up to two orders of magnitude) compared to the same flood model implemented serially in a CPU-based environment [8]. Application of Flood2D-GPU in this study uses HEC-RAS dam breach hydrographs, Manning’s n map generated from the land use/cover data provided along with the digital elevation model (with dam burnt into the topography). The methodology of dam breach hydrograph is explained in previous section. To develop spatially variable Manning’s n surface map, the National Land Cover Dataset (NLCD) Land cover (LC) codes were used to estimate approximate Manning’s n values based on recommendations from [6]. Table 2 provides the LC code and its assigned Manning’s n value and Figure 2 depicts the resultant spatially variable Manning’s n map. Table 2: Estimated Manning’s values for National Land Cover Dataset NLCD LULC Class Name Open Water Developed, Open Space Developed, Low Intensity Developed, Medium Intensity Developed, High Intensity Barren Land (Rock/Sand/Clay) Deciduous Forest Evergreen Forest Shrub/Scrub Grassland/Herbaceous Pasture/Hay Cultivated Crops Woody Wetlands Emergent Herbaceous Wetlands
LULC Code 11 21 22 23 24 31 41 42 52 71 81 82 90 95
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Manning's n 0.025 0.016 0.030 0.030 0.050 0.025 0.120 0.200 0.070 0.050 0.030 0.035 0.160 0.110
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Figure 2: Land Use Land Cover Derived Manning’s Map for the case study Consequence Assessment (HEC-FIA) HEC-FIA is used to determine the consequences, such as PAR, economic damages, and LOL, using the results of the Flood2D-GPU simulations. [9] presents that “HEC-FIA is a standalone, GIS enabled model for estimating flood impacts due to flooding…that generates required economic and population data…to compute urban and agricultural economic flood damage, area inundated, number of structures inundated, population at risk, and loss of life.” Input requirements for HEC-FIA include a stream alignment, terrain grid, depth grid, arrival time grid, a structure inventory, and timing information for the purpose of evaluating a single event [9]. Based on the information provided by the formulation team, several assumptions and derivations from the provided data are made to develop the required inputs into HECFIA. ArcHydro Tools, an extension of ArcGIS Desktop, is used to create a stream centerline by processing the DEM. The depth and arrival time grids for all simulations are derived from the results of the Flood2D-GPU simulation. The depth grid required for HEC-FIA is the max depth grid; however, additional depth grids for the specified time intervals were derived for the required PAR, LOL, and economic damage results. The arrival time grid is assumed to be the time after failure at which the flood depth will reach two feet of depth. For this study, the structure inventory must include a structure name, occupancy type, damage category, structure value, and population information in order to derive the prescribed results. HEC-FIA utilizes two separate databases, a structure occupancy and global inventory table, to define the structure inventory against which all consequence estimations are made. The structure occupancy table consists of general occupancy types, such as residential, commercial, and industrial, and, for this study, is populated using occupancy types using the database that comes with Federal Emergency Management Agency’s (FEMA) HAZards United States – Multi-Hazard (HAZUS-MH) Tool [1]. The structure occupancy type table includes the content-to-structure value ratio for defining content values for each structure, links to depth-damage curves for structure and content values, and risk elevations for defining at what elevation an individual is at risk of fatality based on age. HEC-FIA only considers 314
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two age groups, those under and over the age of 65; however, to ascertain consequence results for populations under the age of 14, a separate HEC-FIA model is created for each simulation. The assumption is that anyone under the age of 14 would be accompanied by an adult and either carried vertically to safety or is mobile enough to move vertically to safety. The global inventory table consists of each structure, or parcel for this study, and specific parameters related to each structure such as depth-damage curves for the structure and contents, structure and content value, first floor elevations, and number of stories. This information is either user-defined or derived from other sources such as the DEM or through a relationship with the structure occupancy type table. For instance, the depth-damage curves for the structure and contents are adopted from the structure occupancy type table by way of the general occupancy type assigned to each structure by the user. The structure appraised values, population values for the various age groups, structure name, and number of stories is defined by the user in ArcGIS Desktop and imported into HEC-FIA. First floor elevations are derived from the DEM and increased by an assumed three feet to account for foundation height. The most time intensive process is related to the development of the global inventory. Spatially joining the parcel dataset layer with the census block layer in ArcGIS Desktop to compute the number of parcels per census block provides the ability to uniformly distribute the necessary population values for the age groups of interest, under 14, between 14 and 65, and over 65 years of age, into each parcel. The uniformly distributed population values are spatially joined back to the parcel dataset to attain an age group specific population value for each parcel. With no structure values and limited information regarding structure condition and square footage provided by the formulation team, the structure values are assumed for each occupancy type provided in the parcel dataset through collaboration with a US Army Corps of Engineers (USACE) economist to achieve reasonable values for each structure occupancy type (Table 3). Each structure is assigned a generic name based on the occupancy type. The structure name, population, structure values, and HAZUS occupancy type are compiled within ArcGIS Desktop and imported into HEC-FIA’s global inventory. Table 3: Estimated Structure Value Assumptions Zone A-1 C-1 C-2 O-1 P R-1 R-2 R-3 RA-1 R-D R-LT R-T SU-1 SU-2
Description Rural Agriculture - 1-ac Minimum Zone Neighborhood Commercial Zone Community Commercial Zone Office and Institutional Zone Parking Zone Single Family Residential Zone Residential Zone: Houses, Townhomes and Medium Density Apartments Residential Zone: Houses, Townhomes and High Density Apartments Residential and Agricultural Zone, Semi-Urban Area Residential and Related Uses Zone, Developing Area Residential Zone: Houses and Limited Townhomes Residential Zone: Houses and Townhomes Special Use Zone Special Neighborhood Zone, Redeveloping Area
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Estimated Value ($1000) 50 650 650 1000 750 125 1000 2500 75 100 750 500 1500 125
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A time window is needed for each HEC-FIA simulation, which defines a start, end, and warning issuance time for each simulation. The start time is based on the initiation of the breach. The assumption of no warning is made based on the scenario described of no cell phone service and no emergency action plan. [9] assert that warning time plays a critical role in preventing loss of life and property, and as such, is a very sensitive parameter in the HECFIA software. The sensitivity of warning time also emphasizes the importance of a robust warning system especially in situations of high risk dams just upstream of major population centers. PAR, LOL, and economic damages at specific time intervals are required results for this study. Because of the manner in which HEC-FIA treats incremental changes in PAR, LOL, and economic damages, depth grids, derived from Flood2D-GPU results, for each of these time intervals are simulated individually through HEC-FIA to develop these consequence results and post-processed to determine the PAR, LOL, and economic damages incrementally for each time interval. Because this is a labor intensive task, this time interval analysis is only conducted for the four base BPE method simulations. With all of the necessary information pre-processed and imported into HEC-FIA, two simulations, one for over and under the age of 65 and one for under the age of 14, are set up and run for each BPE method simulation, each sensitivity range, and for each required time interval totaling 184 HEC-FIA simulations. The results from these simulations are presented in the Results and Discussion section, and electronic copies of the required models, modeling outputs, GIS results, and spreadsheet results are provided supplemental to this document.
Results and Discussion Two-Dimensional Hydraulic Modelling Results In this report, the two-dimensional hydraulic modeling results using the base condition Froehlich 1995a BPE method are presented results and used in comparisons. Also, a sensitivity analysis of four breach parameters is evaluated and results are compiled and discussed. Additional results from these analyses and from other BPE methods are provided supplemental to this report. The Froehlich 1995a base simulation resulted in flood depths ranging from 0.5 m to nearly 8 m in the downstream areas of the dam. Figure 3 presents the map of the standard deviation in the peak flood depths from the multiple breach hydrographs which were derived from the sensitivity analysis of the four breach parameters. A high variation in the peak flood depths is noticed along the canyon immediately downstream of the dam. In the downstream region away from the mountainous terrain, smaller standard deviation in simulated peak flood depths is noticed that indicates less sensitivity due to the breach parameters. Thus it is likely that variability in consequence estimates including direct flood damages, LOL, PAR in the downstream region may be smaller compared to the upstream region of the floodplain. A similar trend is also observed in the maximum flood velocities for these breach hydrographs.
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Figure 3: Standard deviation map of the peak flow depths from the dam breach hydrographs developed for the four breach parameters using Froehlich 1995a BPE method Figure 4 presents the simulated flood discharge hydrographs at the five required cross-section locations. For the Froehlich 1995a BPE method, the variability of flood discharge from all the sensitivity runs are represented by the error bars. It is observed that a large variability in the magnitude of flood discharge is observed at the beginning of the dam breach. This is attributed to the influence of breach parameter sets used for generating the hydrographs. This is a clear indication that the consequence analysis, especially the LOL, PAR, direct economic damages during the initial 60 minutes of the breach, will be significantly affected by the choice of breach parameters used. With progression of time, the variability decreases at all cross-section locations.
Figure 4: Simulated flood discharge plots at the five cross-section locations for the Froehlich 1995a BPE method. Error bars indicate the variability of discharges due to various breach parameter sets. 317
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Consequence Assessment Results For the purposes of presenting results and making comparisons, the consequence results using the base condition Froehlich 1995a BPE method is considered the basis of the comparative analysis. In addition to this comparative analysis, a sensitivity analysis of four breach parameters are evaluated for each BPE method and results are compiled and discussed in this report. Additional results from these analyses are provided supplemental to this report. The Froehlich 1995a base simulation results in a PAR of 13976 persons with 2223 persons (about 16% of total PAR) under the age of 14 and 2695 persons (about 19% of total PAR) over the age of 65. HEC-FIA estimates a fatality total of 1914 persons under these simulation conditions, which is almost 14% of the total PAR. In reality, the total LOL seems to be inflated and is probably overestimated when considering the resilience and resourcefulness of a typical population. However, the assumption of no warning time appears to be the largest source of the elevated LOL totals, and a sensitivity of this assumption could provide a large range of possible outcomes for LOL. For instance, a warning just 15 minutes in advance of the breach reduces the total estimated LOL by 55% to 862 persons. A warning 1 hour in advance of the breach results in an estimated LOL of 511 persons or a 73% reduction. Direct economic damages are estimated using HEC-FIA using a depth of flooding derived from the depth grid and a first floor elevation for each structure in combination with a depthdamage curve. The total economic damages estimated by HEC-FIA are approximately $361 million dollars; however, this value is based strongly on the structure value assumptions made based on occupancy type and the content-to-structure value ratio derived from the FEMA HAZUS-MH Tool [1]. For the Froehlich 1995a base simulation, Table 4 shows incremental PAR, LOL, and direct economic impacts for specified time intervals. The PAR does not become affected by flooding until 30 minutes after the initiation of the dam breach based on flood wave propagation estimated by Flood2D-GPU and the population provided by the census block dataset. However, during the 30-60 minute time interval following the breach a large percentage of the PAR becomes affected and the greatest LOL occurs during this period. The greatest percentage of the PAR actually becomes affected in the 120-180 minute time interval due to an area of low population in the center of the damage reach. Interestingly, the LOL during the 120-180 minute time interval is not substantial even with the larger PAR due to lower flood depths allowing individuals to retreat vertically to safety within each structure. The decreased flood depths also explain why the direct economic damages are also greatest in the 30-60 minute time interval as opposed to the 120-180 minute time interval. Table 4: Incremental Time Interval Results for the Froehlich 1995a Base Simulation Time Interval (min)
Total PAR
0-15 15-30 30-60 60-90 90-120 120-180 > 180 Total
0 0 2915 512 1410 7638 1501 13976
14-yr and Under PAR 0 0 755 122 168 956 222 2223
65-yr and Over PAR 0 0 210 31 347 1835 273 2695
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LOL
Direct Economic Impact ($US)
0 0 1549 329 5 7 24 1914
$0.00 $0.00 $128,971,979.93 $77,632,731.34 $42,877,015.97 $85,161,297.09 $26,714,802.91 $361,357,827.24
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The majority of the flooding by area occurs at the lower depths of flooding (0 to 1.5m) with a majority of the PAR contained within these same lower depths of flooding. Table 5 illustrates the distribution of PAR and flooded area across the various depth ranges. Table 5: Incremental Depth Range Results of PAR and Flooded Area Peak Flood Depth Range (m) 0 - 0.5 0.5 - 1 1 - 1.5 1.5 - 2 2 - 2.5 2.5 - 3 3 - 3.5 3.5 - 4 4 - 4.5 4.5 - 5 5 - 5.5 5.5 - 6 6 - 6.5 6.5 - 7 7 - 7.5 7.5 - 8 >8 Total
Flooded Area (m²)
Total PAR
14-yr and Under PAR
65-yr and Over PAR
3498463 6433454 5310192 3706161 2874378 2379332 1600888 756803 427789 337724 372655 378761 252956 247030 200874 198091 2305519 31281068
2786 4993 2056 887 424 262 107 68 69 115 217 249 155 282 228 245 827 13976
447 664 267 89 49 35 29 20 19 32 59 71 43 78 61 61 199 2223
464 1079 488 286 126 71 8 4 5 8 15 17 11 22 16 12 62 2695
Supplemental results for the various simulations are provided electronically. When initially choosing a BPE method, the question of which technique provides the most accurate results is raised driving the need for a comparative and sensitivity analysis. The goal of this evaluation is to understand how the uncertainty of these methods and the sensitivity of the critical breach parameters affect the results of a consequence analysis. Table 6 summarizes the consequence results of both the comparative and sensitivity analysis. For the comparative analysis, the results from the Froehlich 1995a, Froehlich 2008, and VTG base simulations are extremely similar while the results of the MLM base simulation are much lower than the other three simulations, which is expected based on the dam failure hydrographs computed using HEC-RAS. The mean results for all parameters is typically only slightly less than the base simulation results. In addition, the base simulation results are typically much closer to the higher end of the range of values from the sensitivity analysis. A closer look at the results of the breach 319
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invert sensitivity analysis shows that consequence results are very sensitive to the breach invert assumption. These breach invert simulations are skewing the range of consequence results on the lower end of the range while also lowering the mean of the sensitivity simulations. The large values for standard deviation for all consequence results can also be attributed to the breach invert simulation runs. For the Froehlich 1995a sensitivity analysis, removing the breach invert sensitivity simulations from the statistical calculations produces a much tighter range for all consequence results with a 66% reduction in total PAR standard deviation, a 68% reduction in LOL standard deviation and an 80% reduction in direct economic impacts standard deviation, which indicates that these reductions in standard deviation are also due to the sensitivity of the breach invert parameter. In terms of sensitivity, breach invert is, above all, the most sensitive breach parameter; however, when comparing the breach hydrograph and consequence results of the various BPE methods, time of formation also drives the shape and peak of the breach hydrograph, which is directly related to flood consequence results. In fact, the difference in breach hydrograph peak and shape between the MLM and other three BPE methods, which is quite significant, is primarily due to the longer time of formation computed for MLM. In interest of space, many of the results discussed are not presented in this document; however, the results are provided supplemental to this report electronically.
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Table 6: HEC-FIA Base Simulation Results for Comparison and Sensitivity Simulation Statistics
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Conclusions This paper presents a solution to the problem statement using Flood2D-GPU coupled with HEC-FIA offering possible hydraulic and consequence outcomes under the hypothetical scenario provided. The comparative analysis of various BPE methods provides recognition of the similarities and differences between the methods. The sensitivity analysis of the breach parameters demonstrates the importance of specific parameters by illustrating the effects of these parameters on not only the hydraulic outputs but also the consequence results. These analyses combined begin the conversation into utilizing the most appropriate BPE method and focusing in on which breach parameters are the most critical to approximate the most accurate hydraulic and consequence results in a study of this nature. The results of the comparative analysis suggests that similar consequence results can be expected when using the Froehlich 1995a, Froehlich 2008, and VTG breach parameter estimation methods but, without additional case studies using this same approach, relative replication of these results is not substantiated. However, all of these approaches are based on straightforward regression relationships and should produce similar results. The MLM BPE method is similar in many aspects to the other three BPE methods with the exception of the inclusion of a volume of material eroded term. Additional case studies could provide insight into the differences of results experienced during this study. In general, the results of the sensitivity analysis follow the expected trends. As breach depth, width, and side slopes increase and the time of breach formation decreases, the breach discharge hydrograph shape becomes more compressed resulting in higher peak discharges, which ultimately results in more significant flood impacts and consequences. Several conclusions can be drawn from these analyses. Final breach invert is very difficult to estimate and is an assumed value not calculated by any of the BPE methods. Unfortunately, the results of this analysis show that the breach hydrograph and consequence results are extremely sensitive to the breach invert assumption. When conducting a risk-based assessment of a dam failure, special care should be taken to ensure that the best estimation of breach invert is assumed based on the sensitivity of breach hydrograph and consequence outcomes to the estimation of this parameter. Ultimately, additional research could provide a better approach to estimating this parameter drastically reducing the uncertainty in these types of analyses. When comparing the breach parameter estimations of SS, ElB, Wb, and tf, using the three BPE methods that produced similar consequence results, with the results of the sensitivity simulations, the breach formation time proves to be a critical parameter in the development of breach discharge hydrographs. Even with a Wb varying between 49 and 146 m, the resulting breach hydrograph shape and peaks are similar due to the similarity in formation time, which ranges from 0.6 to 0.9 hours.
References (arranged in alphabetical order) [1] Department of Homeland Security, Federal Emergency Management Agency (2007), Multi-Hazard Loss Estimation Methodology Flood Model HAZUS-MH MR3 Technical Manual, Washington, D. C. [2] Ferziger, J.H., and Peric, M. (2002). “Computational methods for fluid dynamics, 3rd edition” Springer, New York. 322
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[3] Froehlich, D. C. (1995a). “Embankment dam breach parameters revisited.” Water Resources Engineering, Proceedings of the 1995 American Society of Civil Engineers Conference on Water Resources Engineering, New York, 887–891. [4] Froehlich, D. C. (1995b). “Peak outflow from breached embankment dam.” Journal of Water Resources Planning and Management, American Society of Civil Engineers, 121(1), 90–97. [5] Froehlich, D. (2008). “Embankment Dam Breach Parameters and Their Uncertainties.” Journal of Hydraulic Engineering, 134(12), 1708–1721. [6] Hossain, AKM Azad, Yafei Jia, and Xiabo Chao. (2009). "Estimation of Manning's roughness coefficient distribution for hydrodynamic model using remotely sensed land cover features." Geoinformatics, 2009 17th International Conference. [7] Judi, D. R., Burian, S. J., and McPherson, T. N. (2010). “Two-dimensional fast-response flood modeling: Desktop parallel computing and domain tracking.” Journal of Computing in Civil Engineering, in press. [8] Kalyanapu A. J., Shankar S., Stephens A., Judi D. R., Burian S. (2011). “Assessment of GPU computational enhancement to a 2D flood model” Journal of Environmental Modeling and Software, 26, 1009-1016. [9] Lehman, W. and, Needham, J. (2009). “Consequence estimation for dam failures.” Hydrologic Engineering Center, US Army Corps of Engineers. [10] MacDonald, T. C., and Langridge-Monopolis, J. (1984). “Breaching characteristics of dam failures.” Journal of Hydraulic Engineering, 110(5), 567–586. [11] Patankar, S.V. (1980). Numerical heat transfer and fluid flow, Taylor and Francis. [12] Tingsanchali, T. and Rattanapitikon, W. (1999). “2-D modeling of dambreak wave propagation on initially dry bed.” International Journal of Science and Technology, 4(3), 28-37. [13] United States Army Corps of Engineers. (2010). HEC-RAS River Analysis System Users Manual, Hydrologic Engineering Center, US Army Corps of Engineers. [14] United States Army Corps of Engineers. (2012). HEC-FIA Flood Impact Analysis - Users Manual, Hydrologic Engineering Center, US Army Corps of Engineers. [15] Von Thun, J. L., and Gillette, D. R. (1990). “Guidance on breach parameters.” Internal Memorandum, U.S. Dept. of the Interior, Bureau of Reclamation, Denver, 17. [16] Wahl, T. L. (2004). “Uncertainty of Predictions of Embankment Dam Breach Parameters.” Journal of Hydraulic Engineering, 130(5), 389-397.
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Dam Failure and Consequence Assessment with Standardized USACE MMC Procedures International Commission on Large Dams 12th International Benchmark Workshop on Numerical Analysis of Dams (Theme C: Computational Challenges in Consequence Estimation for Risk Assessment) D. Williams1 and K. Buchanan2 U.S. Army Corps of Engineers, Tulsa, Oklahoma, USA U.S. Army Corps of Engineers, Huntington, West Virginia, USA 1
2
Email: [email protected]
Abstract The U.S. Army Corps of Engineers’ (USACE) Modeling, Mapping and Consequences (MMC) Production Center has submitted an entry for the International Commission on Large Dams (ICOLD) 12th International Benchmark Workshop on Numerical Analysis of Dams (Theme C: Computational Challenges in Consequence Estimation for Risk Assessment). As part of this exercise, standard MMC processes were used to develop hydraulic modeling of a dam break inundation as well as the associated consequences. Modeling techniques included one- and two- dimensional numerical models, use of geographic information systems, and economic and life loss estimation. Breach parameters were developed from standard regression equations. Based on the rules and regulations that USACE has established for the estimation of risk, economic consequences and life loss modeling results from this breach analysis corresponded to a high hazard dam. This analysis demonstrates the standardized MMC process for consistently evaluating all USACE dams while providing the flexibility to analyze projects that range from small and simple to large and extremely complex.
Introduction The Modeling, Mapping and Consequences (MMC) Production Center supports the U.S. Army Corps of Engineers (USACE) Institute for Water Resources (IWR) Risk Management Center (RMC) and the HQ USACE Office of Homeland Security, Critical Infrastructure Protection and Resilience (CIPR) program by analyzing the potential consequences of dam and levee infrastructure failures. The mission of the MMC includes production of hydraulic models, consequences estimates, and inundation maps for USACE dams and levees in support of these programs. All products are developed in accordance with MMC standard operating procedures. The intent of MMC products is to support a risk-based assessment, prioritization, and management framework for USACE CIPR, Dam Safety, and Levee Safety programs. Analysis of the risk associated with dam and levee failure events allows decision makers to prioritize investments to reduce the risk to life and property. MMC dam safety analysis is typically based on five failure and five non-failure dam breach scenarios needed for CIPR and Dam Safety program evaluations. Breach parameters for earthen dams (e.g. average breach width, breach formation time, and side slopes) are calculated using the following empirical equations: MacDonald and Langridge – Monopolis (1984) [1], Von Thun and Gillette (1990) [2], Froehlich (1995) [3], and Froehlich (2008) [4]. Results from all of these methods are compared, and results from a single empirical equation 325
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is selected for the analysis. Parameters from multiple equations are not considered to be interchangeable. Hydraulic modeling is performed using either HEC-RAS [5] (one-dimensional) or FLO-2D [6] (two-dimensional) unsteady flow analysis. The use of FLO-2D in the MMC process is generally restricted to areas where the one dimensional flow assumption utilized in HEC-RAS is not appropriate; such as coastal plains, alluvial fans, flat dense urban areas, and other large flat unconfined floodplains. Both models provide unsteady flow simulation. Steady flow models are not permitted within the MMC process. Following the development of the hydraulic model, the output from the dam failure analysis is processed using ArcGIS [7]. Inundation depth grids and inundation boundary shapefiles are generated, and these files are used in conjunction with structure and population data from tax parcels, point shapefiles, or the Federal Emergency Management Agency (FEMA) HAZUS [8] dataset to develop a HEC-FIA (Flood Impact Analysis) [9] model. The structure information is compared with the extent and timing of the flooding to estimate damages and life loss. This information (in the form of a consequence assessment report), along with an atlas of the downstream inundation, is provided by the MMC to USACE district offices for emergency action planning.
Project Description The MMC is participating in the International Commission on Large Dams (ICOLD) 12th International Benchmark Workshop on Numerical Analysis of Dams (Theme C: Computational Challenges in Consequence Estimation for Risk Assessment). Participants in Theme C are free to select the type and sophistication of modeling used to develop flood inundation areas and consequences estimates. When possible, the standard MMC process [10] was followed in the development of the dam break modeling and consequence estimation for this project. The numerical problem proposed for the workshop consists of estimating the consequences of failure of a dam in a mountain valley above the fictitious city of Hydropolis. The hypothetical dam, which was constructed for flood control, is 3.5 kilometers upstream from the city. Sandy clays and clayey sands are the predominant material within the rolled earth fill dam. With a 61-meter vertical distance from toe to crest, the structure has a high head differential when the reservoir is completely filled. The corresponding volume of the reservoir at the dam crest elevation is 38,276,000 cubic meters [11] (Figure 1).
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Figure 1: Storage-elevation curve for the reservoir. This exercise examines the downstream consequences of a catastrophic failure of the dam resulting from overtopping. Expected outcomes from this study include gridded data (peak flood depth, flood wave arrival time, peak unit flow rate, population at risk, loss of life, flood severity, direct economic impact), tabular data (flooded area, population at risk, loss of life, direct economic impact), and hydrographs (dam failure discharge, cross-sectional hydrographs). All of these products except for the peak unit flow rate are standard deliverables in a MMC dam breach analysis.
Model Development Modeling for this exercise was developed using both one- and two-dimensional techniques that are consistent with MMC standard operating procedures. Initially, only a HEC-RAS hydraulic model was planned for this study. This model computes flood wave parameters at cross-sectional nodes along a one-dimensional channel. Inspection of the terrain grid provided for this study revealed that the use of HEC-RAS for the entire downstream reach would be inappropriate since Hydropolis is located within a relatively flat plain below the base of the mountain range. In this reach, it was determined that a two-dimensional model would be more appropriate; therefore, FLO-2D was also used. Existing terrain downstream from the dam was provided by the Theme C formulation team. This dataset was provided in a gridded format with a cell width of 9.476 meters. No projection was assigned to the terrain dataset, but all geospatial data provided by the formulation team was in the same unknown coordinate system. 327
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The terrain dataset was used to extract cross sections in ArcGIS 10 (HEC-GeoRAS [12]) for use in a HEC-RAS model. The spatial extent of this model was restricted to 200 meters below the dam (modeled as an inline structure) as it was used to generate a dam breach hydrograph for a FLO-2D inflow boundary condition (Figure 2). The HEC-RAS model was run as an unsteady flow simulation with a computation interval of 5 seconds. A breach hydrograph was written to file with a 1-minute output interval, which was sufficient for defining the dam breach hydrograph.
Figure 2: HEC-RAS model geometry for breach hydrograph determination. Parameters from the MacDonald and Langridge – Monopolis empirical breach equation were not specifically considered in this analysis since it is considered to be an envelope equation, although parameters from the other equations were checked to ensure that they fell within the envelope range. The Froehlich (1995), Froehlich (2008), and Von Thun and Gillette equations were carried forward as unique scenarios in the HEC-RAS model. Since each of the three methods estimated a short breach formation time, a computational time step of 5 seconds was used in the HEC-RAS model in order to capture the peak discharge. Following the initial model development with the Froehlich (1995), Froehlich (2008), and Von Thun and Gillette breach parameters, the Froehlich (1995) equation was carried forward as the preferred method. Froehlich (1995) analyzed 63 earthen, zoned earthen, earthen with a core wall, and rockfill dam failures [3]. Data from these events was used to develop regression equations for average breach width ̅ (meters) and breach formation time tf (hours): ̅
(1) (2)
For this exercise, K0 = 1.4 (overtopping failure). The reservoir volume Vw (38,276,344 m3) and breach height hb (61 meters) were provided by the organizers of the benchmark workshop. This method provided a short breach development time coupled with a relatively wide breach width (Table 1): Table 1: Breach parameters for hypothetical dam Formation Time (hours)
Breach Equation 328
Bottom Width (m)
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MacDonald and Langridge-Monopolis Froehlich (1995) Froehlich (2008) Von Thun and Gillette
1.8 0.7 0.6 1.5
5 62 39 177
A breach hydrograph generated in HEC-RAS was then used as the inflow boundary condition in a FLO-2D model (Figure 3). The model assumed a static pool elevation of 272 meters with no inflow. Overtopping of the dam crest was the prescribed mode of failure. Unlike the onedimensional flow model (HEC-RAS), FLO-2D computes flow across a network of grid cells with eight possible directions of flow into or out of each cell. This type of flow distribution makes FLO-2D well-suited for analysis of the flood wave downstream from the dam.
Figure 3: Breach hydrograph resulting from Froehlich (1995) parameters Faster computation time of the FLO-2D model required a grid-cell width larger than 9.476 meters. By choosing an integer multiple of the terrain grid-cell size, conversion between the two could be simplified when resampling. Therefore, a computational grid-cell dimension of 94.76 meters (square) was selected. Elevation values were interpolated for the larger computational grid dimensions when imported into FLO-2D. Surface roughness values were developed by converting a National Land Cover Database (NLCD) shapefile [13] to corresponding Manning’s ‘n’ values [14]. The NLCD shapefile was provided by the Theme C formulation team. Once a new surface roughness shapefile was created, the values were interpolated for the computational grid dimensions when imported into FLO-2D. The FLO-2D model was run with a simulation time of 12 hours and an output interval of 5 minutes. Since the downstream reach is both short and steep, the simulation time was more than sufficient for routing of the breach hydrograph through the entire study area. An output 329
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interval of 5 minutes was necessary to capture the peak of the flood wave at downstream locations. A depth grid, flood wave arrival time grid, and flood severity grid were generated by using MAPPER [6], a post-processing tool for FLO-2D models. Peak unit flow rate is not a FLO2D output. This variable was therefore determined by taking the maximum flow rate for each grid cell and dividing by the computational grid-cell width. Cross-sectional hydrographs, which were required at five locations downstream from the dam, were developed by setting those locations as outflow nodes in sequential FLO-2D model runs. The time-series discharge for each cross-sectional outflow grid cell was then summed to construct a representative hydrograph for each key location. This was necessary because FLO-2D does not generate time-series discharge for grid cells within the computational domain if no channel element is used; it does write this output to file for outflow nodes, however. Hydraulic modeling results were then used for the determination of consequences based on a structure inventory. A structure inventory was developed using three datasets provided for the workshop [15]. These included a parcel shapefile with zoning information, a census block shapefile with population and employment information, and a land use grid. Typically, the economic and life loss analysis in a MMC study includes the utilization of a FEMA HAZUS dataset supplemented by imagery, land use, or parcels. Since this was not provided for the benchmark exercise, non-standard methods were used to estimate consequences. The following assumptions were required for the creation of a structure inventory with population values in HEC-FIA:
Since the dam fails at 11:00pm on a Saturday, all residential populations are in their homes. Due to the day and time, only 10% of the workers in certain job types are working. These included the following jobs: Mining, Quarrying, and Oil and Gas Extraction; Utilities; Manufacturing; Transportation and Warehousing; Administrative and Support and Waste Management; Health Care and Social Assistance; Arts, Entertainment, and Recreation; and Accommodation and Food Services. All workers come from census blocks outside the inundation area (not enough data to model movements from one census block to another). All workers are under the age of 65. All data was up to date and accurate (parcels, blocks, and land use).
Once the structure inventory and hydraulic data for the dam breach had been developed, they were used to develop the HEC-FIA model. Depth grids and arrival time grids were developed from FLO-2D output and imported into the model. The arrival time grids represented the time at which the water depth at each cell reached 0.6 meters (approx. 2 feet). Impact areas were created by reclassifying the arrival time grid into arrival time zones required for reporting results (0-15min, 15-30min, etc.). A polygon was created from that file. Some features had to be manually merged to create a unified impact area polygon for the HEC-FIA model. The structure inventory was created from the parcel file with populations and values developed in the pre-modeling phase. Structure placement was at the centroid of each parcel and was done within HEC-FIA. For estimates of necessary evacuation time for each structure, a hazard boundary shapefile was created representing the boundary where the inundation becomes 0.6 meters (2 feet) or less, which is considered a safe zone. Evacuation was modeled as a path from the structure to the nearest point on this boundary at an average speed of 16 kilometers per hour. 330
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The warning time used for the loss of life calculation was set at 30 minutes after the breach initiation. The problem statement specified that the failure was sudden and unexpected, and that witnesses within the canyon had no mobile phone service. A warning would not be issued until after the water reached the urban area at the end of the canyon, approximately 20 minutes after the failure. The warning curve used was the HEC-FIA default curve for the United States Emergency Alert System, and the mobilization curve was the default HEC-FIA curve included in the program. The HEC-FIA program evaluates damages with depth-damage curves based on occupancy type. Population distribution is also performed by occupancy type. To determine occupancy types, a cross walk table was created that assigned each parcel zone description to a specific occupancy type typically used in a HEC-FIA structure inventory. Populations and jobs were provided for each census block and broken down by category. To facilitate the distribution to individual parcels, the populations were combined into the following categories by census blocks:
Resident Population under age 14 Resident Population under age 65 Resident Population over age 65 Working Population (10% of the workers from industries potentially in operation at 11pm on a Saturday)
In order to determine the amount of population for each structure, each occupancy type was assigned a number of residential households or working households. Then for each census block, the residential populations were divided by the total number of residential households in the block and the working populations were divided by the total number of working households in the block. The resulting residential and working household sizes per census block were then used to create the populations for each parcel based on the number and type of households assigned to the occupancy type of the parcel. For example, if the residential household size in a census block was 2.5, then a single family residential structure in that block would get 2.5 people while a five to nine unit multiple family dwelling in the same census block would get 7 households, or 17.5 people. Table 2 shows the final occupancy types used and the number of assigned households. In the beginning there were more occupancy types included, but the ones listed below represent all of the types included in the final inventory after being clipped to the inundation boundary. The methodology, including the number of households assigned to each occupancy type, is representative of a typical HAZUS based structure inventory used by the MMC. Table 2: Population distribution by occupancy type Occupancy Type COM1 COM4 GOV1 RES1-1SNB RES1-2SNB RES1-3SNB
Occupancy Type Description Average Retail Average Prof/Tech Services Average government services Single Family, 1 Story no Basement Single Family, 2 Story no Basement Single Family, 3 Story no Basement 331
Residential Households 1 1 1
Working Households 1 1 -
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RES3AI RES3CI RES3DI
Multi Family Dwelling, Duplex Multi Family Dwelling, 5-9 Units Multi Family Dwelling, 10-19 Units
2 7 14.5
-
No data was provided on structure values in the parcel dataset. A standard USACE dam failure consequence estimate will often use the National Structure Inventory (NSI) developed by the USACE Hydrologic Engineering Center and based on HAZUS and Census data. The team decided to use average values from the NSI for each occupancy type to determine structure values. The NSI data is broken down by county, so a national average would have required significant effort. Instead of using a national average, the team reviewed data on housing values from the 2011 American Community Survey and determined that the state of Maine had the closest housing value to the national average. Data from the NSI for each county in Maine was combined and average values for each occupancy type were calculated. Vehicle and content values were also assigned (Table 3). Table 3: Average values by occupancy type Occupancy Type COM1 COM4 GOV1 RES11SNB RES12SNB RES13SNB RES3AI RES3CI RES3DI
Occupancy Type Description Average Retail Average Prof/Tech Services Average government services Single Family, 1 Story no Basement Single Family, 2 Story no Basement Single Family, 3 Story no Basement Multi Family Dwelling, Duplex Multi Family Dwelling, 5-9 Units Multi Family Dwelling, 10-19 Units
Average Structure Value $545,888
Average Content Value
Average Vehicle Value
$545,888
$21,819
$550,027
$550,027
$22,707
$481,998
$481,998
$27,020
$122,061
$61,030
$14,825
$121,957
$60,528
$14,766
$127,135
$63,568
$15,175
$206,625
$103,312
$19,795
$753,610
$376,805
$26,195
$1,288,765
$644,383
$29,931
Since the parcel shapefile appeared only to have information about zoning and not specific information about structures, the team assumed that not every parcel had a structure on it. Also, some parcels had “unknown” zone information so those parcels were removed from the final inventory. Information in the census blocks and the land use file also needed to be evaluated to inform the structure inventory development. One particular issue involved a large number of parcels in the middle section of the study area. These parcels appeared to be subdivided into mostly single family home lots. However, the census block data showed no residential population in those areas. Figure 4 shows census blocks labeled with total residential population with the parcels overlain on top in blue. The red census blocks have zero population.
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Figure 4: Comparison of parcels and census block populations The land cover file showed no development in the area in question except for the roads themselves; no development is shown to indicate structures. Figure 5 shows development as red or pink. The roads in the center area are listed as Developed-Open Space and everything other than roads is listed mostly as grassland or shrub (brown and pale green colors).
Figure 5: Land use downstream of the dam This data can be interpreted in several ways. For example, it may be a situation where a subdivision has been planned for future development, and the land has been parceled off and a 333
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road system developed but no houses have been built and occupied. Alternatively, it could be a scenario where the parcel data was recently updated while the land cover and census data are several years old and do not reflect the recent development. With no way to ground truth the data through aerial imagery or site visits the team chose to adopt the first interpretation, assuming that all data was current and structures had not yet been built in that area. The large group of central parcels was removed from the final inventory by deleting all residential parcels in census blocks with no residential population. In addition, all parcels more than 100 meters outside the dam failure inundation area were deleted. This minimized the parcels to only those necessary for the HEC-FIA structure inventory.
Results and Discussion The hydrograph that was computed from Froehlich (1995) breach parameters had a peak discharge of 30,917 m3/sec at the dam site. The time to peak discharge occurred 31 minutes after breach initiation began. Nearly all of the reservoir volume was evacuated within 90 minutes of the breach initiation. Discharge immediately downstream from the dam remained confined to the mountain valley. This part of the reach, in which the flood wave was limited to a maximum width of 400 meters, continues downstream for approximately 3,000 meters. Minimal attenuation of the flood wave occurred through the valley, which was verified by developing cross sections downstream to the base of the higher terrain (Figure 6). At the lower end of the valley, the peak discharge of the cross-sectional hydrograph had only decreased to 26,474 m3/sec.
Figure 6: Breach hydrographs at selected locations downstream from the dam 334
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When the flood wave reached the more highly populated area below the base of the mountain, a change in the slope and layout of the surrounding terrain resulted in a more pronounced attenuation and lateral dispersion of the inundated area. This behavior of the flood wave is consistent with the transition from a steep mountain valley to a relatively flat, unconfined plain such as an alluvial fan or coastal area. This attenuation was highlighted by peak unit flow rates, which ranged from 300 m2/sec immediately downstream from the dam to less than 1 m2/sec for most of the area in the broader flood plain. Peak depths of the flood wave ranged from 24.5 meters immediately downstream from the dam to less than 1 meter at the downstream computational boundary (Figure 7). Much of the populated area fell within the 1- to 3- meter range, although some structures in the lower part of the mountain valley were inundated by depths as high as 12 meters. Arrival times for the flood wave were only 30 minutes for the lower part of the mountain valley. The population residing within the broader flood plain below the valley fell within an arrival time zone of 1 to 1.5 hours. The most distant reach of the modeled area coincided with an arrival time of 4 hours.
Figure 7: Peak inundation depths (meters) below the dam Initial results from the HEC-FIA model are displayed below in Tables 5 and 6 (by arrival time and by depth of flooding) as required by the benchmark reporting requirements. Table 4: Consequences by arrival time Time Interval (min) 0-15 15-30 30-60 60-90 90-120 120-180 > 180 Total
Total Population At Risk 3,951 7,612 12,672 24,235
14-yr and Under Population at Risk 1,002 933 1,952 3,887
65-yr and Over Population at Risk 282 1,855 2,166 4,303
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Loss of Life 1,844 34 11 1,889
Direct Economic Impact ($US) $142,504,368 $266,968,912 $248,115,360 $657,588,640
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Table 5: Consequences by depth Peak Flood Depth Range (m)
Flooded Area (m²)
Total Population At Risk
14-yr and Under Population at Risk
65-yr and Over Population at Risk
0 - 0.5 0.5 - 1 1 - 1.5 1.5 - 2 2 - 2.5 2.5 - 3 3 - 3.5 3.5 - 4 4 - 4.5 4.5 - 5 5 - 5.5 5.5 - 6 6 - 6.5 6.5 - 7 7 - 7.5 7.5 - 8 >8 Total
3,699,606 11,646,576 23,248,253 12,625,355 6,151,044 2,990,216 2,128,172 1,670,211 1,068,576 511,839 377,144 448,981 233,470 152,654 287,348 296,328 1,562,455 69,098,227
2,749 4,249 9,162 3,183 1,353 540 153 129 173 133 259 299 250 136 286 502 679 24,235
399 660 1,401 446 160 85 13 28 49 32 71 78 67 33 69 131 167 3,887
511 687 1,626 716 373 122 55 19 15 9 17 27 15 9 19 43 43 4,303
Conclusion This exercise demonstrates the practices and procedures that the MMC uses to analyze the large inventory of dams that are owned and operated by USACE. Most of these dams are larger and more complex than the hypothetical dam that was analyzed in this project. Some include complex reaches that extend for hundreds of miles downstream with travel times that last for days or weeks. Elements of each of these dams and their downstream reaches are unique, and it is important to emphasize that the sophistication of MMC models, while incorporating a standardized process, varies by project. Since the reach in this exercise was both steep and short, flood wave travel times to the downstream boundary of the model were very quick. As a result of these conditions, comparisons of the inundation areas resulting from the Froehlich (1995), Froehlich (2008), and Von Thun and Gillette equations were very similar. Modeling with dam breach parameters from all three equations resulted in a flood wave with a steep peak and fast travel time through the confined mountain valley below the dam followed by lateral attenuation of the flood wave in the broad plain below the base of the mountain range. Breach parameters from Froehlich (1995) were carried forward in the modeling used for consequences analysis. Estimation of economic damages and life loss was challenging because FEMA HAZUS data, a standard input for MMC HEC-FIA models, was not available for the benchmark exercise. Therefore, average values were used, which may or may not be representative of Hydropolis. Total damages of $657M and a life loss of nearly 2,000 persons were estimated for this dam breach.
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Based on the rules and regulations that USACE has established for the estimation of risk, these results correspond to a high hazard dam. The procedures that were used in developing the hydraulic modeling and consequences estimation for the benchmark exercise were consistent with USACE MMC processes. These guidelines, while providing consistency in the analysis of all USACE owned and operated dams, also provide enough flexibility to analyze projects that range from small and simple to large and extremely complex.
Acknowledgments Funding for this project was provided by the U.S. Army Corps of Engineers, Office of Homeland Security. Internal review of the numerical modeling and consequences estimation results was provided by the U.S. Army Corps of Engineers, Hydrologic Engineering Center.
References [1] MacDonald, T.C., and Langridge-Monopolis, J. (1984). Breaching characteristics of dam failures. Journal of Hydraulic Engineering, 110(5): 567-586. [2] Von Thun, J.L. and Gillette, D.R. (1990). Guidance on breach parameters. Unpublished internal memorandum, U.S. Bureau of Reclamation, Denver, CO, March 13, 1990. [3] Froehlich, D.C. (1995). Peak outflow from breached embankment dam. Journal of Water Resources Planning and Management, 121(1): 90-97. [4] Froehlich, D.C. (2008). Embankment dam breach parameters and their uncertainties. Journal of Hydraulic Engineering, 134(12): 1708-1721. [5] U.S. Army Corps of Engineers (2010). HEC-RAS River Analysis System (Version 4.1) [Computer software]. Davis, CA: Hydrologic Engineering Center. [6] O’Brien, J.D. (2009). FLO-2D (Version 2009) [Computer software]. Nutrioso, AZ: FLO Engineering. [7] ESRI (2011). ArcGIS Desktop (Release 10) [Computer software]. Redlands, CA: Environmental Systems Research Institute. [8] Federal Emergency Management Agency (2012). HAZUS-MH (Version 2.1) [DVD]. Washington, DC: FEMA. [9] U.S. Army Corps of Engineers (2012). HEC-FIA Flood Impact Analysis (Version 2.2) [Computer software]. Davis, CA: Hydrologic Engineering Center. [10] U.S. Army Corps of Engineers (2013). MMC standard operating procedures for dams. Vicksburg, MS: Modeling, Mapping and Consequences Production Center. [11] International Commission on Large Dams (2013). Part 1 – Hydraulic Modelling and Simulation. Graz, Austria: 12th International Benchmark Workshop on Numerical Analysis of Dams (Theme C). [12] U.S. Army Corps of Engineers (2011). HEC-GeoRAS Extension for ArcGIS 10 (Version 10.0) [Computer software]. Davis, CA: Hydrologic Engineering Center. [13] Homer, C.G., Huang, C.C., Yang, L., Wylie, B., and Coan, M (2004). Development of a 2001 national landcover database for the United States. Photogrammetric Engineering and Remote Sensing, 70(7): 829-840. [14] Mattocks, C., and Forbes, C. (2008). A real-time, event-triggered storm surge forecasting system for the state of North Carolina. Ocean Modelling, 25: 95-119. [15] International Commission on Large Dams (2013). Part 2 – Consequence Estimation. Graz, Austria: 12th International Benchmark Workshop on Numerical Analysis of Dams (Theme C).
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OPEN THEME
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Behavior of an arch dam under the influence of creep, AAR and opening of the dam/foundation contact E. Robbe1 1
Hydro Engineering Center of EDF,Savoie-technolac, 73373 Le Bourget du Lac, FRANCE E-mail: [email protected]
Abstract This paper summarizes the behavior analysis of a thin, 37m high, arch dam built in 1956. At first, the behavior of this dam was quite difficult to understand: irreversible displacements occur during the years following the first filling, and then, from 1960 until now, others displacements were observed and difficult to read considering usual behavior of arch dam. In order to understand it and to evaluate the stress level of the concrete, finite-element analyzes are leading. First linear, then non-linear analysis taking increasingly into account dam/foundation contact opening, swelling, creep and damage are used to answer the problem and understand the historic behavior of this dam
Introduction The subject of this paper is an arch dam with a particularly complex behavior observed. The purpose of the study is to understand this behavior and to evaluate the level of stress of the dam. Increasingly complex finite-element analyzes are used (from linear to non-linear) and calibrated on the observed displacements of the dam. Stress levels are also compared to crack patterns observed on the structure. In this presentation, geological conditions and safety evaluation of the gravity abutment will not be discussed.
Presentation of the dam
Figure 1: view of the dam
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The studied dam is a thin arch, with a gravity abutment on the right side. 37 m high above its foundation made of gneiss, the dam is divided into 6 cantilevers (about 17m wide) for the arch part and 2 cantilevers for the right abutment. Figure 1 shows the design of the dam: the thickness goes from 2 m on the crest to 5.4 m at the bottom, with a sloping upstream face in the lower part only. The valley is quite wide (L/H about 3) and clearly asymmetrical. The dam has been built from 1954 to 1956. In 1961, works has been done in order to grout 2 parallels faults discovered on the left abutment and to build an apron to protect the foundation in case of overflowing.
Observed behavior This part describes the behavior of the dam monitored with points on the downstream face of the dam from the first filling in 1956 until 2001. These points are represented on Figure 2. From 2001 until now, pendulums are used in order to follow the dam’s behavior. A statistic analysis of the monitoring data is used in order to evaluate the part of the hydrostatic and the seasonal effects on the dam’s displacements. Once theses reversible effects evaluate, the irreversible effects can be estimate. Regarding these data, the behavior can be separated in 2 phases: During the 5 years after the first filling of the dam (1956-1960) 40 years from 1961 until 2001 Behavior after first filling The analysis of the radial displacements of the dam shows the following behavior: At the crest, the central part moves downstream while the top of the sides cantilevers move upstream, All the points at the central cantilever records downstream displacements. Long-term behavior The radial displacements of the monitored points located on the downstream face of the dam are represented on the Figure 2 for approximately 40 years. Theses displacements can be described the following way: The crest of the dam moves upstream continuously (0.3 mm/year for the central cantilever) The lower part of the arch goes first downstream during 10 years then upstream after 10 years of stabilization. The velocity of the downstream displacement is close to the one observed on the crest. A first global interpretation of these observations is that the concrete is under the influence of swelling, which can explains the upstream displacements of the crest. For the lower part, creep is more important during the first 10 years, that explains the first downstream displacements, and decreases gradually and therefore swelling become leading.
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Figure 2: irreversible radial displacements (from 1956 to 2001)
Linear analysis A finite-element linear analysis of the dam is realized with Code_Aster [1]. The physical properties for concrete and rock are chosen by comparison between the monitored data and the results of the FE analysis. The calibration is done considering hydrostatic load and thermal load in order to represent correctly the behavior of the dam in winter and summer for example.
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Figure 3: Mesh of the dam and foundation The linear analysis shows that the particular design leads to an unconventional behavior of the dam: the lower part works as cantilever while arch-effects appear in the upper part. That leads to high vertical tensile stress at the heel of the dam (close to 8 MPa in winter) and in the middle of the downstream face where seepage are already observed on a lift joint. Because of theses high tensile stresses, the FE is upgraded in order to take into account the opening of the dam/foundation contact and the lift joint, and to evaluate their consequences.
Non-linear analysis including dam/foundation contact The Figure 4 presents the joint elements introduced in the mesh in order to simulate the nonlinear behavior: dam/foundation contact under the 6 cantilevers, dam/foundation contact under the right gravity abutment, horizontal lift joint at the change of inertia of the dam, contraction joints. Joint elements open under normal tensile stress but share behavior is still elastic-linear whatever the opening-state of the joint. Except for vertical contraction joints, uplift is taking into account. In particular, spread of the uplift under the dam with joint opening is used.
Figure 4: mesh of the joint elements 344
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Initial shape With joint element
Elastic behaviour
Figure 5: comparison of the deformed shapes at the crest (elastic, with joint elements) The introduction of the joint elements modifies the deformation at the crest of the dam (Figure 5) Theses irreversible displacements are equivalent to the ones observed during the first years after the first filling of the dam and presented earlier. The analysis shows that the dam/foundation contact is open at the heel of the dam, particularly in winter (5 mm). That leads to an increase of the uplift under the central cantilever. This is confirmed by a recently installed piezometer which records the almost full uplift during winter.
Non-linear analysis including AAR and creep If the previous section evaluates the behavior of the dam after few years, the recorded displacements during the following 40th years need to be taken in account in order to estimate correctly the stress state of the dam. Considering the displacements recorded, the behavior of the dam is under the influence of swelling and creep and therefore, the model used have to be able to represent correctly these phenomena. In this purpose, the model developed by Grimal [2] for AAR on concrete dam is particularly adequate. This visco-elasto-plastic orthotropic damage model includes chemical pressure induced by AAR and takes into account the influence of creep on the behavior. It has been used to evaluate the behavior of Chambon's dam [2] and showed the model’s capability to reproduce displacements with acceptable accuracy. The main developments brought by this model concern interactions between AAR pressure and long term strain (creep) on the one hand, and the swelling anisotropy induced by oriented cracking on the other hand. Here, this model is coupled to the joint elements presented earlier. The calibration of the model is realized the following way: first, the kinetic of the chemical pressure induced by AAR and the volume of AAR gel are chosen in order to fit the displacement monitored on the crest of the dam then, the parameters for creep are chosen in order to fit displacements of the lower part of the dam The result of this calibration is presented on Figure 6.
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Figure 6: irreversible radial displacement estimated and calculated Considering the complexity of the dam's behavior, the calibration is considered acceptable despite the fact that the central cantilever's behavior isn't perfectly represented. However, the model is able to represent the change of direction of the lower part of the dam (displacements downstream then upstream) while the crest is moving continuously upstream. The influence of the long term behavior of the dam is analyzed regarding the evolution of the dam/foundation contact opening (Table 1) and of the stress state of the concrete (Figure 7). For the first point, the model shows a closure of the dam/foundation contact, particularly in summer while an opening is still likely possible (3.3 mm). That can explain why the full uplift is recorded only in winter under the dam. The analysis of the stress state of the dam shows an important increase of the compression stress close to the left abutment: from 5 to 10 MPa between 1960 and today. On the right side, the gravity abutment brings more flexibility, which leads to lower stress values. In-situ stress measurements with flat jack are under consideration in order to confirm the stress values calculated. Table 1: evolution of the opening of the dam/foundation contact with swelling
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Opening of the dam/foundation contact (mm)
Opening of the dam/foundation contact (mm)
1961
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Figure 7: main arch stress on the upstream face in summer: without (1961) and with swelling (2011)
Conclusion At the beginning of this study, the dam's behavior was difficult to understand: multiple mechanisms were involved and therefore, the problem could not be solved with basic interpretations. As often, model increasingly complex allow to separate each phenomena (opening of the dam/foundation contact, swelling, creep) and to understand their influences on the general behavior of the dam. The last model is used to evaluate the impact of the longterm behavior on the stress state of the dam and to decide what need to be done to assure the safety of the structure.
References [1] Code_Aster software, www.code-aster.org [2] Grimal E., Bourdarot E (2013). Modeling AAR on concrete dam. ICOLD 81th Annual Meeting Symposium, Seattle, 2013, pp. 99-109.
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Need for transient thermal models, with daily inputs, to explain the displacements of arch gravity dams I. Escuder-Bueno1, D. Galán2 and A. Serrano3 1
Instituto de Ingeniería del Agua y Medio Ambiente. Universitat Politècnica de València (Spain) http://www.ipresas.upv.es 2
División de Seguridad de Presas. Canal de Isabel II Gestión (Spain) http://www.gestioncanal.es 3
iPresas, SPIN-OFF UPV. web: http://www.ipresas.com E-mail: [email protected]
Abstract La Aceña Dam, with 66 meters of maximum height, belongs to the typology of concrete gravity dams, and is operated as part of the water supply infrastructures of a Spanish major city. Many instrumentation records were available but among them, the ones provided by four direct pendulums outstood by its quality and consistency. The range of the values of the movements registered by those pendulums, of almost 4 cm and of totally elastic nature (showing no irreversible movements), set some interpretation challenges. The apparent incapacity of the models to reproduce the observed behaviour was used as a starting point for the diagnosis of the main sources of uncertainty: the nature of the foundations and the state of the joints, among others. These aspects have constituted the developmental axis of a series of works that have led to a number of effective and efficient actions on the dam during the last years. However, an updated transient thermal model with daily inputs on external and internal measured temperatures, coupled with a mechanical model has shown an spectacular improvement in the explanation of the displacements recorded by the instrumentation.
Software and constitutive models The first numerical model of global simulation of the stress-deformation behaviour of the dam of la Aceña was done as part of the works “First review and general safety analysis”, carried out by the consultancy firm OFITECO in 2005 for CANAL DE ISABEL II, owner of the dam. The model was done with SAP2000NL and intended, among others, to capture some existing radial movements that have been registered by the four direct pendulums and that were higher that expected. Based on the results of these works, Francisco Blazquez Prieto and Ignacio Escuder presented the case as one of the problems to be tackled by independent calculation teams during the “Ninth International Benchmark Workshop on Numerical Analysis of Dams”, organized by the Dam Calculation Committee of the International Commission on Large Dams (ICOLD) in Saint Petersburg, Russia, in June 2007. According to Escuder and Blazquez (2007), despite the great diversity of software (SAP 2000NL, CANT-SD, COQ-EF, SOFiSTiK, MERLIN, DIANA y ANSYS) and thermal models (three of which were transient), none of the seven teams that worked on the records of the radial movements tracked by the pendulums from January 1999 till September 2001, was able to predict the magnitude of the recorded displacements when using realistic values of stiffness of the dam body and foundations. 349
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In 2012, a re-evaluation of the model was done, which is presented in the present paper. The distinctive feature and more promising fact of this re-evaluation when adopting different software and a new behaviour model, was the fact having a much longer list of data (now running until 2011) that included daily temperature measurements (whereas at the time of the Benchmark Workshop, the experts had to work on monthly measurements given that daily measurements were missing in 1999 and an important part of 2000). With regard to thermal modelling, it is worth mentioning that the 2005 model, though done with a SAP 2000NL Licence, a very appropriate code for structures in general, could not guarantee to make the most of the data, mainly due to the limitations of the thermal module of the package (stationary nature). Upon consideration of this aspect, it was decided to develop the numerical models with a different software package, FLAC (Itasca, 2004), which would allow a transient analysis of temperature transmission and thus provide the best-adjusted results realistically attainable. FLAC (Itasca, 1994) is a finite differences code (explicit scheme) that allows the simulation of the behaviour of soils, rocks, concrete structures, etc. as well the interaction among these elements. The program is based on a scheme of Lagrangian calculation in which each element behaves accordingly to a specifically prescribed stress-displacement relation, as a response to the applied forces and the existing constrains in its boundaries. FLAC is doted with an internal programming language (FISH) that allows the definition of each calculation organization (i.e., complex construction sequences) and calculations of very different nature (i.e., for each constitutive relation). The main potential of the base code is the capacity of the software to model tensional states through the use of pre-set constitutive models and others purposely defined by the user. For la Aceña case, a mechanical elastic and linear model was adopted and coupled with a transient thermal model. The stiffness parameters that define the mechanic model are Young’s Modulus (E) and Poisson’s Ratio (), which are somehow equivalent and that can express the straindisplacement relations accordingly to the Volumetric Deformation Coefficient (K) and Shear Deformation Coefficient (G). With regard to the thermal module of FLAC 3D, it must be noticed it incorporates diffusion models as well as advection ones. The simulation of the first type of model, used for the case of la Aceña, allows the analysis of heat transmission through the materials along time and the development of thermally induced stresses and displacements. The process of heat diffusion that the program solves through the optional thermal module is defined by the following differential equation:
2T 2T 2T T k· 2 2 2 q v z y t x
·C·
(1)
Where ρ is the density of the medium, C the specific heat, k is the conductivity and qv is the source of hydration heat. In the development of the model, all involved parameters (density, conductivity and specific heat) have been assumed as being constant and independent of the temperature. Indeed, since the variations of concrete properties with temperature are very small, they can be neglected without concern. With respect to the boundary conditions, the external parameters of the dam are considered as convective surfaces, that is, as behaving accordingly to the following relation: 350
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qn h·T Te
(2)
Where qn is the component of normal flux pointing towards the outside of the considered boundary expressed in W/m2, h is the convective heat transfer coefficient expressed in W/m2ºC, T is the temperature of the boundary surface and Te is the temperature of the surrounding fluid, with both temperatures expressed in ºC. In the coupled thermal-mechanical model, the solution of the thermal stresses requires a reformulation of the incremental relations of stress-displacements that FLAC 3D achieves through a subtraction of the part of movement due to the change of temperature from the total incremental deformation. Finally, the laws of movement defined in a continuous medium are transformed through the former approximation into a system of equations that are applied to the nodes that represent the structure. These equations correspond to the laws of Newton applied to these discretized nodes and result in a system of ordinary differential equations that can be solved numerically from the explicit method of finite differences with respect to time. Summing up, it can be stated that the code FLAC gathers the necessary conditions to tackle successfully the study of La Aceña dam behaviour, taking advantage of the existence of thermal and movements records of a level of detail and quality very unusual in such structures.
Simulation methodology The present analysis simulates the stress-deformation behaviour of the dam of La Aceña in the period of 1999-2011, which is the one for which movement data were available through pendulum readings. In particular, two radial measurements were available (crest and horizontal gallery) from 4 pendulums (pendulums from 1 to 4). The mechanical properties used in the model are the same employed in former studies of the dam and are shown hereafter (Table 1): Table 1: Dam and concrete mechanical properties Young’s Modulus (dam) Poisson’s Ratio (dam) Density (dam) Young’s Modulus ( (foundation) Poisson’s Ratio (foundation) Density (foundation)
20 · 109 Pa 0.2 2405.7 kg/m³ 10 · 109 Pa 0.2 2242.6 kg/m³
The main difference between the current analysis and those carried out in the past is that better data are now available. In particular: 1. Daily records of the external temperatures in the dam since October 2000. Before, only average monthly temperatures were available. 2. Daily data from the three thermometers located in the concrete since January 2008. Prior to this date there was no data about dam concrete temperatures. 3. Data on the reservoir water temperature (year 2011, monthly periodicity).
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Higher periodicity (daily) of pendulums readings since July 2004 thanks to the installation of automatic readers. In order to understand the effect the different hypotheses have on the results, several models of the dam have been done, through the combination of the following options: Geometry: 2d and 3d Foundation: With and without foundation Temperatures: simplified hypothesis and improved hypothesis Since 2008 there exists a register of the temperatures measured in three points inside the concrete that have helped to improve the original thermal hypothesis in the terms indicated hereafter, and that correspond to what has been named “improved hypothesis”: The temperature of the concrete in contact with the air is equal to the ambiance temperature plus the effect of solar radiation (∆T = 10 °C). The temperature of concrete in contact with the water has been assumed to be constant and equal to 5° C except for the superior layer that acts as a transition element between the ambiance temperature and water. Specific heat: 3.5 W/m K, thermal conductivity: 650 J/kg K, convection coefficient: 16 W/m² K and linear thermal expansion coefficient: 10-5 Figure 1 shows the global geometry of the model and Figure 2 the location of the pendulums whose movements, measured in the crest, were used to check the goodness of the adjustment. Figure 3 represents the whole of the external variables (water level and external average temperature) as well as the pendulums measurements.
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nsulting Group, Inc.
odel Perspective 1 2012 Rotation: X: 40.000 Y: 0.000 Z: 30.000 Mag.: 1 Ang.: 22.500
shown
Group, Inc. USA
Figure 1: Complete geometrical model
Figure 2: Location of the four direct pendulums
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Summary of the results and conclusions The main conclusions that can be extracted from the current analysis of the stress-deformation behaviour of the dam of La Aceña through the simulation methodology described along the former sections, established in order to make the most of the whole set of records of thermal monitoring and movements, are: The elaborated models reproduce in a reasonably satisfactory way the thermal behavior of the structure as well as the movements observed in the dam. The fact of having carried out a transient thermal simulation for each day, capturing the impact of the direct sun exposure on the downstream side of the dam and estimating with accuracy the water temperature have been key elements to the goodness of the obtained numerical model. Figure 4 shows a summary of the average errors incurred by the different models. As it can be observed, the best model is the so called “reference one” (3D, with foundation and the hypothesis of improved temperature).
Promedio de errores absolutos
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Impact of no cosidering foundation
Impact of no cosidering 3D
Sin cimiento
2d
P
Impact of no updating thermal parameters
P
P
P
0 Modelo de referencia
Hipótesis de temperatura inicial
Mezcla 2d - 3d
Figure 4: Average errors in mm for each model (shown for each pendulum, from P1 in darker red to P4, in a lighter grade.) It can be stated that thanks to the recently developed numerical models a new tool is at hand to interpret the strain-deformation behaviour of the dam of La Aceña. This tool can complement the auscultations that are currently being done on the dam and that have provided an excellent level of information about both the thermal and movement behaviour. Also, the numerical tool has permitted the satisfactory and consistent reproduction of the stressdeformation behaviour auscultated until the end of 2011 and can act in the future as a way of contrasting the impact of new actions planned on the dam as well as a form of detecting potential behaviour change trends. Figure 5 shows the comparison between the reference model (with the hypothesis of improved temperature) and the model with the original hypothesis of temperature.
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Figure 5: Extent of the improvement of the thermal model (reference model in red, nonupdated thermal model in blue)
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Acknowledgements Some of the calculation routines have been developed within the following Research Project: “Integration of the components of anthropic risk in the systems of dams and reservoirs global safety management” (Incorporación de los componentes de riesgo antrópico a los sistemas de gestión integral de seguridad de dams y embalses BIA2010-17852), funded by the Spanish Ministry of Science and Innovation and with funds FEDER.
References [1] I. Escuder and F.Blazquez, Analysis of the Elastic Behaviour of an Arch Dam, Hydropower and Dams, Issue 5, (2007)
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The rehabilitation of Beauregard Dam: the contribution of the numerical modeling A. Frigerio1 and G. Mazzà1 1
Ricerca sul Sistema Energetico – RSE S.p.A., via R. Rubattino 54, Milan, ITALY [email protected]
Abstract The paper describes the case-history of the Beauregard dam (Italy), a concrete arch-gravity structure 132 m high built between 1951 and 1960 on the Dora di Valgrisenche River. The design reservoir volume was about 70 mil. m3. The geological and geotechnical investigations carried out since the dam construction and deepened in the last decade have underlined that a Deep-Seated Gravitational Slope Deformation (DSGSD) is located on the left slope. Since the first fillings of the reservoir, the interaction between the DSGSD and the dam was recognized to have relevant implications on the dam structural safety. For that reason, the Italian Dam Authorities in 1969 prescribed a limitation of the reservoir level with a corresponding reduction of the its volume to 6.8 million m3, about 1/10 of the initial design volume. The studies, which include a detailed analysis and thorough interpretation of the monitoring data over a time span of more than 50 years, have allowed to gain insights into the understanding of the DSGSD behavior and its interaction with the dam. To interpret the already experienced effects of the bank movements against the dam and to forecast the future possible trends, the numerical modeling activities have played a key role. The solution chosen to guarantee a long-term safety operation foresees the demolition of the upper part of the dam in order to drastically reduce the cracking pattern of the dam body caused by the compression of the vault for the DSGSD movement.
Introduction The Beauregard dam, located in the Aosta Valley, Italy (Figure 1) was completed in 1958. Operated by the Italian Electric Energy Company (ENEL) up to July 2001, the entire scheme was then acquired by CVA (Compagnia Valdostana delle Acque).
Figure 1: Location of the Beauregard dam on the Italian territory and aerial view of the dam
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The 132m high arch-gravity dam has a crest length of 408m, it is 45.6m thick at the foundation level and 5m thick at the crest level. With a maximum operating design level of 1770m asl, the total reservoir volume was estimated in 70 mill. m3. The filling of the reservoir was undertaken in stages between 1958 and 1968. Since the dam construction, the monitoring system installed on the left slope abutment showed the presence of a clear relationship between the reservoir level and the rate of movement of the Deep Seated Gravitational Slope Deformation (DSGSD) located on this left slope (Figures 2, 3, and 4).
Figure 2: Displacement vectors measured from surface targets
Figure 3: Horizontal displacement distribution along the plumb-lines installed in the lower portion of the slope close to the dam
Figure 4: Zones of maximum shear strain rate: overall slope (safety factor = 1.401.45)
Coherently with the left bank movement, the dam was observed to deflect upstream, due to the trust of the slope against the vault, having as a consequence the appearance of cracks on the downstream face (Figure 5). As a consequence, in 1969 the operational reservoir level was lowered down to 1710m asl, corresponding to a reservoir volume of 6.8 mill. m3, as enforced by the Italian Dam Authorities.
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Figure 5: The dam was observed to deflect upstream with cracks appearing on the downstream face and openings of some upstream vertical joints The continuous monitoring of the DSGSD and of the dam, carried out by ENEL, first, and by CVA, in recent years, has allowed to operate the hydroelectric power scheme under closely controlled conditions. Moreover, some works carried out to control the superficial water flows due to melting snow and rain, in order to limit their filtration into the sliding body, have allowed to reduce the yearly speed of the slope from 1-2 cm to few mm. However, the interaction between the DSGSD and the dam has been recognized to have relevant effects on the long-term dam behavior as the cracking pattern was continuously progressing in time. Figure 6 shows the results of an on-site investigation carried out with a tomographic system that put into evidence the areas of the dam where cracking is particularly important, i.e. along the peripheral joint and close to the downstream dam toe.
Figure 6: P-wave velocity tomogram: (left) downstream face; (right) main cross section The strong interaction between the DSGSD and the dam has been recognized to have relevant possible implications from the civil protection as well as energy production points of view, and posed important territorial and environmental issues. For this reason the Owner decided to start a further study, making also reference to the support offered by numerical modeling, with the aim to find the most suitable and long-term solution of the problem.
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The contribution of the numerical modelling Synthetically, the main aims of the numerical model were: The interpretation of the dam behavior experienced since its first fillings, The calibration of the mechanical parameters of the dam-rock system in terms of comparison between the observed dam behavior and model results, The forecast of the future dam behavior at short-middle term adopting the calibrated model, The support to the designer for the analysis of possible technical solutions to guarantee the long term operation of the dam. The numerical simulations of the dam were carried out using the FEM code ABAQUS/Standard [1]. A three-dimensional finite element model of the concrete arch dam including a proper portion of the surrounding rock mass foundation was generated in order to carry out numerical analyses (Figure 7). In the rock mass foundation the active portion of the DSGSD has been taken into account as well as the shear surface, modeled in terms of an interface whose behavior is described by a frictional law. The nodes where the DSGSD movements were imposed are outlined with a red point in Figure 7.
Figure 7: Finite Element mesh of the dam and the surrounding rock mass A linear elastic isotropic constitutive law was assumed for the rock mass foundation whose mechanical parameters were defined on the basis of the results provided by the more recent geotechnical studies. The Concrete Damaged Plasticity (CDP) constitutive law, available in ABAQUS library, was adopted to describe the concrete behavior (Figure 8). The model assumes that the uniaxial tensile and compressive responses of concrete are characterized by damaged plasticity. Under uniaxial tension the stress-strain response is linear-elastic until the failure value is attained; beyond this value a strain softening relationship follows. In compression the softening behavior occurs after an initial stress hardening. On the basis of the results of the indirect tensile tests, two different sets of material parameters were assigned to the concrete of the dam body and the "pulvino".
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Figure 8: The Concrete Damage Plasticity constitutive law (Fenves, 1998) has been assumed for concrete and pulvino of the dam Several preliminary analyses were carried out to identify the numerical model (calibration phase) in order to reproduce the real dam behavior dealing with the interaction between the structure and the slope sliding. The movement of the left slope abutment was imposed by appropriate displacement boundary conditions applied at the nodes marked in red (as explained above, Figure 7) and the hydrostatic load was firstly considered taking into account the variation relevant to the first fillings; later, a constant reservoir level at 1710 m asl was assumed, according to the present operational conditions. The identification process was mainly based on: The measured displacements along the dam crest (Figure 9); The monitoring points of two plumb lines, located inside the dam body in the main cross section and in the fourth block towards the left abutment respectively; The crack pattern of the structure as determined by tomographic investigations (Figure 10, to be compared with Figures 5 and 6). As shown in Figures 14-15, the identified numerical model exhibits a good agreement with the deformation of the dam crest as well as the.
Figure 9: Deformation of the dam crest: (left) based on monitoring; (right) based on computations
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Figure 10: Localisation of damage computed with the 3D numerical model: (left) on the dowstream face of the dam; (right) on the main cross section Afterwards, the identified numerical model was used to predict the future behavior of the structure considering a continuous increase of the landslide movement according to the trend foreseen from the past behavior. The main purpose of the analyses was to assess if the dam might undergo to local or global instability phenomena such as snap-back. The curve nr. 1 in Figure 11 allows to exclude snap-back instability for the overall structure because beyond the peak value the reaction force on the main cross section of the dam decreases gradually as the landslide movement increases. Although the measured displacement towards the upstream direction of the middle point of the dam crest (about 0.21 m) is close to the peak value of the reaction force curve, it has to be bear in mind that the structure is subjected to an imposed deformation rate due to the landslide movement. For this reason the structure will be able to follow the softening branch of the reaction force curve avoiding sudden failure when the peak value will be attained.
Figure 11: Reaction force on the main cross section of the dam vs the upstream-downstream displacement of the middle point of the crest dam
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The dam rehabilitation Different possible long-term solutions were considered by the designer (Studio Marcello, Milan) who was in charge of the rehabilitation program. Numerical modeling was widely adopted in order to analyze the different solutions which main aim was the reduction of the trust effect of rock sliding against the dam. Among the different solutions, the two more widely analyzed were those shown in Fig. 12.
Figure 12: Two of the possible long-term solutions adopted for the rehabilitation of Beauregard dam The partial demolition solution (right picture of Figure 12) was the one finally decided by the rehabilitation designer. The demolition of the upper part of the dam will lower the crest elevation from 1772m to 1720m asl. The estimated volume of the demolished concrete will be about 150,000 m3. The demolished material will be used to fill the two volumes located at the upstream and downstream dam toes (Figure 13). Additional works have been carried out in the frame of the rehabilitation interventions, among which the improvement of the hydraulic scheme with the construction of a gated spillway and a discharge tunnel.
Figure 13: Cross section of the dam: final configuration after the demolition
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The methodology chosen for the demolition is blasting. The demolition started in the Spring of 2013 and will end in 2014. Figures 14 and Figure 15 show some phases of the dam demolition that is in progress. Just to give some figures about each demolition step, the case shown in Figure 15 is relevant to an explosion carried out with about 850 ton of explosive located in 120 boreholes with a diameter of about 8 cm and a length of 6m. The demolished concrete shown in the picture regards a volume of about 6m high x 7m thick x 50m wide.
Figure 14: Different sequences of the dam demolition: the dam before the starting of works (up-rigth); the preparation of one explosion step (up-left and down)
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Figure 15: Different sequences of the dam demolition
Conclusion The case history of Beauregard dam is an emblematic case that has posed to the engineers in charge of operation and safety very critical choices, considering that the construction of new dams, and structures in general, represents their main mission. In the Beauregard case it has been deemed necessary to proceed, as sometimes happen for the human life, with a sort of a limb amputation in order the save the dam life. The decision has been difficult and it has been necessary to investigate very deeply and for long time before all possible aspects affecting the system, deciding which could have been the most reasonable solution from the engineering point of view. The choice has been taken considering the different facets of the problem: safety, mainly, environmental protection, energy production. Environmental as well as economic and financial aspects are not in the aim of the present paper and could be widely shown in a future paper. Analogously, also some other technical aspects relevant to the civil works that are under construction at the downstream toe and facing to check the presence of possible leakages, will be discussed in further papers.
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The present paper has the aim to put into evidence the support that numerical modeling can give to decision makers in a very complex problem like the one proposed by Beauregard dam.
Acknowledgements The Authors wish to express their gratitude to Lorenzo Artaz and Morena Colli of CVA who has given the permission to present the present work.
References [1] Hibbitt, et al, ABAQUS/Standard Theory Manual, Release 6.6, 2005. [2] G. Barla et al., The Beauregard dam (Italy) and the deep-seated gravitational deformation on the left slope. Hydropower 2006, Kunming, China, October 2006.
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Earthquake Assessment of Slab and Buttress Dams H. B. Smith1,2 and L. Lia2 1
2
Norplan AS, Oslo, Norway Department of hydraulic and environmental engineering, NTNU, Trondheim, Norway E-mail: [email protected]
Abstract This paper outlines work on the examination of the dynamic properties and seismic safety of slab and buttress dams [1]. The work has been carried out with a linear elastic numerical model established in the Finite Element Method program Abaqus. One single dam section in a typical Norwegian slab and buttress dam with heights of 12 and 25 meters has been modeled in regards to varying reservoir water level and lateral bracing. Abaqus has been used for the frequency analysis and the dynamic time-history analysis. The natural vibration modes of the dam represent movements in separate directions. Through a seismic event the greatest response will be represented by the buttress deflection in direction along the dam axis (axial direction). The resistance towards earthquakes will depend significantly on individual stability of elements rather than the global stability. In particular, the tensile stresses occurring in the buttress when deflecting in the axial direction are found to be a potential failure mechanism. Providing lateral bracing by struts positively influences the response of the slab and buttress dam in seismic events. When lateral bracing is provided, the ability of the dam to transfer the inertial forces to the abutment is important.
Introduction The American-Norwegian engineer Nils Ambursen developed the slab and buttress dam, the Ambursen dam, in the early 20th century. Several of these dams were built until the 1970s, especially in North America and Norway. Figure 1 presents the Håen Dam, a typical Norwegian slab and buttress dam with a structural height of 12 meters. [2] [3]
Figure 1: The Håen Dam, view of downstream side with insulating wall The reinforced concrete structure consists of the buttresses and an upstream inclined slab. The typical section and dimensions of Norwegian slab and buttress dams are presented in Figure 2. Insulating walls and lateral bracing, such as struts, are not always present. The upstream
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slab is often constructed as continuous over two or three sections, with joints where the moment is zero. The resulting moment diagrams are as presented in Figure 3. Earthquake safety regulations were introduced for Norwegian dams in 2010. Due to the recently updated regulations and the low number of slab and buttress dams in earthquake prone regions, there is uncertainty regarding the dynamic assessment of existing slab and buttress dams in Norway. The slab and buttress dam is proven to be highly efficient for static loading. Based on the experiences from registered dam breaches, earthquake incidents and upgrades, the slab and buttress dam needs to be checked for earthquake loads in both horizontal directions. Structural slenderness of buttresses and slabs makes the structure particularly vulnerable to axial loading. Load distribution and interaction between the various elements could cause the strength and stability of individual elements to be more critical than the global stability. When dynamic loads are considered, several possible failure mechanisms are identified.
Figure 2: Typical cross section [4]
Figure 3: Moment distribution in continuous slabs [4]
Finite Element Model Based on a literature review, existing computational methods and site inspections of slab and buttress dams, a linear elastic numerical model was established in the Finite Element Method (FEM) program Abaqus. The importance of various structural details in a seismic context has been analysed and utilised for the modeling of a single dam section in typical Norwegian slab and buttress dams with heights of 12 and 25 meters. Two models have been used for the analysis regarding height variation, reservoir water level and presence of lateral bracing. Dynamic properties, global response and impact on the structural elements have been analysed through eigenfrequency analysis and dynamic response-history analysis. For convenience the results presented in this paper focus on the base case of a 25 meters high dam section. The dam section was modeled, as presented in Figure 4, as a single buttress and continuous slab with symmetrical
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boundary conditions in the slab’s axial direction at the mid-span. The dimensions of the modeled section are:
5 meters c/c between buttresses Crest width of 2.5 meters 5V:4H inclination of the upstream slab 5V:1H downstream inclination of the buttress Both the slab and buttress thickness increases 20 mm per meter from t = 300 mm, at the dam crest
The model was partitioned in order to provide nodes in the locations of all lateral bracing. The generated mesh gave a total of 2676 shell elements and 2863 nodes. The foundation rock is expected to be of good quality and is therefore idealised with no mass and with great stiffness compared to the slender slab and buttresses. The slab and buttress dams are normally designed without transmission of reinforcement to the foundation or between the structural elements. The joint between slab and buttress is designed with a shear key to avoid displacement in the axial direction. As these joints will allow some rotation, all connections are modeled as hinged. Lateral Bracing When present, struts and insulating walls will provide lateral bracing for the buttresses in addition to the upstream slab. Insulation walls are introduced to reduce the temperature gradient through the slab. With typical thickness of t = 120 mm lightly reinforced, low concrete quality, the actual contribution of the insulation wall in a seismic event is not included. Only the effect of lateral bracing on the dynamic properties of the slab and buttress dams’ by struts is therefore investigated. Struts are normally simply supported between the buttresses, i.e. only compression forces are transferred Figure 5: Selected placement of through the struts. The struts are modeled by massless struts elastic springs that ensure the stiffening effect in the axial direction. Working both in compression and tension, the springs were modeled only on one side of the buttress in the chosen locations presented on Figure 5. Loads The applied static loading includes gravity and hydrostatic water pressure. Uplift, sediment loads and downstream water pressure are neglected, in addition to temperature loads. Seismic loads are applied according to Eurocode 8 [5] regulations. This gave a maximum peak ground acceleration for mainland Norway of 1.6 m/s2 horizontally and 1.0 m/s2 vertically. The applied acceleration time histories, presented in Figure 6, are developed for ground type A and adjusted for Norwegian geological and seismic conditions. The acceleration time histories are applied along the foundation in all three directions simultaneously. The acceleration time history in the axial direction is also applied along the edges of the slab. 371
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Because of the upstream inclination, the horizontal hydrodynamic pressure is simulated according to Zangar’s theory of added mass [7]. The overlaying masses of water are assumed to be directly accelerated by the dams’ vertical accelerations [8]. The applied hydrodynamic added mass was decomposed from the horizontal and vertical contributions to work perpendicular to the upstream slab, and applied as inertia point masses at different levels, according to the presented distribution from Figure 7. Because of the flat slab, no hydrodynamic interaction was taken into account in the axial direction.
a) Applied in the upstream-downstream direction
b) Applied in the axial direction
c) Applied in the vertical direction
Figure 6: Utilized acceleration time histories [6]
Figure 7: Distribution of applied hydrodynamic added mass
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Material Properties, Strengths and Prescribed Damping The concrete is modeled with elastic properties corresponding to a B25 concrete, described in Eurocode 2 [9]. The struts are modeled as elastic springs, with corresponding stiffness k = EA/L, acting both in tension and compression. The concrete strength was determined in accordance with Eurocode 2 [9] regulations, adjusted for a 50 % increased dynamic tensile strength, according to Raphael [10]. The struts’ capacity was determined by calculations of buckling and compression capacity. Viscous Rayleigh damping, embedded in the concrete material properties and the hydrodynamic point masses, are used to model all damping in the system, including damwater-foundation interaction and frictional dissipation. The additional stiffness proportional damping from the struts was not taken into account. Because the model represents the structure’s elastic behaviour, a damping ratio of two percent was considered appropriate. It should be noted that an increased damping ratio would reduce the systems response.
Results Frequency Analysis The frequency analysis described the structures’ natural vibration modes with movements in separate directions. In all analysed cases, the first mode is governing for the axial direction. Figure 8 describes the vibration mode shape and variation of period for oscillations in the slab and buttress dam section with water at crest level for different section heights and with or without lateral bracing. a)
b)
c)
Figure 8: The first vibration mode a) Mode shape without lateral bracing b) Mode shape with struts present c) Variation of the vibration period The first mode describing movement in the upstream-downstream and vertical direction was also governing in these directions for all cases. Figure 9 describes the mode shape and variation of period for the slab and buttress dam section with or without presence of water at crest level, for different section heights without lateral bracing.
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a)
b)
c)
Figure 9: The first vibration mode describing movement in the upstream-downstream direction a) Mode shape without water b) Mode shape with water at crest level c) Variation of the vibration period Dynamic Response-History Analysis Through the seismic event, the models were never exposed to compressive stresses close to the concrete design capacity. The maximum principal stresses are therefore presented. It is assumed that cracking of concrete will occur when the design dynamic tensile strength fctd,dyn = 1,5 MPa, is exceeded, marked by gray in the graphic presentations. The typical deflected shape and maximum principal stress state during a seismic event are presented in Figure 10, for the 25 meters high dam section with water at crest level and with or without lateral bracing in the form of struts. a) Without lateral bracing
b) With lateral bracing
Figure 10: Typical deflected shape during seismic event, deformation scale factor of 100
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The maximum principal stresses on each side of the buttress with water at crest level, are presented on Figure 11 without lateral bracing and on Figure 12 with struts.
Figure 11: Maximal principal stresses on both sides of the buttress without lateral bracing during a seismic event
Figure 12: Maximal principal stresses on both side of the buttress with lateral bracing during a seismic event. The time-history plot of transferred inertia forces to the highest loaded strut in the dam section of 25 meters height is presented in Figure 13. Representing both the struts on each side of the buttress, the plot describes the absolute value of the forces through the given elastic spring.
Figure 13: Time-history plot of the highest loaded strut in the 25 meters high dam section.
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Discussion Dynamic Properties The distinct differences in the structures vibration modes in different directions, combined with the variation of the vibration period, leads to the following observations: Dam height
Water level Lateral bracing
An increase in the section height results in a greater mass while the dam stiffness decreases both in the axial and the upstreamdownstream direction. Thus, a greater vibration period in all directions is obtained. At higher water levels, the additional hydrodynamic mass increases the vibration period in the upstream-downstream direction. Increased stiffness in the axial direction due to lateral bracing reduces the vibration period in the same direction.
Seismic Safety Dam safety requirements for slab and buttress dams include overturning and sliding stability in addition to the required structural strength for all cross sections. The dam safety should be considered in a seismic event and retain its integrity during the earthquake. The seismic safety has been evaluated based on global stability and occurring stress levels in the buttress. The struts are additionally considered. As the upstream slab is designed for tensile stresses, it is not further considered. Global Stability The maximum fluctuations of the reaction forces acting in different directions on the dam’s sections would, in combination with one another, constitute a danger to the system's global stability. Nevertheless, the maximum fluctuations occur at various times throughout the event and are only applicable in very short time periods. The system will not have time to react, and there will be no danger of collapse even if the theoretical factor of safety against sliding or overturning in a given time goes below 1.0. The consequences may be minor deformations if exceedance of capacity is repeated. The theoretical safety factor against sliding through the seismic event for the 25 meters high dam section with water at crest level is presented in Figure 14.
Figure 14: Variation in theoretical safety factor against sliding for a 25 meters high dam section with water at crest level.
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Also considering the rest capacity when assuming a frictional factor of tan = 1, the risk of sliding is not the critical case. Nevertheless, the theoretical safety factor against sliding for slab and buttress dams is reduced as the dam height increases. The upstream inclination keeps the dam stable against overturning throughout the seismic event, as tensile stresses never occur along the buttress foundation. Stresses in the Buttresses A lower section height reduces both the response and stress level of the buttress. The presence of lateral bracing prevents the buttress deflection out of its own plane, and largely reduces the stresses. When struts are present, the buttress is bent around its connection points, and the maximum stresses are found in those locations. The occurring tensile stresses in the numerical model have to be seen in relation with the presence of reinforcement. Before 1955 it was common to construct the slab and buttress dams without reinforcement in the buttresses [11]. In addition to tensile stresses, construction joints would endanger the buttress stability in such a case. For dams constructed later than 1955, a minimum amount of reinforcement will always be present. Even though tensile stresses can cause cracking of the concrete, the strength and ductile nature of reinforcement will nevertheless increase the safety level compared to an unreinforced buttress. Struts The impact on the struts from a single section must be seen in relation to the struts compression capacity, found to be dimensioning over the buckling capacity. The transferred inertia forces are accumulated towards the abutments through the struts from neighbouring sections. When beams are not continuous over the dam length, unfavourable moments can be introduced into the buttresses. Other Potential Failure Mechanisms The following potential failure mechanisms should also be taken into consideration; the connections between elements, shear capacity and punching shear capacity of the buttresses, stress accumulation around cutouts, strength of the foundation and bearing capacity of the struts. Uncertainties – Sources of Error Because only one single dam section has been modeled, several assumptions have been required. Among others, the influence of adjacent sections, especially of different heights, is of importance. As the results show a non-linear response and because reinforcement is not taken into account, the response and maximum occurring tensile stresses in the numerical model can only be indicative.
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Conclusions The results demonstrate how the dam vibration modes represent movements in separate directions. An increased dam section height results in an increased response because of the greater oscillating mass. Meanwhile, a raised water level in the reservoir increases the period for oscillation in the upstream-downstream direction. Through a seismic event, the greatest response will be represented by the buttress deflection in the axial direction. The resistance towards earthquakes will be mainly dependent on the individual stability of elements rather than the global stability. In particular, the tensile stresses occurring in the buttress, as a result of deflection in the axial direction, is found to be a potential failure mechanism. Providing lateral bracing by struts positively influences the response of the slab and buttress dam in seismic events. For the Norwegian slab and buttress dams in particular, the contribution of the existing insulating walls is considered to be small but positive. When lateral bracing is provided, the ability to transfer the inertial forces to the abutment is important. In addition to the assessment of simplified analytical methods, it is recommended that further numerical studies should focus on the slab and buttress dams’ nonlinear response, on the importance of global dam geometry, and verification of the numerical input parameters by physical vibration tests of an existing slab and buttress dam.
References [1] Smith. Earthquake Assessment of Slab and Buttress Dams, Master thesis at the Department of Hydraulic and Environmental Engineering, NTNU, Trondheim, Norway, 2013. [2] Various authors. Ambursen dams - Selected articles from the Ambursen Hydraulic Construction Company. R. M. Rudolph, Approximate Date 1912. [3] NORUT Narvik AS. Innovativ forvaltning av betongdammer, 2009. In Norwegian. [4] Guttormsen. TVM4165 - Vannkraftverk og vassdragsteknikk. Tapir akademiske forlag, 2006. Compendium of the Department of hydraulic and environmental engineering, NTNU. In Norwegian. [5] Standards Norway. Eurocode 8: Design of structures for earthquake resistance - Part 1: General rules, seismic actions and rules for buildings, 2004. NS-EN 19981:2004+NA:2008. [6] Kaynia. Communicated by. NTNU and NGI, Oslo, Norway. [7] Zangar. Hydrodynamic pressures on dams due to horizontal earthquake effects. Engineering monographs, No. 11, 1952. [8] Scott et al. Dam Safety Risk Analysis, Best Practices Training Manual. U.S. Department of the Interior, Bureau of Reclamation in cooperation with the U.S. Army Corps of Engineers, 2011. Version 2.2. [9] Standards Norway. Eurocode 2: Concrete structure - Part 1-1: General rules and rules for buildings, 2004. NS-EN 1992-1-1:2004+NA:2008. [10] Raphael. Tensile strength of concrete. Journal of the American Concrete Institute, 2:158 – 165, 1984. [11] Kleivan, Kummeneje and Lyngra. Concrete in Hydropower Structures. Hydropower Development. Norwegian Institute of Technology, Division of Hydraulic Engineering, 1994. 378
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Solution of dam-fluid interaction using ADAD-IZIIS software V. Mircevska1, M. Garevski1, I. Bulajic2 and S. Schnabl3 1
Institute of Earthquake Engineering and Engineering Seismology, univ. “Ss. Cyril and Methodius”, Box 101, 1000 Skopje, R.Macedonia 2 Mining institute, Batajnicki put 2, 11000 Belgrade, Serbia 3 Faculty of Civil and Geodetic Engineering, University of Ljubljana, 1000 Ljubljana, Slovenia E-mail: [email protected]
Abstract Fluid dam interaction has a remarkable impact on the dynamic response of dams and could play an important role in assessment of their dynamic stability. This is particularly emphasized when dams are subjected to strong seismic excitations. The phenomenon has been for the first time physically explained and mathematically solved by Westargaard. The very first dynamic analysis based on application of “Added Mass Concept”; underestimated the random earthquake nature in assessment of the hydrodynamic effects. In the recent years, various BEM-FEM and FEM-FEM techniques have been developed to account for many significant parameters that influence the accuracy of calculated hydrodynamic effects. This paper presents a BEM-FEM orientated solution based on the use of the matrix of hydrodynamic influence as a very effective tool for analyses of extensive domains of fluiddam-foundation rock systems for two major reasons: the computation time is far more effective than that in direct or iterative coupling methods and stability of the solution. The presented analyses are based on the use of genuine software originally written for static and dynamic analysis and design of arch dams.
Introduction The ADAD-ver.3 computer program [1], originally written for static and dynamic analysis and design of arch dams, is under development for the last several years. It implements an analytical procedure for the three-dimensional dynamic analysis of arch dams including the effects of dam-water interaction (water incompressibility), soil-structure interaction and the nonlinear behavior of the of contraction joint manifested by partial joint opening and closing as well as tangential displacement. The process of generation of mathematical model runs parallel and interactively with the process of design of the particular dam. The program gives an option for computer design of the dam body [2,3], whose embedment is in accordance with topology of the terrain. Program offers automatic pre-processing for generation of finite element mesh of dam and part of the foundation mass to account for the dam-foundation interaction phenomenon, as well as effective way of generation of boundary element mesh, sufficiently accurate in following the topology of the terrain to account for the fluid structure interaction. The program use sensitivity search analysis to detect the “most adequate” location of the truncation surface [4], where non-reflecting truncation boundary conditions should be imposed. The truncation surface should be located in a way to define the required completeness of the wave field where expansion of P compressive and dilatational waves takes place followed by scattering and radiational effects. Its further displacement away from
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the dam should have a negligible impact on the calculated magnitude of the hydrodynamic effects. The conducted analyses are based on an original and simple FEM–BEM fluid–structure interaction solution embedded in the ADAD–IZIIS software. This solution eliminates the difficulties of “direct” and “iterative” coupling methods by analyzing independently the two physically coupled sub-domains. The interaction effect is obtained in an uncoupled way computing the matrix of hydrodynamic influence by applying the concept of virtual work of “unit accelerations”. The suggested method does not belong to direct coupling or to iterative coupling methods, yet with its computational steps it offers a two-way coupling by transferring the fluid forces to the structure and the structural accelerations back to the fluid. The paper presents BEM-FEM oriented solution of the fluid-dam interaction along with the boundary element discretization of the reservoir domain. The dam was subjected to El Centro earthquake excitation with duration of 7 sec., scaled to the pick acceleration of 0.3g. The dam properties are the following: Dam height H=130m; Young’s modulus E = 31.5 GPa; mass density ρ = 2450 kg/m3; Poisson’s ratio ν = 0.2; the acoustic wave velocity in water c = 1440 m/s.
Numerical Model of Arch Dam and Fluid Domain The process of generation of the discrete mathematical model of the arch dam and the fluid domain runs parallel with the process of design of the dam body. This process starts by digitization of topographic data of the terrain and the shape of the main central cantilever, Figure 1. The developed pre-processing within the ADAD–IZIIS software enables each topographic isoline to be mathematically presented by a set of equations of a second order, i.e., curves passing through three neighboring digitalized points on it. Each isoline is stored in the computer by means of a certain number of polynomials. The program gives an option for computer design of the dam body [2,3], whose embedment is in accordance with topology of the terrain. Arch dam body can be modeled in a form of few circular segments as well as in the form of a parabola. During modeling, it is possible to observe the shape of the arches at all elevations along with the corresponding tables containing their geometrical parameters and to observe their mutual position in order to control overlapping. Figure 2, presents the mathematical model of the dam generated automatically, using adopted shapes of the arcs at all selected elevations in accordance with the topology of terrain. The model is formed by 199 substructures. The substructures are automatically digitized into a certain number of finite elements that are not presented in the figure. The model contains 6294 finite elements and 11170 external nodes. The bend of the system is 3111. The model posses 2013 contact elements that are involved at the connections between substructure’s blocks. The contact elements are generated automatically and the contact element mesh refinement is in accordance of the model analyst request. ADAD-IZIIS software offers a very efficient and accurate modeling of the 3D fluid boundaries according to the downstream topology of the canyon terrain. The boundary elements at the extrados of the arch dam are directly extracted from the general FE model while the boundary element mesh that represents the boundaries of the reservoir, i.e., both banks, the reservoir bottom, the water mirror and the reservoir end, very accurately follows the shape of the topographic isolines. Generation is simply by giving the number of planes that intersect the terrain along with their distances from the uppermost point of the crown cantilever. In case of highly irregular and twisted terrain, more section planes and a more refined mesh should be used in order to model the complexity of the terrain in the most accurate way, which is undoubtedly easily feasible and with shorter computational time if BEM technique is engaged. However, for the concrete configuration of the terrain and detected “most adequate” location of the truncation surface situated at the downstream 380
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distance of 210m from the dam, the number of boundary elements used in the model for accurate modeling of the 3D fluid boundaries is 1600, figure 3.
Figure 1: a) Topology of the terrain b) Shape of the central cantilever
Figure 2: Mathematical model of a dam (substructures and construction joints)
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Figure 3: Boundary element mesh of the fluid domain Fluid-structure interaction is affected by the irregularity of the terrain in the near surrounding of the dam-fluid interface. The topology of the terrain dictates the “most adequate” location of the truncation surface where non-reflecting truncation boundary conditions should be imposed. The detection of the “most adequate” location of the truncation surface is an important task in development of a reservoir model due to the fact that the HDP intensities on the dam-fluid interface are sensitive to the extent and type of waves generated by the boundaries. The truncation surface should be located in a way that its further displacement away from the dam has a negligible impact on the calculated HDP intensities. The program offers sensitivity search analysis to detect the most adequate location of the truncation surface. For the presented BE model in figure 3., the elapsed CPU time for performing such analysis is approximately 15 min. The numerical procedure is based on the conventional BEM. Laplace differential equation that governs the incompressible and inviscid fluid motion is used. The procedure is conducted over a rigid dam-canyon-walls assemble. It follows a horizontal acceleration of 1g applied in downstream direction. The acoustic elastic P waves were generated as a result of the vibration of the considered upstream dam face and the rigid canyon’s walls. The expansion of generated waves and the way of their propagation as compressive or dilatation waves depends not only on the specified boundary conditions but on the shape of the reservoir boundaries in respect to the direction of the seismic excitation. Three different types of truncation boundary conditions were considered: a) stationary type of truncated boundary conditions, i.e., perpendicular acceleration at all the points on the truncation surface is set to zero; b) hydrodynamic pressure at all points on the truncation surface set to zero; and c) non-reflecting boundary condition. Sixteen different location of the truncation surface were considered and analyzed. The curve associated with the TBC that allows dissipation of the outgoing waves shows mostly decreasing trend until reaching the meeting point of the curves of TBC type (a) and TBC type (b), at distance of L=180 m (L=1.6-1.7Hw) away from the dam. Further on, this curve remains almost horizontal, indicating unaltered value of HDP. This means that it is irrelevant whether TS is positioned at a greater distance than L=180m, since the effect of the amount and type of generated waves is negligible with further increase of the model length. However, this effect is remarkable along length L<180m, wherefore placing TS closer to the dam means overestimation of the HDP. In accordance with the applied direction of ground acceleration and due to the irregular configuration of the terrain in the vicinity of the dam, the right bank generates mostly dilatation waves while the left bank generates compressive waves. For the concrete topological conditions, the considered depth of impounded water, according the results presented in [4], the “most adequate” location of the truncation is selected at a distance of L=1.9Hw=210m.
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Figure 4: Variation of the normalized hydrodynamic pressure magnitudes as a function of the considered 16 locations of the truncated surface and different truncation boundary conditions (selected node at the bottom and at the middle of the crown cantilever)
BEM-FEM Solution of Fluid-Dam Interaction Fluid–structure interaction is the interaction of a moveable and/or deformable structure that is immersed in a fluid and/or contains a fluid. A model that captures such an interaction must use two-way coupling model, where the fluid motion affects the structure’s motion and the structure’s motion affects the fluid’s motion. Coupling should provide compatible link of both media which means equilibrium and correct transition of the physical variables at the interface. There exist various algorithms for coupling the merits of both BEM and FEM numerical methods, direct [5-8] and iterative coupling methods [9-12]. ADAD-IZIIS software [1] is based on BEM-FEM oriented solution of the coupled structure and the incompressible and inviscid fluid. The solution of the coupled system is accomplished by calculating in advance the matrix of hydrodynamic influence utilizing the concept of virtual work of “unit accelerations”. This matrix is stored in the system and recalled in any time step of the dynamic response of the dam. Hence the solution of the coupled systems is actually separated and mutually independent. Hydrodynamic forces are obtained as a product of the matrix of hydrodynamic influence and the vector of manifested total accelerations along the normal at any interface node. The interaction effect at the fluid–solid interface is enforced by adding the matrix of hydrodynamic forces to the classic equation of dynamic motion of the dam, eq. (3). The governing equation for solving the small amplitude irrotational motion of the impounded incompressible and inviscid fluid is governed by the three-dimensional Laplace’s equation as follows: 2W 2W 2W 2 2 0 x 2 y z
(1)
where W(x,y,z) is a function of the potential in the fluid domain. The equation (1) has to be amended by the specified “essential” and “natural” type of boundary conditions that exists at the boundaries of the analyzed fluid domain. Applying BEM technique, Brebbia [13], the discretization of boundary surfaces is by an assemble of eight nodded quadratic “Serendipity” type of boundary elements as follows:
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(2)
where: i=1,NBE; NBE is a number of nodes in the boundary ele ment model. The differential equation of motion of a discrete system written in an incremental form for the "i"-th time increment is the following: ̂ ̈
̂ ̇
̂
̂
̂
(3)
where: ˆ FHD is a vector of hydrodynamic force increment and ˆ Pi is a vector of seismic i force increment. In eq. (3), the vector of hydrodynamic force increment eq, 4 is calculated by use of previously defined matrix of hydrodynamic influence and directly added to the vector of seismic force increment.
ˆ FHD Wij anjtot i
i, j 1, NPT
(4)
where: Wij is a matrix of hydrodynamic influence; anjtot is the absolute acceleration along the boundary element normal; NPT is the total number of nodes at dam-fluid, bottom-fluid and banks-fluid interfaces, where the natural type of boundary condition exists Figure 5, presents the time history response of relative displacement velocity and acceleration for the selected node at the dam crest, top of the crown cantilever, where the extremes of the response occurred. Obviously, fluid structure interaction based on BEM-FEM numerical solution modifies the extreme of the response acceleration at the dam crest by 38% in respect to the dam responce with empty reservoir. Additionally calculated are hydrodunamic effects by use Westergaard added mass concept. It gives lower modification of the dam responce, i.e, the exterem of the relative responce acceleration at the dam crest is modified by 31% Nothe that, the time of extreme occurance is not coinside. Westergaard added mass concept do not give recogition to the impact of the dam flexibility on the amount of generated energy in the fluid domain and therefore on the intensity of the manifested FSI effects. The flexibility property of the dam and the influence of the reservoir domain alter the behavior of the fluid significantly and consequently the coupled system has a stronger response. The figure 6 shows the izolines of distribution of the principal stress G3 that acts along the arches with and without included hydrodynamic effects, whereat hydrodynamic effects are calculated according to added mass method and coupled BE-FE method. The stress extreme is increased by 15% if added mass method is used and 49% if coupled BE-FE method is used. Despite the influence of the terrain irregularities on the amount of energy transferred to the fluid domain, this effect so far, has not been analyzed in detail by exsisting software regardless whether boundary element method (BEM) or finite element method (FEM) is used for fluid discretization. ADAD-IZZIS software gives an opption for taking into account the influence of terrain irregularities on the magnitide of calculated hydrodymanic effecs. Figure 7., presents a snapshot of hydrodynamic pressure distribution over the interface, at time T=4.95sec. It is obtained under the assumptions that the topology of the canyon has a regular shape as indicated in the drawing.
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Figure 5. Modification of the dam response at the dam crest, relative displacement in (m) ; relative velocity in (m/sec) and relative acceleration in (m/sec2). FSI effects defined by use of BEM-FEM solution and Westergaard added mass method
a)
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b)
c) Figure 6: Distribution of principal stress G3 at the extrados face at t=2.42 s a) empty reservoir b) FSI using BEM-FEM regular terrain c) FSI using added mass method
Figure 7: Snapshot of hydrodynamic pressure distribution at time T=4.95 sec 386
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Conclusion ADAD-IZIIS software is based on BEM-FEM oriented solution of the coupled structure and incompressible and inviscid fluid domain. The software offers process of generation of the mathematical model that runs parallel and interactively with the process of design of the arch dam body. Automatically are generated the finite element mesh of the dam and part of the foundation mass for accounting the phenomena of dam-foundation interaction, and boundary element mesh that presents the boundaries of the fluid domain for accounting the fluid structure interaction. The solution of the coupled system is accomplished by use of matrix of hydrodynamic influence utilizing the concept of virtual work of “unit accelerations”. Comparison of the calculated hydrodynamic effects using both BEM-FEM solution and added mass method is made. The added mass method provides acceptable results only in the range of Westergaard restricted hypothesis. Since it neglects the dam flexibility and water compressibility and does not require any discretization of the reservoir domain wherever these features have an impact on the magnitude of hydrodynamic effects there will be discrepancy of the obtained resultants.
References [1] Mircevska V., Bickovski V. (2008). ADAD-IZIIS software: Analysis and Design of Arch Dams, IZIIS- univ. “Ss. Cyril and Methodius”, User’s Manual. [2] Houard L., Boogs (1997). Guide for Design of Arch Dams. A Water Resources, Technical Publication, Engineering Monograph No 36., 1-17. [3] Wiliam, P., Greager, and Joel D., Justin (1984). Engineering for Dams, John Wiley & Sons [4] Mircevska V., Bickovski V., Aleksov I., and Hristovski V. (2013). Influence of irregular canyon shape on location of truncation surface. Engineering Analysis with Boundary Elements, Vol. 37, pp. 624–636. [5] Estorff von O., Prabucki M.J. (1880). Dynamic response in the time domain by coupled boundary and finite elements. Computational Mechanics Vol. 6, pp. 35–46. [6] Pavlatos G.D. Beskos D.E. (1994). Dynamic elastoplastic analysis by BEM/FEM. Engineering Analysis with Boundary Elements Vol. 14, pp. 51–63 [7] Czygan O., Estorff von O. (2002). Fluid–structure interaction by coupling BEM and nonlinear FEM. Engineering Analysis with Boundary Elements Vol. 26, pp. 773–779. [8] Yu G. Mansur, W.J., Carrer J.A.M., Lie S.T. (2001). A more stable scheme for BEM/FEM coupling applied to two-dimensional elastodynamics. Computers & Structures Vol. 79, pp. 811–823. [9] Lin C-C, Lawton EC, Caliendo JA, Anderson LR.(1996). An iterative finite elementboundary element algorithm. Computers & Structures Vol. 39(5), pp. 899–909. [10] Feng YT, Owen DRJ. (1996). Iterative solution of coupled FE/BE discretization for platefoundation interaction problems. International Journal for Numerical Methods in Engineering Vol. 39(11), pp. 889–901. [11] Soares Jr D, von Estorff O, Mansur WJ. (2004). Iterative coupling of BEM and FEM for nonlinear dynamic analyses. Computational Mechanics Vol. 34, pp. 67–73. [12] Estorff von O., Hagen C. (2006). Iterative coupling of FEM and BEM in 3D transient elastodynamics. Engineering Analysis with Boundary Elements Vol. 30, pp. 611–622 [13] Brebbia C.A. (1978).The boundary element method for engineers, Pentech Press
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Influence of Surface Roughness on Sliding Stability Tests and numerical modeling Ø. Eltervaag1, G. Sas2 and L. Lia3 1
Department of Structural Engineering, Faculty of Engineering Science and Technology, Trondheim, Norway 2 Northern Research Institute, Narvik, Norway 3 Department of Hydraulic and Environmental Engineering, Faculty of Engineering Science and Technology, Trondheim, Norway E-mail: [email protected]
Abstract Lightweight concrete dams slide when the shear capacity of one or more sliding planes in the dam’s structure or foundation is exceeded. Several laboratory shear tests were carried out on concrete rock samples. The two materials were mated trough teeth-sawed interfaces with different inclination profile. This paper presents the results of the numerical modeling of those tests. The results of the shear tests were compared to the predictions of the model used in the Norwegian guidelines. It has been found that the model used in the guidelines do not predict the shear capacity accurately. Through finite element analyses a better representation of the tests has been achieved, especially regarding the influence of roughness.
Introduction In Norway, the stability of a dam is reconsidered every 15 to 20 years, depending on the consequence class for the given dam. In the past 40 - 50 years, since the large hydropower development époque in Norway, the level for required safety has increased. Consequently some dams constructed during that period no longer are considered to have the sufficient safety and expensive upgrading is needed.
Figure 1: Profiles of test samples [1]
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Two of the samples with geometry 1 and 4 were casted with bond between the rock and concrete. The remaining ten samples had no bond. During casting of concrete a plastic film was attached on the rock surface of those ten samples to prevent bonding.
Theory According to the Norwegian regulations [2] the sliding stability is assessed using the shear friction method, and is expressed by a factor of safety (FS). For a horizontal sliding plane this factor is found from the following equation. FS
c A V tan
(1)
H
In equation (1) c is the cohesive parameter, A is the shearing plane area, V and H are the vertical and horizontal forces acting on the plane respectively. The frictional factor of the plane surfaces is expressed by tan (ϕ). Thus, the factor of safety is calculated from the averaged normal stress and friction factor of the sliding plane. The shear friction method is based on the Mohr-Coulomb criterion [3] for describing the shear capacity of the sliding plane. This criterion states that the shear capacity, τ, is linearly dependent on the applied normal stress, σ, trough a material specific frictional parameter tan(ϕ) plus a cohesive parameter, c. This can be expressed as follows:
c tan
(2)
In the 1960’s it was recognized that the failure envelope for rock mass (rocks with joints and faults) was curved. One of the major contributions to this understanding was when Patton [4] derived a bi-linear failure criterion from experiments with “saw-toothed” rock specimens shown in Figure 2. Patton observed that sliding occurred at lower levels of normal stress than what was needed to cut off the saw-toothed geometry. This failure mechanism can be described on the form: tan(b i) (3) Were ϕb is the material specific friction angle, simply denoted ϕ in the Mohr-Coulomb relation, equation (2), and i is the angle of the asperities, called the asperity inclination or dilation angle. When the normal stress exceeded a certain value the saw teeth were cut off at their base. Patton [4] explained this as a change of governing failure mode, from a sliding failure along the material interface, to a failure in the material itself. The shear capacity regarding this material failure is described by:
cx tan(r )
(4)
Where cx is the bond of the failing material, and ϕr is the residual friction angle of the failure plane. The shear envelope obtained from these expressions is shown in Figure 2 .
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Figure 2: Patton's bi-linear failure criterion [4] According to Patton [4] the bi-linear failure envelope illustrates that there are two possible failure mechanisms for the rock specimens studied. The first mechanism is sliding over the asperities (saw-teeth) and occurs at low normal stresses. The second mechanism is shearing through the asperities and occurs at relatively high normal stresses. These different failure modes have been further described by Johansson [5]. Johansson [5] developed a conceptual model to describe sliding failure of one idealized quadratic asperity. Three failure modes were identified; sliding along the side of the asperity facing the load, shear-failure along the base of the asperity, and tensile failure in the rock-base of the asperity. A sketch of an idealized asperity is shown in Figure 3.
Figure 3: Principle sketch of an idealized 2D asperity [5] To describe sliding along the loaded face of the asperity (the left side in Figure 3) Johansson [5] uses Patton’s formulation, eq. (4), for shear capacity for low normal stresses (transformed from stresses to forces).
T N tan(b i)
(5)
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The equation for shear failure along the base of the material is derived based on the MohrCoulomb criteria, eq. (2) and Patton’s equation for high normal stress, eq. (4).
T ci Lsp 2 N tan(i )
(6)
For a tensile failure to occur in the rock-base, the vertical tensile stresses must exceed the tensile capacity of the rock. To calculate the average tensile stress in the rock Johansson [5] assesses the moment-equilibrium about point O in Figure 3.
T
(3 ci 4 ti ) Lasp 2
(7)
2 tan(i)
Where σti is the tensile strength and σci is the compressive strength of the rock.
Method For this work the finite element software ATENA-Science developed by Cervenka Consulting has been used [6]. The geometry of tested samples is modeled as a 2D plane stress problem, left side of Figure 5. To avoid numerical instability when applying the loads directly on concrete, the surrounding steel shear box was modeled also, right side and medallion in Figure 4. In Atena the boundary conditions were applied as fixed lines along the outside of the lower steel frame, the loading was introduced by a vertical line load along the top of the steel frame, and horizontal displacement along the left edge of the upper steel frame. Monitoring points were added at the left edge of the upper steel frame and at the right edge of the concrete material to enable load/displacement curves.
Figure 4: 2D geometrical model of a test sample with ten degree asperity angle ATENA uses interface elements to model contact between two parts of a model. To model the behavior of the interface elements ATENA uses the Penalty Method [6]. The physical properties of these interface elements are governed trough the Mohr-Coulomb criterion presented in eq. (1). To obtain input values of the interface properties the results from the tests are compared to the failure criteria described above. However, these classic formulations present large errors compared to the test results. Reliable input parameters were obtained by applying a hybrid formulation where the shear capacity is determined from the actual failure 392
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mechanism for the tests; sliding over or shearing off the asperities, eqs. (5) or (6). This hybrid formulation for shear capacity of the samples can be expressed as follows, V
c A N tan(b i) cos i(1 tan i tan b )
(8)
where i is the residual inclination angle of the asperities, which may differ from the original inclination angle if the asperities are sheared off. The basic friction angle is material constant and therefore equal for all the samples. Thus, the basic friction angle can be found from a sample without asperities, and is determined to be 34.4˚. The cohesive parameter is found by inserting the known values from the results of each of the tests into eq. (8). These values are listed in Table 1. The physical explanation of this parameter is believed to be wear of the interface as cut-off of micro asperities (micro roughness) as shown in eq. (6). The cut-off shows a decreasing effect as the samples slide against each other. To account for this a softening behavior of the cohesive parameter is introduced. In ATENA the effect of the asperities (macro roughness) is included trough the geometry of the interface. In ATENA the loading is applied incrementally. First the normal pressure was applied as a vertical line-load. The line-load for the different tests has been found by distributing the applied experimental vertical force presented in [1] over the length of the samples (240mm). The applied line-loads are presented in Table 1. Table 1: Interface and loading input parameters Sample 1.1 1.2 1.4 2.1 2.2 2.3 3.1 3.2 3.3 4.2
Test results [1] i [˚] δHmax [mm] 40 15.23 40 15.44 40 26.34 20 24.99 20 21.98 20 20.07 10 33.16 10 32.94 10 33.23 0 33.23
N [kN] 27.52 46.91 68.49 27.57 47.77 68.36 24.82 45.98 67.65 67.23
i [˚] 0 0 0 20 20 20 10 10 10 0
Input used in ATENA δH “Cohesion” Q [kN/m] [mm] [kPa] 20 1245 114 20 1720 195 20 2371 285 20 282 115 20 781 199 20 1552 285 20 309 103 20 382 192 20 536 282 20 62 280
Then the horizontal force is applied trough prescribed deformation of ten millimeter at the left edge of the upper steel frame, see Figure 4. To achieve plane horizontal displacement (avoid rotation) of the upper part master-slave boundary conditions are introduced along the upper edge in this interval. Interval three and four are identical to interval two, except that the master-slave boundary conditions are removed. The standard incremental and iterative Newton–Raphson method for material nonlinear structural analysis was used in the numerical simulations, based on the finite element method. The specimens were modeled with a mesh of 8-node serendipity plane stress finite elements. A Gaussian integration scheme with 2 x 2 integration points was used for all the concrete elements, [6,7].The mesh is shown in Figure 5.
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Figure 5: Mesh of test 2.2 Both the concrete and rock materials are modeled using a fracture-plastic model that combines constitutive sub-models for tensile and compressive behavior, as presented in the ATENA user manual [6], see Figure 6. This fracture model employs the Rankine failure criterion and exponential softening, with the hardening/ softening plasticity component based on the Menétrey– William failure surface [6]. The concrete post-cracking tensile behavior was simulated by a softening function in combination with the crack band theory [6].
ef c ef
ft
d
fct
E0
c
t
0
eq
unloading
wc = 5.14
Gf
loading
U
Gf fct
f
ef c
wc
Figure 6: Constitutive model for concrete (left) and softening function (right) [6] The decision for modeling the rock as concrete is based on the behavior of the actual rock samples during compressive standardized tests. It was observed that this behavior is more in line with the available material models for concrete than rock. The steel shear box was modeled as a linear elastic material. Theoretical background of the above mentioned constitutive models are given in the ATENA Theory manual [7]. Input parameters for the rock and concrete material are given in Table 2.
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Table 2: Material parameters for concrete material models Material Young’s Modulus [GPa] Poisson’s Ratio Tensile strength [MPa] Compressive Failure Stress [MPa] Fracture Energy [MN/m] Plastic strain Onset of crushing [MPa]
Concrete 37 0.2 4.1 58 1.03x10 -4 -0.00147 -8.61
Rock 100 0.2 2.317 280 1.05x 10-5 -0.000296 -1.54
Critical compressive displ. [m]
-0.0005
-0.0005
23
23
3
Density [kN/m ]
In Table 1it is seen that the geometry of tests 1.1 to 1.4 is modeled with zero inclination angle. The reason for this is that these tests showed a shearing failure mode, where the asperities were cut off, therefore there were no sliding between asperities but material failure. A total shearing failure mode has been hard to obtain from the numerical models. The main reason for this is that the analyses crash after the material starts to crack due to instability problems with zero or negative Jacobian for the stiffness matrix. This error message is an indication of an ill-conditioned system of equations [8]. Perhaps a more refined mesh with more integration points could have been used. Due to low computational capacities this was not possible at the time when this work was carried out. In Figure 7 the concrete material failure is shown. The blue field in the left part indicates that the tensile strength of the material is reached.
Figure 7: Material failure for test 1.4. Due to the numerical instability problems only the peak shear capacity was obtained from these analyses. However, this capacity is governed by the concrete material parameters alone, not by the interface parameters. In the test report [1] only the compressive strength was given for the rock and concrete materials. Thus, the tensile strength and onset of crushing have been generated automatically by ATENA. To overcome these problems the tests in series 1 have been modeled using the residual inclination angle of the asperities (zero) according to eq. (8).
Results The results from the numerical analyses have been monitored by recording the horizontal displacement and reaction-force along the loaded edge of the model. The load-displacement curves from the analyses (labeled ATENA) are presented in the same diagrams as the curves from the actual tests (labeled LTU after the lab where the tests were conducted). In total, eleven of the twelve shear tests were analyzed. Due to space limitation in Figure 8 below, the graphical results are shown for selected analyses only. Full description of all results are presented in [9].
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Figure 8 Load displacement diagrams showing tests and numerical results In general the procedure of assuming cut-off of micro roughness as cohesion gives a good representation of the peak shear capacity of the actual tests. When coupled with a softening behavior of the cohesive parameter, the capacity along the failure development and the residual shear capacity are represented sufficiently accurate. Since the input parameters are found from each specific test the finite element models should produce results that match both the peak and residual shear capacity from the tests exact. However, studying the results, it becomes evident that some errors still occur. The difference between the peak shear capacity from the laboratory tests and the numerical analyses are listed in Table 3.
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Table 3: Error in peak shear capacity from the numerical models. Sample
Vmax[kN] test
Vpeak [kN] ATENA
1.1 1.2 1.4 2.1 2.2 2.3 3.1 3.2 3.3 4.1H 4.2
90.53 131.19 183.42 50.74 100.59 162.83 35.38 58.70 85.43 240.00 49.56
89.91 125.80 173.21 50.81 100.57 162.08 34.65 57.56 83.65 225.75 49.52
[kN] 0.62 5.39 10.41 0.07 0.02 0.75 0.73 1.14 1.78 14.25 0.04
Error [%] 1 4 6 0.1 0.02 0.5 2. 2 2 6 0.1
The error-% is calculated with respect to the actual value from the tests [1]. The average error is approximately 2% with a maximum of 6% for two analyses. Compared to the results from the Mohr-Coulomb and Patton criteria in Table 4 this is a significant improvement. Table 4: Error in peak shear capacity from hand calculations. From tests [1] Sample I [˚]
N [kN]
V [kN
Mohr-Coulomb V [kN] Error [%]
Patton V [kN]
1.1 1.2 1.3 2.1 2.2 2.3 3.1 3.2 3.3
27.52 46.91 68.49 27.57 47.77 68.36 24.82 45.98 67.65
90.53 131.19 183.42 50.74 100.59 162.83 35.38 58.70 85.43
361.60 380.99 402.57 20.32 35.21 50.39 18.29 33.89 49.86
113.73 193.86 283.04 41.19 71.89 102.88 26.06 48.28 71.04
40 40 40 20 20 20 10 10 10
332 190 119 59 65 69 48 42 42
Error [%] 26 48 54 18 28 37 26 18 17
Although the errors in Table 3 are not large, their cause needs to be addressed. One possible source is the inaccurate recording of the residual capacity in the tests. Due to the fact that when concrete was cracking the analysis was not always stable, the residual capacity only describes the capacity as long as the interface asperities are intact. This explains why the residual capacity is not well represented for test series 2 where some deformation of the asperities was registered in the tests. For test series 1 this source of error is avoided as the interface is modeled with the inclination of the residual sliding plane from the actual tests. It is hard to determine why stable results after material failure in the concrete was not obtained, but it is believed that a fine mesh and more refined material models would improve this. Not all the required input parameters for the material models were obtained from the physical tests, thus some default values were used in the modeling. With lacking parameters, a full parametrical study would be needed to refine the materials. The results from the analyses show that the actual shear capacity of the tests are better represented through numerical modeling than the formulations available for hand calculations. Especially the opportunity to represent the softening behavior after the peak shear capacity is reached, is a large benefit.
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Conclusion Through finite element modeling, more sophisticated analyses regarding stability towards sliding might be obtained. This allows more sustainable design of structures subjected to sliding instability. However, it must be noted that due to the limited amount of samples analyzed and the scale effects encumbered with shear capacity, further work is needed to utilize the potential of finite element modeling of stability towards sliding for full scale dams. A full parametric study is needed to improve the method.
Acknowledgements Acknowledgements are given to the professor Kjell H. Holthe at the Norwegian University of Engineering Science and Technology and Dr. Dobromil Pryl at Cervenka Consulting. Acknowledgements are also given to EnergiNorge for financial support.
References [1] Liahagen, S., (2012) Stabilitet av betongdammer - Ruhetens påvirkning på skjærkapasiteten mellom betong og berg. Master’s Thesis. NTNU,IVM, Trondheim, unpuplished. [2] NVE, Norwegian Water Resources and Energy Directorate, (2005), Retningslinje for betongdammer. [3] Coulomb, C. A. (1776). Essai sur une application des regles des maximis et minimis a quelquels problemesde statique relatifs, a la architecture. Mem. Acad. Roy. Div. Sav., vol. 7, pp. 343–387. [4] Patton, F.D (1966). Multiple modes of shear failure in rock. 1st ISRM Congress, September 25 – October 1, 1966, Lisbon, Portugal. LNEC [5] Johansson, F., (2009). Shear Strength of Unfilled and Rough Rock Joints in Sliding Stability Analyses of Concrete Dams. Ph.D. Thesis. Royal Institute of Technology (KTH), Stockholm. [6] Cervenka, V. Cervenka, J. Zednek, J. Pryl, D. (2013) ATENA program Documentation, Part 8: User’s Manual for ATENA_GiD Interface. http://www.cervenka.cz/assets/files/atena-pdf/ATENA-Science-GiD_Users_Manual.pdf [7] Cervenka, V., Jendele, L., Cervenka. J., (2012) ATENA Program Documentation, Part 1: Theory. http://www.cervenka.cz/assets/files/atena-pdf/ATENA_Theory.pdf [8] Pryl, D., Cervenka, V. (2013) ATENA Program Documentation, Part 11: Troubleshooting manual. http://www.cervenka.cz/assets/files/atena-pdf/ATENATroubleshooting.pdf [9] Eltervaag, Ø. (2013) Sliding stability of Lightweight Concrete Dams – Development of numerical models. Master’s Thesis. NTNU, IKT, Trondheim. Unpublished.
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Seismic Analysis of a Concrete Arch Dam Considering Concrete Heat Generation Damage Effects P. Dakoulas1 1
University of Thessaly, Volos, GREECE E-Mail: [email protected]
Abstract This study evaluates the seismic performance and safety of Tavropos arch dam (Greece) considering the dynamic canyon-dam-water interaction. Tavropos dam has a height of 83 m and crest length of 220 m. The numerical model of the arch consists of 17 concrete cantilevers separated by vertical joints, whereas the foundation-abutment model represents a large region of the canyon. The interface behavior at the vertical joints and the horizontal base joint is modeled in three different ways: as bonded surfaces, as surfaces in contact with frictional strength, and as a combination of bonded and contact surfaces. To account for possible prior damage, the study simulated the thermo-mechanical phenomena that took place during dam construction, related to heat generation due to concrete hydration. The seismic analysis is performed against a MCE having a magnitude of 7.5 and peak ground acceleration of 0.47g. The seismic evaluation is based on an extensive parametric investigation of the effects of various key factors, including concrete strength, joint interface behavior, water level, hydrodynamic pressures, ambient temperature, canyon rock flexibility, spillway geometry, earthquake excitation characteristics, etc. It is concluded that the expected overall stability and performance of Tavropos concrete arch dam is satisfactory.
Introduction Lake Plastiras was created after construction of an arch dam (Fig. 1) in Tavropos river, a tributary of Acheloos river in Northern Greece [4]. The double-curvature Tavropos arch dam has a height of 83 m and crest length of 220 m. Construction started in 1955 and was completed in 1959, during which the dam was impounded. Today, Lake Plastiras provides drinking water to the city of Karditsa and nearby villages, and irrigation to the region near Larissa. Its electric power is 130 MW, whereas the lake is an attractive tourist destination with more than 120,000 tourists annually.
Figure 1: Tavropos Arch Dam in Lake Plastiras, Greece 399
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The objective of this work is to conduct a seismic performance and safety evaluation of the data, considering the dynamic canyon-dam-water interaction and current data about the local seismicity based on recent research [11][3][8][9][10].
Figure 2: Finite element discretization of (a) concrete arch dam geometry consisting of 17 parts and (b) Part No.10 geometry at the middle section
Figure 3: Finite element discretization of the canyon geometry (a) model A and (b) model B consisting of full quadratic finite elements for the entire canyon region.
Numerical Model Dam geometry The study is based on a detailed numerical model of the dam and canyon using the code ABAQUS [1]. Fig. 2a illustrates the numerical discretization of the arch dam body. The dam body consists of 17 parts, having a width equal to 12 m, except of the central part, which has width equal to 16 m. The 17 parts are connected with vertical joints. Fig. 2b shows Part 10, located at the mid-section of the dam. As all parts near the central region of the arch, Part 10 consists of 4 concrete blocks: A, B, C and D. Blocks B, C and D at the lower upstream side were designed as to increase the stability of the dam in the case of a potential strong earthquake during the phase of construction. They are all attached to main block A, but there are horizontal joints between blocks B and C, as well as between blocks C and D to reduce seismic tensile stresses. To improve numerical efficiency, only 2 out of the 4 actual horizontal joints of each part were discretized, without any loss of accuracy. All dam parts are
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discretized using solid hexahedral elements C3D8, having 8 nodes and 8 integration points. A total number of about 4400 elements were used for the modeling of the arch dam geometry. Canyon geometry The discretized canyon geometry has length of 600 m, width of 600 m and depth of 250 m. It consists of two parts: (a) the first part discretizes the irregular geometry near the foundation of the dam. It is discretized by using quadratic, reduced integration hexahedral elements C3D20R, each having 20 nodes, whereas the irregular rock surface and dam foundation excavation, are discretized using quadratic modified tetrahedral elements C3D10M, which are robust, accurate and suitable for frictional contact behavior; (b) for the rest of the canyon, two different models have been utilized: (1) model A (Fig. 3a) consists of hexahedral solid elements C3D8, having 8 nodes (2) model B (Fig. 3b) consists of quadratic hexahedral elements C3D20R, having 20 nodes. About 30500 and 27000 elements were used for canyon models A and B, respectively. Damage plasticity model for concrete The plastic-damage constitutive model for cyclic loading by Lee and Fenves [6] is used here for modeling the behavior of concrete. The model takes into account the effects of strain softening, distinguishing between the damage variables for tension and compression. It incorporates a degradation mechanism that represents the effects of damage on the elastic stiffness and the recovery of stiffness after crack closure. Three values of compressive strength have been adopted, based on large cubic specimens (20cm x 20cm x 20cm) obtained during construction and tested after 365 days. The use of large size specimens is important to avoid the negative effect that the large diameter aggregates may have on strength prediction. The three values of compressive strength are: (A) the mean value fc =46 MPa (B) the mean value minus three standard deviations, fc 3 ≈42 MPa and (C) f c =32 MPa, i.e. an unlikely, much lower value to account for possible additional uncertainties. For f c =46 MPa, the stress-strain behavior during uniaxial loading-unloading-reloading of a cube of concrete, subjected separately to compression and tension tests is given in Figure 4. For the cyclic tension tests, the softening behavior and the effect of the accumulated damage are taken into account, whereas for the cyclic compression, these effects are ignored in the present study. Table 1: Concrete Properties Property Density, ρ [kg/m3] Compressive strength, fc [MPa] Dynamic tensile strength, ftd [MPa] Static Young’s modulus, E [GPa] Dynamic Young’s modulus Ed, [GPa] Poisson’s ratio, ν Coefficient of thermal expansion, a [°C-1] Specific heat, c [J/(kg °C)] Conductivity, k [W/(m °C)] Heat transfer coefficient [W/(m2 °C)] Hydration heat, q(t) [kcal/kgcement] (t : in days)
Case A 2350 46 6.4 31 42 0.15 10-5 879 2.5 16
Case B 2350 42 6.0 31 42 0.15 10-5 879 2.5 16
Case C 2350 32 5.3 31 42 0.15 10-5 879 2.5 16
q(t ) t /(0.0653 0.0175t )
Joint behaviour The behavior at the vertical joints and the horizontal joint at the dam base is modeled in three different ways: as bonded surfaces, as surfaces in contact with frictional strength, and as a 401
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combination of bonded and frictional surfaces. The bonded joints are considered to have the tensile strength of concrete of the adjacent blocks. The surfaces with only frictional strength, have a coefficient of friction equal to 0.5 for vertical joints and 0.8 for the horizontal joints.
Figure 4: Damage plasticity concrete model: stress-strain relation during loading and unloading in uniaxial (a) tension and (b) compression. Canyon rock properties The canyon rock material is fine-layered limestone containing sparse intermediate layers of radiolarites. During the static and dynamic analyses of the dam-canyon system, the canyon rock is considered to behave as a linearly elastic material. Table 2 summarizes the canyon rock properties. Two scenarios are examined for the dynamic analyses, namely, case R1 and case R2, having shear wave velocities equal to Vs = 2800 and 2000 m/s, respectively. Table 2: Canyon Rock Properties Property Density, ρ [kg/m3] Static Young’s modulus, E [GPa] Dynamic Young’s modulus, Ed [GPa] Poisson’s ratio, ν Shear wave velocity, Vs [m/s] Hysteretic damping, ξr
Case R1 2450 16 48 0.25 2800 0.03
Case R2 2450 16 25 0.25 2000 0.03
Effect of heat generation during construction It is of interest to examine concrete behavior immediately after construction, especially for high performance concrete, as cracking may develop due to deformation caused by heat that is generated during cement hydration reactions [7][2]. Potential cracking generated during or immediately after construction of various dam segments might have developed weak areas, which could affect its seismic performance during earthquakes. To examine the effects of premature cracking, a numerical simulation of the dam construction process is conducted, modeling the cement-hydration heat generation, radiation, convection, conduction and ambient temperature variation for a period of about 24 months. Fig. 5 plots the distribution of the temperature in a section of the dam after 200 days, whereas Fig. 6 plots the temperature variation with time at points A, B, C, D and E, as well as the ambient temperature. The maximum temperature occurs in the middle point (C) of the dam wall and has a value of 50.5 C at 200 days from the construction start. Examination of the major principal plastic strain caused by the transient temperature differences within the dam showed that the concretehydration heat did not cause any significant cracking within the main body of the dam. Any 402
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such cracking within the dam body is practically either zero or negligible. Some thermal cracking may have developed within the upstream aseismic concrete blocks (see Fig. 2b), but it has absolutely no effect on the performance and stability of the dam in its current state.
Figure 5: Distribution of temperature within the dam body at 200 days since construction start
Figure 6: Variation of temperature at points A, B, C, D and E during construction versus time
Seismic analysis Dynamic characteristics of the dam
The fundamental natural frequency of the dam for full reservoir and canyon rock with shear wave Vs =2800 m/s is 3.97 Hz, whereas for Vs =2000 m/s, it reduces to 3.81 Hz. Earthquake excitation The seismicity of the region has been studied by Panagiotopoulos and Papazachos [10]. For the present study, a Maximum Credible Earthquake of magnitude M 7.5, with an epicentral distance from the dam site about R = 15 km, is considered. The Maximum Probable Earthquake is based on a magnitude of M 7.0 and minimum distance R =15 km. The seismic excitation considered consists of three different acceleration records. The first two have been recorded at rock sites (Lucerne and Pacoima Dam records). The third record is synthetic and has response spectra that match the Eurocode 8 design spectra for rock sites. All records have been base-line corrected and scaled to a peak horizontal outcrop-rock acceleration equal to 0.47g and peak vertical outcrop-rock acceleration equal to 0.30g. Fig. 7 plots the horizontal and vertical acceleration time histories of the synthetic record. Fig. 8 plots the acceleration response spectra for the horizontal component and the Eurocode spectra for rock site. The input excitation at the canyon base is computed through de-convolution of the selected 403
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motions. To be able to account for the radiated seismic energy, appropriate dashpots are placed at the canyon boundary and the excitation is imposed in terms of seismic stresses.
Figure 7: Synthetic excitation at outcrop rock (a) horiz. acceleration (b) vertical acceleration
Figure 8: Synthetic horizontal acceleration spectra and Eurocode spectra for rock sites Hydrodynamic pressures The hydrodynamic pressures acting on the upstream side of the dam have a significant effect on the earthquake behavior of the dam. They are considered here by using the added-mass formulation proposed by Zangar [11], which was implemented in code ABAQUS [1]. The amount of mass b at a node is equal to:
z 1 z z z b h Cm (2 ) (2 ) w A 2 h h h h
(1)
where h = depth of the reservoir, Cm = coefficient based on the angle of upstream surface to the vertical, z = depth of node below water surface, w = water density and A = area around node contributing to hydrodynamic forces.
Results and discussion The objective of the parametric analysis is to investigate the effect of various factors on the seismic performance of the Tavropos Arch Dam. More specifically, the parametric study examines effect of concrete strength variation, seasonal temperature variation, reservoir water level, canyon rock flexibility, joint behavior, hydrodynamic pressures, spillway geometry, and earthquake excitation characteristics. Due to lack of space, only representative results of the most likely scenario are presented in this article, whereas additional results will be published elsewhere. The results presented below are based on a concrete compressive strength of 46
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MPa, dynamic tensile strength of 6.4 MPa, critical summer temperature variation, maximum reservoir water level (792 m), and canyon rock S-wave velocity 2800 m/s. Both models A and B (shown in Fig. 3) are utilized for the analysis. First, the overall response and the stability of the dam are considered. Relative displacements Results from two scenarios are presented here for the evaluation of the response and stability of the dam. The first scenario assumes that the vertical joints and the base joint behave as bonded surfaces. Figs. 9a and 9b plot the relative u/d displacement at mid-crest of the dam subjected to the synthetic record for bonded joints, using the two canyon models A and B, respectively (Fig. 3). The computed relative displacement by the two models is rather similar, with the peak value about 4.5 cm. There is no residual relative displacement at the end of shaking. The peak relative displacement for the other two excitations (not shown here) are about 4 cm. Fig. 10a plots the distribution of the u/d relative displacement for bonded joints using model A at a moment of peak response (t = 6 s). The second scenario is quite conservative as it assumes that the joints have zero tensile strength and behave as surfaces in contact having only frictional strength. Fig. 10b plots the residual relative u/d displacement for the case of un-bonded frictional joints, allowing opening and sliding during shaking. The maximum relative displacement in this case is less than 5 cm, which is considered as safe with regard to the stability of the dam for this very conservative scenario.
Figure 9: Time history of relative u/d displacement at mid-crest assuming bonded joints obtained from (a) model A and (b) model B
Figure 10: (a) Peak relative u/d displacement at t = 6 s assuming bonded joints and (b) Residual relative u/d displacement at the end of shaking assuming un-bonded frictional joints. Stresses and strains Fig. 11 shows the distribution of major principal stress, 1 , at the moment of maximum response of the dam subjected to the Synthetic record based on Model A (Fig. 3). As shown in the figure, high tensile stresses develop at the upper middle part of the arch dam, reaching in some elements the concrete tensile strength for a very small time increment (e.g. 0.01s) and causing some limited plastic tensile strain. These tensile stresses occur when the upper part of the dam has a relative displacement towards upstream with respect to its base. In this case, the 405
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combined effect of dam and water inertia (negative hydrodynamic pressure) cause the upper part of the dam to deform towards upstream. This upstream relative displacement becomes maximum at t=5.32 s for the Synthetic record (see point A in Figure 9a). Such deformation causes an extension of the arch dam along its length, which is maximum at the upper-middle area of the arch, resulting to relatively high, short-duration tensile hoop stresses. It is noted that the tensile stresses become maximum at the upstream surface of the central crest area of the arch, whereas they are quite smaller on the downstream side. Figs. 12a and 12b plot the major principal stress time history at mid-crest, evaluated at the upstream surface element using the two models A and B, respectively. The results show that 1 reaches the tensile strength momentarily in Fig. 12a, but it remains below the tensile strength in Fig. 12b.
Figure 11: Distribution of major principal stress 1 at a moment of maximum tension of the dam subjected to the Synthetic record excitation (time t = 5.32 s)
Figure 12: (a) Major principal stress and (b) minor principal stress time histories at midcrest (upstream surface element)
Figure 13: Accumulated major principal plastic strains at the end of shaking
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Despite the large tensile stresses that may develop momentarily during shaking, the accumulated major principal plastic strain 1p at the end of shaking is very limited. Fig. 13 plots the distribution of 1p at the end of shaking for the Synthetic record based on model A. Its maximum value is only about 5 10-5, located at the upstream face of the mid-crest (concrete part 10). In reality, it is expected that the value of the tensile plastic strain 1p will be even smaller than the values shown in Fig. 13. This is because the actual tensile strength at the joints is expected to be about 0.7 f t , where f t is the tensile strength of the dam concrete. Thus, in areas of higher tensile stress concentration, as at the mid-crest area, any cracks most likely will occur at the weakest points, i.e. the vertical joints. As shown in other analyses not presented here, if the entire vertical joints rupture before shaking (i.e. the joint tensile strength is zero), the tensile stresses developing within the mid-crest area of the arch dam are significantly reduced, due to the instantaneous, local opening of the vertical joints. Therefore, the accumulated major principal plastic strains near the mid-crest area are much smaller. Note that the results in Figs. 11, 12a and 13 are based on model A. By using the more refined model B, it is shown that the tensile stresses developing in the mid-crest area are slightly smaller and do not exceed the tensile strength. Thus, no plastic strains develop in the dam body for the presented case of concrete with compressive strength equal to the mean value f c =46 MPa. Figure 14 plots the distribution of the minor principal stress 3 (maximum compression) at a moment of maximum response, corresponding to time t = 6 s (point B in Figure 9a). This occurs when the dam base and canyon moves upstream, while the dam is pushed by the concrete and water inertia (positive hydrodynamic stresses) towards downstream, thereby increasing the compressive stresses in the mid-crest area of the arch. The maximum value of the compressive stresses at the mid-crest area during shaking is less than 12 MPa, i.e. the compressive stresses are very small compared to the compressive strength of concrete in uniaxial load ( f c =46 MPa). In this case, the minimum available factor of safety against compressive failure FSc fc /(1 3 ) during shaking is larger than 2.5. Thus, no plastic deformations occur due to compressive stresses. The compressive stresses for the other two records are smaller than those obtained for the Synthetic record.
Figure 14: Distribution of minor principal stress 3 at a moment of maximum compression of the dam subjected to the Synthetic record excitation (time t = 6.00 s)
Conclusions Evaluation of the results of all parametric studies leads to the general conclusion that, for the maximum earthquake intensity anticipated, the dam is safe against any instability and its overall performance is satisfactory. More specifically, the main conclusions are the following:
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1. The concrete-hydration heat generated during construction did not cause any significant cracking within the main body of Tavropos dam. 2. The maximum seismic relative displacement of the dam for the case of grout-bonded joints is small (about 3 to 4.5 cm). There are no permanent relative displacements at the end of shaking. 3. For the extreme scenario in which the vertical joints and the joint at the base are already ruptured before the earthquake, the relative displacement of the dam at the end of shaking is limited to values less than 5 cm. Thus, even for this extreme scenario, the stability of the 17 cantilever dam parts is fully assured. 4. Based on the results of all analyses, the maximum compressive stresses developing during seismic shaking range between -10 MPa to -16 MPa and do not cause any plastic strains. 5. The magnitude of tensile stresses within the concrete parts depends on the behavior of the vertical joints. The results of the parametric analysis show that the tensile plastic strains (or cracking) developing within the dam body are very small. Therefore, the performance of the dam with respect to the development of tensile cracking during shaking is satisfactory. 6. The summer environmental temperature condition is the most critical one, as it results to higher tensile plastic strains within the dam body. 7. Among the four reservoir water elevations investigated, the most critical one is the maximum water level (792 m), which results to relatively higher tensile plastic strains within the dam body. 8. The stiffer rock, having a wave velocity Vs =2800 m/s, yielded consistently larger values of tensile plastic strains compared to those obtained for Vs =2000 m/s.
Acknowledgment The financial support by the Public Power Corporation is gratefully acknowledged.
References [1] ABAQUS (2012). Users’ Manual, Simulia, Pawtucket, Rhode Island. [2] Azenha, M., Faria, R., Ferreira, D. (2009), Identification of early-age concrete temperatures and strains: Monitoring and numerical simulation, Cement & Concrete Composites, 31, 369–378. [3] Chopra, A. (2008). Earthquake analysis of Arch Dams: Factors to be considered, 14th World Conference on Earthquake Engineering, October, 2008, Beijing, China. [4] Dakoulas, P. Seismic analysis of Tavropos Arch Dam, Research Report to Public Power Corporation, University of Thessaly, Volos, Greece [5] Escuder, I.B. and Blazquez, F.P. (2007). Lessons learned in the analysis of the elastic behavior of an arch-gravity dam performed by seven independent engineering teams, 9th Bench. Work. Num. Analysis of Dams, ICOLD, St. Petersburg, Russia, June 22-24. [6] Lee J., and Fenves, G.L (1998). A plastic-damage concrete model for earthquake analysis of dams, Journal of Earthq. Eng. & Struct. Dynamics, 27: 937-596. [7] Meghella, M. and Frigerio, A. (2009). Theme A: Initial strain and stress development in a thin arch dam considering realistic construction sequence, 10th Benchmark Workshop on the Numerical Analysis of Dams, ICOLD, Paris, France, Sept. 16-19. [8] Mills-Bria, B., Nush, L., and Chopra, A. (2008). Current Methodology at the Bureau of Reclamation for the Nonlinear Analysis of Arch Dams Using Explicit FE Techniques, 14th World Conference on Earthquake Engineering, October, 2008, Beijing, China.
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[9] Mojtahedi, S. and Fenves, G. (2000). Effect of contraction joint opening on Pacoima Dam in the 1994 Northridge earthquake, California Strong-Motion Instrumentation Program Data Utilization Report, CSMIP/00-05 (OSMS 00-07), Sacramento, Calif. [10] Panagiotopoulos, D. and Papazachos, K.B. (2008). Active tectonics of Thessaly and seismicity of Karditsa, 1st Conference on Development, Karditsa, Greece. [11] US Bureau of Reclamation (2006). State-of-Practice for the Nonlinear Analysis of Concrete Dams at the Bureau of Reclamation, USBR Report, Colorado, USA.
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Earthquake safety assessment of arch dams based on nonlinear dynamic analyses S. Malla1 1
Axpo Power AG, Parkstrasse 23, CH-5401 Baden, SWITZERLAND E-mail: [email protected]
Abstract For the earthquake safety assessment of two major arch dams in Switzerland, various linear and nonlinear dynamic analyses were performed. For these analyses, a safety evaluation earthquake (SEE) with a return period of 10,000 years was considered in each case. Even though the input motion has a PGA of less than 0.2 g, the crest region would be subjected to amplified accelerations significantly larger than 1.0 g. Moreover, high tensile stresses occurring in the crest region are likely to exceed the tensile strength of the vertical contraction joints and the horizontal lift joints. Consequently, these joints could open during the earthquake shaking, possibly resulting in an upper portion of a dam block becoming fully detached from the rest of the dam. The dynamic analysis of possibly detached portions in the crest region showed that they could undergo some sliding and rocking motions. However, they would remain stable during and after the earthquake. Based on the results of the nonlinear dynamic analyses, it is concluded that the earthquake loading could cause some limited damages, but an uncontrolled release of reservoir water is unlikely to occur. Therefore, the investigated dams satisfy the safety requirements for the SEE.
Introduction A program of systematic assessment of earthquake safety of all significant dams in Switzerland is scheduled to be completed by the end of the current year 2013. A document containing state-of-the-practice guidelines for this investigation program was issued by the Swiss Federal Office of Energy in 2003. The primary goal of this program is to ensure the safety of the downstream population against loss of life and property damage (Panduri et al., 2012). For this purpose, it must be shown that the safety evaluation earthquake (SEE) would not cause a dam failure resulting in an uncontrolled release of reservoir water. Depending on the height and the reservoir volume, a dam is categorized as class 1, 2 or 3 and it has to be checked for the SEE ground motion with a return period of 10,000, 5,000 or 1,000 years, respectively. At the time of the design of most of the existing dams, the earthquake action was usually considered as a pseudo-static loading and typically a horizontal seismic coefficient of 0.1 g was assumed. This is clearly inadequate even in regions with relatively low seismicity. In practically all major arch dams, the crest region would experience horizontal accelerations significantly exceeding 1.0 g in the event of the SEE, even when the ground motion at the rock level has a horizontal peak ground acceleration (PGA) of the order of 0.20 g only. In a linear elastic analysis, the earthquake shaking would typically cause high horizontal stresses in the arch direction in the crest region of the dam. As the tensile strength of the vertical contraction joints is usually quite low, these joints are likely to open during the earthquake shaking. Once the vertical joints are open, the dam blocks would behave as vertical cantilevers, leading to high vertical tensile stresses. Since the tensile strength of 411
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horizontal lift joints is usually substantially lower than that of monolithic mass concrete, a horizontal crack may form, causing the upper portion of a dam block to become fully detached from the rest of the dam. Such a detached portion would be subjected to rather high accelerations and it could thus undergo sliding and rocking motions during the earthquake shaking. As long as a possibly detached portion remains stable during and after the earthquake, an uncontrolled release of reservoir water could be ruled out and the safety requirements for the SEE would be satisfied. To investigate the dynamic stability of possibly detached upper portions in the crest region and to analyze the behavior of vertical contraction joints in an arch dam during an earthquake, nonlinear dynamic analyses have to be carried out. In this paper, two illustrative examples are presented to show how earthquake safety of arch dams can be assessed on the basis of results of 2D and 3D nonlinear dynamic analyses.
Methodology for simplified dynamic stability analysis A simplified analysis of the seismic stability of possibly detached concrete blocks in the central upper portion of an arch dam can be performed in the following steps (Malla and Wieland, 2006; Wieland and Malla, 2012): (i) Linear elastic dynamic time history analysis of a three-dimensional (3D) finite element (FE) model of dam-reservoir-foundation system; (ii) Selection of detached concrete blocks for dynamic stability analysis based on envelopes of principal dynamic tensile stresses and absolute accelerations; (iii) Obtaining time histories of radial and vertical components of absolute acceleration at the base of each detached concrete block from the results of analysis step (i); (iv) Setting up 2D FE model of each detached concrete block using contact elements to simulate the cracked lift joint (gap elements may be employed to prevent the concrete block from moving beyond the downstream face of the dam in view of the geometrical constraints in an arch dam); (v) Nonlinear dynamic analysis of rocking-sliding response of each detached concrete block subjected to input base acceleration obtained in step (iii); (vi) Evaluation of the maximum sliding movement and the maximum crack opening displacements at the upstream and downstream edges of the base of each detached concrete block for at least three different earthquake ground motions; and (vii) Assessment of the dynamic stability of detached concrete blocks based on results of step (vi). The seismic safety of the Roggiasca and Gigerwald arch dams in Switzerland was checked by employing this simplified procedure. The main results of this analysis are presented and discussed in the following sections. Furthermore, nonlinear 3D FE analysis of the Gigerwald arch dam was performed to investigate contraction joint openings during the earthquake. All the dynamic calculations were performed with the help of the general-purpose FE software ADINA (ADINA R & D, 2008). For the dynamic analyses, the three components of earthquake excitation were simulated by artificially-generated spectrum-compatible acceleration time histories. The hydrodynamic pressure was modeled as added masses acting perpendicular to the upstream face of the dam.
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Earthquake safety assessment of Roggiasca arch dam The 68 m high Roggiasca arch dam was completed and first impounded in 1965. With a crest thickness of 2.5 m and a maximum thickness at the base of 7.5 m only, the dam has a relatively high Lombardi slenderness coefficient of 21. At the location of this dam, the 10,000-year SEE has a horizontal PGA of 0.17 g. The linear seismic response of the Roggiasca dam was analyzed using a 3D FE model, which comprised the dam, the foundation rock, the sediment deposit on the upstream side and the soil fill on the downstream side. The main results of this 3D analysis performed for 3 different ground excitations (designated as earthquakes 1, 2 and 3) are listed in Table 1. The largest accelerations and dynamic stresses were obtained at the middle and also at the two quarter points of the dam crest. The results showed that the upstream sediment deposit and the downstream soil fill would not play a significant role in the dynamic behavior of the dam. Table 1: Main results of 3D linear elastic earthquake analysis of Roggiasca dam subjected to SEE ground motion under full reservoir condition Dynamic response (envelope) Relative crest displacement (mm) Along-stream direction Across-stream (left-right) direction Vertical direction Absolute crest acceleration (g) Along-stream direction Across-stream (left-right) direction Vertical direction Principal tensile stress (MPa) Principal compressive stress (MPa)
Earthquake 1
Earthquake 2
Earthquake 3
16.2 8.4 2.0
15.9 7.9 2.1
18.7 8.2 2.1
1.45 0.62 0.41
1.34 0.64 0.39
1.27 0.72 0.35
6.1
5.6
6.6
-6.3
-6.0
-6.2
The magnitude of the largest dynamic tensile and compressive stresses in the crest region of the dam is about 6 MPa (see Table 1 and Figure 1). Even higher elastic stresses are computed at the upstream and downstream edges of the dam-rock interface due to the stress singularities at these reentrant corners.
Figure 1: Principal stress vectors due to earthquake 1 (without static loads) at time t = 7.19 s when the highest tensile stress occurs under full reservoir condition 413
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The largest compressive stress in the dam body under the combination of the static and dynamic loads is about -10 MPa, which is not a problem for the dam concrete with a static compressive strength of 46 MPa. In spite of the compressive stresses due to the static loads (self-weight and hydrostatic pressure), relatively high horizontal tensile stresses of up to about 4 MPa oriented in the arch direction still occur in the crest region during the earthquake shaking. If the thermal stresses in winter and the effect of the ongoing chemical expansion of the dam concrete would also be considered, the tensile stresses would be even higher. In order to investigate the stability of possibly detached concrete blocks during the earthquake excitation, simplified 2D FE models shown in Figure 2 were employed. As the Roggiasca arch dam is rather thin, any detached block in the crest region would be quite slender.
Figure 2: 2D FE models of 7 m and 13 m high detached blocks in crest region of Roggiasca arch dam (gap elements prevent any movement beyond the downstream face) In a dynamic analysis involving rigid body motions, it is considered prudent to use only the stiffness-proportional part of the Rayleigh damping model, as the mass-proportional part corresponds to external viscous dampers connected to the nodes of the model (Hall, 2006). Three different Rayleigh damping models were considered in the stability analysis (see Table 2). Model A corresponds to that used in the 3D linear dynamic analysis. In model B, only the stiffness-proportional part is kept. In model C, the stiffness-proportional part is further substantially reduced, which is very conservative. Table 2: Rayleigh damping models used for 2D dynamic stability analysis of detached cantilever blocks in crest region of Roggiasca dam Rayleigh damping model
Parameter -1
Parameter (s)
A
(s ) 2.20
B
0.00
0.00124
C
0.00
0.00020
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0.00124
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The results of the dynamic stability analysis listed in Table 3 show that the detached portions in the crest region have a tendency to undergo only rocking with virtually no sliding, a behavior explained by the slender form. Even the rocking motion is quite small and causes dynamic crack openings of only a few millimeters at the base of the detached portion. Table 3: Main results of 2D dynamic stability analysis of detached portions in crest region of Roggiasca dam subjected to SEE ground motion under full reservoir condition Rayleigh damping model
Earthquake
Max. horiz. crest displacement (mm)
Max. sliding displacement (mm)
(a) 13 m high detached block above crack at level 942 m a.s.l. A 1 -4 0.0
Max. crack opening, u/s face (mm)
Max. crack opening, d/s face (mm)
0.0
0.7
B
1
-5
0.0
0.0
0.8
C
1
-6
0.0
0.0
1.1
B
2
-11
0.0
0.0
2.2
C
2
-14
0.0
0.0
3.0
B
3
-11
0.0
0.0
2.2
C
3
-17
-0.2
0.0
3.8
(b) 7 m high detached block above crack at level 948 m a.s.l. A
1
-15
-0.1
0.0
5.6
B
1
-16
-0.1
0.0
6.1
C
1
-79
-0.2
0.1
30.7
B
2
-22
-0.1
0.0
8.5
C
2
-128
-0.4
0.0
49.8
B
3
-25
-0.1
0.0
9.7
C
3
-171
-0.7
0.1
66.7
The results of the dynamic stability analysis show that the detached portions remain dynamically stable even when subjected to peak horizontal accelerations about twice as large as the pseudo-static overturning acceleration. This behavior can be explained by the fact that acceleration spikes would have a relatively high frequency of about 5 Hz corresponding to the dominant natural frequency of the dam. Hence, such a spike would exceed the pseudo-static overturning acceleration only for a very short duration of less than one-tenth of a second, which is too short to produce any significant block rotation. A review of literature on dynamic overturning of rigid blocks also confirms that acceleration peaks would have to be many times larger than the pseudo-static overturning acceleration for a block with dimensions of the order of a few meters to be toppled by a dynamic base excitation at such a frequency, as illustrated in Figure 3 (Shi et al., 1996; Zhang and Makris, 2001).
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Figure 3: Boundaries between stable and unstable regions of dynamic rocking motion based on analytical solutions for half and full sinusoidal base excitations (Shi et al., 1996, Zhang und Makris, 2001) Note: A ground excitation with an amplitude of 1.5 g and a frequency of 5 Hz corresponds to the red square (p/p = 22.4 und ap/g = 3.2) and the red triangle (p/p = 29.6 und ap/g = 4.5) in the case of, respectively, the 7 m and 13 m high detached portions.
Earthquake safety assessment of Gigerwald arch dam The 147 m high Gigerwald dam was completed and first impounded in 1976. The dam thickness varies from 7 m at the crest to about 22 m at the base. The main results of the linear elastic dynamic analysis of a 3D FE model of this doublecurvature arch dam for 3 different earthquakes are listed in Table 4. The crest region would be subjected to amplified horizontal accelerations as high as about 2.0 g during the 10,000-year SEE with a horizontal PGA of 0.19 g. Table 4:Main results of 3D linear elastic earthquake analysis of Gigerwald dam subjected to SEE ground motion under full reservoir condition Dynamic response (envelope) Relative crest displacement (mm) Along-stream direction Across-stream (left-right) direction Vertical direction Absolute crest acceleration (g) Along-stream direction Across-stream (left-right) direction Vertical direction Principal tensile stress (MPa) Principal compressive stress (MPa)
Earthquake 1
Earthquake 2
Earthquake 3
48.0 15.3 6.6
45.6 16.0 7.4
44.5 18.0 6.9
1.83 0.84 0.75
2.04 0.86 0.78
1.98 0.80 0.71
8.7
9.8
8.7
-10.3
-9.5
-9.6
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The earthquake shaking produces dynamic tensile stresses of nearly 10 MPa in the central crest region of the dam. Even after the combination with the compressive stresses due to the hydrostatic water load, high horizontal tensile stresses of up to about 6 MPa would still remain in the arch direction. In reality, such tensile stresses cannot develop due to the presence of the vertical contraction joints, whose tensile strength is normally quite low. The next step was to perform the dynamic analysis of a 3D FE model in which all 23 vertical contraction joints were simulated as frictional contact surfaces and the dam concrete was assumed to be linear elastic (uncracked). This analysis was performed for 3 different input motions. Figures 4 to 6 depict some results obtained for earthquake 2. This analysis showed that the seismic shaking would cause relative sliding displacements of nearly 1 cm between the adjacent blocks and the contraction joints would open by maximum about 4 mm. The largest compressive stress in the dam obtained in the nonlinear analysis approaches nearly -19 MPa (excluding the corner singularity at the dam-rock interface), which is not a problem for the dam concrete. In comparison, the maximum compressive stress in the case of the linear analysis is about -16 MPa. In spite of the joint displacements, the dynamic displacements and accelerations of the dam computed in the nonlinear analysis do not deviate significantly from those in the corresponding linear analysis. The main difference lies in the absence of any significant horizontal tensile stresses in the arch direction in the central crest region in the case of the nonlinear analysis owing to the presence of the contraction joints. During the brief openings of the contraction joints, the upper portion of a dam block acts temporarily as a cantilever, due to which relatively high transitory vertical tensile stresses exceeding 6 MPa appear on the downstream face of the dam, as depicted in Figure 6. Hence, horizontal cracks are likely to form, especially at the lift joints, possibly resulting in the detachment of the uppermost portion of a central block from the rest of the dam body.
Figure 4: Time histories of opening and sliding displacements of vertical contraction joint 11/12 at crest level due to earthquake 2 under full reservoir condition 417
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Figure 5:Maximum opening and sliding displacements of vertical contraction joints at crest level due to earthquake 2 under full reservoir condition
Figure 6: Stresses due to cantilever behavior during opening of vertical contraction joints The safety of possibly detached portions in a central dam block subjected to the SEE shaking was assessed using simplified 2D models (see Figure 7). This analysis showed that an 8 m high detached portion could slide by up to about 50 cm towards the reservoir during the SEE and the rocking motion would result in crack opening displacements of up to about 7 cm. However, the detached block would remain stable during and after the earthquake and the earthquake damage would not lead to an uncontrolled release of water.
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Figure 7: 2D FE model of 8 m high detached portion of a central block of Gigerwald arch dam (gap elements prevent any movement beyond the downstream face) For a less conservative dynamic stability assessment, a nonlinear analysis was also performed using a 3D model in which the uppermost 8 m of the central dam block was assumed to be detached from the rest of the dam along the vertical contraction joints at the sides and an assumed horizontal crack along a lift joint. As shown in Figure 8, the 3D analysis showed that such a detached block would slide by up to 16 cm towards the reservoir, which is only about one-third of the result obtained in the corresponding more conservative 2D analysis. The substantially smaller sliding displacement in the 3D analysis can be attributed mainly to the additional frictional resistance at the vertical contraction joint on each side, an effect that could not be taken into account in the simplified 2D approach. The maximum crack opening also decreased to about 3 cm in the 3D analysis.
Figure 8: Sliding displacement of 8 m high detached concrete block in central cantilever of Gigerwald arch dam subjected to earthquake 2 under full reservoir condition
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Conclusions The results of the earthquake safety assessment of the Roggiasca and Gigerwald arch dams can be summed up as follows: 1. The ground excitation during the 10,000-year SEE produces relatively high accelerations and tensile stresses in the central upper portion of a large arch dam. 2. The tensile strength of the vertical contraction joints and the horizontal lift joints is usually significantly lower than that of the monolithic concrete. Thus, these joints may open during the earthquake shaking, possibly resulting in the upper portion of a dam block becoming fully detached from the rest of the dam. 3. Any detached uppermost part of a dam block in the investigated dams may be subjected to some sliding and rocking motions, but it will remain stable during and after the earthquake. 4. The earthquake loading could cause limited damages in the analyzed dams, but it will not lead to an uncontrolled release of reservoir water. Hence, the investigated dams satisfy the safety requirements for the SEE.
References [1] ADINA R & D (2008). ADINA User Interface, Command Reference Manual, Vol. I: ADINA Solids & Structures Model Definition, Report ARD 08-2, Watertown, Massachusetts, USA. [2] Hall, J.F. (2006). Problems encountered from the use (or misuse) of Rayleigh damping. Earthquake Engineering and Structural Dynamics. Vol. 35, pp. 525-545. [3] Malla, S. and Wieland, M. (2006). Dynamic stability of detached concrete blocks in arch dam subjected to strong ground shaking. Proceedings of the 1st European Conference on Earthquake Engineering and Seismology (ECEES), Geneva, Switzerland, 3-8 September 2006. [4] Panduri, R., Droz, P., Malla, S., Radogna, R., Wieland, M. and Darbre, G.R. (2012). Ongoing seismic safety assessment of Swiss dams. Proceedings of the 24th International Congress on Large Dams, ICOLD, Kyoto, Japan, 6-8 June 2012. [5] Shi, B., Anooshehpoor, A., Zeng, Y. and Brune, J.N. (1996). Rocking and overturning of precariously balanced rocks by earthquake. Bulletin of the Seismological Society of America, Vol. 86, No. 5., pp. 1364-1371. [6] Wieland, M. and Malla, S. (2012). A simple method for seismic stability analysis of detached concrete blocks and estimation of contraction joint openings in large arch dams. Hydropower and Dams, Vol. 19, Issue 3, pp. 113-119. [7] Zhang, J. and Makris, N. (2001). Rocking response of free-standing blocks under cycloidal pulses. Journal of Engineering Mechanics, ASCE, Vol. 127, No. 5, pp. 473-483.
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proceedings 12th International
BENCHMARK WORKSHOP
ON NUMERICAL ANALYSIS OF DAMS, 2nd - 4th October, 2013, Graz – AUSTRIA Edited by
Gerald Zenz and Markus Goldgruber
Heft / Volume 39
proceedings ICOLD 12th benchmark workshop
I C O L D
Austrian National Committee on Large Dams Stremayrgasse 10/II, A-8010 Graz, AUSTRIA, Phone: ++43/316/8861, [email protected], www.atcold.at
Published by the Austrian National Committee on Large Dams with the support of the Austrian Reservoir Commission (Staubeckenkommission)
BENCHMARK WORKSHOP
ON NUMERICAL ANALYSIS OF DAMS, 2nd - 4th October, 2013, Graz – AUSTRIA Edited by
Gerald Zenz and Markus Goldgruber
Heft / Volume 39
proceedings ICOLD 12th benchmark workshop
I C O L D
Austrian National Committee on Large Dams Stremayrgasse 10/II, A-8010 Graz, AUSTRIA, Phone: ++43/316/8861, [email protected], www.atcold.at
Published by the Austrian National Committee on Large Dams with the support of the Austrian Reservoir Commission (Staubeckenkommission)
PROCEEDINGS OF THE ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS 2nd - 4th OCTOBER, 2013, GRAZ – AUSTRIA
Edited by
Gerald Zenz Institute of Hydraulic Engineering and Water Resources Management Graz University of Technology
Markus Goldgruber Institute of Hydraulic Engineering and Water Resources Management Graz University of Technology
Published by
ATCOLD – AUSTRIAN NATIONAL COMMITTEE ON LARGE DAMS © 2014
All rights reserved
The authors are responsible for the content of their contributions. The texts of the various papers in this volume were set individually by typists under the supervision of each of the authors concerned.
Cover pictures copyright by Graz University of Technology Layout, Design, Cover Artwork by Institute of Hydraulic Engineering and Water Resources Management Printed by Graz University of Technology
Preface For the International Commission on Large Dams (ICOLD) the “Committee on Computational Aspects of Analysis and Design of Dams” is the general Organizer of Benchmark Workshops. This 12th Benchmark Workshop is held in the city of Graz, situated in the South of the Alps on both sides of the river Mur. Six universities with over 40,000 students are in addition responsible for the young and dynamic charm of our city. Since over 200 years Graz University of Technology is one of them. Advanced numerical tools with user friendly interfaces are available and widely used for structural analyses. Such numerical analyses require a solid theoretical background of the applicability of methods to be used. On the other hand, the results gained need a careful interpretation with respect to the underlying assumptions and their practical relevance. ICOLD Benchmark examples of generalized engineering problems are devoted to bridge the gap between numerical analyses, the interpretation of results and their theoretical as well as practical relevance. Since 1991 eleven benchmark workshops were organized for different numerical problems in the field of concrete and fill dams under static and dynamic loading conditions. The results of these benchmark workshops are made available to the dam engineering community on the internet and in proceeding. Results are published in ICOLD bulletins:
Bulletin N 94 Computer Software for Dams Bulletin N 122 Computational Procedures for Dam Engineering Bulletin N 155 Guidelines for Use of Numerical Models in Dam Engineering
This 12th Benchmark Workshop provides an excellent opportunity for engineers, scientists and operators to present and exchange their experiences and the latest developments related to the design, performance and monitoring of dams. There are three example topics and an open theme formulated and discussed: Theme A: Formulators:
Fluid Structure Interaction, Arch Dam - Reservoir at Seismic Loading Gerald Zenz, Markus Goldgruber
Theme B: Formulators:
Long Term Behavior of Rockfill Dams Camilo Marulanda, Joan Manuel Larrahondo
Theme C: Formulators:
Computational Challenges in Consequence Estimation for Risk Assessment Yazmin Seda-Sanabria, Enrique E. Matheu, Timothy N. McPherson
The support to this Benchmark Workshop from the Members of the Committee on Computational Aspects of Analysis and Design of Dams and especially from the Formulators is gratefully acknowledged. The contributions from the Core Organizing Team – Markus Goldgruber and Harald Breinhälter – are very much appreciated. Finally, I want to thank the participants for their scientific contributions herein. Gerald Zenz Chairman
Table of Content Conference Organizations
9
Acknowledgements
11
Overview of Contributions Theme A
13
Fluid Structure Interaction: Arch Dam ‐ Reservoir at Seismic Loading - Benchmark Problem Description
15
- Result Comparison of the Participants
27
- Papers
65
G. Maltidis and L. Stempniewski
67
W. Kikstra, F. Sirumbal and G. Schreppers
77
G. Faggiani and P. Masarati
87
A. Tzenkov, A. Abati and G. Gatto
99
M. Chambart, T. Menouillard, N. Richart, J.-F. Molinari and R. M. Gunn
111
A. Popovici, C. Ilinca and R. Vârvorea
123
R. Malm, C. Pi Rito, M. Hassanzadeh, C. Rydell and T. Gasch
139
M. Brusin, J. Brommundt and H. Stahl
149
S. Shahriari
165
A. Frigerio and G. Mazzà
167
A. Diallo and E. Robbe
177
Theme B
189
Long Term Behavior of Rockfill Dams - Benchmark Problem Description
191
- Papers
201
F. Ezbakhe and I. Escuder-Bueno
203
5
Table of Content Theme C
211
Computational Challenges in Consequence Estimation for Risk Assessment - Benchmark Problem Description
213
- Result Comparison of the Participants
227
- Papers
239
M. Davison, M. Hassan, O. Gimeno, M. van Damme and C. Goff
241
M.S. Altinakar, M.Z. McGrath, V.P. Ramalingam, D. Shen, Y. Seda Sanabria and E.E. Matheu 255 L. Mancusi, L. Giosa, A. Cantisani, A. Sole and R. Albano
271
O. Saberi, C. Dorfmann and G.Zenz
283
D. McVan, J. Ellis, G. Savant and M. Jourdan
293
B. A. Thames and A. J. Kalyanapu
309
D. Williams and K. Buchanan
325
Open Theme
339
Choice of Contributor - Papers
339
Behavior of an arch dam under the influence of creep, AAR and opening of the dam/foundation contact E. Robbe
341
Need for transient thermal models, with daily inputs, to explain the displacements of arch gravity dams I. Escuder, D. Galán and A. Serrano
349
The rehabilitation of Beauregard Dam: the contribution of the numerical modeling A. Frigerio and G. Mazzà
359
Earthquake Assessment of Slab and Buttress Dams H. B. Smith and L. Lia
369
Solution of dam-fluid interaction using ADAD-IZIIS software V. Mircevska, M. Garevski, I. Bulajic and S. Schnabl
379
Influence of Surface Roughness on Sliding Stability Tests and numerical modeling Ø. Eltervaag, G. Sas and L. Lia
389
Seismic Analysis of a Concrete Arch Dam P. Dakoulas
399
Earthquake safety assessment of arch dams based on nonlinear dynamic analyses S. Malla
411
7
Conference Organizations The conference was organized by Graz University of Technology, Graz, Austria Conference Chairman Gerald Zenz (Graz University of Technology, Austria) Committee on Computational Aspects of Analysis and Design of Dams Chairman I. Escuder‐Bueno (Spain) Vize‐Chairman G. Mazza (Italy) Technical Advisory Team RESTELLI, F. (Argentina) ZENZ, G. (Austria) CURTIS, D. (Canada) CHEN, S. (China) MARULANDA, C. (Colombia) VARPASUO, P. (Finland) TANCEV, L. (Form. Yug. Rep. of Macedonia) FROSSARD, E. (France) BEETZ, U. (Germany) DAKOULAS, P. (Greece) NOORZAD, A. (Iran)
MEGHELLA, M. (Italy) UCHITA, Y. (Japan) ANDERSEN, R. (Norway) POPOVICI, A. (Romania) GLAGOVSKY, V. (Russia) MINARIK, M. (Slovakia) HASSANZADEH, M. (Sweden) GUNN, R. (Switzerland) MATHEU, E. (United States) CARRERE, A. (Honorary Member) (France) FANELLI, M. (Honorary Member) (Italy)
Local Organizing Committee Graz University of Technology Institute of Hydraulic Engineering and Water Resources Management Head Gerald Zenz Organizer Markus Goldgruber
9
Acknowledgements The editors are grateful to the members of the following organizations for their support: ATCOLD ICOLD Benchmark Workshop Problem Formulation Teams International Water Power & Dam Construction
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
THEME A
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Theme A Fluid Structure Interaction Arch Dam – Reservoir at Seismic loading
Formulators: Dipl. Ing. Markus Goldgruber Univ. Prof. Dipl. Ing. Dr. techn. Gerald Zenz
Institute of Hydraulic Engineering and Water Resources Management Graz University of Technology
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Introduction Advanced numerical tools with user friendly interfaces are available for structural analyses. Such numerical analyses require a solid theoretical background of the applicability of methods to be used. On the other hand, the results gained need a careful interpretation with respect to the underlying assumptions and their practical relevance. ICOLD Benchmark examples of generalized engineering problems are devoted to bridge the gap between numerical analyses, the interpretation of results and their theoretical as well as practical relevance. Challenges of the analyses of concrete dams are always the definition of material parameters, the spatial discretization and the appropriate simulation of loading sequences. Additionally, specific attention is paid on the structural integrity and entire safety under seismic loading conditions. To account for this problem, the interaction of the dam and the reservoir is topic of this theme A. By means of the Finite Element Method linear and nonlinear analyses under dynamic excitation are carried out. However, for the required and appropriate simulation of the dam reservoir interaction different approaches are used. With respect to future nonlinear dynamic analyses, these simulations herein shall be in the time domain only. A common approach to take the dynamic water interaction into account is to use an added mass approach. A more sophisticated possibility is the use of Acoustic or Fluid Elements. The simulations of earthquake excitation of arch dams have shown that the analyzed stresses in the structure could vary significantly based on the interaction modeling. The added mass approach is still a widely used technique but tends to overestimate the stresses and therefore it is conservative in contrary to other techniques. This benchmark now intends to compare different modeling techniques and will show the amount of deviations. All investigations are carried out for an artificially generated symmetric arch dam and simplified loading and boundary conditions. Universities, engineering companies and regulatory bodies are invited to contribute to the benchmark and take part in the discussion of results gained. Focus of this benchmark example The focus of this benchmark is to carry out the Dynamic Fluid Structure Interaction for a large arch dam. Every participant may choose his own order of details in modeling. The main goal of this example is the application of different approaches like: Added mass technique (Westergaard, Zangar,…) Acoustic Elements (compressible, incompressible) Fluid Elements (compressible, incompressible) Further on, the usage of different Boundary Conditions is possible for: Reservoir - Foundation o Reflecting (on the bottom and the sides) o Non-reflecting (at the end of the reservoir) The modeling of the block joint opening – due to tensile stresses and nonlinear effects - is not focus of this benchmark example. However, to carry out this analysis in the time domain will provide the opportunity for further non-linear analyses.
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General basic assumptions The following general basic assumptions and boundary conditions for the investigations should be used: Same spatial discretization (Model/Mesh) of the Structure, Foundation and Reservoir Same Material Parameters Acceleration-Time-History in X-,Y-,Z-Direction Reservoir is infinite in length (non-reflecting) Rayleigh Damping Results to be compared – Visualization Based on these basic assumptions and results gained the contributors are encouraged to intensify and focus their effort to achieve results with higher profound physical justification and explain the differences. (E.g.: different spatial discretization, more appropriate modeling of the interaction; different length of the reservoir; need for nonlinear effects). An interpretation of the evaluated results from an engineering point of view should be given.
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Model and Geometry An Arch Dam, Foundation and Reservoir Model layout for the benchmark has been generated and is available for downloading. Arch Dam Model Symmetric Geometry Total Height: 220 Meters Valley width (crest): ~ 430 Meters Valley width (bottom): ~ 80 Meters Arch Dam Geometry The Arch Dam Geometry has been generated with the Program “Arch Dam Design”, which was developed as part of the Master-Thesis by DI Manuel Pagitsch.
Arch Dam Model
Plan View
View from the upstream
Main Section
Figure 1: Arch dam geometry Foundation Model Symmetry is used for the foundation too. 500 Meters Height: Length: 1000 Meters Width 1000 Meters
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500m
1000m
1000m Figure 2: Foundation geometry Reservoir Model Length: assumed minimum of 460 Meters (> 2x Height of the Dam) Modeling the interaction with Acoustic- or Fluid Elements
Figure 3: Reservoir geometry
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Acceleration Time History Transient Acceleration (amax ≈ 0.1g) X-,Y-,Z- Direction Artificially generated time history
Figure 4: Reservoir geometry
Material Parameters The Material properties are defined for isotropic and homogenous conditions. Rock mass Density: 0 kg/m3 Poisson - ratio: 0,2 Youngs - modulus: 25000 MPa Water Density: 1000 kg/m3 Bulk - modulus: 2200 MPa Dam
Density: 2400 kg/m3 Poisson - ratio: 0,167 Youngs - modulus: 27000 MPa
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Mesh Properties Two different Meshes of the entire system are provided for investigations, as these are a coarse and a fine mesh. If desired, the parts can also be provided as ABAQUS/CAE Model File, ACIS- or IGES-Files, if a specific mesh is intended to be discretized. Coarse Mesh Arch Dam Total number of nodes: 2083 Total number of elements: 356 312 quadratic hexahedral elements of type C3D20R (ABAQUS CAE) 44 quadratic wedge elements of type C3D15 (ABAQUS CAE) Foundation Total number of nodes: 11608 Total number of elements: 2340 quadratic hexahedral elements of type C3D20R (ABAQUS CAE) Reservoir Total number of nodes: 12493 Total number of elements: 2640 quadratic hexahedral elements of type C3D20R (ABAQUS CAE)
Figure 5: Coarse mesh of the dam, foundation and reservoir
21
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Fine Mesh Arch Dam Total number of nodes: 13733 Total number of elements: 2736 quadratic hexahedral elements of type C3D20R (ABAQUS CAE) Foundation Total number of nodes: 13298 Total number of elements: 2700 quadratic hexahedral elements of type C3D20R (ABAQUS CAE) Reservoir Total number of nodes: 12493 Total number of elements: 2640 quadratic hexahedral elements of type C3D20R (ABAQUS CAE)
Figure 6: Fine mesh of the dam, foundation and reservoir
22
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Elements and Node Numbering in ABAQUS/CAE The provided input-files are containing a list of the nodes and elements of the mesh and also predefined “node sets” for the different sections which should be investigated. The Node numbering of ABAQUS/CAE is plotted in the following figures.
Figure 7: Node numbering of wedge and brick elements in ABAQUS/CAE These figures are showing the node numbering for the two different element types which are used in the provided input-files.
23
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Loading The following loading sequence is intended to be used. Gravity Hydrostatic Water Pressure (full supply water level = Crest Height) Seismic Loading o Modal Superposition or o Direct time integration (Implicit/Explicit)
Results Following results should be evaluated and plotted. Eigenfrequencies (1 – 10) The evaluation of the first 10 Eigenfrequencies of the structure, including the interaction with the reservoir, should be provided. Mode Shapes (1 – 10) The evaluation and plotting of the first 10 Mode-Shapes of the structure, including the interaction with the reservoir, should be provided. Hoop Stresses, Vertical Stresses and Min./Max. Principal Stresses Evaluation of the different stresses should be done for Static Loads Seismic Loads (Min., Max.) 3 different sections (Main Section and ~45 degrees on the left and right hand side)
Left Section
Main Section
Right Section
Figure 8: Evaluation sections of the arch dam
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Figure 9: Evaluation examples for the stresses Radial Deformation Evaluation of the Radial Deformation should be done for Static Loads Seismic Loads (Min., Max.) Main section
Figure 10: Evaluation examples for the radial deformation
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
RESULT COMPARISON OF THE PARTICIPANTS THEME A
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Results Comparison Theme A Fluid Structure Interaction Arch Dam – Reservoir at Seismic loading Participants, Programs and Approaches Overall there are 11 participants from 9 different countries (Swiss, Netherlands, France, Germany, Sweden, Italy, Iran, Romania and Austria) who were contributing to the workshop and decided to solve the problem. Each participant had the opportunity to choose his preferred numerical program and modelling technique to account for the fluid structure interaction. The prevailing boundary conditions, which are the same for each participant, are defined in the section “Benchmark Problem Description Theme A”. The “Reference Solution” (REF) in the diagrams and tables doesn’t claim to be the optimum solution. It shows the results of the simulations done at the Institute of Hydraulic Engineering and Water Resources Management by Markus Goldgruber. The following table lists the participants and their used programs and approaches. The informations in the last column should point out some specific differences between the participants which may influence the results and are worth mentioning. Some of the participants have provided results of more than just one simulation, but for the comparison just one of these has been used. All the other results of the approaches and models can be found in their papers in the following section (Papers – Theme A) All participants had to evaluate eigenfrequencies, mode shapes, deformations and stresses. In the case of dynamic simulations one will get minimum and maximum values. Therefore, in the diagrams in the results section every participant has three lines, the minimum (left line) and the maximum line (right line) which indicate the minimum and maximum values out of the time history records and the line for static loading (middle line). This middle line indicates the static value out of the sum of the two load cases, gravity and hydrostatic water load. To retain the overview in the diagrams, the minimum, maximum and static values are not explicitly mentioned in the legend. The diagrams for the deformations and the upstream hoop stresses in the main section were plotted additionally for static and dynamic loading separately.
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Table 1: Participants, Programs and Approaches Finite Element Method Program Acoustic ABAQUS Elements
Mesh
A
G. Maltidis
B
W.Kikstra, F. Sirumbal, G. Schreppers
DIANA
Acoustic Elements
Coarse
C
G. Faggiani, P. Masarati
CANT-SD
Acoustic Elements
Coarse
D
A. Tzenkov, A. Abiati, G. Gatto
DIANA
Acoustic Elements
Coarse
E
M. Chambart
DIANA
Fine
Edyn = Esta * 1.25
F
A. Popovici, C. Ilinca, R. Vârvorea
Added mass (Westergaard)
Same as participant B, Construction steps for loadcase deadweight
ABAQUS
Added mass (Westergaard)
Coarse
G
R. Malm, C. Pi Rito, C. Hassanzadeh, C. Rydell, T. Gasch
ABAQUS
Acoustic Elements
Fine
H
M. Brusin
FENAS ECCON IPP
Added mass (Westergaard)
Fine
I
S. Shahriari
ANSYS
Added mass (Westergaard)
Coarse
J
A. Frigerio, G. Mazzà
COMSOL
Acoustic Elements
Coarse
A.Diallo, E. Robbe
Incompressible CODE_ASTER Finite Element Coarse added mass
K
REF M. Goldgruber
ABAQUS
Acoustic Elements
30
Coarse
Some Additional Informations 7.5% critical damping Compressible Fluid, Hybrid FrequencyTime Domain (HFTD) method
Coarse
Infinite Elements at the boundaries, Acceleration-TimeHistory applied on the bottom of the model Construction steps for loadcase deadweight Use of the full Westergaard formula (Period/Frequency dependent) Method to calculate the added mass matrices representing the fluid-structure interaction with a potential approach
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Results Eigenfrequencies
Figure 1: Eigenfrequencies 1 – 10(Column Chart) Table 2: Eigenfrequencies 1 – 10 (Table) Mode
Participant A B C D E F G H I J K REF
1 1.47 1.57 1.54 1.57 1.43 1.54 1.51 1.26 1.28 1.54 1.57 1.54
2 1.54 1.60 1.55 1.62 1.47 1.56 1.54 1.32 1.33 1.55 1.62 1.54
3 1.55 2.36 2.05 2.36 2.21 1.93 1.90 2.01 1.91 2.09 2.35 2.05
4 2.11 2.94 2.22 2.94 2.61 2.30 2.22 2.36 2.37 2.22 2.95 2.29
5 2.33 3.04 2.41 3.04 2.81 2.48 2.42 2.50 2.38 2.33 3.03 2.54
31
6 2.46 3.72 2.83 3.72 3.27 3.04 2.96 3.00 2.91 2.51 3.72 2.96
7 2.61 3.88 2.98 3.87 3.56 3.12 3.01 3.17 2.98 2.83 3.85 3.21
8 2.97 4.56 3.37 4.56 4.09 3.29 3.28 3.65 3.61 2.96 4.56 3.36
9 3.25 4.78 3.40 4.76 4.37 3.61 3.59 3.70 3.62 3.19 4.88 3.76
10 3.37 4.80 3.79 4.80 4.37 3.71 3.76 3.88 3.85 3.37 5.13 3.91
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Mode Shapes
Figure 2: Mode Shapes 1 – 10; Participants A – E 32
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Figure 3: Mode Shapes 1 – 10; Participants F – K
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HEIGHT (m)
Deformations – Main Section
220 200 180 160 140 120 100 80 60 40 20 0 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
DEFORMATION (m) A
B
C
D
E
REF
HEIGHT (m)
Figure 4: Deformation – Main Section (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
DEFORMATION (m) F
G
H
I
J
K
Figure 5: Deformation – Main Section (F – K)
34
REF
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Deformations (Only Static Load) – Main Section
220 200 180 160 140 120 100 80 60 40 20 0 0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
DEFORMATION (m) A
B
C
D
E
REF
HEIGHT (m)
Figure 6: Deformation (Only Static Load) – Main Section (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
DEFORMATION (m) F
G
H
I
J
K
REF
Figure 7: Deformation (Only Static Load) – Main Section (F – K)
35
0.09
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Deformations (Only Dynamic Load) – Main Section
220 200 180 160 140 120 100 80 60 40 20 0 -0.15 -0.13 -0.11 -0.09 -0.07 -0.05 -0.03 -0.01 0.01 0.03 0.05 0.07 0.09 0.11
DEFORMATION (m) A
B
C
D
E
REF
HEIGHT (m)
Figure 8: Deformation (Only Dynamic Load) – Main Section (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -0.15 -0.13 -0.11 -0.09 -0.07 -0.05 -0.03 -0.01 0.01 0.03 0.05 0.07 0.09 0.11
DEFORMATION (m) F
G
H
I
J
K
REF
Figure 9: Deformation (Only Dynamic Load) – Main Section (F – K)
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Hoop Stresses – Main Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -14.00 -12.00 -10.00 -8.00
-6.00
-4.00
-2.00
0.00
2.00
4.00
6.00
4.00
6.00
HOOP STRESS (MPa) A
B
C
D
E
REF
HEIGHT (m)
Figure 10: Hoop Stresses – Main Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -14.00 -12.00 -10.00 -8.00
-6.00
-4.00
-2.00
0.00
2.00
HOOP STRESS (MPa) F
G
H
I
J
K
REF
Figure 11: Hoop Stresses – Main Section – Upstream (F – K)
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Hoop Stresses (Only Static Load) – Main Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -8.00
-7.00
-6.00
-5.00
-4.00
-3.00
-2.00
-1.00
0.00
1.00
2.00
HOOP STRESS (MPa) A
B
C
D
E
REF
HEIGHT (m)
Figure 12: Hoop Stresses (Only Static Load) – Main Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -8.00
-7.00
-6.00
-5.00
-4.00
-3.00
-2.00
-1.00
0.00
1.00
2.00
HOOP STRESS (MPa) F
G
H
I
J
K
REF
Figure 13: Hoop Stresses (Only Static Load) – Main Section – Upstream (F – K)
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Hoop Stresses (Only Dynamic Load) – Main Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -10.00 -8.00 -6.00 -4.00 -2.00
0.00
2.00
4.00
6.00
8.00 10.00
HOOP STRESS (MPa) A
B
C
D
E
REF
HEIGHT (m)
Figure 14: Hoop Stresses (Only Dynamic Load) – Main Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -10.00 -8.00 -6.00 -4.00 -2.00
0.00
2.00
4.00
6.00
8.00 10.00
HOOP STRESS (MPa) F
G
H
I
J
K
REF
Figure 15: Hoop Stresses (Only Dynamic Load) – Main Section – Upstream (F – K)
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Hoop Stresses – Main Section – Downstream
220 200 180 160 140 120 100 80 60 40 20 0 -10.00
-7.50
-5.00
-2.50
0.00
2.50
5.00
HOOP STRESS (MPa) A
B
C
D
E
REF
HEIGHT (m)
Figure 16: Hoop Stresses – Main Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -10.00
-7.50
-5.00
-2.50
0.00
2.50
HOOP STRESS (MPa) F
G
H
I
J
K
REF
Figure 17: Hoop Stresses – Main Section – Downstream (F – K)
40
5.00
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Hoop Stresses – Left Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -10.00
-8.00
-6.00
-4.00
-2.00
0.00
2.00
4.00
6.00
HOOP STRESS (MPa) A
B
C
D
E
REF
HEIGHT (m)
Figure 18: Hoop Stresses – Left Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -10.00
-8.00
-6.00
-4.00
-2.00
0.00
2.00
4.00
HOOP STRESS (MPa) F
G
H
I
J
K
REF
Figure 19: Hoop Stresses – Left Section – Upstream (F – K)
41
6.00
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Hoop Stresses – Left Section – Downstream
220 200 180 160 140 120 100 80 60 40 20 0 -12.00
-9.50
-7.00
-4.50
-2.00
0.50
3.00
HOOP STRESS (MPa) A
B
C
D
E
REF
HEIGHT (m)
Figure 20: Hoop Stresses – Left Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -12.00
-9.50
-7.00
-4.50
-2.00
0.50
HOOP STRESS (MPa) F
G
H
I
J
K
REF
Figure 21: Hoop Stresses – Left Section – Downstream (F – K)
42
3.00
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Hoop Stresses – Right Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -9.00
-7.00
-5.00
-3.00
-1.00
1.00
3.00
5.00
HOOP STRESS (MPa) A
B
C
D
E
REF
HEIGHT (m)
Figure 22: Hoop Stresses – Right Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -9.00
-7.00
-5.00
-3.00
-1.00
1.00
3.00
5.00
HOOP STRESS (MPa) F
G
H
I
J
K
REF
Figure 23: Hoop Stresses – Right Section – Upstream (F – K)
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Hoop Stresses – Right Section – Downstream
220 200 180 160 140 120 100 80 60 40 20 0 -12.00
-10.00
-8.00
-6.00
-4.00
-2.00
0.00
2.00
HOOP STRESS (MPa) A
B
C
D
E
REF
HEIGHT (m)
Figure 24: Hoop Stresses – Right Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -9.00
-7.00
-5.00
-3.00
-1.00
1.00
3.00
5.00
HOOP STRESS (MPa) F
G
H
I
J
K
REF
Figure 25: Hoop Stresses – Right Section – Downstream (F – K)
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Vertical Stresses – Main Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -6.00
-4.00
-2.00
0.00
2.00
4.00
6.00
8.00
10.00
VERTICAL STRESS (MPa) A
B
C
D
E
REF
HEIGHT (m)
Figure 26: Vertical Stresses – Main Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -6.00
-4.00
-2.00
0.00
2.00
4.00
6.00
8.00
VERTICAL STRESS (MPa) F
G
H
I
J
K
REF
Figure 27: Vertical Stresses – Main Section – Upstream (F – K)
45
10.00
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Vertical Stresses – Main Section – Downstream
220 200 180 160 140 120 100 80 60 40 20 0 -12.00
-10.00
-8.00
-6.00
-4.00
-2.00
0.00
2.00
4.00
VERTICAL STRESS (MPa) A
B
C
D
E
REF
HEIGHT (m)
Figure 28: Vertical Stresses – Main Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -12.00
-10.00
-8.00
-6.00
-4.00
-2.00
0.00
2.00
VERTICAL STRESS (MPa) F
G
H
I
J
K
REF
Figure 29: Vertical Stresses – Main Section – Downstream (F – K)
46
4.00
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Vertical Stresses – Left Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -4.00
-2.00
0.00
2.00
4.00
VERTICAL STRESS (MPa) A
B
C
D
E
REF
HEIGHT (m)
Figure 30: Vertical Stresses – Left Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -4.00
-2.00
0.00
2.00
4.00
VERTICAL STRESS (MPa) F
G
H
I
J
K
REF
Figure 31: Vertical Stresses – Left Section – Upstream (F – K)
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HEIGHT (m)
Vertical Stresses – Left Section – Downstream
220 200 180 160 140 120 100 80 60 40 20 0 -10.00
-8.00
-6.00
-4.00
-2.00
0.00
2.00
VERTICAL STRESS (MPa) A
B
C
D
E
REF
HEIGHT (m)
Figure 32: Vertical Stresses – Left Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -10.00
-8.00
-6.00
-4.00
-2.00
0.00
VERTICAL STRESS (MPa) F
G
H
I
J
K
REF
Figure 33: Vertical Stresses – Left Section – Downstream (F – K)
48
2.00
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Vertical Stresses – Right Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -4.00
-2.00
0.00
2.00
4.00
6.00
VERTICAL STRESS (MPa) A
B
C
D
E
REF
HEIGHT (m)
Figure 34: Vertical Stresses – Right Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -4.00
-2.00
0.00
2.00
4.00
6.00
VERTICAL STRESS (MPa) F
G
H
I
J
K
REF
Figure 35: Vertical Stresses – Right Section – Upstream (F – K)
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Vertical Stresses – Right Section – Downstream
220 200 180 160 140 120 100 80 60 40 20 0 -10.00
-8.00
-6.00
-4.00
-2.00
0.00
2.00
VERTICAL STRESS (MPa) A
B
C
D
E
REF
HEIGHT (m)
Figure 36: Vertical Stresses – Right Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -10.00
-8.00
-6.00
-4.00
-2.00
0.00
2.00
VERTICAL STRESS (MPa) F
G
H
I
J
K
REF
Figure 37: Vertical Stresses – Right Section – Downstream (F – K)
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Minimum Principal Stresses – Main Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -14.00
-12.00
-10.00
-8.00
-6.00
-4.00
-2.00
0.00
2.00
MIN. PRINCIPAL STRESS (MPa) A
C
D
E
REF
HEIGHT (m)
Figure 38: Minimum Principal Stresses – Main Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -14.00
-12.00
-10.00
-8.00
-6.00
-4.00
-2.00
0.00
MIN. PRINCIPAL STRESS (MPa) F
G
H
I
J
K
REF
Figure 39: Minimum Principal Stresses – Main Section – Upstream (F – K)
51
2.00
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
Minimum Principal Stresses – Main Section – Downstream
HEIGHT (m)
220 200 180 160 140 120 100 80 60 40 20 0 -14.00
-12.00
-10.00
-8.00
-6.00
-4.00
-2.00
0.00
2.00
MIN. PRINCIPAL STRESS (MPa) A
C
D
E
REF
HEIGHT (m)
Figure 40: Minimum Principal Stresses – Main Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -14.00
-12.00
-10.00
-8.00
-6.00
-4.00
-2.00
0.00
2.00
MIN. PRINCIPAL STRESS (MPa) F
G
H
I
J
K
REF
Figure 41: Minimum Principal Stresses – Main Section – Downstream (F – K)
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ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
HEIGHT (m)
Minimum Principal Stresses – Left Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -12.00
-10.00
-8.00
-6.00
-4.00
-2.00
0.00
2.00
MIN. PRINCIPAL STRESS (MPa) A
C
D
E
REF
HEIGHT (m)
Figure 42: Minimum Principal Stresses – Left Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -12.00
-10.00
-8.00
-6.00
-4.00
-2.00
0.00
MIN. PRINCIPAL STRESS (MPa) F
G
H
I
J
K
REF
Figure 43: Minimum Principal Stresses – Left Section – Upstream (F – K)
53
2.00
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HEIGHT (m)
Minimum Principal Stresses – Left Section – Downstream
220 200 180 160 140 120 100 80 60 40 20 0 -20.00 -18.00 -16.00 -14.00 -12.00 -10.00 -8.00 -6.00 -4.00 -2.00 0.00
2.00
MIN. PRINCIPAL STRESS (MPa) A
C
D
E
REF
HEIGHT (m)
Figure 44: Minimum Principal Stresses – Left Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -20.00 -18.00 -16.00 -14.00 -12.00 -10.00 -8.00 -6.00 -4.00 -2.00 0.00
2.00
MIN. PRINCIPAL STRESS (MPa) F
G
H
I
J
K
REF
Figure 45: Minimum Principal Stresses – Left Section – Downstream (F – K)
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HEIGHT (m)
Minimum Principal Stresses – Right Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -10.00
-8.00
-6.00
-4.00
-2.00
0.00
2.00
MIN. PRINCIPAL STRESS (MPa) A
C
D
E
REF
HEIGHT (m)
Figure 46: Minimum Principal Stresses – Right Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -10.00
-8.00
-6.00
-4.00
-2.00
0.00
MIN. PRINCIPAL STRESS (MPa) F
G
H
I
J
K
REF
Figure 47: Minimum Principal Stresses – Right Section – Upstream (F – K)
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HEIGHT (m)
Minimum Principal Stresses – Right Section – Downstream
220 200 180 160 140 120 100 80 60 40 20 0 -20.00 -18.00 -16.00 -14.00 -12.00 -10.00 -8.00 -6.00 -4.00 -2.00 0.00
2.00
MIN. PRINCIPAL STRESS (MPa) A
C
D
E
REF
HEIGHT (m)
Figure 48: Minimum Principal Stresses – Right Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -20.00 -18.00 -16.00 -14.00 -12.00 -10.00 -8.00 -6.00 -4.00 -2.00 0.00
2.00
MIN. PRINCIPAL STRESS (MPa) F
G
H
I
J
K
REF
Figure 49: Minimum Principal Stresses – Right Section – Downstream (F – K)
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HEIGHT (m)
Maximum Principal Stresses – Main Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -3.00
-1.00
1.00
3.00
5.00
7.00
9.00
11.00
MAX. PRINCIPAL STRESS (MPa) A
C
D
E
REF
HEIGHT (m)
Figure 50: Maximum Principal Stresses – Main Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -3.00
-1.00
1.00
3.00
5.00
7.00
9.00
11.00
MAX. PRINCIPAL STRESS (MPa) F
G
H
I
J
K
REF
Figure 51: Maximum Principal Stresses – Main Section – Upstream (F – K)
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HEIGHT (m)
Maximum Principal Stresses – Main Section – Downstream
220 200 180 160 140 120 100 80 60 40 20 0 -2.00
-1.00
0.00
1.00
2.00
3.00
4.00
5.00
MAX. PRINCIPAL STRESS (MPa) A
C
D
REF
E
HEIGHT (m)
Figure 52: Maximum Principal Stresses – Main Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -2.00
-1.00
0.00
1.00
2.00
3.00
4.00
5.00
MAX. PRINCIPAL STRESS (MPa) F
G
H
I
J
K
REF
Figure 53: Maximum Principal Stresses – Main Section – Downstream (F – K)
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HEIGHT (m)
Maximum Principal Stresses – Left Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -3.00
-1.00
1.00
3.00
5.00
7.00
MAX. PRINCIPAL STRESS (MPa) A
C
D
E
REF
HEIGHT (m)
Figure 54: Maximum Principal Stresses – Left Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -3.00
-1.00
1.00
3.00
5.00
7.00
MAX. PRINCIPAL STRESS (MPa) F
G
H
I
J
K
REF
Figure 55: Maximum Principal Stresses – Left Section – Upstream (F – K)
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Maximum Principal Stresses – Left Section – Downstream
HEIGHT (m)
220 200 180 160 140 120 100 80 60 40 20 0 -2.00
-1.00
0.00
1.00
2.00
3.00
4.00
MAX. PRINCIPAL STRESS (MPa) A
C
D
E
REF
HEIGHT (m)
Figure 56: Maximum Principal Stresses – Left Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -2.00
-1.00
0.00
1.00
2.00
3.00
4.00
MAX. PRINCIPAL STRESS (MPa) F
G
H
I
J
K
REF
Figure 57: Maximum Principal Stresses – Left Section – Downstream (F – K)
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HEIGHT (m)
Maximum Principal Stresses – Right Section – Upstream
220 200 180 160 140 120 100 80 60 40 20 0 -3.00
-1.00
1.00
3.00
5.00
7.00
MAX. PRINCIPAL STRESS (MPa) A
C
D
E
REF
HEIGHT (m)
Figure 58: Maximum Principal Stresses – Right Section – Upstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -3.00
-1.00
1.00
3.00
5.00
7.00
MAX. PRINCIPAL STRESS (MPa) F
G
H
I
J
K
REF
Figure 59: Maximum Principal Stresses – Right Section – Upstream (F – K)
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HEIGHT (m)
Maximum Principal Stresses – Right Section – Downstream
220 200 180 160 140 120 100 80 60 40 20 0 -2.00
-1.00
0.00
1.00
2.00
3.00
4.00
MAX. PRINCIPAL STRESS (MPa) A
C
D
E
REF
HEIGHT (m)
Figure 60: Maximum Principal Stresses – Right Section – Downstream (A – E) 220 200 180 160 140 120 100 80 60 40 20 0 -2.00
-1.00
0.00
1.00
2.00
3.00
4.00
MAX. PRINCIPAL STRESS (MPa) F
G
H
I
J
K
REF
Figure 61: Maximum Principal Stresses – Right Section – Downstream (F – K)
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Discussion of Results Eigenfrequencies It was to calculate the first 10 Eigenfrequencies of the structural response, including the interaction with the reservoir under full reservoir conditions. The results are summarized in Figure 1 to Figure 3 and Table 2. The criteria to compare the results are only the Eigenfrequency – no shape deformation, nor mass contribution to the different modes and directions are considered. For commenting in detail, this information would be valuable. The lower frequencies compared to the others of the first two modes of participants H and I are due to the use of the added mass approach. The overestimation of the additionally excited mass leads to slightly lower Eigenfrequencies, while the remaining are around 1.5 Hz. Noticeable is that the participants who used the program Diana (B, D and E) are getting higher frequencies starting from the third. This counts also for the participant K, who used the Open Source software Code_Aster. All other participants are getting more or less the same Eigenfrequencies for the first 10 modes. Deformation The comparison of the deformations is done for the main (middle) section only. The static loading accounts for dead weight and water loading together. No temperature loading is accounted for. The static deformation reveals, that many results show almost the same behavior, except those from participants D, E, H and K. The higher static deformation of participants D, H and K are due to modelling of the construction stages. The dynamic deformations are, as expected, varying in a wider range. Especially worth mentioning is also the result by participant G, he, as the only one, used infinite elements on the vertical boundary and applied the acceleration-time-history record on the bottom of the model, which could be the reason for in general higher values. The lower values of radial deformation of participant E are a result of the higher young’s modulus used. Stresses The comparison of the stresses of each of the participant and each of the diagrams are focused on essential aspects. Therefore, the discussion is kept general and just the quality of some graphs and values is discussed, but not the quantity in detail. Every participant has used his own preferred program, modelling technique and approach, so it’s to await that different results are gained. As it was up to the participant to use immediate or stepped construction sequences, the stress distribution differs. A 0.5[MPa] difference for the static loading, at a stress level of 6[MPa], one might accept, but not larger (Figure 12 and Figure 13). Worth mentioning are the results by participant G. He used, as already mentioned in the discussion of the deformations, infinite elements and applied the acceleration time history just on the bottom of the model. So the same as for the deformation counts here, the stresses in contrast to the others are far the highest in almost each diagram and beyond awaited results. Participants B and D used the program Diana with the Hybrid Frequency-Time Domain Method (HFTD-Method), which takes frequency dependent properties, such as compressibility of fluid, reservoir-bottom absorption and far-field reflection, into account. Both of them got similar results compared to the others, which prove the usability of this sophisticated analysis method on the one hand – but shows the applicability of less elaborated models, under these assumptions, too.
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Participant K was the only one who used the Open Source software Code_Aster. Such software, which is mostly used at research facilities and universities, is license free, but often more difficult to apply than commercial ones. Nevertheless, the provided results are matching with the results from the other participants. It is worth mentioning, that the added mass approach (according to Westergaard) is able to provide, comparable results, under the circumstances of this benchmark. All of the three contributors (E, H and I) using this approach are in the range of expected results.
Conclusion The comparison of all participants’ shows, that despite of the same boundary conditions (Model of the Geometry, Finite Element Mesh, Load cases, Material properties, Linear analysis, etc.) still assumptions are taken and required to carry out the analyses. These additional assumptions are starting with the application of the construction sequence, increase of material properties for dynamic loading, abutment boundary properties, application of dynamic loading and some specific assumptions based on the program used. Best practice examples and recommendations are published in ICOLD Bulletins and are available for engineers. However, for any specific problem to be solved, the assumptions for an analysis applied need to be reconfirmed in the light of the entire problem. In general it is astonishing, to see the large differences between the results of individual. Everybody had the opportunity to choose his preferred modelling technique to account for the fluid structure interaction, but most of the contributors used either added mass technique or acoustic elements. In practice it is still common to use an added mass approach according to Westergaard. Normally this assumption yields conservative results in contrary to modelling with acoustic elements. The solution of participant e.g. I, who has used Westergaards’ formula with its fully, frequency dependent extension, shows very similar results to those analyzed using “higher” constitutive models. According to results of the participants using either the coarse or fine mesh (described in the former section) had just a marginal influence on the frequencies, deformations and stresses within the structure. The purpose of choosing this arch dam example (220m in height, totally symmetric) wasn’t just for evaluating the influence of different modelling techniques, but also for engineers, scientists and operators to have a kind of reference solution. The diagrams and tables of the results of all participants should help to quantify and compare frequencies, deformations and stresses of such a structure. Concluding, everybody should be aware of the fact, that results of such simulations should be treated critically, because mistakes in modelling and application cannot be excluded. Usually, reference solutions from former comparable projects for validation should be used to proof the results for plausibility.
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PAPERS THEME A
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Fluid Structure Interaction Arch Dam – Reservoir at Seismic Loading G. Maltidis1 and L. Stempniewski2 1
Federal Waterways Engineering and Research Institute, Kußmaulstrasse 17, 76187 Karlsruhe, GERMANY 2 Karlsruhe Institute for Technology, Institute of Concrete Structures and Building Materials, Department of Concrete Structures, Gotthard-Franz-Str. 3, 76131, Karlsruhe, GERMANY E-mail: [email protected]
Abstract The fluid structure interaction is an important issue that must be taken into account for the analysis and design of hydraulic structures. Since the first attempts to calculate the hydrodynamic pressures on structures analytically (Westergaard, von Kármán, Mononobe, Housner, Chwang, Zangar) the engineers and researchers have the last years a very useful tool, the finite and boundary element method, in order to analyze complicated structures taking into account different sophisticated phenomena. However, even nowadays, the common praxis is to use the early developed techniques, because of their simplicity and capability of implementation in the most finite element codes.
Introduction Since 1933, the hydrodynamic pressures on oscillating structures, which are in contact with water, are taken into account with the simplified assumption that the water is incompressible and the structure is star using the added mass approaches, first proposed by Westergaard for vertical star surfaces and later extended by Zangar for inclined surfaces. Although these approaches apply under conditions which hardly are met, they are widely used also nowadays because of their simplicity in incorporating them in finite element codes. However, the result of analysis with the added mass approach may come out to be very conservative leading to wrong decisions. The modeling of the water with finite solid element around the 1980’s gave the opportunity for the analyst to take account some phenomena, as the water compressibility but raised other numerical problems as such type of modeling of water is suffering many times of hourglass making the analysis instable. The use of acoustic elements seems to be the more beneficial, as there are hardly numerical problems, and most of the phenomena, which take place for a dynamic fluid structure interaction can be modeled. With acoustic elements the analyst can consider the water compressibility, the wave absorption at the infinite end of the reservoir and the impedance of wave radiation at the reservoir sediments.
Hydrodynamic Pressures Added mass approaches The most well-known added mass approach is the one of Westergaard (1933)[1]. Westergaard proposed the following formula for the computation of hydrodynamic pressures as added masses under the restrictions that the reservoir is infinite, the upstream surface of the dam is vertical and the dam is rigid: γ 7 m Hy w A (1) 8 g
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where m the added mass, H and y the height and the depth of the reservoir respectevily, γw the density of the water, g the gravitational acceleration and A the contributing area around the node.
Figure 1: Graphical representation of Westergaard’s and Zangar’s calculation models. Zangar (1952)[2], using an electric analogue, extended the added mass approach of Westergaard for inclined upstream surfaces of the dam, introducing reductive factors dependent on the angle of inclination. y y y y γ m 0,5 H Cm (2 ) (2 ) w A (2) H H H g H where Cm a coefficient based on the angle of inclination and the other parameters as Westergaard’s formula. Fluid Elements The fluid elements are solid elements to which the characteristics of the water are applied. The incompressibility or the water as well as the null shear resistance are introduced with a Poisson number equal with 0,5 or close to this value for the finite element programs. The bulk modulus of the water is K=2,2 GPa. The modeling of the water with solid elements causes numerical instabilities because of the introducing of zero energy modes (hourglass modes). This effect can be mitigated with the use of hourglass control and by applying the free surface boundary condition for the vertical node displacements [9]. Moreover, a nonlinear material behavior with tension cut off or a contact interaction which allows only compression to be transmitted will avoid unrealistic tension stress of the dam caused by the water. Acoustic Elements The acoustic elements are used to model the fluid behavior of the air. They have no shear and tension resistance and they transmit only pressures [10]. With assignment of the water bulk modulus they model the water behavior very good. Numerous boundary conditions can be assigned to the acoustic elements, which model natural phenomena such as wave absorption at the far end of the reservoir, sloshing of the free surface, wave impedance at the reservoir’s bottom due to sediments etc. For the acoustic elements no special numerical care has to be taken except for assigning the boundary conditions. Model aspects For this benchmark two models (one with coarse mesh and one with fine mess) are investigated. The mesh of the reservoir is the same for both cases. The foundation was considered massless, so no further care was taken for wave absorption or deconvolution of the seismic motion. Because of the massless foundation with no radiation absorption of the 68
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seismic waves and due to the lack of further non-linearities of the dam’s material and of the contact interfaces, a big enough structural damping is applied. As presented in [6] for a big range of frequencies the total Rayleigh damping is between 8 and 10 %. Due to the linear finite element analysis a 10% viscous damping is used by [6]. Here, because of the small peak ground acceleration of 0,1g a value of 7,5% of structural damping was chosen in order to determine the Rayleigh stiffness damping factor a and the Rayleigh mass factor β. For the reservoir hydrodynamic pressures, two added mass approaches and one reservoir modeling with acoustic elements were investigated. The generalized Westergaard’s [11] and the Zangar’s added mass approaches were used. The added masses were given via a user subroutine which defines user elements in Abaqus [5]. The two models with acoustic elements differ only in the wave absorption’s method of the far field. The first uses acoustic infinite element whereas the latter impedance boundary condition. The impedance condition can be given either as element based or as surface based condition. Moreover a boundary condition is given at the reservoir free surface constraining the dynamic acoustic pressures to be zero. The surfaces of the rock and the dam are tied with the surfaces of the reservoir.
Figure 2: The model with two different meshes and two reservoir modeling approaches: left fine mesh and acoustic elements and right coarse mesh and added mass elements.
Tied with the dam
Free surface condition
Tied with the rock Boundary impedance or acoustic infinite elements
Figure 3: The reservoir with its boundary conditions.
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Table 1: Material Parameters Density (kg) Poisson Ratio Young Modulus (MPa) Bulk Modulus (MPa)
Rock 0 0,2 25000 -
Water 1000 2200
Dam 2400 0,167 27000 -
Analysis’ methods The seismic analyses were carried out using the time history with direct integration and the modal time history. From computational time the modal time history is a little bit faster than the time history with direct integration. The time histories were gives as nodal acceleration to the boundaries of the rock. A baseline correction offered by Abaqus was also applied to them. Results The results of the analyses are presented at the next tables and diagrams. The first table shows the ten first modes for the dam, with empty and with full reservoir modeled by the different methods described before. The diagrams show due to lack of space only some of the results containing the minimal and maximal vertical, minimum principal and maximum principal stresses of the dam for the different reservoir models and for the different dam mesh. The results are given for the paths along the height of the dam, for the upstream and the downstream sections. For convenience abbreviations were introduced to the diagrams (e.g. “dti” refers to direct time integration, “West” to Westergaard’s added mass, “ac” to acoustic element, “imp” to impedance boundary condition for the acoustic elements, “inf” to acoustic infinite elements, “modal” to modal dynamic analysis). The results for the fine mesh model are given with dashpot line in order to differ easier than the ones of the coarse mesh model. The analysis with the infinite elements had more computational cost than the analysis with the impedance condition. In order to obtain similar results to the impedance boundary condition with the use of infinite acoustic elements, care must be given in the definition of the infinite elements’ thickness. There are trivial differences when the analyst uses the improved rather than the planar non-reflecting condition offered by Abaqus. The results of the fine model with acoustic elements for the reservoir gave too conservative results. The author believes that these results for the given meshes of dam and reservoir are not correct due to violation of the contact condition, according to which the slave surface nodes must be finer than the master surface nodes.
Figure 4: The frequencies for the two models (coarse left, fine right) and for the four reservoir models.
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Table 2: Ten first modes for the coarse model Mode Nr.
Empty
Acoustic
Westergaard f T (Hz) (sec)
f (Hz)
T (sec)
f (Hz)
T (sec)
f=1,92
T=0,52
f=1,47
T=0,68
f=1,29
f=2,03
T=0,49
f=1,54
T=0,65
f=2,91
T=0,34
f=1,55
f=3,59
T=0,28
f=3,63
Zangar
f (Hz)
T (sec)
T=0,78
f=1,39
T=0,72
f=1,32
T=0,76
f=1,44
T=0,69
T=0,65
f=1,96
T=0,51
f=2,12
T=0,47
f=2,11
T=0,47
f=2,30
T=0,43
f=2,49
T=0,40
T=0,28
f=2,33
T=0,43
f=2,44
T=0,41
f=2,65
T=0,38
f=4,29
T=0,23
f=2,46
T=0,41
f=2,86
T=0,35
f=3,07
T=0,33
f=4,50
T=0,22
f=2,61
T=0,38
f=3,08
T=0,32
f=3,33
T=0,30
f=4,80
T=0,21
f=2,97
T=0,34
f=3,57
T=0,28
f=3,82
T=0,26
f=5,19
T=0,19
f=3,25
T=0,31
f=3,73
T=0,27
f=4,05
T=0,25
f=5,52
T=0,18
f=3,37
T=0,30
f=3,77
T=0,27
f=4,07
T=0,25
1
2
3
4
5
6
7
8
9
10
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Table 3: Ten first modes for the fine model Mode Nr.
Empty
Acoustic
Westergaard f T (Hz) (sec)
f (Hz)
T (sec)
f (Hz)
T (sec)
f=1,91
T=0,52
f=1,47
T=0,68
f=1,30
f=2,03
T=0,49
f=1,54
T=0,65
f=2,90
T=0,35
f=1,54
f=3,57
T=0,28
f=3,62
Zangar
f (Hz)
T (sec)
T=0,77
f=1,41
T=0,71
f=1,33
T=0,75
f=1,45
T=0,69
T=0,65
f=1,98
T=0,50
f=2,14
T=0,47
f=2,00
T=0,50
f=2,32
T=0,43
f=2,50
T=0,40
T=0,28
f=2,29
T=0,44
f=2,48
T=0,40
f=2,68
T=0,37
f=4,27
T=0,23
f=2,46
T=0,41
f=2,91
T=0,34
f=3,11
T=0,32
f=4,48
T=0,22
f=2,53
T=0,40
f=3,12
T=0,32
f=3,37
T=0,30
4,78
T=0,21
f=2,96
T=0,34
f=3,62
T=0,28
f=3,85
T=0,26
f=5,17
T=0,19
f=3,13
T=0,32
f=3,81
T=0,26
f=4,12
T=0,24
f=5,49
T=0,18
f=3,27
T=0,31
f=3,86
T=0,26
f=4,13
T=0,24
1
2
3
4
5
6
7
8
9
10
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The next diagrams give some representative comparisons between results for the different reservoir models and analysis’ methods.
Figure 6: The vertical stresses for the different reservoir models at the downstream main section.
Figure 7: The vertical stresses for the different reservoir models at the upstream main section.
Figure 8: Comparison between the coarse and fine model for the vertical stresses.
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Figure 9: The hoop stresses for the different reservoir models at the downstream main section.
Figure 10: The hoop stresses for the different reservoir models at the upstream main section.
Figure 11: The radial deformations for the different reservoir models at the main section.
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Figure 12: Comparison between the modal dynamic analysis and the direct time integration for the coarse model with the Zangar’s approach at the downstream (left) and upstream (right) main section.
Conclusion The earthquake analysis of an arch dam-reservoir-foundation system was performed with different modeling aspects according to the formulators’ directions. The results show very near values for the two added mass approaches, with the one of Zangar to be a little bit more favorable than the one of Westergaard. The acoustic elements models with the two nonreflecting approaches give identical results. The coarse and fine models differ only in the base stresses due to the coarser mess of the coarse model and some deviations are noticed at the added mass models. Although the modal dynamic analysis is much faster than the direct time integration, delivers conservative results.
Acknowledgements This paper consists a part of the research project „Earthquake Analysis and Design of Hydraulic Structures“, which is funded by the Federal Waterways and Research Institute of Germany in cooperation with the Institute of Concrete Structures of the Karlsruhe Institute for Technology. The contribution of both participated Institutes and persons involved is highly acknowledged.
References [1] Westergaard, H. M. (1933). Water pressures on dams during earthquakes. Transactions of the American Society of Civil Engineers, American Society of Civil Engineers, New York, New York, Paper 1835, 1933. [2] Zangar, C. N. (1952). Hydrodynamic Pressures on dams due to horizontal earthquake effects. U.S. Departmant of Interior, Bureaus of Reclamation, Engineering Monographs No.11 [3] Chwang, A.T., Housner, G.W. (1978). Hydrodynamic pressures on sloping dams during earthquakes. Part 1. Momentum Method. Journal of Fluid Mechanics, vol. 2, part 2, pp. 335-341 [4] Chwang, A.T. (1978). Hydrodynamic pressures on sloping dams during earthquakes. Part 2. Exact Theory. Journal of Fluid Mechanics, vol. 2, part 2, pp. 343-348 [5] ABAQUS (2011),User’s manual, Version 6.11. Dassault Systèmes Simulia Corporation, Providence RI, USA.
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[6] US Bureau of Reclamation (2006), State-of-Practice for the NonlinearAnalysis of Concrete Dams at the Bureau of Reclamation, USBR Report, Colorado, USA. [7] United States Society on Dams (2008). Numerical models for seismic evaluation of concrete dams. Review, evaluation and interpretation of results. USSD, Denver, USA. [8] Zienkiewicz, O. C., Bettess, P. (1978). Fluid-Structure dynamic interaction and wave forces. An introduction to numerical treatment. International Journal For Numerical Methods In Engineering, Vol. 13, 1-16. [9] Wilson, E.D., Khalvati,M. (1983). Finite elements for the dynamic analysis of fluid-solid systems. International Journal For Numerical Methods In Engineering, Vol. 19, 16571668. [10] Matthew Muto, Nicolas von Gersdorff, Zee Duron, Mike Knarr (2012). Effective Modeling of Dam-Reservoir Interaction Effects Using Acoustic Finite Elements, in Proceedings of Innovative Dam and Levee Design and Construction for Sustainable Water Management, 32nd Annual USSD Conference, New Orleans, Louisiana, April 2327, 2012, Pages 1161-1168. [11] Kuo, James Shaw-Han, (1982). Fluid-structure interactions: added mass computations for incompressible fluid. UCB/EERC-82/09, Earthquake Engineering Research Center, University of California, Berkeley, 1982-08.
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HFTD Analysis of an arched dam at seismic loading W. Kikstra1, F. Sirumbal1 and G. Schreppers1 1
TNO DIANA BV, Delftechpark 19a, 2628 XJ Delft, NETHERLANDS E-mail: [email protected]
Abstract Hybrid Frequency-Time Domain (HFTD) method is applied to analyze the response of a damfoundation-reservoir system. With this method the effect of frequency dependent properties such as compressibility of fluid, reservoir-bottom absorption and far-field reflection can be considered. HFTD results are compared with transient Newmark-type results and effect of frequency dependent properties is found to reduce amplitudes and stresses in the dam for the chosen bottom absorption in the reservoir.
Introduction and analysis procedure The benchmark case study of the arch dam-reservoir interaction at seismic loading was modeled with DIANA software. The formulation used by DIANA to couple the Finite Element equations of motion for Fluid-Structure Interaction (FSI) problems is a mixed displacement – scalar potential approach, which defines the solid variables in terms of displacement degrees of freedom (DOF), and the fluid variables in terms of pressure DOF. This definition of the fluid domain using Acoustic Finite Elements is called the Eulerian pressure formulation. One of the advantages of this type of formulation is the simple description of the fluid domain using a single scalar pressure variable ( ). This reduces considerably the number of variables of the system since only one DOF per node is required to describe the motion of the fluid domain. Taking into account that for dam-reservoir interaction problems the fluid motion is not substantial but small, considerable simplifications can be made in its equation of motion formulation. Those simplifications are a consequence of the following hypotheses assumed in DIANA FSI models: Small displacement amplitudes Small velocities (convective effects are omitted) Inviscid (viscous effects are neglected) Small compressibility (variation of density is small) No body forces in the fluid
Based on these hypotheses, the scalar fluid wave equation of motion is defined by Eq.(1).The wave speed (c), defined in terms of the fluid density ( ) and bulk modulus ( ), is defined by Eq.(2). ̈
(1) (2)
√
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In the same way, the FSI condition expressed in Eq. (3) relates the fluid gradient pressure in the normal direction ( ) to the interface surface (ΓI) with the structure acceleration vector ( ̈ ). ̈
(3)
on ΓI
In addition to the fluid-structure interface (ΓI) condition defined in Eq. (3), Figure 1 shows three types of boundary conditions which can be defined in DIANA FSI models.
Figure 1: Fluid and solid domains, fluid-structure interface and boundary conditions It is possible to specify two types of boundary conditions for the free surface of the reservoir (Γs). The first and simplest one is a consequence of neglecting the effect of the surface waves, prescribing a pressure equal to zero in the horizontal top free surface, as expressed in Eq. (4). This essential or Dirichlet type of boundary condition is the one used in the benchmark case study. Additionally, in DIANA it is possible to define a second type of boundary condition which takes into account the pressure caused by the free surface gravity waves. (4)
on Γs
Two types of infinite extent boundary condition (Γe) are considered in the benchmark case study. The first one defined by Eq. (5.a) prescribes the hydrodynamic pressure equal to zero at a distance equal to the reservoir length. The second one defined by Eq. (5.b) is a radiation boundary theoretically located at an infinite distance from the dam, which ensures that no incoming waves enter into the system (only outgoing waves). on Γe
(5.a)
̇ on Γe
(5.b)
Finally, two types of bottom boundary condition (Γb) are considered in the benchmark case study. The first one is setting the gradient of the pressure in the normal direction equal to zero, as expressed in Eq. (6.a). The second one defined by Eq. (6.b) is radiation boundary which takes into account the energy absorption of the bottom materials in terms of the wave 78
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reflection coefficient ( ), defined as the ratio between the amplitudes of the incident pressure wave over the reflective pressure wave. on Γb
(6.a) ̇ on Γb
(6.b)
In this paper, the Hybrid Frequency-Time Domain method is applied to solve the fluid equation of motion in the frequency domain. In this way, the hydrodynamic pressure amplitude vector is obtained in terms of the structural displacement amplitude vector. On the other hand, the structure non-linear equation of motion defined in the time domain is formulated based on relative displacement. In this way, the earthquake ground acceleration vector is introduced as external loading. The non-linear internal force of the structure is defined as the difference between the linear internal force and an unknown pseudo force vector, which is substituted into the non-linear equation to obtain the HFTD pseudo-linear equation of motion of the dam in the time-domain. Nevertheless, the HFTD method solves the pseudo-linear equation of motion in the frequency domain, and therefore it is required to use the Discrete Fourier Transform (DFT) to pass from one domain to the other. The fluid contribution is defined by additional mass and additional damping which both are defined in the frequency domain, and thus can be defined as frequency dependent properties. As a consequence the HFTD method can account for the effects of non-linear material behavior, compressibility of fluid, radiation at infinite extend and reservoir-bottom absorption, whereas standard transient analysis with Newmark-type of time-integration schemes cannot consider these effects together.
Model definition Because in the mesh that has been provided by the benchmark formulators some node connection incompatibilities were found, a new mesh has been defined using the provided geometry specifications as shown in Figure 2.
Figure 2: Components and geometrical characteristics of the foundation-dam-reservoir interaction system
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Element with second order displacement interpolation were used for foundation and dam and element with linear fluid-pressure interpolation were used for the reservoir. Mesh characterstics are listed in Table 1 and material characteristics are listed in the Tables 2-4. Component
Table 1: Characteristics of the Finite Element Model Mesh Type
Dam
Solid 3D
Foundation
Solid 3D
Reservoir
Flow 3D
Parameter Modulus of elasticity
DIANA element name CHX60 CTP45 CHX60 CTP45 CTP15H CHX20H
Number of Elements
Number of Nodes
712
3601
4896
23339
2670
11950
Table 2: Material parameters for the concrete dam
Poison modulus
DIANA variable name YOUNG
Value / Type 2.7 x 1010
Units N / m2
POISON
1.67 x 10-1
-
3
Density
DENSIT
2.4 x 10
Rayleigh damping
RAYLEI
5.71199 x 10-1 1.447 x 10-3
Parameter Conductivity
Table 3: Material parameters for the reservoir fluid
kg / m3 -
DIANA variable name CONDUC
Value / Type 1
Units -
Sonic speed
CSOUND
1.483 x 103
m/s
Density
DENSIT
1.0 x 103
kg / m3
Wave reflection coefficient for infinite extent boundary Wave reflection coefficient for bottom absorption boundary
ALPHAB
0
-
ALPHAB
5.0 x 10-1
-
Table 4: Material parameters for the foundation soil Parameter Modulus of elasticity Poison modulus Density
DIANA variable name YOUNG POISON
Value / Type 2.5 x 1010 2.0 x 10-1
Units N / m2 -
DENSIT
0
kg / m3
The transient responses of the two analysis cases shown in Figure 3 are determined and studied. Case I corresponds to the frequency independent system, for which the fluid of the reservoir is assumed to be incompressible ( ) and the infinite extent and bottom boundary conditions of the reservoir are defined by Eqs. (5.a) and (6.a), respectively. This frequency independent analysis case is solved with both HFTD and Newmark methods, with the objective of assessing the accuracy of HFTD method. On the other hand, Case II corresponds to a frequency dependent system in which fluid compressibility and reservoir radiation boundary conditions are included. Reservoir bottom absorption ( ) and radiation boundary of infinite extent, defined by Eqs. (6.b) and (5.b), respectively, are 80
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included in the analysis. As it was previously explained, the transient analysis of this frequency dependent system cannot be solved by the standard Newmark method, only by the HFTD method. Non-linear material behavior of the dam-structure is not considered in these cases.
Figure 3: Foundation-dam-reservoir interaction analysis cases
Results Prior to the time domain analysis, eigenvalues of the dam-foundation system including interaction with the fluid reservoir were determined. The first 10 mode-shapes and corresponding eigenvalues are presented in Figure 4. Case I was solved using both Newmark time integration and HFTD analysis. Figure 5 shows the amplitude of the displacement of the crest at the main section. Agreement is so close that the vertical stress, hoop stress and radial displacement results for Case I will only be displayed for the Newmark time stepping analysis. The envelopes for hoop stresses, vertical stresses and radial displacements along with their static values are given in figures 6 to 8, respectively. On the other hand, the frequency dependent properties of the reservoir and reservoir boundaries of Case II introduce more damping in the system which should lead to lower responses to the earthquake loading. Figure 9 shows that the crest amplitude has considerably lower peaks compared to Case I. The same magnitude reduction is observed in the envelopes of hoop stresses, vertical stresses and radial displacements given in Figures 10 to 12, respectively. A 2D study to discriminate the effects of fluid compressibility, radiation boundary and bottom absorption shows that reservoir fluid compressibility increases the response, that there is minimal influence of the radiation boundary if sufficient reservoir length is modeled and that bottom absorption damps the response of the dam-reservoir system [3]. For all the analysis results presented (Eigen-analysis, Newmark and HFTD), the effect of selfweight gravity and hydrostatic pressure loads were taking into account as initial conditions.
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Figure 4: Mode-shapes and eigenvalues of the dam-foundation-reservoir interaction system
Figure 5: HFTD vs Newmark. Crest amplitude [m] at main section 82
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Figure 6: Vertical stress [Pa] against elevation [m] for left, main and right dam sections at upstream and downstream dam face for incompressible reservoir.
Figure 7: Hoop stress [Pa] against elevation [m] for left, main and right dam sections at upstream and downstream dam face for incompressible reservoir.
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Figure 8: Radial displacement [m] against elevation [m] for left, main and right dam sections at upstream and downstream dam face for incompressible reservoir.
Figure 9: Compressible vs Incompressible. Crest amplitude [m] at main section
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Figure 10: Vertical stress [Pa] against elevation [m] for left, main and right dam sections at upstream and downstream dam face for compressible reservoir.
Figure 11: Hoop stress [Pa] against elevation [m] for left, main and right dam sections at upstream and downstream dam face for incompressible reservoir.
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Figure 12: Radial displacement [m] against elevation [m] for left, main and right dam sections at upstream and downstream dam face for compressible reservoir.
Conclusion The given foundation-dam-reservoir system was analyzed for the full duration of the earthquake. With the HFTD method implemented in the standard version of DIANA for the case of frequency independent properties the same results could be reproduced as with implicit time stepping with Newmark’s method. With HFTD the effect of frequency dependent properties such as compressibility of fluid, reservoir bottom absorption and infinite extend reflection have been analyzed and quantified, resulting in an interesting method to be applied specially in the dynamic analysis of dam-reservoir interaction models.
References [1] Veletsos, A.S., and Ventura, C.E. (1985). Dynamic analysis of structures by the DFT method. J. Struct. Eng. ASCE, Vol.111, 2625-2642. [2] TNO DIANA BV (2011). DIANA User’s Manual – Release 9.4.4, Delft, The Netherlands. [3] Sirumbal, F. (2013) Numerical modeling of dam-reservoir interaction seismic response using the Hybrid Frequency-Time Domain ( HFTD ) method Masters thesis, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands. [4] Darbre, G.R. (1996). Nonlinear dam-reservoir interaction analysis. Proc. Eleven World Conf. on Earthquake Engineering, Acapulco, Paper No. 760. [5] Zienkiewicz, O.C., and Bettes, P. (1978). Fluid-structure dynamic interaction and wave forces. An introduction to numerical treatment. Int J Numer Meth Eng, Vol. 13, 1-16.
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Finite Element Modelling of Seismic Fluid-Structure Interaction for a large Arch Dam G. Faggiani1 and P. Masarati1 1
Ricerca sul Sistema Energetico - RSE SpA, via R. Rubattino 54, 20134 Milano, ITALY E-mail: [email protected]
Abstract The linear dynamic fluid-structure interaction at seismic loading for the artificially generated large arch dam provided for the Theme A was modelled using the approach of acoustic compressible elements, with both the coarse and fine meshes provided by the formulators of the 12th ICOLD Benchmark Workshop. The effects of incompressible fluid (theoretical hypothesis of the added mass models) and of the partial absorption of the hydrodynamic pressure waves at the reservoir boundary (bottom and sides) were also investigated. Simulations were carried out using the RSE in-house FEM code CANT-SD, specifically designed for dynamic linear and non-linear analyses of dam-reservoir systems. The coarse and fine meshes showed not dissimilar results, except at dam-foundation interface. The results of the analyses confirmed that incompressible models could result relatively conservative and highlighted the benefits of the approach of acoustic elements, mainly the possibility to take into account the damping effect on the fluid boundary.
Introduction Seismic safety assessment of large arch dams is actually a very important matter. Both dam-reservoir interaction and non-linear mechanisms due to the contraction joint opening and sliding could greatly affect the mechanical behaviour of the dam: therefore they must be properly considered in order to obtain reliable numerical simulations under strong earthquakes. Theme A of the 12th ICOLD Benchmark Workshop on Numerical Analysis of Dams [1] is aimed at comparing the different available approaches to model the dynamic fluid-structure interaction: to this purpose the structural response to a seismic loading of an artificially generated large arch dam has to be investigated. The analyses have been carried out using the RSE in-house FEM code CANT-SD [2], specifically designed for linear and non-linear dynamic (seismic) analyses of dam-reservoir systems. This code is currently used at RSE for safety assessment of concrete dams, and it was adopted to deal with some themes proposed in previous ICOLD Benchmark Workshops [3] [4] [5] [6] [7] [8]. Regarding the main simulation options required to effectively address the present Theme A, CANT-SD models the fluid reservoir by means of acoustic elements and solves the transient dynamic coupled problem using an implicit direct time integration method.
Geometrical and physical model The two FEM parabolic meshes, Coarse and Fine, considered in the simulations are reported in Figure 1 and 2. Only the dam meshes provided in the Theme A have been adopted unchanged. The fluid domain was obtained by extruding the upstream face of the dam mesh for a length 3 times the total height of the dam (220 m): the resulting mesh was only joined to 87
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the upstream dam face, not to the rock. Foundation meshes were modified in order to obtain coincident meshes at concrete-rock interface as necessary for the proper operation of CANTSD.
Figure 1: Arch dam FEM model – Coarse mesh
Figure 2: Arch dam FEM model – Fine mesh A monolithic behaviour of the dam body has been considered, as no construction joints have been modelled. Dam concrete and foundation rock, assumed to behave linear-elastically, were characterized by the physical-mechanical parameters provided by the formulators and reported in Table 1 along with the properties of the fluid. A 5% structural damping ratio (ξ) was assumed in the analyses: damping matrix is expressed as linear combination of mass and stiffness matrices according to Rayleigh formulation. The calibration of structural damping ratio was based on the frequency range resulting from modal analysis, such that the damping was almost constant in this range (Figure 3). Table 1: Material properties Parameter Modulus of elasticity (MPa) Bulk modulus (MPa) Poisson’s ratio (-) Density (kg/m3)
Rock mass 25000 0.2 0
88
Water 2200 1000
Dam concrete 27000 0.167 2400
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0.150
ξstiffness ξmass ξRayleigh
Damping ratio [-]
0.125 0.100 0.075
0.050 0.025 0.000
0
1
2
3
4
5
6
7
8
9 10 11
Frequency [Hz]
Figure 3: Rayleigh viscous damping The dynamic fluid-structure interaction, modelled in CANT-SD following the classic acoustic approach [9], is here briefly summarized. The hydrodynamic pressure in the compressible fluid of the reservoir is governed by the wave equation: 2 p 2 p 2 p 1 2 p x 2 y 2 z 2 C 2 t 2
(1)
where C = √(k/) is the velocity of sound in the fluid, k the bulk modulus and the density. The boundary conditions are (Figure 4): free surface p0
upstream face of the dam
open boundary
̈ ̇
bottom and sides ̇ where n is the outward normal, ̈ the normal acceleration and q the damping coefficient on the bottom and sides. The upstream face of the dam results the only surface of interaction between structure and fluid: the accelerations at the dam face represent the “actions” of the dam on the reservoir, which in turn “reacts” through the hydrodynamic pressures exerted on the dam face. The last boundary condition accounts for the partial absorption of hydrodynamic pressure waves [10] [11]: this damping phenomenon is mainly caused by the layer of sedimentary material possibly deposited in the reservoir, but could be significant also in cases of few or no accumulated sediments.
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Upstream face of the dam Free surface
Bottom and sides Open boundary
Figure 4: Boundaries of reservoir Finite element subdivision of the reservoir leads to the discretized form of the outlined acoustic problem, i.e. a system to be coupled with the structural one (that includes the hydrodynamic pressure loads). The fluid-structure coupled problem is governed by the following system: {
̈ ̈
̇ ̇
(2) ̈
The first subsystem governs the mechanical behaviour of the dam and the (unknown) pressure p represents an applied load; the second governs the acoustic behaviour of the reservoir and the (unknown) acceleration ü represents an assigned boundary condition. The seismic analysis proposed in Theme A was performed for three physical models of the reservoir behaviour, differing in fluid compressibility and/or boundary absorption. The first physical model (Base Case) fully respects the requirements of the formulators: the provided value of the bulk modulus of water (2200 MPa) was adopted, boundary absorption was neglected (q=0 on bottom and sides, i.e. reflecting condition) and the non-reflecting condition was considered at the end of the reservoir. The other physical models have the aim to examine, for the particular given scenario, the following interesting aspects: the effect of the incompressibility hypothesis and the importance of the boundary absorption. A brief hint of both aspects follows, along with the necessary choices for the two additional simulations. The assumption of fluid incompressibility is connected with the “added mass models”, which assimilate the action of the fluid to that of “some kind of mass” (physical mass or a mass matrix) attached to the upstream face of the dam. It is worth noting that the comparison reported in this paper just concerns the incompressibility hypothesis, and can be therefore only applied to the rigorous “added mass matrix” model, clearly defined in [9] and referred to as “Finite Element Added Hydrodynamic Mass Model” in [12]. No investigation was made about the effects of any possible change on this matrix (i.e. for computational convenience), nor any comparison was made with Westergaard-type techniques, however strongly discouraged in [12] for arch dams. 90
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A key simplified parameter [13] that determines the significance of water compressibility is the ratio Ωr=f1r/f1 of the fundamental frequency of the reservoir to that of the dam-foundation system without water: the higher this ratio, the less the importance of fluid compressibility. Although the limit Ωr=2 is valid only for gravity and not for arch dams, as clearly stated in [13], it has often been used, at least as a broad clue, for arch dams too [12]. The fundamental frequency f1r of the reservoir can be roughly evaluated as C/4H, where C is the velocity of sound in water and H is the water depth: for the analysed system the value is f1r=1.7 Hz. The fundamental frequencies of the dam-foundation system without water were calculated and resulted 1.9 Hz for anti-symmetric mode shape and 2.0 Hz for symmetric mode shape. The value of the ratio Ωr, about 0.9, indicates that compressibility should probably be important. The physical model to investigate the effects of compressibility (Incompressible Case) was developed using a value of the bulk modulus one hundred times greater than that of the Base Case (k=2200×100 MPa) and assuming the reflecting boundary condition at the end of the reservoir. Two modal analyses were carried out, the first with k=2200×100 MPa and the second with k=2200×10000 MPa to verify that the value of the bulk modulus assumed in dynamic analysis was great enough to simulate incompressibility: no significant differences were found in the results. The coupled mechanical-acoustic approach allows taking into account the effect of the partial absorption of hydrodynamic pressure waves on the boundary of the reservoir [10] [11], impossible to simulate using any added mass model. To assign the boundary absorption, the damping coefficient q must be quantified: to this purpose it is convenient to express it by means of the wave reflection coefficient α (the ratio of the amplitude of the reflected hydrodynamic pressure wave to the amplitude of the incident one) [11]: 1 (1 ) q (3) C (1 ) The wave reflection coefficient α represents a more physically meaningful description of the phenomenon: α=1 corresponds to a reflecting (rigid) boundary, α=0 corresponds to a nonreflecting (transmitting) boundary, -1<α<0 corresponds to an even major damping behaviour. The value of α can be determined on the basis of field investigations [12] [14] [15]. The results reported in [14] indicate values of α, measured at seven concrete dam sites, varying over a range from -0.55 to 0.66: three of these values were negative, due to thick layers of soft sediments. A value of 0.82 was determined at the dam site investigated in [15]: a rock site with very little or no accumulated sediments. Making reference to the reported results of field investigation, a value α=0.5 was considered sensible to examine the importance of the boundary absorption in the seismic analysis: the corresponding physical model (Damped Case) was obtained from the Base Case using a damping coefficient q=0.000225. Table 2 summarizes all the simulations: Base Case (B) was performed both with the coarse and fine meshes, while Incompressible (I) and Damped (D) Cases only with the coarse mesh. Table 2: Summary of simulations Case Base Incompressible Damped
Coarse mesh B I D
91
Fine Mesh B -
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Loadings The numerical analyses simulated the effects of the following loadings/actions: 1. dead weight 2. hydrostatic pressure with water level at crest height 3. seismic loading The seismic loading was provided in the Theme by means of artificially generated acceleration time histories in the three coordinate directions X (upstream), Y (cross-stream) and Z (vertical): the elastic response spectra (5% damping ratio) are reported in Figure 5. It’s worth noting that the given time histories represent a quite moderate intensity earthquake. The acceleration time histories were only assigned to the bottom and sides of the foundation, not to the boundary of the reservoir too. The transient dynamic coupled problem was solved using an implicit direct time integration method (HHT) [16]; an integration step of 0.002 s was chosen, in order to well represent frequencies up to 25 Hz. 0.35
upstream
0.3
cross-stream vertical
Acceleration [g]
0.25 0.2
0.15 0.1 0.05
0
0
0.5
1
1.5 2 2.5 Period [s]
3
3.5
4
Figure 5: Elastic response spectra
Results The following sections report and discuss the results of Base Case with both coarse and fine meshes and of Incompressible and Damped Cases. The stress state is represented either by diagrams showing vertical and hoop static and dynamic (envelope) stresses, or by contour plots showing principal stresses envelopes, expressed in megapascal, positive if tensile. The displacements of the dam, expressed in metres, are positive if directed downstream. Eigenfrequencies and mode shapes The modal analysis allowed the computing of the natural resonant frequencies of the damreservoir system and the corresponding mode shapes. Table 3 reports the first four eigenfrequencies for both the coarse and the fine mesh. The values obtained with the two meshes were almost identical: as expected, the refined discretization exhibited a slightly more flexible behaviour than the coarse one. These frequencies corresponded to the first anti92
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symmetric and the first three symmetric mode shapes, shown in Figure 6 and 7 for the coarse and fine mesh respectively. The first four eigenfrequencies occurred in a range of periods of about 0.45÷0.65 s, matching the response spectra in their descending branch. Table 3: Natural frequencies Mode 1 2 3 4
Eigenfrequencies [Hz] Coarse Mesh Fine Mesh 1.547 1.540 1.551 1.549 2.052 2.050 2.229 2.222
Figure 6: Mode Shapes (Base Case, Coarse Mesh)
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Figure 7: Mode Shapes (Base Case, Fine Mesh) Seismic simulation with Coarse and Fine Meshes The transient dynamic analysis supplied the stress-strain state of the system due to the application of the seismic loading. The simulations developed for the Base Case with Coarse (CB) and Fine (FB) meshes showed how the use of a more refined mesh does not lead to significant differences in the results, though involving quite a higher computational effort (about 5 times). Vertical and hoop stresses in dam main section are reported in Figure 8 and 9 for upstream and downstream faces respectively, both for fine and coarse meshes: stresses were evaluated for static and seismic loads. Vertical stresses resulted always compressive upstream, except at dam foundation interface, while downstream tensile vertical stresses, up to 1 MPa, occurred in the upper part of the dam, above 610 m a.s.l., caused by the seismic loading. Due to the seismic loading, a variation of vertical stress of about 1÷1.5 MPa was observed both upstream and downstream. Figure 8 highlights that vertical stresses calculated with the coarse mesh are underestimated near the dam-foundation interface, where tensile stress concentrations are usual in many arch dams. Hoop stresses both on upstream and downstream faces were compressive, confirming that the arch effect was activated and the dam behaved according to its monolithic scheme. The effect of the seismic loading, greater in the upper part of the dam, involved a stress variation of about 4 MPa upstream (at 675 m a.s.l.) and 2 MPa downstream (at about 695 m a.s.l.). Figure 10 and Figure 11 report the tensile stress envelope and the dynamic maximum displacement contour plots for the two analysed meshes. Tensile stress on the downstream face was essentially vertically oriented: due to the seismic loading the upper central part of the dam exhibited stresses up to 1.6 MPa. Figure 11 shows a downstream displacement of about
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715 C Static 695 F Static 675 CB Max FB Max 655 CB Min 635 FB Min 615 595 575 555 535 515 495 6.00 4.00 2.00 0.00 -2.00 -4.00 -6.00 -8.00 Vertical Stresses [MPa]
Height [m a.s.l.]
Height [m a.s.l.]
8 cm, due to the seismic loading. A similar upstream displacement was observed for the minimum envelope. The contour plots allowed the overall comparison between simulations with coarse and fine meshes and confirmed that the spatial trend of the stress-strain state was comparable in the whole dam body. 715 695 675 655 635 615 595 C Static 575 F Static 555 CB Max FB Max 535 CB Min 515 FB Min 495 1.00 -1.00 -3.00 -5.00 -7.00 -9.00 -11.00 Hoop Stresses [MPa]
715 695 675 655 635 615 595 C Static 575 F Static 555 CB Max FB Max 535 CB Min 515 FB Min 495 6.00 4.00 2.00 0.00 -2.00 -4.00 -6.00 -8.00 Vertical Stresses [MPa]
Height [m a.s.l.]
Height [m a.s.l.]
Figure 8: Base Case - Vertical (left) and hoop (right) stresses on the upstream surface 715 695 675 655 635 615 595 C Static 575 F Static 555 CB Max FB Max 535 CB Min 515 FB Min 495 1.00 -1.00 -3.00 -5.00 -7.00 -9.00 -11.00 Hoop Stresses [MPa]
Figure 9: Base Case - Vertical (left) and hoop (right) stresses on the downstream surface
Figure 10: Base Case - Maximum principal stress – Coarse (left) and Fine (right) meshes, downstream view. 95
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Figure 11: Base Case - Maximum dynamic displacement – Coarse (left) and Fine (right) meshes, upstream view
715 695 675 655 635 615 595 575 555 535 515 495 4.00
Static B Max B Min I Max I Min D Max D Min
2.00 0.00 -2.00 -4.00 Vertical Stresses [MPa]
Height [m a.s.l.]
Height [m a.s.l.]
Incompressible fluid and absorption effects Based on the comparison discussed in the previous section, the effects of incompressible fluid (Incompressible Case - I) and of the reservoir boundary absorption (Damped Case – D) on the dynamic response of the dam were investigated only with the coarse mesh. The results of the analyses are summarized and compared with those of the Base Case (B) in Figure 12 and 13, where vertical and hoop stresses in dam main section are reported for upstream and downstream face respectively. The general trend of stresses for the Base Case was already illustrated in the previous section. The curves representing stresses for Incompressible and Damped Case generally laid respectively outside and within those of the Base Case, confirming that the incompressible model could result relatively conservative and that the reservoir boundary absorption could reduce the earthquake response of the dam. In the analysed situation, relevant to an earthquake of very moderate severity, the difference among the three models could look worthless. However these differences could become highly significant if a greater seismic loading were considered. Referring to the hoop stresses, a 2 times amplified earthquake would result in the occurrence of tensile stresses in the arcs (Figure 14) involving the transition from a monolithic to an independent cantilever separated by vertical joints behaviour, if these stresses act in the dam for a significant height (starting from the crest). Figure 14 illustrates that for incompressible model tensile stresses resulted about 3÷5 times higher than for the damped model, involving the upper 80 m of the dam, a double height if compared with damped model.
-6.00
715 695 675 655 635 615 595 575 Static B Max 555 B Min I Max 535 I Min 515 D Max D Min 495 1.00 -1.00 -3.00 -5.00 -7.00 -9.00 -11.00 Hoop Stresses [MPa]
Figure 12: Comparison among Base, Incompressible and Damped Case - Vertical (left) and hoop (right) stresses on the upstream surface 96
715 695 675 655 635 615 595 575 555 535 515 495 4.00
Height [m a.s.l.]
Height [m a.s.l.]
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Static B Max B Min I Max I Min D Max D Min
2.00 0.00 -2.00 -4.00 Vertical Stresses [MPa]
-6.00
715 695 675 655 635 615 595 Static 575 B Max B Min 555 I Max 535 I Min D Max 515 D Min 495 1.00 -1.00 -3.00 -5.00 -7.00 -9.00 -11.00 Hoop Stresses [MPa]
715 695 675 655 635 615 595 B Max 575 I Max 555 D Max B-ampli2 535 I-ampli2 D-ampli2 515 Static 495 6.00 4.00 2.00 0.00 -2.00 -4.00 -6.00 -8.00 Hoop Stresses [MPa]
Height [m a.s.l.]
Height [m a.s.l.]
Figure 13: Comparison among Base, Incompressible and Damped Case - Vertical (left) and hoop (right) stresses on the downstream surface 715 695 675 655 635 615 595 B Max 575 I Max D Max 555 B-ampli2 535 I-ampli2 D-ampli 515 Static 495 8.00 6.00 4.00 2.00 0.00 -2.00 -4.00 -6.00 -8.00 Hoop Stresses [MPa]
Figure 14: Comparison among Base, Incompressible and Damped Cases for an amplified earthquake- Hoop stresses on upstream (left) and downstream (right) surface
Conclusion Theme A of the 12th ICOLD Benchmark Workshop on Numerical Analysis of Dams, dealing with the linear dynamic fluid-structure interaction at seismic loading, has been approached by using CANT-SD, a RSE in-house FEM code for dynamic linear and non-linear analyses of dam-reservoir systems. The dynamic fluid-structure interaction was modelled with the approach of compressible acoustic elements. The analyses with the reflecting boundary condition on the bottom and sides of the fluid domain, performed with both coarse and fine meshes to test the effects of different spatial discretization, showed that the use of a more refined mesh does not lead to significant differences in the results, though involving quite a higher computational effort. Simulations considering incompressible acoustic elements or reservoir boundary absorption were also performed: the results of these analyses confirmed that the incompressible model could result relatively conservative and that reservoir boundary absorption could significantly reduce the earthquake response of the dam. The use of an incompressible model instead of a compressible one, capable to account for damping effect on the fluid boundary too, speeded up the transition from a monolithic structural scheme to a different one with independent cantilevers and joints. 97
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Acknowledgements This work has been financed by the Research Fund for the Italian Electrical System under the Contract Agreement between RSE S.p.A. and the Ministry of Economic Development General Directorate for Nuclear Energy, Renewable Energy and Energy Efficiency in compliance with the Decree of March 8, 2006.
References [1] Zenz G., Goldgruber M. (2013). Theme A formulation. Fluid Structure Interaction. Arch Dam – Reservoir at Seismic loading. 12th ICOLD Benchmark Workshop on Numerical Analysis of Dams, Graz, Austria. [2] Masarati P., Meghella M. (2000). The FEM computer code CANT-SD for non-linear static and dynamic analysis of dams. Enel.Hydro rep. n. 6045, Milano, Italy. [3] Faggiani G., Frigerio A., Masarati P., Meghella M.(2011). Finite element modelling of concrete swelling effects on Kariba dam. 11th ICOLD Benchmark Workshop on Numerical Analysis of Dams, Valencia, Spain. [4] Meghella M., Masarati P. (2007). FEM analyses for the interpretation of the structural behaviour of La Aceña dam. 9th ICOLD Benchmark Workshop on Numerical Analysis of Dams, St. Petersburg, Russia. [5] Meghella M., Frigerio A., Masarati P. (2005). Evaluation of AAR on the behaviour of Poglia dam adopting two different approaches. 8th ICOLD Benchmark Workshop on Numerical Analysis of Dams, Wuhan, Hubei, P.R. China. [6] Meghella M., Mazzà, G. (2003). Safety evaluation against sliding of a gravity dam with curved shape. 7th ICOLD Benchmark Workshop on Numerical Analysis of Dams, Bucharest, Romania. [7] Bon E., Chillè F., Masarati P., Massaro C. (2001). Analysis of the effects induced by alkali-aggregate reaction (AAR) on the structural behaviour of Pian Telessio dam. 6th ICOLD Benchmark Workshop on Numerical Analysis of Dams, Salzburg, Austria. [8] Bolognini L., Masarati P., Bettinali F. Galimberti C. (1994). Non-linear analysis of joint behaviour under thermal and hydrostatic loads for an arch dam. 3rd ICOLD Benchmark Workshop on Numerical Analysis of Dams, Paris, France. [9] Zienkiewicz O. C. (1977). The Finite Element Method, 3rd Edition, McGraw-Hill. [10] Fok K.L., Chopra A.K. (1986). Earthquake analysis of arch dams including dam-water interaction, reservoir boundary absorption and foundation flexibility. Earthquake Engineering and Structural Dynamics, Vol. 14, pp. 155-184. [11] Fenves G., Chopra A.K. (1983). Effects of reservoir bottom absorption on earthquake response of concrete gravity dams. Earthquake Engineering and Structural Dynamics, Vol. 11, pp. 809-829. [12] US Army Corps of Engineers – USACE (2003). Engineering and Design - Time-History Dynamic Analysis of Concrete Hydraulic Structures. EM 1110-2-6051. [13] Fok K.L., Chopra A.K. (1986). Frequency response functions for arch dams: hydrodynamic and foundation flexibility effects. Earthquake Engineering and Structural Dynamics, Vol. 14, pp. 769-795. [14] Ghanaat Y., Redpath B.B. (1995). Measurement of reservoir-bottom reflection coefficient at seven concrete dam sites. QUEST Structure Report No. QS95-01. [15] Ghanaat Y., Hall R.L., Redpath B.B. (2000). Measurement of dynamic response of arch dams including interaction effects. Proc. of 12WCEE2000. [16] Hilber H.M., Hughes T.J.R., Taylor R.L. (1977). Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Engineering and Structural Dynamics, Vol. 5, pp. 283-292. 98
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ICOLD 12-th International Benchmark Workshop on Numerical Analysis of Dams Theme A: Fluid Structure Interaction Arch Dam ‐ Reservoir at Seismic Loading A. Tzenkov1, A. Abati1 and G. Gatto1 1
STUCKY SA, Rue du Lac 33, PO Box, 1020 Renens, SWITZERLAND E-mail: [email protected]
Abstract One of the main concerns regarding the numerical dynamic analysis of arch dams is the proper modelling of the fluid-structure interaction between the dam and the impounded water. There are several approaches to this, which enables accounting for the hydrodynamic pressures on the upstream face of the dam with different precision and, respectively, with different computing effort. This work investigates the impact of the hydrodynamic approach opted for on the computed stresses and displacements of an example 220-m high doublecurvature arch dam. It is shown that, for this particular benchmark problem, it is important to consider the compressibility of water.
Introduction The hydrodynamic phenomena occurring on the interface between a dam and the impounded water may have significant effect on the structural response of the dam. The structure (the dam wall and its foundation) and the fluid (the impounded water) are two different physical systems that interact with each other and thus present a coupled problem. According to the classification given in [1], the latter is a Class I coupled problem in which the coupling occurs on the interfaces between the domains. A milestone procedure to account for the hydrodynamic effects on dams was established by Westergaard in 1933, [2] by introducing the concept of added mass. Although Westergaard’s approach was limited within the assumptions of rigid dams with vertical upstream faces, infinitely long reservoirs, and incompressible fluids, it enabled accounting for the hydrodynamic effects in the everyday engineering practice. Using the electric analog method, Zangar [3] improved Westergaard’s approach by establishing a family of parabolas by means of which it is possible to compute the hydrodynamic pressure on rigid dams with sloping upstream face. With the advent of the computer and the increased utilization of the numerical methods, it became possible to account for the effects due to (1) the fluid compressibility, (2) the hydrodynamic pressure waves’ partial absorption by deposited sediments, (3) the foundation inertia and damping, as well as for non-linear dam behaviour (4). Frequency domain [4][5], time domain [6], and hybrid frequency-time domain (HFTD) [7] procedures have been developed and used over the past decades. An inherent limitation of the frequency domain approach is that it presupposes linear structural behaviour; on the other hand, it enables readily considering effects (1), (2) and (3). In contrast, in the time-domain, nonlinear structural behaviour can be accounted with reasonable computational effort, but it is more difficult to simulate the other hydrodynamic phenomena. This work investigates the differences of the stresses and the displacements computed by 99
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
means of the FEM in using three different approaches for modelling the hydrodynamic effects. The analyses are performed on the current Benchmark example 220 m high doublecurvature arch dam by means of the computer program DIANA [8]. First, two linear seismic analysis are carried out in the time domain by using added masses computed by the Westergaard formula and by means of a fluid-structure interaction analysis in the assumption of incompressible fluid. Next, a linear analysis is performed by means of the HFTD approach and in modelling the fluid’s compressibility. In addition, a nonlinear seismic analysis is performed in the time-domain, by means of a fluid-structure interaction analysis for incompressible fluid. Details of the procedures utilized, the results of the computations, and a discussion on the results are presented in the following sections.
Finite Element Model The finite element model of the example arch dam – foundation – reservoir system is created based on the geometry and the spatial discretization given by the Formulator of the current Theme A [9]. The only difference is that the reservoir model’s length is 5 times the dam height. Following the conditions of [9], the investigations have been done on two meshes: a coarse one, and a finer one. The structural system of the coarse mesh is modelled by 2516 hexahedral and wedge isoparametric solid finite elements (356 for the dam and 2160 for the foundation). The reservoir is modelled by 1956 3-D flow elements, while the dam-reservoir interface is represented by 177 fluid-structure interface elements. All the elements are based on quadratic interpolation. The total number of nodes of the coarse mesh is 21547. The fine mesh has 78426 nodes; the dam and the foundation are modelled by 2848 and 10080 elements respectively; however, it yields almost the same result regarding the structural response as the coarse mesh. Translational supports in the three global directions are specified as structural boundary conditions on the bottom and side surfaces of the foundation model. The material parameters of the dam and the foundation are the same as the ones prescribed in [9]. The sonic wave velocity =1483 m/s is associated with the reservoir fluid elements. Finally, water density of 1000 kg/m3 is specified for the fluid-structure interface elements. Westergaard Added Mass (WG) The added masses computed by means of the Westergaard formula [2] are applied on the nodes of the upstream face of the dam by means of CQ24TM boundary surface elements [8]. 178 such elements are defined for the coarse mesh; they are 712 for the finer model of the system. A distributed translational mass material model is associated to the boundary surface elements, which allows precise automatic calculation of the added masses. The total mass assembled for the coarse mesh finite element model is TM=0.14E+11 kg (for massless rock). Without added mass, TM=0.471E+10 kg. Fluid - Structure Interaction with Incompressible Fluid (FSI) As already mentioned, the solution is conducted in the time domain. To render the system frequency independent, the fluid is specified as incompressible by setting tending to infinity, assigning hydrodynamic pressure at the far-field and at the free surface of the reservoir, and setting the hydrodynamic pressure gradient equal to zero in the normal direction of the reservoir bottom. In this case, the hydrodynamic effect is represented by a consistent mass matrix that is added to the mass matrix of structural system. Fluid Structure Interaction with Compressible Fluid (HFTD) In this case, the system is frequency dependent. Sonic wave velocity c=1483 m/s is associated with the reservoir fluid elements and radiation boundary condition is specified for the fluid far-field, Equation (1): 100
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
(1) ̇
Fully reflecting boundary is assumed for the reservoir bottom, Equation (2): (2) The hydrodynamic pressure is set
at the free surface of the reservoir.
Analysis Performed The load-cases specified in [9] are the self-weight of the dam, the hydrostatic pressure and the seismic loading. The present study is performed by means of phased analysis, which allows modelling the loading history. Thus, first are computed the stresses due to the self-weight of the dam. The hydrostatic pressure is applied next. Finally, the seismic loading is applied in the three directions of the supports using the base acceleration time-histories given in [9] and multiplied by a factor of 0.1 , where is the gravity acceleration. Static Analysis The dam construction stages are modelled in an approximate, but realistic way by means of phased activation of the dam elements. It is done in order to obtain correct strain and stress state due to the self-weight of the dam body prior to the application of the hydrostatic loading. In the linear analyses, the hydrostatic pressure is activated in a single phase as an instantaneous loading, whereas in the nonlinear analysis it is applied at ten steps corresponding to ten consecutive levels of filling of the dam reservoir. Eigenvalue Analysis The results of the computed eigenfrequencies for the cases of empty and full reservoir with added masses defined by the Westergaard formula (coarse and fine mesh) and by a FEM incompressible fluid-structure interaction analysis are presented in Table 1. Note that the general coordinate system axes are as follows: X-axis is the stream direction (from u/s to d/s), Y-axis is the cross-stream direction (from right to left), and Z-axis is the vertical direction. Table 1: Eigenrequencies and Effective Mass Percentage (Empty, WG, WG fine mesh, FSI) Mode 1 2 3 4 5 6 7 8 9 10
Empty Reservoir , Hz X, % Y, % 1.931 0.0 18.4 2.040 30.6 0.0 2.929 15.4 0.0 3.623 0.1 7.5 3.643 13.2 0.0 4.313 0.0 36.2 4.550 3.1 0.0 4.824 4.5 0.0 5.203 0.0 20.8 5.578 0.0 0.1 Σ 66.9 83.0
Z, % 0.0 1.0 0.7 0.1 15.0 0.0 3.4 61.7 0.0 0.0 81.8
Full Reservoir, WG WG FM , Hz X, % Y, % Z, % , Hz 1.314 0.0 8.5 0.0 1.305 1.340 26.4 0.0 0.2 1.336 2.004 12.4 0.0 0.1 1.992 2.362 15.7 0.0 1.1 2.352 2.519 0.0 2.4 0.0 2.498 2.961 0.0 6.6 0.0 2.944 3.184 0.8 0.0 0.1 3.153 3.703 1.4 0.0 0.1 3.680 3.903 0.0 0.7 0.0 3.860 3.938 7.1 0.0 1.6 3.914 Σ 63.8 18.1 3.3
101
FSI , Hz 1.572 1.622 2.362 2.944 3.040 3.719 3.869 4.555 4.764 4.803
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Mode 1,
1.527 Hz
Mode 2,
1.622 Hz
Mode 3,
2.362 Hz
Mode 4,
2.944 Hz
Mode 5,
3.040 Hz
Mode 6,
3.719 Hz
Mode 7,
3.869 Hz
Mode 8,
4.555 Hz
Mode 9,
4.764 Hz
Mode 10,
4.803 Hz
Figure 1: Dam Mode Shapes, Incompressible Fluid-Structure Interaction Analysis 102
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Envelopes of Min
80 Domain Analysis Hybrid Frequency Time The HFTD analysis is performed assuming linear structural behaviour and for only one time segment comprising the100whole duration of the seismic input. Envelopes of Max
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Results
140
The following line types are used to designate the type of analysis carried out: 160
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200
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4
6
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8 10 12 14 16 18 20
Figure 2: Analysis Type Designation
Radial Displacement (cm)
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Figure 3: Main Section Hoop Stresses (U/S on left; D/S on right)
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0
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4 -2
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Direct Time History Analyses As already discussed, two linear and a non-linear direct time history analyses are performed. The analyses are carried method of Hughes, Hilbert and Taylor with -0.3. 0 out by the The Rayleigh proportionality constants are computed so as to give a modal damping ratio of 5% in the first and the20 twelfth vibration modes. The nonlinear analysis considers only the effects due to opening/closing at the contraction joints (it is assumed that the joints are 40 provided with strong shear keys). The contraction joints are modelled by structural interface elements CQ48I to which 60 is associated a nonlinear elastic interface material model.
ICOLD - 12th INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS
715
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Hoop Stress (MPa)
Figure 4: Left Section Hoop Stresses (U/S on left; D/S on right)
0
715 Envelopes of Max
20
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-8
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Figure 5: Right Section Hoop Stresses (U/S on left; D/S on right) 104
2
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Upstream
Downstream
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-8
-6
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Elevation (m asl)
Envelopes of Min
6
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Figure 6: Main Section Cantilever Stresses
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-8
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Figure 7: Left Section Cantilever Stresses 105
4
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Figure 8: Right Section Cantilever Stresses
Downstream
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Principal Stress (MPa)
Figure 9: Main Section Minimum and Maximum Principal Stresses (U/S on left; D/S on right) 106
Reservoir Depth (m)
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Minimum and Maximum Principal Stresses Upstream
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Figure 10: Left Section Minimum and Maximum Principal Stresses
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Downstream
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Figure 11: Right Section Minimum and Maximum Principal Stresses 107
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Figure 12: Radial Displacements (Left Section on left; Main Section, Right Section on right)
18
80
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Figure 13: Time History of the Crown Displacement for Different Analyses
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Discussion and Conclusion The maximum static compressive stress is the hoop stress at ¾ the dam height on the u/s face of the main section. It amounts to approximately 7 MPa, which is within the admissible limits regarding the compressive strength of concrete. The maximum static radial displacement reaches 8.2 cm at the top of the central cantilever. The maximum compressive stresses during the ground motion reach approximately 12 MPa in the hoop direction on the u/s face. Seismic tensile hoop stresses varying between 1 MPa and 2 MPa are computed at the top part of u/s face of the main section; they disappear if the contraction joint opening/closing is modelled. The maximum seismic tensile vertical stresses exceed 2 MPA at ¾ the dam height on the upstream face of the main section. The maximum compressive and tensile stresses occurring during the earthquake are below the admissible limits. The maximum amplitude of the dynamic vibrations is about 8 cm with respect to the initial displaced shape of the dam. The hydrodynamic effect modelling approaches investigated in the present study lead to similar results regarding the structural response of the example arch dam. In general, the Westergaard added masses approach yields higher compressive and tensile stresses, as well as higher radial displacements. The compressible fluid analysis results in lower stresses with respect to the incompressible fluid assumption, which is due to the increased damping of the coupled system. In fact, according to [11], if the natural dam frequencies are significantly lower than this of the dam reservoir, , the behaviour of the latter is similar to the behaviour of incompressible fluid. In our case 1.98 Hz; this explains to a certain extent the obtained differences in the results with compressible and incompressible fluid.
References [1] Zienkiewicz, O.C., Taylor R.L. (2000). The Finite Element Method, Fifth edition, Vol. 1: The Basis, Butterworth-Heinemann, Oxford. [2] Westergaard, H.W., (1933). Water pressures on dams during earthquakes. American Society of Civil Engineers (ASCE), Proceedings. [3] Zangar, C.N., (1952). Hydrodynamic pressures on dam due to horizontal earthquake effects. Engineering Monograph No.11, U.S. Bureau of Reclamation. [4] Tan, H., and Chopra, A.K., (1995). Earthquake analysis of arch dams including damwater foundation rock interaction. Earthquake Eng. Struct. Dyn., 24(11), 1453–1474. [5] Wang, J. T., and Chopra, A. K. (2010). Linear analysis of concrete arch dams including dam-water-foundation rock interaction considering spatially varying ground motions. Earthquake Eng. Struct. Dyn., 39(7), 731–750. [6] Kuo, J. (1982). Added mass computations for incompressible fluid. Report UCB/EERC82/09, Earthquake Engineering Research Center, University of California at Berkeley. [7] Chavez, J.W., Fenves, J.L. (1993). Earthquake analysis and response of concrete gravity dams including base sliding. Report UCB/EERC-93/07, Earthquake Engineering Research Center, University of California at Berkeley. [8] TNO DIANA (2013). DIANA User’s Manual, Release 9.4.4. Delft, the Netherlands. [9] Graz University of Technology, Institute of Hydraulic Engineering and Water Resources Management (2013). 12-th International Benchmark Workshop on Numerical Analysis of Dams, Theme A. [10] Hughes, T. (1987). The Finite Element Method – Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Inc. [11] Sécurité des ouvrages d’accumulation (2003). Documentation de base pour la vérification des ouvrages d’accumulation aux séismes, rapports de l’OFEG, série Eaux, version 1.2.
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Dynamic Analysis of an Arch dam with FluidStructure Interaction Use of an Open source code AKANTU M. Chambart1, T. Menouillard1, N. Richart2, J.-F. Molinari2 and R. M. Gunn1 1
STUCKY SA, Rue du Lac 33, CH-1020 Renens, SWITZERLAND 2 EPFL ENAC LSMS, CH-1015 Lausanne, SWITZERLAND E-mail: [email protected], [email protected]
Abstract In this contribution, the fluid-structure interaction is modeled using the added mass technique and the incompressible fluid model. Results show that the added mass technique is more conservative. Different meshes are compared with this later method, demonstrating that the problem is not mesh dependent. Finally, the computation times obtained with two different software are compared, showing the efficiency of the new open-source software Akantu
Introduction One of the exercises proposed by the ICOLD for 12TH INTERNATIONAL BENCHMARK WORKSHOP ON NUMERICAL ANALYSIS OF DAMS consists in the dynamic analysis of an arch dam under a seismic loading. Since the geometry, the material properties and the loading are imposed to the participants, the focus is put on another modelling aspect which is the fluid-structure interaction. The methods proposed to model the fluid-structure interaction starts with the simplest ones, the added mass technique to the most sophisticated ones where the fluid is explicitly modeled as a compressible body. In our practice, as an engineering company, our choice for one method versus another is often governed by the gain in accuracy versus the loss in time. Most of the time in projects, the added mass approach is used because it is the fastest and usually sufficient to fulfill the authorities demands. This benchmark is an interesting opportunity to compare different methods in terms of results but also in terms of computation time. The comparison is carried out using the finite element software DIANA [1], simply using some of the proposed methods (added mass and incompressible fluid). As a second step, the results obtained with DIANA using the added mass technique are compared with the ones given by a new open-source software AKANTU [2] developed at the Swiss Ecole Polytechnique Fédérale de Lausanne. Akantu is a quite innovative object-oriented program written in C++. The objective using this latter software is to optimize the computation time, in order to get reliable results in shorten time, or run sophisticated computations in an acceptable amount of time. In order to estimate the mesh dependency of the problem, the results with two meshes are compared with the added mass technique. Therefore the need in future for High Performance Computing can be assessed.
Numerical Modelling Model and Mesh Figure 1 presents different meshes used with the Westergaard theory for dealing with the hydrodynamic pressure, and also the mesh with the fluid. There are a coarse and a fine mesh, whose numbers of elements are respectively 2874 and 22280.
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Figure 1: View of the meshes: coarse and fine, and with the fluid. Materials In this problem, three different materials are involved: the concrete of the dam, the rock of the foundation, and the water of the reservoir. Their density and stiffness are the valuable data in linear elastic problem. Table 1 presents the materials parameters. The density of the rock is numerically taken to be zero in order to avoid wave propagation at the boundary of the rock foundation. In addition, a dynamic Young’s modulus is used, whose value is 125% the static one [4]. Table 1: Materials properties. Properties Concrete Rock Water Density 2400 kg/m3 0 kg/m3 1000 kg/m3 Poisson ratio 0.167 0.2 Young’s modulus 27 GPa 25 GPa Loadings Static loadings. The self-weight of the concrete is given by the density of the concrete and the gravity. The hydrostatic pressure on the upstream face of the dam is given by the water level, i.e. 715 m asl. Dynamics loadings. The acceleration history of the seismic input is given through three accelerograms presented in Figure 2, and these three accelerograms are simultaneously applied at the boundary of the rock foundation. Their corresponding absolute displacement histories are also presented in Figure 2. Depending on the FEM program used, the boundary conditions may have to be given in displacement or in acceleration.
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Figure 2: Acceleration and displacement time histories. Fluid-structure interaction Two alternatives are explored in this section. First, the most basic development to deal with hydrodynamic pressure on a surface is the method developed by Westergaard [1]; it takes the hydrodynamic effect into account by adding masses on the upstream surface of the structure. There is no need to mesh the fluid, thus the size of the system is restricted to the size of the solids. The added mass is defined according to [1] as: (1) Where w is the density of the water, hw the total height of water and h the distance from the bottom of the reservoir to the current point where mw is evaluated. Figure 3 illustrates the Westergaard equation.
Figure 3: Westergaard theory on the upstream face of a dam. Second, a method [3] where the fluid, considered incompressible, is explicitly meshed as a 3D part of the system is also considered. Solid elements are quadratic, whereas fluid elements are only linear. Therefore special interface elements are required on the upstream face of the dam 113
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to link the fluid to the solid domains. In addition, special boundary elements are involved to deal with the bottom absorption, the free surface and the far field surface of the reservoir. A method dealing with compressible fluid is not developed in this paper.
Vibration modes The vibration modes have been evaluated with different meshes and the Westergaard method for describing the hydrodynamic pressure. A very good agreement between the results of the different meshes was observed. The reason is that the linear elastic material model describes similar dynamic behavior of the structure, not depending on the mesh size. Figure 4 presents the deformed shape of the first 10 modes within the coarse mesh.
Figure 4: Deformed shapes of the first 10 vibrations modes. Table 2 presents the vibration frequencies and the mass percentage of the dam in each direction obtained with the coarse mesh and Westergaard theory. Mode 1 is a left-right motion with 20% of the mass of the dam. Mode 2 is the major vibration with 35% of the mass in the upstream-downstream direction. Table 2: Frequencies and mass percentage for the coarse mesh with Westergaard. Mode Frequency Mass percentage Mass percentage Mass percentage US/DS Vertical Left-Right [Hz] 1.48 1% 0% 20% 1 114
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1.50 2.27 2.67 2.92 3.38 3.73 4.25 4.54 4.64
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A damping effect of 5% is taken through Rayleigh coefficients. They are evaluated from the knowledge of two modes (1 and 13) containing 80% of total mass. The Rayleigh damping matrix C is defined by two parameters and as: (2) C=M+K where M is the mass matrix, K the stiffness. With the selected modes (1 to 13), the coefficients are: =0.726286 and =0.002346. Figure 5 shows the damping as a function of the frequency; the crosses on the graphs are the different vibration modes, and the damping of frequencies 1 and 13 are exactly 5%. As it has been selected, 80% of the mass of the dam has a damping between 4 and 5%. The modes with high frequency, i.e. greater than 10 Hz, are minor modes in terms of mass percentage. 10 9 8
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Results Table 3 presents the different computations run and the corresponding mesh size. With DIANA, two simulations with Westergaard and one with the fluid were run, whereas only two simulations were run with AKANTU. All simulation had 2000 steps for the dynamics. Table 3: Table of the different computations run with the two codes. Mesh Nodes Elements Elapsed time CPU time [s]
DIANA Westergaard Coarse Fine 3614 25057 2874 22280 1h10 7h00 13’291 s 40’752 s
With Fluid Coarse 14478 3632 4h00 34’841 s
AKANTU Westergaard Coarse Fine 3614 25057 2874 22280 0h13 5h40 2’819 s 74’000 s
The computations are run for both codes in parallel with the same number of processors available. However for Akantu the computation is performed on a powerful laptop while for Diana a workstation optimized to numerical simulations is used. The results show that Akantu
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is more efficient since the computation time is divided by 5 for the coarse mesh and by 1.25 with the fine mesh. The evaluation of the hoop, vertical stresses and radial displacement are performed on the three following sections denoted by Left, Mid and Right sections as shown in Figure 6.
Figure 6: View of the different sections for post-processing the results. Vertical stresses The vertical stress is a relevant result within the dam. Thus Figure 7 presents the vertical stress on the upstream and downstream faces of the Left section of the dam: static, maximum and minimum results are presented with the coarse and fine mesh and the Westergaard theory, with Diana and Akantu. Both meshes give similar results too for static, maximum and minimum vertical stresses. The vertical stress is always negative in static, but its maximum reaches positive value in some parts during the earthquake. All four results obtained with the Westergaard theory agree well. However the stress envelops obtained with the most sophisticated computation, i.e. incompressible meshed fluid, is quite smaller. The Westergaard theory seems conservative.
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Figure 7: Vertical stresses in the different sections. Hoop stresses The hoop stress is the other relevant results in terms of stress in an arch dam. Figure 8 presents the hoop stress at the different sections on the upstream and downstream faces of the dam: static, minimum and maximum stresses are presented for the different computations. There is a good agreement between the results obtained by both meshes and the Westergaard theory. One observes that the hoop during the earthquake is mainly negative, so the arch dam is in compressive stress state in static and also during the earthquake.
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Figure 8: Hoop stresses in the different sections. Principal maximum stresses Figure 9 presents the maximum principal stress at the different sections on the upstream and downstream faces of the dam: static, minimum and maximum stresses are presented for the different computations. There is a good agreement between the results obtained by both meshes and the Westergaard theory. One observes that the minimum of the maximum principal stress during the earthquake is close to the static maximum principal stress. Indeed the maximum principal stress can only increase due to the dynamic loading from the static case. This maximum principal stress behaves like the vertical stress in the arch dam during dynamics.
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Figure 9: Principal maximum stresses in the different sections. Principal minimum stresses Figure 10 presents the minimum principal stress at the different sections on the upstream and downstream faces of the dam: static, minimum and maximum stresses are presented for the different computations. There is a good agreement between the results obtained by both meshes and the Westergaard theory. This minimum principal stress behaves like the hoop stress in the arch dam during dynamics.
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Figure 10: Principal minimum stresses in the different sections. Radial displacements The radial displacement quantifies the motion of the dam, in static and during the Earthquake. Figure 11 presents the radial displacement of the different sections: static, minimum and maximum obtained with the coarse and fine mesh and the Westergaard theory with both software and also the displacement obtained with the incompressible fluid. There is a good agreement between these two results in static and in dynamic. The static radial displacement at the top of the Mid section is about 8 cm, whereas the maximum during the Earthquake is about 16 cm, and the minimum about 3 cm.
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Figure 11: Radial displacements.
Conclusion This benchmark allowed to investigate the field of how, in term of numerical method, to deal with the hydrodynamic pressure on the upstream face of an arch dam. This has then shown that the Westergaard theory, which consists in simply adding mass on nodes of the upstream face of the dam, is clearly not mesh dependent. Second, the method dealing with the mesh of the incompressible fluid, gives smaller envelops in terms of maximum and minimum of displacement and stresses of the arch dam. In addition, the development of the Westergaard theory in an open source code did not show any relevant difficulties and the computation time can be significantly decreased.
References [1] Water pressures on Dams during Earthquakes, H.M. Westergaard, Transactions, ASCE 98:418-472, 1933. [2] Fluid-Structure Interactions: added mass computations for incompressible fluid, James Shaw-Haw Kuo, University of California, Berkeley, August 1982. [3] Hybrid Frequency Time Domain – Validation, TNO DIANA Report, 2008-DIANAR003, 2010. [4] Sécurité des ouvrages d’accumulation – Documentation de base pour la verification des ouvrages d’accumulation aux séismes, rapports de l’OFEG, série Eaux, version 1.2, mars 2003. 121
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Study on Arch Dam– Reservoir Seismic Interaction A. Popovici1, C. Ilinca1 and R. Vârvorea1 1
Technical University of Civil Engineering, 124 Lacul Tei Bd. 020396-Bucharest, ROMANIA E-mail: [email protected]
Abstract The arch dam – reservoir seismic interaction is investigated using ABAQUS 6.11 and DESARC 3.1 software. DESARC computer code offers the advantage of simplicity and computation speed due to the degrees of freedom based on the stresses (Ritter modified method) being very recommended for arch dams preliminary structural analysis. The coarse mesh given by formulator was used for investigation in ABAQUS and 12 arches equally spaced on dam height were used in DESARC. The water effect was considered according to added mass procedure as well as acoustic elements. All analyses were performed in the linear elastic field. The results are presented in compliance with formulator requests: eigenfrequencies and mode shapes, hoop stresses, vertical stresses, min./max. principal stresses and radial displacements in three different sections for static and seismic loads. A special attention is paid to compare the results concerning arch dam – reservoir seismic interaction in different hypotheses applying two software.
Introduction The effects of different hypotheses on arch dam - reservoir seismic interaction are investigated in this paper based on data provided in Theme A by formulator, Graz University of Technology – Institute of Hydraulic Engineering and Water Resources Management [1]. The arch dam is a symmetrical structure with the followings main characteristics: maximum height 220 m chord length at dam’s crest 430 m valley width at bottom 80 m The analyses are carried out using ABAQUS 6.11 [2] and DESARC 3.1 [3],[4] software. The finite element mesh of the arch dam structure-foundation-reservoir system used in ABAQUS corresponds with alternative coarse mesh given by formulator (Figure 1). The main features of the coarse mesh are as follows: arch dam – total number of nodes 2083 total number of elements 356 (312 C3D20R and 44 C3D15) foundation – total number of nodes 11608 total number of elements 2340 C3D20R reservoir – total number of nodes 12493 total number of elements 2640 C3D20R
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Figure 1: Finite element mesh of the arch dam structure-foundation-reservoir system used in ABAQUS code Other hypotheses used in analyses performed with ABAQUS were the followings: the dead weight were applied on monolithic structure the reservoir influence on dam seismic response was considered by added mass procedure excepting the free vibration analysis for full reservoir using acoustic elements C3D20A for reservoir mesh; the water was considered as incompressible with shear modulus tends to zero; the seismic response was evaluated for empty and full reservoir by modal superposition and direct time integration; all analyses were based on assumption of linear elastic behavior of materials; three accelerograms in direct time integration were applied on the faces of foundation, respectively x, y and z directions; the fraction of critical damping in direct time integration was 5%, α and β coefficients in linear Rayleigh model were computed for ω1 and ω2 , resulting α=0.3900 and β=0.0064. DESARC is an interactive, fast and reliable computer code very recommended for preliminary fast static and dynamic structural analysis of arch dams. The simplicity and computation speed of DESARC is due to the degrees of freedom based on the stresses (Ritter modified method and not on displacements. The program can handle symmetric arch dams both with circular and parabolic midline, rotationally symmetric structure with liquid inside or outside, cooling towers etc. All the geometrical parameters (thickness, radii, half lengths and cantilever shape) are expressed under assumption of parabolic variation with elevation. Accordingly the real arch dam shape provided by formulator was equated as assumptions presented above (Figure 2)
Figure 2: Equating real dam shape under assumptions of DESARC
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A number of 12 arches of the dam was considered under analysis (Figure 3). In may be remark from figures 2 and 3, resulting structure under DESARC assumptions has the shape very close to the actual dam.
Figure 3: Geometrical features of the arch dam equated under DESARC assumptions Other hypotheses used in analyses performed with DESARC were the followings: the dam-foundation interaction was considered by Vogt coefficients (Er/ Ec=0.90, Er=25000 MPa and Ec=27000 Mpa); the reservoir influence on dam seismic response was considered by added mass procedure; the hydrodynamic pressures collinear with the direction of horizontal upstream – downstream earthquake (Phd,0) were evaluated with Westergaard relationship. For other directions (Phd,α) was applied the following relation: (1)
Phd,α = Phd,0 . cos α
α being the angle between earthquake direction and the normal to the surface at the point considered; the dead weight were applied on isolated cantilevers and monolithic structure, too.
Some aspects concerning mathematical models The effect of the water in the reservoir, under the assumption of an incompressible ideal fluid, is usually calculated by use of the added mass procedure. This is a mathematical artifice used in order to simplify the analysis of the structure-liquid seismic interaction. The added mass is determined from the hydrodynamic forces {Ph(t)} and is attached to the mass of the structure. Unlike the dead mass of a structure, the added mass acts only on hydrodynamic force direction, namely the direction of the normal to the surface it is applied. Assuming that the directions of hydrodynamic forces, of earthquake and of degrees of freedom of the structure are similar (Figure 4a), the added mass matrix [Mh] is determined with relation: {
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̈ } is total acceleration response to normal direction at surface in the considered where { ̈ point. In this case the analysis of the structure-liquid seismic interaction is performed as for usual structure according to formula: ] { ̈ }
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In the general case when the directions of the normal to surface, of the earthquake and of the structure degrees of freedom are different, the added masses computed according to (2) relationship must be projected successively on the normal to surface direction and on the degrees of freedom of the structure (Figure 4b).
Figure 4: Assessment of added masses The dynamic equilibrium equations which include the structure-liquid seismic interaction are written in this general case as follows: [ [
] [ ] [ ] [ ] [
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where [rc,n] has dimensions equal with the number of the degrees of freedom of the system and contains on diagonal the cosine directors between the normal to surface in the nodes of the system mesh and earthquake direction and [rn,x,y,z] has dimensions corresponding to the number of the degrees of freedom of the system and contains on diagonal the cosine directors between the normal to surface in the nodes of the system mesh and directions of the dynamic degrees of freedom of the system.
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Results of analyses The results are provided as requested by the formulator. In the Table 1 are presented the natural circular frequencies (ω, rad/s) and respectively natural frequencies (f, cycles/s ) values of the first 10 mode shapes including interaction with the reservoir modeled with acoustic elements C3D20A and added mass procedure computed with ABAQUS and the value of the frequency of the fundamental mode shape including interaction with the reservoir by added masses computed with DESARC. It may remark that natural frequencies computed with ABAQUS by added mass procedure are smaller with about 2…21 % relative to their counterparts having reservoir interaction modeled with acoustic elements. In figure 5 is illustrated the fundamental mode shape of the arch dam in hypothesis of the full reservoir with water elevation at the dam crest. Table 1: Eigenfrequencies ABAQUS
DESARC Eigenfrequency-added mass (Rad/s) (Hz) (Rad/s) (Hz) 7.756 1.235 10.557 1.681 7.837 1.248 12.139 1.933 14.865 2.367 15.047 2.396 16.460 2.621 17.942 2.857 19.430 3.094 19.594 3.120 22.935 3.652
Eigenfrequency-acoustic elements Mode no. (Rad/s) (Hz) 1 9.726 1.548 2 9.843 1.567 3 12.160 1.935 4 14.458 2.301 5 15.584 2.480 6 19.109 3.041 7 19.573 3.115 8 20.695 3.294 9 22.687 3.611 10 23.280 3.705
Figure 5: Fundamental mode shape view from downstream Figure 6 illustrates three accelerograms provided by formulator for earthquake analyses (maximum acceleration 0.1g) and respectively figure 7 response spectra corresponding to accelerograms used for modal superposition analyses (spectral analyses).
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Figure 6: Accelerograms provided by formulator for earthquake analyses
Figure 7: Response spectra (x,y,z) computed from the accelerograms provided by formulator and used in spectral analyses Radial displacements on x, y and z directions in dam central sections at crest and base levels computed with ABAQUS in direct time integration method are illustrated in figure 8. Hoop stresses, vertical stresses and radial displacements in the dam central section downstream/upstream faces generated by combined dead weight + hydrostatic pressures are illustrated in figures 9a,b. The results in spectral analysis and direct time integration, in the dam central section, downstream/upstream faces, full reservoir case are comparatively presented in figures 10a,b. The min./max stresses in the dam central section, downstream/upstream faces resulted during direct time integration under combined actions of dead weight + hydrostatic pressures + three-dimensional earthquake 0.1g are illustrated in figures 11a,b
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Figure 8: Radial displacements in dam central section at crest (top) and base (bottom) computed with ABAQUS
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Figure 9a: Combined dead weight + hydrostatic pressures – Hoop and vertical stresses, radial displacements on downstream face, central section
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Figure 9b: Combined dead weight + hydrostatic pressures – Hoop and vertical stresses, radial displacements on upstream face, central section
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Figure 10a: Spectral analysis and direct time integration, central section, downstream face, full reservoir
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Figure 10b: Spectral analysis and direct time integration, central section, upstream face, full reservoir.
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Figure 11a: Min./max stresses, direct time integration, dead weight+ hydrostatic pressure + three-dimensional earthquake 0.1g
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Figure 11b: Min./max., direct time integration, dead weight+ hydrostatic pressure + threedimensional earthquake 0.1g
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Comments on results of analyses In compliance with some results illustrated in the above captions and other ones resulted from the analyses carried out the followings comments may be pointed out: DESARC3.1 a very simply, friendly and fast computer code in all applications led to results close to those provided by much more sophisticated ABAQUS6.11 computer code. As a result DESARC computer code is very recommended for arch dams preliminary structural analysis. Usually, in engineering practice dead loads are applied on arch dam isolated cantilevers, taking into account that during dam construction the contraction joints are not grouted. However, even under these conditions, some of the dead weight of cantilevers is transferred to arches, this quota depending of the dam shape and valley opening. Coming back to the arch dam provided by formulator it may remark from stresses and displacements to dead loads and hydrostatic pressures computed with DESARC in both hypotheses of isolated cantilevers and monolithic structure (fig.8a,b). that the effects of isolated cantilever hypothesis in relation with monolithic structure are not important. Accordingly, the analyses with ABAQUS, for simplicity, were carried out only in case of monolithic structure. The dam displacements due to dead weight reach about 1 cm to upstream at the crest level. Dead weight is mainly transferred on cantilever. At the bottom of the central section the vertical stresses reach -7 MPa compression at upstream toe and -1.5 MPa compression at downstream heel. The hypothesis with isolated cantilever is a conservative one. On arches, the maximum stress reaches -2 MPa compression at the crest level. Hydrostatic pressure as independent load generates maximum radial displacement of 7.5 cm in central section, dam crest elevation. The vertical stresses vary between 1 MPa tension and 4 MPa compression at downstream face and, respectively between 8 MPa tension at the dam upstream toe and -2 MPa compression at upstream face. The hoop stresses vary between 0.5 MPa tension, -3.7 MPa compression at downstream face, respectively between 0.5 MPa tension, -7 MPa compression at upstream face. The big vertical tensile stress generated by hydrostatic pressure at the dam upstream toe is reduced to 1.5 MPa by vertical compressive stress in the same point due to dead weight. Based on results concerning natural periods values computed in full reservoir hypothesis (Table 1) (fundamental period 0.809 s ABAQUS, added mass, 0.645 ABAQUS, acoustic elements)it may conclude the hydrodynamic forces computed with Westergaard formula to generate added masses are higher than hydrodynamic forces developed by acoustic elements. A comparison between seismic responses computed in the full reservoir hypothesis by spectral analysis and direct time integration with ABAQUS points out a good correlation between correspondent displacements response their maximum reaching about 13 cm but generally significant differences between correspondent stresses. For instance in the dam central section at downstream face the principal stresses vary between 1.50 MPa (tension) and -4 MPa (compression) in direct time integration and respectively 0,9 MPa and -9.50 MPa in spectral analysis. Maximum displacement obtained in direct time integration at combination static loads (dead weight + hydrostatic pressures) + earthquake (full reservoir) reaches 18 cm to downstream at the crest level in central section. Max/min vertical stresses on downstream face varies between 0.8 MPa and -6.0 MPa . On upstream face at the dam upstream toe the vertical stress reaches tension of 7 MPa. It is a vulnerable point for cracking. On horizontal direction in the same point the hoop stresses vary between 3.5 MPa tension and -2.5 MPa compression. It may conclude that excepting dam upstream toe where exists cracking risk, the dam withstand earthquake action without notable incident. As a general conclusion based on data presented above, the seismic response parameters of arch dams, especially seismic stress state differ significantly function of method of analysis.
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Unfortunately, the scarcity of recordings concerning arch dams behavior during strong earthquakes make difficult to validate a method of analysis using field recordings from.
References [1] Graz University of Technology THEME A Fluid Structure Interaction Arch Dam – Reservoir at Seismic Loading. 12-th International Benchmark Workshop on Numerical Analysis of Dams, Graz – Austria, 2013. [2] ABAQUS 6.11. Abaqus / CAE User's Manual. United States of America: Abaqus Inc, 2009. [3] Fanelli M., Giuseppetti G., Rabagliati U. Il metodo di Ritter Modificato. L’Energia Eletrica, No.3, 1978. [4] Fanelli M., Fanelli A. Designer’s Guide DESARC 3.1 for Windows ISMES S.p.A., Bergamo, 1995. [5] Popovici A., Popescu C. Dams for water storage (in Romanian) Editura Tehnica, Bucharest, 1992. [6] Priscu R., Popovici A., Stematiu D., Stere C. Earthquake Engineering for Large Dams Editura Academiei (Romania) and John Willey & Sons Ltd. (U.K.), Chichester, 1985. [7] Popovici A., Abdulamit A., Toma I. Moldoveanu T. .Assessment of the seismic safety of dams based on „in site” measurements and back analyses. Proceedings XXI-st International Congress on Large Dams, Montreal, 2003.
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Concrete arch dam at seismic loading with fluid structure interaction R. Malm1,2, C. Pi Rito2, M. Hassanzadeh3,4, C. Rydell1,3 and T. Gasch1,3 1
KTH Royal Institute of Technology, Concrete Structures, Stockholm, Sweden. 2 SWECO Infrastructure, Hydropower and Dams, Stockholm, Sweden 3 Vattenfall Engineering, Stockholm, Sweden. 4 Lund University, Building Materials, Lund, Sweden. E-mail: [email protected]
Abstract A concrete arch dam have been analyzed during seismic loading with a model based on acoustic elements to describe the water and infinite elements as quiet boundaries to prevent wave reflection. The results have also been compared with a simplified model based on Westergaards added mass approach. The simplified model is only used, in this study, for comparison with the more advanced model with acoustic elements. Therefore the results from this simplified model are just used as a rough estimate of the induced stresses and displacements. Despite this, the simplified Westergaard model gives similar results compared to the more advanced model with acoustic elements for the water and infinite elements for the boundaries. The largest difference between the models often occurs at the nodes in the base of the arch dam, which may be due to poor discretization. Generally, the Westergaard added mass gives higher maximum principal stresses at the base on the upstream side than the acoustic model, while often underestimating the min principal stresses at the base on the downstream side. Both models show high tensile stresses near the base of the arch dam that would result in cracks.
Finite element models The studied geometry was given by [1], and in this paper, the mesh denoted as fine mesh in [1] has been used. All numerical analyses in this paper have been performed with the commercial software Abaqus ver.6.12. The model consists of a foundation with dimensions 1000 x 1000 x 500 m built up of 2700 second order brick elements with reduced integration. In one of the models presented in this paper, infinite elements have been used for the outer surfaces of the foundation in order to remove reflecting waves on the model edges, see Figure 1 a). The arch dam is 220 m high, with a width of 430 m at the crest and 80 m at the base. The arch dam consists of 2736 second order brick elements with reduced integration (8 integration points). The geometry of the reservoir is 460 m long (i.e. two times the dam height). In the following analyses, two different models to account for the water will be presented. In a simplified model, the reservoir has been replaced with nodal masses corresponding to the Westergaard added mass approach [2]. In the more advanced model, the reservoir has been included by means of acoustic elements and infinite acoustic elements to account for an infinite long reservoir. Loads and boundary conditions All analyses have been performed in two steps; static analysis of response due to gravity and hydrostatic water pressure, and dynamic implicit analysis (transient dynamic analysis) to calculate the response from induced ground accelerations due to the seismic load.
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In all analyses, the gravity load has only been assigned to the elements that constitute the arch dam. The rock is according to the instructions, [1], defined with zero density. In the simplified model, based on the Westergaard added mass approach, additional nodal masses are introduced on the upstream surface of the arch dam. However, it is important that the nodal masses are not assigned a gravity load since that would affect the static behavior of the dam. The hydrostatic water pressure has been assigned to the upstream surface of the arch dam, and to all rock surfaces subjected to the water pressure, as seen in Figure 1 b). The maximum hydrostatic pressure is defined as 2.1582 MPa (220 m x 9.81 m/s2 x 1000 kg/m3). In the more advanced model, with acoustic elements, an additional boundary condition with zero acoustic pressure on the free surface of the water has been defined.
a) b) Figure 1: a) Geometry and element types of the model, b) applied hydrostatic water pressure. The ground accelerations are illustrated in Figure 2. In the dynamic analysis, a constant time step of Δt = 0.01 s have been used. The upper frequency that can be captured in the analysis, i.e. the Nyquist frequency, is thereby 50 Hz. According to the frequency analyses that were performed, the cumulative mass is nearly 100 % for frequencies up to 30 Hz and thereby the chosen time increment is considered sufficient. In addition, the time history signal of the earthquake is sampled with Δt = 0.01 s and thereby the highest reproducible frequency in the sample is 50 Hz. Acceleration [G]
0.05 0 -0.05
Acceleration [G]
0.05 0 -0.05
Acceleration [G]
Horizontal 1 EQ
0.05 0 -0.05
0
2
4
6
8
10 12 Time [s] Horizontal 2 EQ
14
16
18
20
0
2
4
6
8
14
16
18
20
0
2
4
6
8
14
16
18
20
10 12 Time [s] Vertical EQ
10 Time [s]
12
Figure 2: Transient time history signals of the ground accelerations. In the static analysis, all nodes on the bottom surface of the foundation were constrained for all translations, i.e. in x-, y- and z-direction. In the seismic analysis, these boundary conditions are replaced, where all nodes on the bottom surface were assigned prescribed ground accelerations according to Figure 2. The sides of the foundation have not been assigned boundary conditions in neither the static nor the dynamic analysis. The reason for this is that 140
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these surfaces will be defined with infinite elements in the model with acoustic elements. It is not possible to define prescribed conditions in the form of accelerations, velocities or displacements on nodes that are part of infinite elements if the infinite elements are to function as quiet boundaries. One approach to overcome this would for instance be to calculate corresponding stresses from the induced accelerations. However, considering the large geometry of the rock, it is likely that seismic wave incoherence would influence the results. It is therefore unlikely to assume that the ground vibrations would be identical on all sides. In addition, the rock is defined with zero mass, and therefore it will not produce mass inertia forces. Hence, the approximation adopted here, where the boundary is applied only to the bottom nodes, is considered to be valid for this case. Material properties and damping All material properties have been defined by [1]. Rayleigh damping have also been used in all presented analyses. The damping ratio for concrete has been assumed equal to 4 % according to Regulatory Guide 1.61, [3]. The damping of the water is usually assumed to be 0.5 %; this has been judged to be negligible and therefore not included in the analyses. The lower frequency of the Rayleigh damping have been defined corresponding to the frequency where 5 % of the cumulative effective mass is active, which in this case is f1 = 1.27 Hz. This ensures minimal underestimation of response in the low frequency range according to [4] and [5]. The upper frequency have been defined corresponding to the frequency where 80 % of the cumulative effective mass is active, which in this case is f2 = 9.76 Hz. The corresponding damping values ξ1 and ξ2 have been calculated so that the minimum damping in the interval f1 – f2 is 1 % less than the target damping value, i.e. ξmin = 3 %. Based on this approach, the obtained Rayleigh damping coefficients are α = 0.66314 and β= 0.0013503 with the corresponding curve illustrated in Figure 3. The average damping in the interval f1 – f2 is 3.6 %, i.e. slightly lower than the target value of 4 % and thereby conservative. 0.1
Damping
0.08 0.06 0.04
m = 5 %
0.02 0
0
5
m = 80 % 10
15 20 Frequency (Hz)
25
30
Figure 3: Rayleigh damping curve (α = 0.66314 and β= 0.0013503) The Rayleigh damping coefficients have been defined in the material definition for concrete but also for the rock. Since the rock have zero density, only the stiffness proportional part of the Rayleigh damping (β= 0.0013503) will be active. The Rayleigh damping curve has been verified against a modal analysis with uniform damping of 4 %, where the Rayleigh damping gave slightly conservative result. Model 1: Westergaard added mass approach According to Westergaard [6], the hydrodynamic forces exerted on a dam due to earthquake ground motions are equivalent to inertia forces of a volume of water attached to the dam moving back and forth with the dam while the rest of the reservoir water remains inactive. The influence of the reservoir on a 2D rigid monolith with vertical upstream face is included by introducing the impulsive mass of the water and thereby, altering the dynamic properties of 141
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the structure. Kuo [7], further developed the method to also account for the curvature of an arch dam, this have however not been considered in this paper. The reason for this is that the simplified added mass model is only used, here, for comparison with the more advanced model with acoustic elements. Therefore, the results from this simplified model are just considered as a rough estimate of the induced stresses etc. A MatLab script have been developed to calculate the tributary surface area for each node on the upstream surface, and based on this the nodal mass has been calculated according to [6]. The calculated nodal masses are illustrated in Figure 4, where the colors in the figure represent the added weight (in kg) for each node on the upstream surface of the arch dam. 750
Westergaard added mass
4.26e+006
700 3.55e+006 650 2.84e+006 600 2.13e+006 550 1.42e+006 500
450
710000
-200
-150
-100
-50
0
50
100
150
200
Figure 4: Nodal masses based on Westergaard added masses Model 2: Acoustic and infinite elements In the more advanced model, the water has been included by means of 2640 second order acoustic elements. These elements are based on an acoustic formulation with the fluid wave velocity, i.e. the variation in pressure, as the independent variable. These elements do not include any terms for body forces and therefore, the hydrostatic pressure needs to be included as a pressure load on the structure, as previously shown in Figure 1 b). The finite element mesh of the acoustic elements is stationary at all nodes except at the boundaries of the fluid domain, i.e. the fluid-structure interface. For boundaries adjacent to a structural domain, the nodes of the acoustic medium can be prescribed to follow the nodes of the structural domain, giving a pressure change in the acoustic medium. A pressure boundary which prescribes zero acoustic pressure on the free surface has been defined, due to the lack of displacement DOF. This gives no actual displacement of the free surface but is correct in the sense of wave propagation in the medium. In order to account for an infinite long reservoir, 264 additional infinite acoustic elements are defined on the upstream side of the reservoir. In order to prevent reflecting waves on the model edges, 510 infinite elements have been defined for the surfaces on the foundation rock as illustrated in Figure 1 a). The infinite elements provide quiet boundaries and are based on Lysmer and Kuhlemeyer [8] for dynamic response. The damping constants for longitudinal and shear waves are calculated as (1) d p c p 2G
d s cs G
(2)
E E G 2 1 1 1 2 , where, and E is the elastic modulus, ν is poisons ratio and cp and cs are the longitudinal and shear wave speed. The infinite rock elements were defined with the same properties as rock (see [1]), with the addition of a defined density of 2600 kg/m3, required for the damping. The bottom surface
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was defined without infinite elements, due to the fact that the seismic ground motion was applied in the base. The length of the infinite rock elements are 1000 m, i.e. chosen as the total length of the original foundation. The infinite elements are active in static analyses, by providing horizontal stiffness to the foundation.
Eigenfrequencies and modes
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Figure 5: The 10 first eigenmodes obtained from the acoustic model.
g) Mode 7 f) Mode 6
a) Mode 1
b) Mode 2
c) Mode 3
h) Mode 8
i) Mode 9
d) Mode 4
e) Mode 5
j) Mode 10)
The 10 first mode shapes obtained from the acoustic model are illustrated in Figure 5.
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In some cases, the acoustic model gives two closely spaced frequencies with the same mode shape. In Table 1, these are therefore presented together as one mode. In addition, some modes in the acoustic model are mainly relevant to the water, and hence these are not presented here. It can however be mentioned that the first mode corresponding to the water is about 0.7 Hz. The frequencies for each mode is also presented from the Westergaard added mass approach and the original model without water for comparison in Table 1. Table 1: Calculated eigenfrequencies for the ten first modes. Frequency (Hz) Mode
Acoustic elements
Added mass
W.o. water
1
1.5114
1.3730
2.0002
2
1.5431
1.3291
1.9012
2.054
2.8785
3
1.9056
1.9850
4
2.2255
2.3534
5
2.4254
2.4665
6
2.9626
2.9997
2.5633
3.5491
7
3.0084
3.1169
2.5782
3.5998
8
3.2818
3.3150
2.8773
4.1744
3.2224
4.4779
9
3.5886
10
3.7748
As seen in the table, the acoustic model gives frequencies that are in-between the case without water and the case with Westergaard added mass. Notable is also that the first and second mode have change place in the acoustic model compared to the others models.
Stresses Stresses are presented for three sections, Left section (-45°), Main section (0°) and Right section (+45°) from the center of the arch, as defined by [1]. The following stresses are presented; maximum and minimum principal stresses, hoop stresses and vertical stresses. For each of these stresses, the max and min envelope for each node on the upstream (US) or downstream (DS) surface are presented. This means that the presented stresses are not occurring at the same time; instead these are the maximum or minimum stresses that occur for the whole time period for each node on the upstream or downstream side of each of the studied sections. As seen in Figure 6 there is quite similar results between the simplified Westergaard added mass method and the method with acoustic and infinite elements. The minimum principal and hoop stresses show, generally, better resemblance between the two models than the max principal and vertical stresses. The largest difference between the models often occurs at the nodes in the base of the arch dam, which may be due to poor discretization. Generally, the Westergaard added mass gives higher maximum principal stresses at the base on the upstream side than the acoustic model, while underestimating the min principal stresses at the base on the downstream side. Both models show high tensile stresses near the base of the arch dam that would result in cracks. It can from this comparison not be drawn any conclusion that the simplified Westergaard added mass approach would be more conservative than the method with acoustic and infinite elements. The analyses also show that, despite the
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symmetric shape of the arch dam, there is a difference between the results at the left and right section, as seen in Figure 7.
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Figure 6 Stresses in the main section (0°) according to the model with acoustic elements and the simplified Westergaard model, on US and DS surface respectively.
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Figure 7: Stresses in the left (-45°) and right section (+45°) according to the model with acoustic elements on US and DS surface respectively.
Displacements The max and min envelope of the relative radial displacements are presented in Figure 8 for the Main section (0°). In figure, the relative displacement presented i.e. the rigid body displacement due to the earthquake has been removed. In the figure, a comparison between the acoustic model and the simplified Westergaard model is shown. The models give similar results, but the acoustic model gives larger deflections. x-dir (radial) upstream 700 680 660
Elevation (m)
640 620 600 580 560
Max (Westergaard) Min (Westergaard) Max (Acoustic) Min (Acoustic) Static
540 520 500 -0.05
0
0.05 0.1 Deflection (m)
0.15
Figure 8: Displacements in the main section (0°) on US surface.
Discussion A concrete arch dam have been analyzed during seismic loading with a model based on acoustic elements to describe the water and infinite elements to describe quiet boundaries to prevent wave reflection. The results from this model have also been compared to a simplified model based on Westergaards added mass approach. The simplified model is only used for comparison and should therefore be considered as a rough estimate of the induced stresses and displacements. The performed analyses have showed that there are several factors that have been assumed in this study that influence the results, such as Rayleigh damping – the choice of damping ratio for the different materials and especially the choice of corresponding frequencies influence the results. Seismic excitation – in these analyses, only the bottom of the foundation is subjected to the prescribed seismic excitation. Effects such as seismic wave incoherence have not been considered. Infinite boundaries – in the analyses infinite boundaries have defined for both the foundation but also for the reservoir. The results from the two models are quite similar, where the minimum principal and hoop stresses generally are more similar between the two models than the maximum principal and 147
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vertical stresses. The largest difference between the models often occurs at the nodes in the base of the arch dam, which may be due to poor discretization of the mesh at these points. Generally, the Westergaard added mass gives higher maximum principal stresses at the base on the upstream side than the acoustic model, while often underestimating the min principal stresses at the base on the downstream side. Both models show high tensile stresses near the base of the arch dam that would result in cracks. The analyses also show that, despite the symmetric shape of the arch dam, there is a difference between the results at the left and right section.
Acknowledgements The research presented was carried out as a part of "Swedish Hydropower Centre - SVC". SVC has been established by the Swedish Energy Agency, Elforsk and Svenska Kraftnät together with Luleå University of Technology, The Royal Institute of Technology, Chalmers University of Technology and Uppsala University. www.svc.nu. The presented work was financially supported by SVC, Sweco Infrastructure, KTH Royal Institute of Technology and Vattenfall Engineering. The authors would like to thank these organizations for making our participation in this benchmark workshop possible.
References [1] ICOLD, 2013. 12th International Benchmark Workshop on Numerical Analysis of Dams. Theme A – Fluid Structure Interaction. Arch Dam – Reservoir at Seismic loading. Graz University of Technology. [2] Westergaard, H. M. (1933). Water Pressures on Dams During Earthquakes. Transactions, American Society of Civil Engineers, Vol 98. [3] US NRC (2007). Damping values for seismic design of nuclear power plants. Regulatory Guide 1.61. U.S. Nuclear Regulatory Commission. [4] ASCE 4-98, 1998, “Seismic Analysis of Safety-Related Nuclear Structures and Commentary,” American Society of Civil Engineers, Reston, Virginia. [5] ASCE/SEI 43-05, 2005, “Seismic Design Criteria for Structures, Systems, and Components in Nuclear Facilities,” American Society of Civil Engineers, Reston, Virginia. [6] EM1110-6051 (2003). Time-History Dynamic Analysis of Concrete Hydraulic Structures. US Army Corps of Engineers. [7] Kuo, J.S.-H. (1982). Fluid-structure Interactions: Addedd Mass Computations for Incompressible Fluid. Report No. UCB/EERC-82/09. Earthquake Engineering Research Center, University of California, Berkeley. [8] Lysmer, J., and R. L. Kuhlemeyer, “Finite Dynamic Model for Infinite Media,” Journal of the Engineering Mechanics Division of the ASCE, pp. 859–877, August 1969.
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Fluid Structure Interaction Arch Dam – Reservoir at Seismic loading A solution using FEnas M. Brusin1, Dr. J. Brommundt1 and H. Stahl1 1
AF-Consult Switzerland Ltd., Täfernstrasse 26, CH-5405 Baden, SWITZERLAND E-mail: [email protected]
Abstract This paper represents a contribution to the ICOLD – 12th International Benchmark Workshop on Numerical Analysis of Dams, THEME A - Fluid Structure Interaction Arch Dam – Reservoir at Seismic loading. The formulated problem, modeling of a 220 m high and 430 m wide arch dam under static and seismic loads, was addressed by the finite element code FEnas. FEnas is a Swiss code originally developed at the Imperial College London. The interaction between the dam and the water in the reservoir was implemented using the added mass technique in the general form following Westergaard. The simulations performed included a comparative analysis of two meshes with different spatial discretization. The computed results show that the impact of the discretization is rather small. Moreover, the results of the analyses have been compared to typical deformations observed at Swiss dams and similar studies on existing dams. All results here are in good agreement with preceding studies of this kind.
Introduction Problem Definition The problem to be analyzed and solved here is exhaustively described in the problem description provided by the workshop organizers [1], who are referenced as the “formulator” in the following. The stresses and deformations of a dam interacting with the water in its completely filled reservoir under earthquake load shall be simulated. As the title of the benchmark workshops says, this interaction is the focus of this study. Consequently, typical further challenges in studies of this type are handled in a pragmatic way, e.g. all simulations assume linear elastic material behavior. The earthquake is represented by three correlated time series of acceleration in the three dimensions, lasting 20 seconds in total. Two FE meshes are provided, one in coarse and one in fine resolution. Finite Element Code FEnas The finite element analysis of the dam was performed using FEnas ECCON IPP/ Version 2.9.3 2012/08/24. FEnas which was initially developed at the Imperial College London and was continuously enhanced and integrated with a graphical user interface by Walder & Trueb Engineering AG, Switzerland [2]. FEnas is capable of solving a wide range of static and dynamic load problems including consideration of thermal impact. The dimension of the problems solvable in FEnas is constrained by aspects of computation only. Nonlinearity in terms of material behavior and geometry can be considered. The typical field of application of FEnas is in design of buildings and concrete structures, including tunnels and dams. The FEnas element library includes all element types necessary for this type of analyses. However, acoustic or fluid elements are not included and cannot be modeled.
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Modelling Concept and Implementation General Assumptions and Approach The software package FEnas allows to analyze the problem of interaction of fluids and structures under seismic impact using the added mass technique. This technique is considered to be a conservative approach and gives satisfactory results in engineering practice [3]. The mass of water which oscillates along with the structure is added as a discrete mass at all element nodes of the upstream face of the arch dam as described below. All analyses were carried out for the two meshes provided by the formulator, i.e. a coarse and a fine mesh. For each mesh the following three calculations were performed: Linear static analysis for the self-weight load and hydrostatic pressure load. Calculation of eigenvectors of oscillation for the first 10 tones, including the added mass of water. Linear dynamic analysis by using direct time integration, also including the added mass of water. For further analysis of the results a superposition of the results of static and dynamic analyses was performed as described below. All approaches to the modeling, loads, boundary conditions, and material parameters are identical for the simulations with the two meshes. Considering this, if not stated otherwise, the following discussions refer to the fine mesh; references to the coarse mesh are explicitly marked. Mesh Properties The mesh was delivered by the formulator of the benchmark competition. Two adaptations were performed: 4. The discretization of the terrain of the coarse mesh did not coincide with the discretisation of the dam at their mutual interface. One line of elements had to be corrected to create a consistently meshed interface. 5. The discretization of the terrain of the fine mesh required the same adaptation. Moreover, the discretisation of the terrain was not adapted to the refined discretisation of the dam. Therefore, the topmost layer of elements of the terrain mesh was refined to gain a consistent discretisation of dam and terrain at their interface. We consider the impact of these adaptations as minor. Their big advantage is that no code internal treatment and definition of the interfaces with different discretisation is needed but the meshes can be used as such. Due to the selected approach with the added mass, the developed finite element mesh for the reservoir provided by the formulator was not taken into account. The table below provides information about the mesh parameters. Figures of the mesh discretization are given further below. Table 1: Mesh parameters of the adapted meshes Total number of nodes elements Hexahedron-quadratic elements with 20 nodes wedge-quadratic elements with 15 nodes
Coarse mesh Dam Terrain 2083 11608 356 2340 312 2340 44 0
Fine mesh Dam Terrain 13733 15062 2736 3120 2736 2700 0 420
Material Parameters and Boundary Conditions The formulator provided the following linear elastic isotropic material characteristics. 150
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Table 2: Material parameters Parameter
Density Poisson – ratio Young’s – modulus
Dam 2400 kg/m3 0.167 27000 MPa
Rock Mass (Terrain) massless 0.2 25000 MPa
Considering that the outer limits of the modeled terrain are sufficiently far away from the dam, the boundary conditions at the outer surface and the bottom of the terrain were determined as springs with large stiffness that practically prevents any movement in all three directions (quasi fixed bearing). Modeling approach for added mass technique The mass and its distribution are relevant when calculating the natural frequencies of a structure as well for the dynamic analysis during the earthquake. The specific mass is calculated from density, and is in FEnas automatically taken into account in a physical manner. In order to take into account the interaction of fluids and structure, it is necessary to add a mass of water which oscillates together with the dam. Water joined to the body dam is added to the points of the upstream face as an additional concentrated mass according to Westergaard’s approach in its generalized form [3, 4]. Westergaard showed that the hydrodynamic pressures exerted on the face of the dam due to the earthquake ground motion is equivalent to the inertia forces of a body of water attached to the dam and moving back and forth with the dam while the rest of reservoir water remains inactive. He suggested a parabolic shape for this body of water with a base width equal to 7/8 of the height. In the generalized Westergaard method [3, 4] normal hydrodynamic pressure Pn at any point on the curved surface of the dam is proportional to the total normal acceleration shown in figure below, i.e.: (1) ̈ i
7 w H i (H i Zi ) 8
(2)
Figure 1: Westergaard hydrodynamic pressure [3] Where: ̈ - Total normal acceleration at point i i - Westergaard pressure coefficient at point i 151
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w - Mass density of water H i - Depth of water at vertical section that includes point i Z i - Height of point i above the base of the dam The normal pressure Pn at each point is then converted to an equivalent normal hydrodynamic force by multiplying by the tributary area associated with that point. This procedure is semiautomated in FEnas, the user has to define the surface load according to Westergaard, while the conversion from surface pressure to added mass at nodes is done automatically by the program. Since the Westergaard coefficient depends on the depth of water in front of the upstream face, which varies with dam side, the value H is taken in accordance with the colors on the next figure, one average value of H correspond for each color.
Figure 2: Westergaard added masses technique in FEnas (left: zones of similar H, right, hydrodynamic forces as vectors, colours show magnitude of force in kN/m2)
Modelling Approach for Self-Weight and Hydrostatic Load Deadweight or self-weight refer to the same load case. Application of this load case generally mimics the construction schedule. Here, a pragmatic two step modeling approach is implemented. Superposition of the two steps represents deadweight. In the first step, every second cantilever is modeled massless with a very low Young’s modulus, whereas all other cantilevers are modeled by their correct properties (cf. Table 2). In the second step, this procedure is repeated exchanging the material parameters between the two sets of cantilevers. For the further analysis, the joints are considered as grouted and the entire construction retains hydrostatic pressure as arch dam. Hydrostatic load has been defined as surface load perpendicular to the surface of all elements at the upstream surface. The magnitude of the surface load depends on the water level with respect to the vertical position of the element. Modelling Approach for Determining the Eigenfrequencies The formulator required calculation of the first 10 eigenfrequencies of the structure; this includes the interaction with the reservoir, i.e. the eigenfrequencies of the dam together with the added mass of water are to be calculated. The lowest natural frequencies are determined by resolving the characteristic value or eigenvalue problem. Here, only iterative methods can be used to solve eigenvalues problems. FEnas uses the “subspace iteration method”. The modes of oscillation are estimated for a number of start vectors (the subspace) and then iteration is used for continuous improvements. The minimum number of start vector required is equal to the number of requested eigenvalues; the maximum number is equal to the number of degrees of freedom of the entire FE structure.
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Modelling Approach for Transient Seismic Loads Linear dynamic analysis was performed using direct time integration (time history analysis). As method for integration of the dynamic equations of motion we have used the Newmark method. The accelerograms given by the formulator were used as such and are plotted in the following figure.
Figure 3: Accelerograms in x, y, z direction The acceleration values are given on the interval of 0.01 s. All performed simulation used time steps with the same duration. With a total of 2000 time steps of 0.01 s, the total duration of the earthquake is 20 s. The dynamic analysis is taking into account the Rayleigh damping effect. Raleigh dumping is based on assumption that damping is proportional to the stiffness and mass matrix of the structure. Here, a damping of 5% for the first two eigenfrequencies was considered.
Results Eigenfrequencies Table 3: Eigenfrequencies Mode No. 1 2 3 4 5 6 7 8 9
Coarse mesh Fine mesh Frequency [Hz] Period [s] Frequency [Hz] Period [s] 1.25 0.8000 1.26 0.7937 1.34 0.7463 1.32 0.7576 2.05 0.4878 2.01 0.4975 2.33 0.4292 2.36 0.4237 2.47 0.4049 2.50 0.4000 3.01 0.3322 3.00 0.3333 3.15 0.3175 3.17 0.3155 3.66 0.2732 3.65 0.2740 3.70 0.2703 3.70 0.2703 153
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10
3.88
0.2577
3.88
0.2577
Mode Shape
Figure 4: Radial deformation [mm] with mode shapes of the eigenfrequencies exaggerated by a factor of 17000 Stress Analysis All evaluations of results of stresses were done for vertical stresses, hoop stresses, and where considered useful for principal stresses 1 and 3. Note that the vertical stresses shown are not 154
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vertical with respect to a global coordinate system but vertical with respect to the element coordinate system. This means that the stresses shown are parallel to the element and dam surface. Stresses due to Static loads Stresses of static loads are presented separately for the load of self-weight and for the combination of self-weight and hydrostatic pressure load. They are presented for the upstream and downstream surface allowing checking if the model results are plausible.
Figure 5: Vertical and hoop stresses [MPa] due to self-weight
Figure 6: Vertical and hoop stresses [MPa] due to self-weight and hydrostatic pressure 155
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Seismic stress-analysis depending on the time Monitoring was performed for the vertical and hoop stresses for selected elements on the upstream and downstream face of the dam. There were selected three elements of both surfaces in three different heights at main cross section defined by the formulator, which is considered later (cf. Figure 10). On the figures below you can see the position of the elements at which stresses are monitored. Each of the figures shown below includes minimum (MIN) and maximum (MAX) stress observed at the selected elements (each of the selected elements includes 20 nodes). Note that these are the stress due to seismic loading only, self-weight and water load are not directly considered, water to a certain extend as the seismic loading affects the Westergaard masses as well.
Figure 7: Monitored elements on main cross section
Figure 8: Time series of stresses observed at monitored elements on upstream surface
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Figure 9: Time series of stresses observed at monitored elements on downstream surface Evaluation of the stresses for three different cross sections For three cross-sections, one in the center and two at each side halfway between the center and the right respectively left end of the dam, stresses over dam height are shown for the upstream and downstream surface. The three examined cross-sections are marked on the figure below.
Left section
Main section
Right section
Figure 10: Layout of analyzed cross sections The following loads respectively load combinations are plotted: Self-weight load - [SW] Self-weight + hydrostatic pressure load – Normal operating conditions – [NOC] Maximum of self-weight + hydrostatic pressure + seismic load – [MAX] Minimum of self-weight + hydrostatic pressure + seismic load – [MIN]
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It should be noted that the last two evaluations, maximum and minimum, represent envelopes of stresses, i.e. this state of stress is artificial, because the maximum / minimum do not occur at all nodes at the same time.
Figure 11: Stresses for main cross section
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Figure 12: Stresses for left cross section
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Figure 13: Stresses for right cross section Radial Deformation The evaluation of results of radial deformation was done considering the following sequences: Radial deformation due to the static loads Seismic radial deformation-analysis depending on the time Radial deformation analysis for three cross-section (cf. above) Radial deformation of the static loads Radial deformations of the static loads are presented separately for the load of self-weight and for the combination of self-weight and hydrostatic pressure load.
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Figure 14: Radial deformation [mm] – self-weight (left), self-weight + hydrostatic pressure (right) Seismic radial deformation-analysis depending on the time Monitoring was performed for the radial deformation for selected nodes on the upstream face of the dam. There were selected three nodes of upstream surfaces in three different heights at the three cross section defined by the formulator (see Figure 10). On the figures below you can see the position of the nodes at which radial deformation are monitored. Note that these are the radial deformations due to seismic loading only, self-weight and water load are not directly considered, water to a certain extend as the seismic loading affects the Westergaard masses as well.
Figure 15: Monitored nodes on upstream surface
Figure 16: Radial deformation for left, main and right cross section respectively
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Radial deformation at the three cross sections For the three cross-sections the radial deformations over height are plotted. Evaluation of the radial deformation is given for the following loads respectively combination of loads: - Self-weight load - [SW] - hydrostatic pressure load – Normal operating conditions – [NOC] - Maximum of hydrostatic pressure + seismic load – [MAX] - Minimum of hydrostatic pressure + seismic load – [MIN] These combinations mimic the observations of a pendulum.
Figure 17: Radial deformation – main cross section
Figure 18: Radial deformation – left cross section
Figure 19: Radial deformation – right cross section
Discussion and Summary Stress-strain analysis of arch dam during seismic activity including reservoir-structure interaction using added masses technique according to generalized Westergaard approach is presented. Interaction of an arch dam with the impounded water leads to an increase in the dam vibration periods. The fact that water moves with the dam increases the total mass that is in motion. This added mass increases the natural periods of the dam, which in turn affects on the effective earthquake inertia forces. The Westergaard method usually gives the largest 162
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added-mass values, which is evident by its increasing the vibration periods the most. However, this does not automatically give the largest stresses, because response of the dam also depends on the characteristics of the earthquake ground motion. Although Westergaard approach is widespread in practice, for different variants of quasi-static and dynamic analysis of seismic effects, the above-mentioned facts lead to the conclusion that the results must be taken with caution. After comparison of two different sizes of spatial discretization of meshes we can conclude, that the coarser mesh also gives satisfactory results which can be used in engineering practice. In fact, the differences are small. Whereas the performed simulations with linear material behavior are not very costly in terms of computation time and required computer power for both meshes, the coarser mesh could be very useful as soon as more complex processes, e.g. non-linear material behavior, are to be considered. Maximum earthquake stresses are located in the central upper portion of the dam as well along the dam-foundation contact zone. Maximum radial deformation occurs in the crest, and decreasing towards lateral sides, as well as towards fixed end. The vertical stresses observed at the upstream surface show some artefacts for the load cases with the earthquake as they are supposed to be zero at the top of the dam like it is observed for self-weight and self-weight with hydrostatic pressure. These non-zero values are considered to be artefacts. The same artefact in a much smaller magnitude is seen at the top of the downstream surface. Stress time histories curves shows several significant tensile stress cycles that can lead to open the contraction joints, especially in crown zone, while on the time histories of radial deformation in addition to timing of deformation could be noted as dominant periods of the first few natural frequencies. The results that are shown have been compared to typical deformations observed at Swiss dams. The overall behavior of the dam here is in good agreement with what AF has seen in reality. The comparison of the results of the earthquake simulations with previous studies of this type performed at AF was positive. We can conclude that the added mass technique in combination with the direct dynamic analysis relatively quickly and easy produce results useful in engineering practice.
Acknowledgements The FEnas developer and support team around Dr. Urs Trüeb provided valuable support in the conversion of the meshes into the FEnas mesh format.
References [1] Graz University of Technology, Institute of Hydraulic Engineering and Water Resources Management, 12th International Benchmark Workshop on Numerical Analysis of Dams, 2.-4. OCTOBER, 2013, GRAZ – AUSTRIA, THEME A - Fluid Structure Interaction Dam – Reservoir at Seismic loading. Download from Arch http://portal.tugraz.at/portal/page/portal/Files/i2130/Icold_bmws2013/Theme_A/Files_an d_Information_Theme_A.zip, 23.07.2013 [2] FENAS, Walder + Trüeb Engineering AG. (2013). Walder + Trüeb Engineering AG. http://www.waldertrueb.ch/index.php?id=23&L=2%3Fcomply, 23.07.2013 [3] Ghanaat, Y. (1993). Theoretical Manual for Analysis of Arch Dams. US Army Corps of Engineers. http://www.dtic.mil/dtic/tr/fulltext/u2/a269682.pdf, 24.07.2013 [4] Kuo, J. (1982). Fluid-structure interactions: Added mass computations for incompressible fluid. Earthquake Engineering Research Center. Berkeley: University of California.
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Fluid Structure Interaction Arch Dam-Reservoir at Seismic Loading S. Shahriari1 1
Institute of Hydraulic Engineering and Water Resources Management, Stremayrgasse 10/2, A-8010 Graz, AUSTRIA E-mail: [email protected]
Abstract In the present paper, a direct time domain procedure is used for dynamic linear analysis of the coupled arch dam-reservoir-foundation system. The hydrodynamic force on the upstream face of the dam is modeled by added mass method and compressible fluid elements for comparison. The concrete and massless foundation rock was assumed to be linear elastic. Connection between dam and foundation is modeled by coupling all DOF of the corresponding nodes. Also, viscous damping was applied to the materials by using RayleighDamping. Numerical results showed that the different modeling techniques of the interaction lead to different response of the system and stress distribution in the dam’s body. Modeling the interaction by added mass method, increases the period of the system as well as overestimating or underestimating the stresses compare to the model with fluid elements.
Conclusion The following conclusions are drawn based on the numerical experiments conducted herein:
Choosing appropriate damping parameters is a very important step in the dynamic analysis of the dam-water-foundation system. Underestimating the system’s damping by choosing higher than 7th angular frequency as the second frequency for calculation of Rayleigh-Damping parameters can lead to overestimating the stresses in the model.
Modal responses of the dam-water-foundation system are calculated using the finite element software, ANSYS. The results showed that the eigenfrequencies from the model with added mass approach were lower than those calculated from the model which utilized acoustic element to model the reservoir.
Direct time integration procedure is used for dynamic analysis of the system and deformations and stresses are calculated for static and dynamic load cases. Maximum deformation due to static loading was 7.4 cm at the crest level followed by the maximum deformation of 13.6 cm after dynamic analysis. The tensions in arch dams are not desirable; therefore, the tension stresses were studied. Evaluation of hoop stresses indicated that there is no significant tension developed in the main section with maximum value of 0.6 MPa. Furthermore, the maximum tension stress value in the U/S face of the right and left sections is higher than the main section(S=3.6 MPa). Vertical stresses evaluation showed that the maximum tension was occurred at the base of the U/S face of the main section and found to be 6.69 MPa. Also, the maximum compression stress is developed (S=12.5 MPa) at 180 m from the foundation level of the U/S face of the main section.
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The Seismic behaviour of an Arch Dam-ReservoirFoundation System A. Frigerio1 and G. Mazzà2 1
Ricerca sul Sistema Energetico – RSE S.p.A., via R. Rubattino 54, Milan, ITALY E-mail: [email protected]
Abstract The paper summarizes the main features and the related results of the linear static, modal and seismic analyses carried out on the concrete arch dam proposed as Theme A of the 12 th ICOLD International Benchmark Workshop. The analyses have been performed by means of the COMSOL Multiphysics software, making reference to the coarse finite elements mesh provided by the Formulators, but some slightly changes. In the modal and dynamic analysis, the water-dam-foundation interaction has been taken into account. The impounded water has been modeled by means of acoustic finite elements, assuming appropriate boundary conditions to simulate the wave reflecting conditions between the fluid and the foundation, as well as the fluid and the upstream face of the dam, and the non-reflecting condition at the end of the reservoir channel. Results have been provided according to the requests of the Formulators in terms of radial displacements, hoop and vertical stresses on three vertical sections of the dam.
Introduction Seismic analyses have been proposed and discussed during some of the first Benchmark Workshops organized by the Committee on Computational Aspects of Analysis and Design of Dams [1,2]. Hence, it appears appropriate the proposal of the present seismic analysis of a large arch dam, [3], in order to verify the possible improvements both on the modeling and on the computational aspects. In the present paper, all the analyses have been carried out with COMSOL Multiphysics, a software that allows modeling and simulating any physics-based system. In particular, the COMSOL Acoustic, [4], and Structural Mechanics Modules, [5], have been used. In the following paragraphs the geometrical and physical model will be described and the results will be provided according to the requests of the Formulators.
Geometrical and physical model The coarse Finite Element mesh provided by the Formulators, [3], has been slightly modified in order to have an exact geometrical match at the interface between the nodes of the fluid and those of the foundation rock. The reservoir and the foundation domains have been extended upstream to avoid possible numerical influences of the non–reflecting water surface at the end of the reservoir on the dynamic response of the dam under seismic loading conditions. Finally, the finite elements of the central vertical sections of the dam have been refined to discretize with a more regular grid the whole mesh of the dam (Figure 1). The displacement field of each finite element of the numerical model has been discretized with quadratic shape functions.
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Figure 1: The coarse mesh used to study the dam-reservoir-foundation system Material properties Linear elastic constitutive models have been assigned to the dam and the foundation rock. The physical-mechanical properties are summarized in Table 1. Table 1: Concrete and rock properties Domain
Material
Dam Foundation
Concrete Rock
Density [kg/m3] 2400 0
Poisson ratio 0.167 0.200
Young modulus [MPa] 27000 25000
The Rayleight damping model has been taken into account to define the dam behavior during seismic loading conditions. Assuming a 5% structural damping ratio, the mass and stiffness damping parameters of the Rayleight formulation are as follows: = 0.94; = 2.65E-03 The reservoir has been discretized by means of acoustic finite elements whose properties are reported in Table 2. Table 2: Water properties Domain Reservoir
Material Water
Density [kg/m3] 1000
Speed of sound in water [m/s] 1500
Bulk modulus [MPa] 2200
Loading and boundary conditions According to the formulation of the Theme A, the following loadings sequence has been considered: 168
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Dead loads Hydrostatic water pressure with the maximum water level equal to the dam crest height (i.e. 715 m a.s.l.) Seismic loads, provided by the Formulators in terms of accelerations along the three Cartesian directions.
Different boundary conditions have been used for the foundation domain depending on the type of the analysis performed. In case of static and modal analyses, symmetric conditions have been applied to the lateral rock walls whereas fixed constraints have been assigned to the basement rock. In seismic analysis, the accelerations - or the equivalent related displacement varying in time - have been applied to the lateral and bottom surfaces of the foundation. In modal and seismic analyses, the fluid domain has been modeled by means of acoustic elements, assigning the following boundary conditions:
Dam-reservoir interface (refer to the COMSOL “node” acoustic-structure boundary, [4]):
(
)
(1)
(2)
where n is the normal to the interface, the water density, p the fluid “acoustic” pressure, qd the dipole source (equal to zero in the present case), utt the acceleration field of the structural domain at the fluid interface and the stresses tensor.
Foundation-reservoir interface (refer to the COMSOL “node” impedance, [4]):
(
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where n is the normal to the reservoir-foundation interface, the water density, p the fluid “acoustic” pressure, qd the dipole source (equal to zero in the present case), Zi the acoustic input impedance assumed equal to: (4)
/q
being q a damping coefficient that characterizes the effects of absorption of the hydrodynamic pressure waves at the boundary, according to the following equation [6]: (5) is the wave reflection coefficient that accounts for the behavior of the absorption of hydrodynamic pressure waves at the boundary, whereas c is the speed of sound in water. According to some literature case studies, has been considered equal to 0.75, [7].
Upstream-reservoir surface (refer to the COMSOL “node” plane wave radiation, [4]):
(
)
(
)
(6)
where n is the normal to the reservoir-foundation interface, the water density, p the fluid “acoustic” pressure, qd the dipole source (equal to zero in the present case), c is the speed of sound in water and Qi the monopole source (equal to zero in the present case).
Free surface (refer to the COMSOL “node” sound soft boundary, [4]):
(7)
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Numerical analyses In the following paragraphs the main features of each numerical analysis will be provided, specifying the assumed hypotheses. Static analysis A linear static analysis has been carried out in order to apply the dead load and the hydrostatic water pressure, thus the water domain has not been included in the geometrical model. The applied boundary conditions are reported in the previous paragraph. Modal analysis The first 10 eigenfrequencies of the dam have been computed taking into account the damreservoir interaction. The applied boundary conditions are reported in the previous paragraph. Seismic analysis Seismic analyses in time domain are generally carried out considering a massless foundation and applying a spatially-uniform ground motion directly at the basement rock. In the present paper, at first an ordinary differential equations problem has been solved, making reference to the foundation domain only, in order to compute the displacements associated to the earthquake motions (Figure 2). In the subsequent dynamic analysis, these displacements have been applied to the bottom and lateral rock walls to calculate the dynamic response of the dam-reservoir-foundation system. This procedure has been chosen because it allows computing easily the relative displacements of the system; anyway, as an alternative, the same results could be attained applying directly the accelerations to the foundation boundaries. The boundary conditions taken into account during the seismic analysis are reported in the previous paragraph.
Figure 2: Acceleration time history (amax 0.1g)
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Results With reference to the results of modal analysis, the first 10 eigenfrequencies and the related modal shapes are reported in Table 3. Table 3: Eigenfrequencies and modal shapes 1
1.54247 [Hz]
6
2.51800 [Hz]
2
1.55194 [Hz]
7
2.83905 [Hz]
3
2.09799 [Hz]
8
2.96151 [Hz]
4
2.22299 [Hz]
9
3.19079 [Hz]
5
2.33002 [Hz]
10
3.37707 [Hz]
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According to the requests specified by the Formulators, the results of the static and seismic analyses are provided in terms of radial displacements [m], hoop and vertical stresses [MPa] on the upstream and downstream faces of three vertical sections of the dam. Table 4: Upstream face – Central section (US-C) – Radial displacements, hoop and vertical stresses
Table 5: Downstream face – Central section (DS-C) – Radial displacements, hoop and vertical stresses
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Table 6: Upstream face – Right section (US-R) – Radial displacements, hoop and vertical stresses
Table 7: Downstream face – Right section (DS-R) – Radial displacements, hoop and vertical stresses
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Table 8: Upstream face – Left section (US-L) – Radial displacements, hoop and vertical stresses
Table 9: Downstream face – Left section (DS-L) – Radial displacements, hoop and vertical stresses
Conclusion
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The paper presents the dynamic response of the dam-reservoir-foundation system according to the basic requests of the Formulators. Only the coarse mesh has been considered as some preliminary analyses have demonstrated that, compared to a high computational effort, results showed only slight differences. A second aspect to be emphasized refers to the boundary conditions assumed for the fluid domain at the rock interface: if a fully reflecting condition is assumed, the stresses computed on the dam are considerably higher than those attained in the case discussed in this paper where a partial absorption conditions has been assumed. This means that experimental investigations should be undertaken in order to define more realistic boundary conditions. A suggestion for the Formulators, in the synthesis phase, refers to the opportunity to make some comparisons between the main outcomes of Theme A and those of the previous Benchmark Workshops, cited above.
Acknowledgements This work has been financed by the Research Fund for the Italian Electrical System under the Contract Agreement between RSE S.p.A. and the Ministry of Economic Development General Directorate for Nuclear Energy, Renewable Energy and Energy Efficiency in compliance with the Decree of March 8, 2006.
References [1] Theme A – Seismic analysis of the Talvacchia dam. (1992). 2nd ICOLD International Benchmark Workshop on Numerical Analysis of Dams. Committee on Computational Aspects of Dam Analysis and Design. Bergamo, Italy. [2] Theme A1 – Earthquake of an arch dam including the nonlinear effects of contraction joint opening. (1996). 4th ICOLD International Benchmark Workshop on Numerical Analysis of Dams. Committee on Computational Aspects of Dam Analysis and Design. Madrid, Spain. [3] Zenz G., Goldgruber M. (2013). Theme A formulation. Fluid Structure Interaction. Arch Dam – Reservoir at Seismic loading. 12th ICOLD Benchmark Workshop on Numerical Analysis of Dams, Graz, Austria. [4] COMSOL Multiphysics. (2013). Acoustic Module – User’s Guide. Version 4.3b [5] COMSOL Multiphysics. (2013). Structural Mechanics Module – User’s Guide. Version 4.3b [6] Ka-Lun Fok, Anil K. Chopra. (1986). Earthquake analysis of arch dams including damwater interaction, reservoir boundary absorption and foundation flexibility. Earthquake engineering and structural dynamics, Vol.1, pp 155-184. [7] Ghanaat Y., Redpath B.B. (1995). Measurement of reservoir-bottom reflection coefficient at seven concrete dam sites. QUEST Structure Report No. QS95-01.
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12th International Benchmark Workshop on Numerical Analysis of Dams Theme A: Fluid Structure Interaction Arch Dam – Reservoir at Seismic Loading A. Diallo1 and E. Robbe1, 1
EDF-CIH, 15 Avenue Lac du Bourget Savoie Technolac, 73370 Le Bourget du Lac, FRANCE E-mail: [email protected], [email protected]
Abstract Dynamic response of arch dams to earthquake ground motion is significantly affected by the interaction between the dam and the impounded water. There are several approaches to take into account this dynamic dam-water interaction. The purpose of this benchmark is to compare different modelling techniques with different levels of precision and therefore with different computing efforts. This paper presents the results of the 3 approaches investigated and the fundamental hypotheses adopted for each them. The first approach is a generalized Westergaard added mass, the second approach is an incompressible finite-element added mass and the third approach is based on a sub structuring method where the fluid is compressible. All investigations are carried out for an artificially generated symmetrical arch dam and simplified loading and boundary conditions. In general, Westergaard added mass yields higher compressive and tensile stresses, as well as higher radial displacement. The sub structuring approach, where the compressibility of the water and the impedance of the foundation are taken into account, yields lower stresses.
Introduction The objective of this paper is to present the seismic analysis of a 220 m high double curvature arch dam undertaken as part of the 12th benchmark study. The results of the analysis are presented for the 3 approaches used to take into account dam-water interaction in accordance with the general assumptions made by the benchmark organizing committee. The analyses of the results focus on the impact of the hydrodynamic approach used on the computed stresses and displacements of the dam. The following approaches are presented: Generalized Westergaard added mass Incompressible finite element added mass Sub structuring method where water compressibility is taken into account. For each approach, the fundamental hypotheses are presented and the physical justification is given.
Description of arch dam analysis - modelling methods The analyses were performed with Code_Aster. A computer program developed by EDF; it offers a full range of multiphysical analysis and modelling methods.
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Materiel parameters Because a modal superposition method is used for the three approaches, the Rayleigh damping cannot be used, rather, a value of modal damping is specified for each mode. This value is fixed at 5% which is the value usually considered in real projects. The mass density of rock is not specified by the benchmark organizing committee because the traditional method assumes that the foundation is massless, however the sub structuring method investigated takes into account the impedance matrix of the foundation rock. It is therefore necessary to specify the mass density of rock; a value of 2400 kg/m3 is considered for the rock in this analysis. The other material properties are considered in accordance with the general assumptions made by the benchmark organizing committee. Although in practice the dynamic modulus of elasticity is usually considered 25 percent greater than static modulus of elasticity, in this analysis the supplied static values are respected. Static analysis The application of the dead load should consider the manner in which the dam was constructed. Arch dams are often constructed as independent cantilever blocks separated by vertical joints. Since these joints are not capable of transferring dead load horizontally until they are grouted, dead loads should be applied to individual cantilever to simulate this condition. This may be accomplished by performing dead load analysis in two steps. First, dead loads are applied to even cantilevers (Set-1) and the stresses are extracted. In second analysis, the dead loads are applied to odd cantilevers (Set-2) separately and the stresses are extracted. The addition of the stresses obtained from the two steps is considered as the initial stress state of the dam and the displacement of the dam due to the dead load is not considered. Water loads due to the hydrostatic pressures of the normal water level are external forces acting on the u/s face of the dam. The hydrostatic pressures are applied to the monolithic arch structure after the construction joints are grouted. Dynamic modelling The methods used in this analysis are based on the modal superposition method. The modal analyses performed for the empty dam show that 90 percent of the mass of the dam is excited by a frequency range between 1.93 and 10 Hz. The maximum of the seismic is also in the same range (1-10 Hz).A significant amplification can therefore be expected. The different approaches used to take into account the fluid-structure interaction are: a) Generalized Westergaard Added-mass The added-mass representation of dam-water interaction during earthquake ground shaking was first introduced by Westergaard [1]. In his analysis of a rigid 2D gravity dam with a vertical upstream face, Westergaard showed that the hydrodynamic pressures exerted on the face of the dam due to the earthquake ground motion is equivalent to the inertia forces of a body of water attached to the dam and moving back and forth with the dam while the rest of reservoir water remains inactive. A general form of the Westergaard added-mass concept which accounts for the 3D geometry [2] can be applied to the earthquake analysis of arch dams. The general formulation is based on the same parabolic pressure distribution with depth used by Westergaard, except that it makes use of the fact that the normal hydrodynamic pressure at any point on the curved surface of the dam is proportional to the total normal acceleration, ̈ : (1) ̈
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(2)
√
Density of water; Westergaard pressure coefficient; H: Water depth; Z: Level of the point on the curved surface of the dam. The normal pressure at each point is then converted to an equivalent normal hydrodynamic force by multiplying by the tributary area associated with that point. b) Incompressible Finite-Element Added Mass – Potential Approach The added-mass representation of the impounded water can be obtained more accurately by a finite-element solution of the pressure wave equation, which fully accounts for the complex geometry of the dam and the reservoir. The impounded water represented by the wave equation is discretized using a finite element mesh of incompressible liquid elements. The solution is obtained by numerical procedures with the following boundary conditions: (3) The hydrodynamic pressures at the water – free surface are assumed to be zero, that is, the effects of surface waves are neglected. The reservoir bottom and sides, as well as a vertical plane at the upstream end of the reservoir model, are assumed to be rigid. For rigid boundaries the normal pressure gradients or the total normal accelerations are zero. The normal pressure gradients at the dam-water interface are proportional to the total normal accelerations of the fluid. The computed pressures for the nodal points on the upstream face of the dam are then converted into equivalent nodal forces, from which an added-mass matrix representing the inertial effects of the incompressible water is obtained.
c) Compressible Water with Absorptive Reservoir Boundary- Substructuring Method The substructuring method consists of dividing the complete system into three substructures: the structure, the water, and the foundation, each of which can be partially analysed independently of the others. The structure is represented by a 3D finite element, which permits modelling of a general geometry and linear elastic material properties. The water domain and the foundation region are represented by boundary elements. The added mass representation of the impounded water described above ignores the effect of water compressibility and reservoir boundary absorption. However, the water compressibility and reservoir boundary absorption can significantly affect the hydrodynamic pressure and hence response of arch dams to earthquakes [3]. Interaction of the dam with the foundation rock leads to an increase in vibration periods, primarily due to the flexibility of the foundation rock. Dam-foundation interaction also decreases the dam response if damping arising from material damping in the foundation rock and radiation damping associated with wave propagation away from the dam are considered in the analysis. Procedures for earthquake response analysis of arch dams including dam-water interaction, water compressibility, reservoir boundary effects and dam-foundation interaction are developed by EDF [4]. In this procedure, the radiative damping and the hysteretic damping of the foundation are also considered. To represent the infinity domain, Green’s functions are used in the fluid domain and in the foundation domain. In the fluid domain, the Helmholtz equation is discretized using boundary elements. 179
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(4) The solution is obtained by numerical procedures with the following boundary conditions: The hydrodynamic pressures at the water – free surface are assumed to be zero, that is, the effects of surface waves are neglected.
The normal pressure gradients at the dam-water interface are proportional to the total normal accelerations of the fluid.
The normal pressure gradients at the foundation-water interface (reservoir boundary) are proportional to the total normal accelerations of the fluid.
The hydrodynamic pressure wave impinging on the reservoir boundary is partly reflected into the water, and partly refracted (absorbed) into the boundary materials. The partial absorption at the reservoir boundary is approximately represented by a reflection coefficient known as “α”, which is the ratio of reflected to incident wave amplitude.
In the foundation domain, the Navier equation is also discretized using boundary elements. To take into account the infinity domain, Green functions are used. (5) The solution is obtained by numerical procedures with the following assumptions:
The rock foundation is assumed homogeneous, isotropic and linear elastic.
A value of 5% is considered for the modal damping.
Results and comparison of the different approaches Modal Analysis The interaction of an arch dam with the impounded water leads to a decrease in the dam vibration frequencies. This is because the dam cannot move without displacing the water in contact with it. The fact that water moves with the dam increases the total mass that is in motion. This added mass decreases the natural frequencies of the dam, which in turn affects the response spectrum ordinate and hence the effective earthquake inertia forces. The flexibility of the foundation rock also decreases vibration frequencies of the dam. Table 1 gives the fundamental modes obtained by the 3 approaches investigated in this analysis. The results show that the Westergaard method gives the largest added-mass value, as evidenced by its decreasing the fundamental frequency the most. However, this does not automatically means that Westergaard approach gives the largest stresses, because the response of the dam also depends on the characteristics of the earthquake ground motion.
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Table 1: fundamental period of the dam Empty reservoir
Fundamental frequency
1.93 Hz
Full reservoir Westergaard Incompressible fluid elements
1.29 Hz
1.57 Hz
Compressible fluid element with 50% wave absorption
1.49 Hz
Stresses and displacement Under static loads, the maximum radial displacement reaches 8.32 cm at the top of the central cantilever. This displacement is only due to hydrostatic pressure acting on the u/s face of the dam. The relatively symmetrical deformation of the arch dam leads to the compression of the lower portion of the d/s face of the dam and to tractions in the abutments on the u/s face of the dam. The maximum static principal stress (compression) reaches 9.32 MPa at lower portion in the abutments on the d/s face of the dam. The minimum static principal stress (tensile) also reaches 4.40 MPa at the lower portion of the u/s of the dam. This tensile stress (4.40 MPa) could lead to the opening of a joint at the contact between dam and the foundation rock. However the contraction joint opening/closing is not modelled. Figures 1 and 2 give the principal stress contours under static loads.
Figure 1: Principal stresses (compression) under static loads
Figure 2: Principal stresses (tensile) under static loads During the ground motion, the maximum displacement of dynamic vibrations is about 6 cm with the respect to the initial displaced shape of the dam. The maximum displacements represented in fig 3 show that the Westergaard method yields higher radial displacements at the crest of the dam. The substructuring method where we take 181
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into account the radiative damping and hysteric damping in the foundation as well wave absorption at the reservoir boundary, yields lower displacements (20% of reduction). 250
Height (m)
200 150 100 50 0 0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
displacement (m) min_westergaard static max_substructuring
min_potentiel_approach max_westergaard
min_substructuring max_potentiel_approach
Figure 3: Maximum displacement of the central cantilever at the d/s face of the dam In general, the Westergaard method yields higher compressive and tensile stresses. Indeed, the seismic tensile vertical stresses at the u/s face of the main cantilever vary from 2.4 MPa for the Westergaard method to 0.2 MPa for the substructuring method at a point at ¾ of the dam height (cf fig4). 250
200
Height(m)
150
100
50
0 3.00
2.00
min_westergaard static max_substructuring
1.00
0.00
-1.00 -2.00 -3.00 Stresses (MPa) min_potential_approach max_westergaard
-4.00
-5.00
min_substructuring max_potential_approach
Figure 4: Vertical stresses at u/s face during the ground motion
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Figure 5 gives maximum hoop stresses at the d/s face of the central cantilever during the earthquake. It shows a tensile stress varying from 1.6 MPa for Sub structuring method to 2.2 MPa for Westergaard and potential approach at the top part of the cantilever; this could lead to the opening of vertical joints. 250
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Figure 5: Hoop stresses at d/s face during the ground motion In general, the ground motion increases the maximum principal stress observed at the lower portion of the abutments on the d/s face of the dam as well the minimum principal stress observed at the lower portion of u/s face of the dam. Thus, the maximum compressive stresses during the ground motion varying from 11.26 MPa for sub structuring method to 12.77 MPa for added mass approach on the d/s face of dam (cf fig6 right_section). At ¾ dam height on the u/s face, the maximum compressive stresses also varying slightly, from 9.82 MPa for sub- structuring approach to 10.45 MPa (cf fig 6 main_section).
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Figure 6: Maximum principal stresses at u/s face of central cantilever and d/s face of right section In figure 7, we notice that the substructuring method increases slightly the tensile stress at the contact between dam and the foundation (2.84 MPa under static loads and 3.14 MPa during earthquake). However this amplification is very significant with added-mass methods, thus with Westergaard added mass, the tensile stress reaches 5.96 MPa during the earthquake and with the potential approach it is about 5.24 MPa. These tensile stresses could lead to the opening of the contact between the dam and the foundation.
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240 220
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4.00
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min_potential max_westergaard
0.00
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Figure 7: Minimum principal stresses (tension) at the u/s face of right section
Influence of reservoir boundary absorption A hydrodynamic pressure wave impinging on the reservoir boundary is partly reflected into the water, and partly refracted (absorbed) into the boundary materials. If the reservoir boundary materials are relatively soft, an important fraction of the reservoir water energy can be absorbed, leading to a major reduction in the dynamic response of the dam. Therefore, the values of the absorption ratio for the design and safety evaluation of dams subjected to earthquake loading should be measured or selected conservatively. The purpose of this section is to show the effect of this absorption on the dynamic response of the dam by studying 3 cases of absorption (0%, 50% and 100% absorption). As expected, wave absorption at the reservoir boundary reduces significantly the dynamic response of the dam. Figure 8 gives the radial spectrum at the crest of the dam and for fundamental frequency (f1=1.49 Hz), we observe that the correspondent pseudo acceleration varies from 1.93g (total absorption) to 4.14 g (without absorption).
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Pseudo-acceleration en (g)
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1.00
0.10
0.01
0.00 0.10
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frequence in (Hz)
10.00
50%- absorption
100.00 total_absoption
Figure 8: radial spectrum at the crest of the dam for 3 cases of absorption Reservoir boundary absorption also decreases compressive and tensile stresses (cf fig 9), as well as the radial displacement (cf fig 10). In figure 9, we observe that the maximum stress varies from 8.95 MPa (total absorption) to 11.26 MPa (without absorption) at ¾ dam height on the u/s face of the main section. Figure 10 gives the radial displacement of the central cantilever for 3 cases of absorption and we can observe that the crest displacement varies from 11.4 cm (total absorption) to 15.9 cm (without absorption). 250
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static max_50%_abs
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Figure 9: Hoop stress of the central cantilever for 3 cases of absorption
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max_without_abs
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Figure 10: Radial displacement of central cantilever for 3 cases of absorption
Conclusion The different approaches to hydrodynamic effect modelling investigated in the present study lead to similar behaviour regarding the structural response of the example arch dam. In general, the Westergaard approach yields higher compressive and tensile stresses, as well as higher radial displacements. The substructuring method where wave absorption is taken into account at the reservoir boundary decreases significantly the dynamic response of the dam due to the increased damping of the coupled system.
Acknowledgements The authors wish to thank Electricité de France (EDF-CIH) who financed this study. Particular thanks to Frederic LAUGIER, head of the Civil Engineering – Structures department for his assistance in preparing the study, and to EDF-R&D for their collaboration.
References [1] Westergaard, H.M (1933), “Water pressure on dam during earthquakes”, transactions American Society of Civil engineers, Vol 98. [2] Kuo, 1982 “Fluid structure interactions: Added Mass Computations for incompressible fluid”. Report No. UCB/EERC-82/09, University of California Earthquake Engineering Research Center, Berkeley. [3] Fok and Choppra 1985 “ Earthquake Analysis and Response of Concrete Dams” Report No UCB/EERC-85/07, University of California Earthquake Engineering Research Center, Berkeley [4] www.code_aster.org
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THEME B
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Theme B Long Term Behaviour of a Rockfill Dam
Formulators: Camilo Marulanda, Ph.D., & Joan Manuel Larrahondo, Ph.D. INGETEC Cra 6 No. 30A-30 Bogotá, Colombia [email protected] & [email protected]
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Introduction Overview: Long Term Rockfill Behhaviour It is widely accepted that rockfill in dam shells exhibit time-dependent, creep-like behaviour ([1]-[3]). Early work by Terzaghi (1960) [2] proposed that such behaviour was due to breakage of rock particles at highly-stressed contacts. Once broken, the rockfill particles would rearrange to attain more stable configurations, thus inducing displacements. Interestingly, most of the experimental evidence available to date still suggests that the main cause of the time-dependent rockfill behaviour is particle breakage and crushing, phenomena enhanced by the presence of water [2]. Furthermore, a number of constitutive models have been proposed to describe such time dependence, including logarithmic, hyperbolic, and visco-elastic relationships between long-term strain and time [1]. Earth-core rockfill dams (ECRD) typically show dramatic deformations upon reservoir first filling [1]. Time-dependent deformation depends on water content of rock particles, such that yield is highest in saturated rock particles, while tends to fade for very dry materials [2]. Since the work by McDowell and Bolton (1998) [2], successive deformation stages are recognized in rockfill-type granular materials subjected to compression stresses as the confining stress is increased. At low stresses, deformation is due to particle rearrangement. From a micromechanical point of view, rockfill deforms mainly because of interparticle sliding and rotation [1]. As the confining stress increases, the granular skeleton becomes gradually blocked, and particle breakage and crushing are triggered. Such mechanism is called “clastic yielding”. Finally, for very high stresses the strain vs. log stress plots have upwarddirected concavity. This is attributed to the comminution limit of small particles. The time-dependent deformation of rockfill can thus be explained by progressive breakage and crushing of stressed particles [1]. Particle breakage takes place as subcritical propagation of preexisting microcracks within loaded rock particles. The velocity of crack propagation is a function of the local stress intensity factor and the prevailing relative humidity. Particle breakage combines with complex localised crushing at interparticle contacts to boost rockfill deformation [1]. Not a large number of databases are available that comprise 25 years or more of instrumentation data from a rockfill dam. The problem proposed in this document consists in reproducing the development of displacements and stresses of a real-case dam during construction, reservoir impoundment, and operation.
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Benchmark Example: Alberto Lleras Dam (Guavio Hydroelectric Project), Colombia Dam Description The case history that is being proposed as benchmark example for the Theme B of the 12th Workshop on Numerical Analysis of Dams is that of the Alberto Lleras Dam, part of the Guavio Hydroelectric Project in Colombia. The Guavio Hydroelectric Project is located in the Cundinamarca region of central Colombia, some 120 km northeast from Bogotá. The project spans between the towns of Ubalá, where the dam is located, and Mambita, where the underground powerhouse is located [4]. The project harnesses the hydropower potential of the lower Guavio River (1100 m nominal head) and comprises the following components designed by INGETEC ([4]-[6]):
The Alberto Lleras Dam: a 245-m high, clay-core rockfill dam, the highest of its kind in South America [3]. The total dam volume is about 17 x 106 m3 which impounds a 14-km long, 950 x 106 m3 reservoir. A 34-m high, 4500 m3/s capacity, ogee-type spillway controlled by radial gates and two 600-m long tunnels A 13.3-km long, 8 m diameter power tunnel and a 505-m high pressure shaft A 234-m long, 17-m wide, 35-m high underground powerhose located 560 m below ground. The total installed generating capacity of the Guavio project is 1600 MW, currently delivered by eight Pelton turbines. A 1160-m long diversion tunnel as well as two other tunnels, 2330 m and 2190 m long, respectively, that divert the Batatas and Chivor Rivers into the Guavio reservoir A 5.2-km long, 8-m diameter tailrace tunnel to return the Guavio river water back to its main course
The Guavio project was built between 1979 and 1995 [2] and first filling took place in February 1992. The average flow of the river at the dam site is 62 m³/s. Figure 1 presents an aerial view of the Guavio Project and the Alberto Lleras Dam. Available instrumentation data span from 1987 through 2013. Specifically, measurements include data from:
78 level survey points on the dam crest and faces 8 Vertical Movement Recorders (RMV) 24 Pneumatic Settlement Sensors (SNA) 40 pressure cells (CP) 26 pneumatic piezometers (PZ)
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Figure 1: General view of the Guavio Hydroelectric Project and the Alberto Lleras Dam [4] The objective for the participants of Theme B is to reproduce, via a numerical model, the displacements observed on the maximum cross section of the Alberto Lleras Dam. The minimum requirements that shall be incorporated in their numerical model are a threedimensional model using the provided simplified topography and the use of the known dam zoning and the construction sequence. Geology Overview The greater Guavio project area comprises Palaeozoic rocks overlain by Mesozoic rocks [5]. The dam site is underlain by Palaeozoic quartzite, argillite, and limestone. At the dam site, the river features a 600-m deep, narrow canyon. The upper half of the canyon at the dam site (left abutment) exhibits Cretaceous rocks. The Palaeozoic formations strike diagonally with respect to the river canyon and dip upstream. Most of these rocks below 30 m of depth are fresh, hard, and depict low permeability [5]. The limestone shows frequent karstic features (caverns) which were tackled via an extensive grouting programme. No geological faults exist at the dam foundation area [5]. The steep slopes of the canyon rendered some relief joints parallel to the river flow. No particular treatment was necessary for these joints.
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Input Data All detailed input data information is provided in this manuscript’s attachments. A description of each database follows. Dam Geometry Attachment 1 contains all drawings related to dam location geometry. Specifically, the following data files are provided:
Maximum longitudinal section and description of dam zones (as-built; Oct. 1989; DWG and PDF files) Plan view and topography (as built; 50-m elevation contours; Nov. 1989; DWG and PDF files) Dam zoning description table (XLS file), including placed volumes and as-built compaction specifications
Construction Sequence Construction of the dam started on June 1983 with foundation excavation work [7]. Fill placement operations initiated on September 1984 and ran through August 1989. In general, during the six rainy months of the year (May-October) rockfill was placed and compacted in the outer portions of the shell. During the remaining six drier months (November-April), fill placement was concentrated in the central portion of the dam, namely the core and confining rockfill [7]. Attachment 2 contains all drawings and data related to construction sequence. The following data files are provided:
Construction sequence plan (maximum longitudinal section; as-built; Nov. 1989) Fill placement sequence summary table (XLS file)
Material Properties The outer portions of the shell rockfill comprised weathered fragments of quartzite, argillite, and limestone [7]. On the other hand, the inner part of the dam below the core consisted in sound, fresh, and clean rockfill. The dam core consists mainly in shale fragments in a silty clay matrix [7] obtained from a nearby quarry named San Pedro. The core material features medium plasticity and fines content >30%. Attachment 3 contains all drawings and data related to material characterization. Specifically, the following data files are provided:
DWG, PDF, and XLS files containing rockfill properties varying with dam depth DWG, PDF, and XLS files containing earth-core properties varying with dam depth One XLS file containing as-built grain size distribution bands for each dam zone One XLS file containing compressibility curves (pre-construction) for the earth core and shell rockfill materials
For illustration purposes, material parameter data that were employed during the geotechnical design of the dam are provided next:
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Earth core and shell rockfill compressibility: an experimental programme was undertaken prior to construction to obtain design compressibility parameters from both the earth core material and the rockfill. 30-cm thick samples of dam-core material were subjected to one-dimensional consolidation tests in a 30-cm diameter oedometer. The earth-core materials featured maximum grain size of 6.5 cm and natural water content during testing. Samples were reconstituted via compaction of five layers using a 30-lb hammer falling freely from a height of 24 in. Between 160 and 200 blows per layer were applied. The rockfill material compressibility was also studied using the same large oedometer as that of the core materials. However, rockfill samples were tested dry and using the same grain size distribution as that of the construction specification (with maximum grain size of 6.5 cm). Samples were reconstituted by compaction of four layers with a vibrating table during 1.5 min each layer. During the loading process, the samples were intentionally inundated at the 16 kg/cm2 loading step. The following table presents the experimental programme initial conditions and results in terms of compressibility coefficient. Attachment 3 also contains the compressibility curves obtained from these tests. Results Max Ave Min Max Ave Min
Water content of the < # 4 sieve fraction (%) 28 24 21 -
Dry density (Ton/m3) 1.84 1.77 1.71 2.25 2.17 2.03
av (10 cm2/kg) 4.7 3.6 2.6 2.1 1.6 0.8 -3
Plasticity of the earth core material (measured values during the dam geotechnical design phase)
Liquid limit (%) Plastic limit (%) Plasticity index (%)
39 26 13
Instrumentation data During construction and particularly after reaching 2/3 of the dam height, large displacements took place within the dam core [7]. As a result, most of the cable of the pneumatic and electric instruments was destroyed. Overall, about 50% of the original instruments installed were lost due to construction operations. Available instrumentation data for this workshop span from 1987 through 2013. Attachment 4 contains all drawings and data related to instrumentation data. Specifically, the following files are provided:
Five folders with XLS data files Two layouts of installed instrumentation (as-builts; plan view and maximum longitudinal section; DWG and PDF; July 1990)
A brief description of available instrumentation data follows. 196
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Movements Surface movements: Data from a total of 63 level survey instruments, labelled 13 through 75 are provided. Survey elevation data span from 1989 through 2013.
Vertical Movement Recorders (RMV) or “crossarm gages”: nine folders are provided containing monthly RMV data, spanning from 1989 through 1995. Additionally, for the 1996-2008 period, a XLS file is provided with data collected in eight of the nine available RMVs. RMV instruments or crossarm gages consist of a series of telescopic pipe sections with alternate sections anchored to the embankment fill by horizontal channel crossarms located at certain depth intervals, typically 3 m ([8] See Figure 2). The crossarms ensure that the coupled pipes move together as compression of the fill progresses. Measuring points are located at the lower end of each inner pipe; depths to such measuring points are usually sounded by a probe with spring-loaded sensing pawls, lowered on a steel tape. To produce a measurement, the probe is lowered just below a given inner pipe and then raised until the paws latch against the lower end.
Pneumatic Settlement Sensors (SNA): two folders; instruments 28-41 with monthly data from 1987 to 1996; instruments 49-58 with monthly data from 1988 to 1995.
Figure 2: RMV (crossarm gage) schematic [8]. Left: telescopic pipe array. Right: measurement probe Pressure Cells (CP) Three folders are provided. All total pressure cell data is reported in kg/cm 2. Monthly data are provided as follows:
CP 1-24: data from 1986 to 1998; CP 1, 3, 5, 6, and 7 are at 1405 masl elevation; CP 8 through 24 are at 1463 masl elevation CP 28-37: 1987 to 1998 at 1523 masl elevation
CP 41-46: data from 1988 to 1997 at 1589 masl elevation
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Pneumatic Piezometers (PZ) One XLS file is provided that contains data collected from the dam foundation piezometers (PZ 82, 83, and 84). Data were collected monthly spanning from 1988 through 2004. 15 XLS data files are provided with data collected from the dam core piezometers (PZ 45 through 49) at 1523 masl elevation. Data were collected monthly spanning from 1987 through 2001.
Requested Results All workshop participants are kindly requested to provide a paper, 15 pages maximum, in which all assumptions are clearly stated, particularly regarding initial and boundary conditions. To facilitate results rating, the participants must also submit their key results under a prescribed format whereby an Excel file under the name ThB_ResXXXX.xls shall be created (XXXX represents a name or acronym of the participant’s organisation). Fill Displacements Displacement contours for three longitudinal sections and one cross section along the dam crest shall be presented. Results at the end of construction, at the condition prior to impoundment, and after reservoir impoundment shall be included. Stresses within the Dam Calculated principal and vertical stresses obtained from the analysis on five locations within the fill are requested. The results shall be presented both at the end of construction and after partial and full impoundment. Creep and Arching Effects A discussion supported with simulation results is requested regarding long-term displacement behaviour including possible arching effects within the dam body. Groundwater Flow The participants are requested to submit the resulting groundwater flow net under 1620 masl reservoir level water level conditions. Acknowledgements The formulators are grateful to EMGESA Colombia for allowing the use of the instrumentation data for this workshop
References [1] Oldecop, L. A. and Alonso, E. E. (2007). Theoretical Investigation of the TimeDependent Behaviour of Rockfill. Géotechnique, V. 57, No. 3, pp. 289–301 [2] Oldecop, L. A. and Alonso, E. E. (2001). A Model for Rockill Compressibility. Géotechnique, V. 51, No. 2, pp. 127-139 [3] Justo, J. L., Durand, P. y Justo, E. (2003). Un Modelo Tridimensional para el Estudio de la Fluencia en Presas de Materiales Sueltos. Rev. Int. Mét. Num. Cálc. Dis. Ing. V. 19, No. 3, pp. 313-330 (in Spanish) [4] INGETEC S.A. (2013). Hydropower Projects – Guavio http://www.ingetec.com.co/experiencia-ingles/textos-proyectos-ingles/proyectohidroelectricos-ingles/guavio-ingles.htm (Accessed 2013)
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[5] Marulanda, A. and Amaya, F. (1989). The Design and Construction of Colombia’s Guavio Dam. Water Power and Dam Construction, December 1989. pp. 41-60 [6] INGETEC S.A. (2013). Hydroelectric Developments. http://www.ingetec.com.co/brochures-ingles/BROCHURE-HIDROELECTRICASINGLES.pdf (Accessed 2013) [7] Amaya, F. and Marulanda, A. (1993). Behavior of Guavio Dam during Construction and First Filling. Proceedings of the International Symposium on High Earth-Rockfill Dams. ICOLD-CSHEE. Beijing, China. [8] Dunnicliff, J. (1988). Geotechnical Instrumentation for Monitoring and Performance. John Wiley & Sons. 577 p.
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PAPERS THEME B
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Solution to Theme B Long Term Behaviour of a Rockfill Dam F. Ezbakhe1, I. Escuder-Bueno1 1
Universidad Politécnica de Valencia, Spain E-mail: [email protected]
Preface The authors acknowledge that the solution herein presented is very simplified and should not be taken as a professional reference by any means. However, with the purpose of enabling the formulator to estimate the order of magnitude of the lack of accuracy and/or physical inconsistencies by using methods as simplified and empirical as those herein presented, the paper may be of certain interest.
Introduction Theme B of the 12th Benchmark Workshop on Numerical Analysis of Dams (ICOLD) consists in the evaluation of the long term behavior of a rockfill dam, analyzing the case of the Alberto Lleras Dam, part of the Guavio Hydroelectric Project in Colombia. The objective is to reproduce, via a numerical model, the behavior of the dam during the construction, impoundment and exploitation, providing the evolution of displacements and stresses within the dam as results. For the solution presented in this paper, the problem has been solved by FLAC3D (Itasca), a finite differences program developed by Itasca Consulting Group, Minnesota. This software uses the finite differences method to analyze the mechanical behavior of a continuous medium in 3D until it reaches an equilibrium state.
Figure 1: View of the Alberto Lleras Dam 203
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Statement of the problem Alberto Lleras Dam is a clay-core rockfill dam which main features are given in the next table: Table 1: Main features of the Alberto Lleras Dam Height Width Dam volume Reservoir volume
245m 950m 7hm3 950hm3
Construction of the dam lasted five years, from September 1984 to August 1989. In general, during the six rainy months of the year (May-October) rockfill was placed and compacted in the outer portions of the shell. During the remaining six drier months (November-April), fill placement was concentrated in the central portion of the dam. The impoundment started on February 1992 and run through September 1992. The dam consists in three rockfill zones, with similar mechanical parameters but different granulometric composition. It includes also two transition zones and a clay core, featuring a medium plasticity clay matrix.
Figure 2: View of the Alberto Lleras Dam The dam site is underlain by quartzite, argilite and limestone, with no geological faults. The formations strike diagonally with respect to the river canyon and are fresh, hard and depict low permeability.
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Features of the implemented model Geometrical model From a simplification of the real topography and geometry of the problem, a 3D geometrical model of the dam was implemented, formed by 7286 nodes and 5800 solid elements. Next figures show the FLAC 3D geometrical model of the dam.
Figure 3: FLAC 3D geometrical model 205
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Mechanical model The mechanical model selected for the solution herein presented is linear-elastic. This model was chosen for its simplicity and wide applicability solving soil and rock mechanics problems. In this model, strain increments generate stress increments according to the linear and reversible law of Hooke: (1) where the Einstein summation convention applies, δ_ij is the Kroenecker delta symbol, and α_2 is a material constant related to the bulk modulus, K, and shear modulus, G, as: (2) New stress values are then obtained from the relation: (3) Next table provides the values that have been adopted for the mechanical model: Table 2: Mechanical parameters
Core Rockfill Foundation
Density (kg/m3) 2010 2750 2500
Bulk modulus (Pa)
Shear modulus (Pa)
7.5·106 8.5·107 4·1010
5.4·106 4.7·107 3·1010
The boundary conditions applied are: - Plane x=-300 (lateral surface). Movements constrained on the x-axis - Plane x=400 (lateral surface). Movements constrained on the x-axis - Plane y=-300 (front surface). Movements constrained on the y-axis - Plane y=1300 (back surface). Movements constrained on the y-axis - Plane z=-300 (underside). Movements constrained on all axis Constructive behaviour The construction process may affect the stress state, particularly if excess pore pressures develop in the soils and do not dissipate completely during the construction stages. In addition, staged modeling of the embankment lift construction also provides a better representation of the stresses in the embankment. In this case, due to the difficulty involved in modeling the actual construction sequence, it has been simplified considering that it takes place in uniform layers of 10m high. Thus, the numerical process is to activate each layer, assigning materials and properties, and calculating the model to reach equilibrium. Post-constructive behaviour In Geotechnical Engineering, time-dependent settlement is associated to the process of consolidation. This behavior would be therefore determined by the rate at which water is able to flow through the ground pores under a certain hydraulic gradient. Boughton (1970) found that this settlement also occurred during the construction process. For calculation purposes, Boughton got a good fit for the form of post-construction deformations 206
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by applying the entire weight of the structure at once. Then, adopting a scale factor, he got a good fit for the magnitude of the deformations. In this case, the scale factor used is 1.05 for dry behavior (from the completion of construction to the start of filling) and 1.11 for the behavior of the wet rockfill (with filled reservoir).
Results Settlements Settlements of the dam body are obtained at three different times: - August 1989 completed the construction of the dam - January 1992 immediately before the first filling - September 1992 1630 masl reservoir Settlements are also obtained in nine point of the dam: Table 3: Settlements in the dam
M-30 M-36 M-48 M-53 M-64 M-66 M-72 SNA-30 SNA-38
aug-89 Settlement Elevation (m) (masl) 0.00 1641.081 0.00 1641.020 0.00 1641.751 0.00 1640.420 1.54 1604.027 1.60 1570.985 1.46 1515.672 3.60 1517.435 3.42 1518.242
jan-92 Settlement Elevation (m) (masl) 0.810 1640.271 0.875 1640.145 0.363 1641.388 0.290 1640.130 1.974 1603.594 1.827 1570.758 1.617 1515.513 4.037 1516.995 3.872 1517.790
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sep-92 Settlement Elevation (m) (masl) 0.971 1640.110 1.043 1639.977 0.470 1641.281 0.368 1640.052 2.070 1603.498 2.032 1570.553 1.715 1515.415 4.593 1516.439 3.991 1517.671
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Figure 4: Settlements in August 1989
Figure 5: Settlements in January 1992
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Figure 6: Settlements in September 1992 Stresses Next figure represents the stresses of the dam after dam construction:
Figure 7: Principal stresses in August 1989 209
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Principal stresses (s1) and vertical stresses (sv) are also obteined in five different points: Table 4: Stresses in the dam @ EOC
Reservoir at EL 1575
Reservoir at EL 1630
ID
s1 [Kpa]
sv [Kpa]
s1 [Kpa]
sv [Kpa]
s1 [Kpa]
sv [Kpa]
CP-11
1594.3
1556.5
1619.1
1599.9
1752.3
1725.6
CP-6
1345.2
1281.2
3015.2
1964.7
3237.5
2482.1
CP-19
1909.4
1554.0
2240.3
1914.9
2806.5
2019.0
CP-22
2871.0
1459.6
2884.9
1947.6
2916.8
1992.0
CP-3
2769.7
2653.6
3438.1
2949.6
3475.0
2969.2
Conclusion In order to study the long dam’s long term behavior, a linear elastic analysis has been realized. This means that a linear analysis is performed for each time step and when the equilibrium is reached the calculus moves to next instant of time. The implemented model considers the dam body as a structure without interfaces and permits the evolution of the displacements and stresses over time. The simulation of this evolution can resembles to some extent to the data collected by auscultation, with a margin of error due to the different simplifications made in the model.
References [1] Escuder, I. “Estudio del comportamiento tenso-deformacional de pedraplenes inundables mediante simulaciones numéricas formuladas en diferencias finitas y calibradas con lecturas de instrumentación”. Doctoral Thesis [2] Itasca Consulting Group, Inc., “Fast Lagrangian Analysis of Continua in 3 Dimensions version 3.1: Theory and Background, Optional features”, Minneapolis, Minnesota, 2006 [3] ICOLD, “Guidelines for use of numerical models in dam engineering”, Bulletin 155
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THEME C
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Theme C
Computational Challenges in Consequence Estimation for Risk Assessment: Numerical Modelling, Uncertainty Quantification, and Communication of Results
Part 1 – Hydraulic Modelling and Simulation Sponsoring Organizations: U.S. Army Corps of Engineers (USA) U.S. Department of Homeland Security (USA) Formulators: Yazmin Seda-Sanabria (Formulation Team Co-Chair) National Program Manager, Critical Infrastructure Protection & Resilience Program, U.S. Army Corps of Engineers, Headquarters, 441 G Street NW (ATTN: CECW-HS), Washington, DC 20314 (USA), Email: [email protected] Enrique E. Matheu (Formulation Team Co-Chair) Chief, Critical Lifelines Branch, Sector Outreach and Programs Division, Office of Infrastructure Protection, U.S. Department of Homeland Security, 245 Murray Lane Arlington, VA 20598-0608 (USA), Email: [email protected] Timothy N. McPherson (Formulation Team Technical Lead) R&D Group Leader, Energy and Infrastructure Analysis, Los Alamos National Laboratory, Los Alamos, New Mexico 87544 (USA), Email: [email protected] Mustafa Altinakar Director and Research Professor, National Center for Computational Hydroscience and Engineering, The University of Mississippi, Brevard Hall Room 327, P.O. Box 1848, University, MS 38677-1848 (USA), Email: [email protected] Mark Jourdan Research Hydraulic Engineer, Coastal and Hydraulics Laboratory, U.S. Army Engineer Research and Development, 3909 Halls Ferry Road, Vicksburg, Mississippi 39180 (USA), Email: [email protected] Michael K. Sharp Technical Director, Geotechnical and Structures Laboratory, U.S. Army Engineer Research and Development, 3909 Halls Ferry Road, Vicksburg, Mississippi 39180 (USA), Email: [email protected] For technical information, please contact Dr. Tim McPherson at [email protected]
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Introduction In the last decade, computational capacity has grown dramatically such that multiprocessor computing techniques are now widely available. This increase in resource availability has allowed the development of a vast array of new models for flooding and consequence assessment. Many of these models are computing flood wave propagation at extremely high temporal and spatial resolutions. When these models are coupled to equally complex models of population mobility, infrastructure impact and economic consequence, simulation frameworks are created that can support a paradigm shift from standard approaches to dam risk analysis. Although computational advances have increased the availability and applicability of novel tools, there is a deficiency in benchmarks on the use of those tools in risk assessments. An obvious application of increased availability and efficiency of computational resources is to conduct probabilistic risk assessments using Monte Carlo techniques, but the application of such approaches entails a wide range of assumptions and technical decisions regarding the management of uncertainty such as variable uncertainty, parameter uncertainty, uncertainty in probabilistic sub-model, measurement error, computational errors, and numerical approximation to name a few. Universities, engineering companies and regulatory bodies are invited to contribute to the benchmark and take part in the discussion of results gained. This document is part 1 in a 2-part series for Theme C. Part 1 pertains to the hydraulic modelling and simulation of the dam breach and subsequent flood wave and provides details regarding the available data, dam geometry and failure, and expected modelling and simulation solution requirements. Part 2 primarily focuses on consequence estimation using the modelling and simulation results from Part 1. Benchmark Focus The numerical problem proposed for the workshop consists of estimating the consequences of failure of a dam near populated areas with complex demographics, infrastructure and economic activity. The dam in question will be near the city of Hydropolis, a virtual testbed for flood risk analysis to be built in preparation for the benchmark study. Theme C participants are free to select the type and sophistication of the simulation engines used to solve the problem, including 1-d, 2-d and 3-d flood simulation tools, Population at Risk (PAR) and Loss of Life (LOL) estimation techniques, and infrastructure and consequence assessment models.
Flood Modelling and Simulation The following sections are intended to provide information regarding the data provided for the dam failure modelling and simulation benchmark. Specifically this information includes the topographic data and dam geometric and construction information. It is not the intent of this benchmark to set requirements as to which modelling and simulation environment should be used. Therefore, the descriptions and data provided are intended to be useful to a wide range of modelling and simulation environments at many levels of fidelity. Dam Information A hypothetical embankment dam was constructed in a mountainous region. The high- hazard dam sits directly above a lightly populated area and 3.5 kilometres away from an urban environment. The front and rear views of this dam are shown in Figure 2-1 and 2-2, respectively.
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The primary function of the dam is flood control for heavy snowmelt and strong monsoonal weather patterns. In addition, the reservoir provides some water supply and recreational activities to nearby communities. The following sections provide more detail regarding the geometry and the construction of the hypothetical dam.
Figure 1: Front view of dam and surrounding topography
Figure 2: Rear view of dam, reservoir, and surrounding topography
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Dam Geometry The hypothetical embankment dam is considered high head (61 m) with moderate storage (38 million cubic meters). An overview of the dam geometry is provided in Table 5 and a crosssectional profile is shown in Figure 3. Table 5: Dam geometric parameters Parameter Description
Value
Dam Location (x, y)
4499.66, 6681.57
Crest Length (m)
360
Crest Width (m)
24
Crest Elevation (m)
272
River Bed Elevation (m)
211
Upstream Embankment Slope (?H:1V)
3
Downstream Embankment Slope (?H:1V)
3
Figure 3: Cross-sectional view of the hypothetical dam The storage capacity at crest elevation is more than 38 million cubic meters. The stagevolume curve for the reservoir is shown in Table 6. Table 6: Reservoir stage-volume curve Elevation (m)
Surface Area (m²)
Volume (m³)
Elevation (m)
Surface Area (m²)
Volume (m³)
211 213
0 898
0 266
243 245
605,559 682,169
6,214,463 7,500,447
215
7,812
7,432
247
762,461
8,944,612
217
25,502
39,871
249
848,052
10,553,044
219
43,821
109,250
251
916,938
12,322,332
221
62,409
216,078
253
978,640
14,218,482
223
85,038
364,294
255
1,035,581
16,233,223
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225
117,544
565,695
257
1,094,197
18,365,080
227
153,732
836,860
259
1,155,316
20,614,116
229
190,728
1,181,477
261
1,216,975
22,987,820
231
222,068
1,594,314
263
1,278,543
25,483,100
233
252,239
2,067,481
265
1,338,586
28,100,382
235
294,797
2,613,155
267
1,398,359
30,837,356
237
351,808
3,254,198
269
1,467,826
33,701,532
239
461,611
4,084,756
271
1,543,126
36,712,416
241
531,463
5,077,507
272
1,584,052
38,276,344
Dam Construction Information Homogenous Dam The dam is a rolled earth fill structure composed of predominantly sandy clays and clayey sands. Compaction was achieved using 4 passes of a 50-ton pneumatic-tired roller on 0.3meter loose lifts. Strength properties for the dam were developed from results from undrained triaxial strength tests. The drained strength parameters were based on 5% axial strain and were selected to represent initial confining stresses up to 478 kPa. The undrained strengths were interpreted from an approximate evaluation of Su/mc ratios estimated from the reported undrained tests. The undrained strength S u was taken as one-half the maximum deviator stress for axial strains up to 10%. The estimated strengths are summarized in Table 7. Table 7: Selected parameters for dam Parameter Description Strength Parameters: c Effective (drained) cohesion in kPa Effective (drained) friction angle
Value
Su
43.09+ 0.175mc
19.15 kPa 14°
Undrained strength in kPa
Stiffness Parameters: Vs1 Shear wave velocity at v = 1 atm Gmax,1 Maximum shear modulus at v = 1 atm Others: Saturated unit weight sat k Permeability
152.4 m/s 46443 kPa 2002 kg/m³ 1.910-6 cm/s
Foundation Foundation stiffness parameters were based on shear wave velocity values without reduction for strain level or changes in effective stress. A simplified stiffness distribution was used to reflect the gradual increase in stiffness with depth. The selected properties are summarized in Table 8.
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Table 8: Selected parameters for foundation Parameter Description
Value Base of dam to Foundation depth of below 3.6 m 3.6 m
Stiffness Parameters: Vs
Shear wave velocity
167 m/s
185 m/s
G
Shear modulus
6.75105 kPa
8.33 106 kPa
Poisson’s ratio
0.25
0.25
Unit weight
2,242kg/m³
2,402 kg/m³
Permeability
9.510-7 cm/s
9.510-7 cm/s
Others: K
Dam Failure Guidance The dam failure for this benchmark takes place when the pool elevation is at crest elevation. The mode of failure will be assumed as an overtopping failure. While it is open for participants to conduct detailed (physically-based) modelling of the breaching process as part of the benchmark, several breach parameter estimation approaches available in the literature are summarized in the appendix of this document. Data Provided In addition to the information provided in this document, gridded data representing the topography and the land use classification are provided. All gridded datasets conform to the same domain and cellsize. The resolution of data provided is considered a base dataset and may be altered if required by the modelling and simulation environment used by the participant. Digital Elevation Data Two digital elevation models (DEM) are provided to benchmark participants: DEM representing pre-dam construction. DEM representing post-dam construction. Participants in the benchmark may decide which DEM is more appropriate for use based on individual requirements for modelling and simulation. Land Use/Cover Benchmark participants are provided a gridded dataset representing the hypothetical land use/cover for the simulation region. These data follow the classification guidance and values provided in the National Land Cover Dataset (NLCD). For completeness, a description of land use classifications is provided in Table 9.
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Table 9: Land use/cover classifications and descriptions NLCD Class 11 12 21 22 23 24 31 41 42 43 52 71 81 82
Description Open Water Perennial Ice/Snow Developed-Open Space Developed- Low Intensity Developed- Med. Intensity Developed- High Intensity Barren Land Deciduous Forest Evergreen Forest Mixed Forest Shrub/Scrub Grassland/Herbaceous Pasture/Hay Cultivated Cropland
Correlation of these classifications to surface roughness for modelling and simulation should be reported by the participant. Flood Modelling and Simulation Reporting Requirements For consistency between benchmark participants, each participant is requested to generate information described in Table 10 from the results of the flood simulation. These data focus on the hydraulics of the simulation. Table 10: Reporting requirements for flood modelling and simulation Required Data Breach Discharge Cross-Section Discharge
Peak Flood Depths Flood Wave Arrival Time Peak Unit Flow Rate Flooded Area
Data Description This hydrograph should show the discharge rate from pre-failure to empty reservoir, in units of m3/ These shall consist of a complete hydrograph for unsteady simulation environments and peak discharges for steady-state simulation environments Gridded dataset representing the peak flood depth, units of meters Gridded dataset with flood wave arrival time in 5-minute intervals Gridded dataset with a value representing the peak unit flow rate in units of m²/s Summation of the total flooded area, units of m². In addition, participants will provide flooded area categorized by range of flood depths at .5 meter intervals
The locations in which participants should provide hydrographs are shown in Table 11
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Table 11: Cross-section locations Cross Section ID 1 2 3 4 5
X Location (m) 4737.18 5971.79 7486.60 9100.84 10716.96
Y Location (m) 6755.39 7053.10 7582.85 7773.91 7397.67
Solution Metadata As described above, Participants are expected to provide gridded data for some of the reporting requirements. These gridded data files must conform to the metadata of the original data provided in the benchmark. These are summarized in Table 12. Table 12: Gridded data metadata requirements Metadata Parameter Left Extent Bottom Extent Right Extent Top Extent Cellsize Columns Rows
Value 0 0 25831.81905 9930.941439 9.4760892 2726 1048
Acronyms and Abbreviations DEM NLCD kPa
Digital Elevation Model National Land Cover Dataset Kilopascal
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Theme C
Computational Challenges in Consequence Estimation for Risk Assessment: Numerical Modelling, Uncertainty Quantification, and Communication of Results
Part 2 – Consequence Estimation Sponsoring Organizations: U.S. Army Corps of Engineers (USA) U.S. Department of Homeland Security (USA) Formulators: Yazmin Seda-Sanabria (Formulation Team Co-Chair) National Program Manager, Critical Infrastructure Protection & Resilience Program, U.S. Army Corps of Engineers, Headquarters, 441 G Street NW (ATTN: CECW-HS), Washington, DC 20314 (USA), Email: [email protected] Enrique E. Matheu (Formulation Team Co-Chair) Chief, Critical Lifelines Branch, Sector Outreach and Programs Division, Office of Infrastructure Protection, U.S. Department of Homeland Security, 245 Murray Lane Arlington, VA 20598-0608 (USA), Email: [email protected] Timothy N. McPherson (Formulation Team Technical Lead) R&D Group Leader, Energy and Infrastructure Analysis, Los Alamos National Laboratory, Los Alamos, New Mexico 87544 (USA), Email: [email protected] Mustafa Altinakar Director and Research Professor, National Center for Computational Hydroscience and Engineering, The University of Mississippi, Brevard Hall Room 327, P.O. Box 1848, University, MS 38677-1848 (USA), Email: [email protected] Mark Jourdan Research Hydraulic Engineer, Coastal and Hydraulics Laboratory, U.S. Army Engineer Research and Development, 3909 Halls Ferry Road, Vicksburg, Mississippi 39180 (USA), Email: [email protected] Michael K. Sharp Technical Director, Geotechnical and Structures Laboratory, U.S. Army Engineer Research and Development, 3909 Halls Ferry Road, Vicksburg, Mississippi 39180 (USA), Email: [email protected] For technical information, please contact Dr. Tim McPherson at [email protected]
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Introduction In the last decade, computational capacity has grown dramatically such that multiprocessor computing techniques are now widely available. This increase in resource availability has allowed the development of a vast array of new models for flooding and consequence assessment. Many of these models are computing flood wave propagation at extremely high temporal and spatial resolutions. When these models are coupled to equally complex models of population mobility, infrastructure impact and economic consequence, simulation frameworks are created that can support a paradigm shift from standard approaches to dam risk analysis. Although computational advances have increased the availability and applicability of novel tools, there is a deficiency in benchmarks on the use of those tools in risk assessments. An obvious application of increased availability and efficiency of computational resources is to conduct probabilistic risk assessments using Monte Carlo techniques, but the application of such approaches entails a wide range of assumptions and technical decisions regarding the management of uncertainty such as variable uncertainty, parameter uncertainty, uncertainty in probabilistic sub-model, measurement error, computational errors, and numerical approximation to name a few. Universities, engineering companies and regulatory bodies are invited to contribute to the benchmark and take part in the discussion of results gained. This document is part 2 in a 2-part series for Theme C. Part 1 pertains to the hydraulic modelling and simulation of the dam breach and subsequent flood wave and provides details regarding the available data, dam geometry and failure, and expected modelling and simulation solution requirements. Part 2 focuses on the consequences of the flood using the modelling and simulation results from Part 1. Benchmark Focus The numerical problem proposed for the workshop consists of estimating the consequences of failure of a dam near populated areas with complex demographics, infrastructure and economic activity. The dam in question will be near the city of Hydropolis, a virtual testbed for flood risk analysis to be built in preparation for the benchmark study. Theme C participants are free to select the type and sophistication of the simulation engines used to solve the problem, including 1-d, 2-d and 3-d flood simulation tools, Population at Risk (PAR) and Loss of Life (LOL) estimation techniques, and infrastructure and consequence assessment models.
Consequence Estimation A full consequence assessment for dam failure generally includes four main categories- public health and safety, economic impact, psychological impact, and governance/mission impact. At a minimum, the consequence assessment should focus on impacts related to human consequences and direct economic impact. This document outlines the requirements for consequence assessment within Theme C of the benchmark, addressing human consequences (e.g., population at risk and loss of life) and direct economic impact. The intent is not to limit the approaches or techniques used by participants but to guide the assessment into categories for comparison. Additional consequence assessments are also encouraged, but will not be used in the assessment of uncertainty.
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Dam Failure Assumptions Over a series of fall rainfall events, the reservoir slowly approaches crest elevation. Due to the frequency of fall rainfall events, there are few people recreating at or near the facility. At 11:00 pm on Saturday evening, the dam fails, releasing a torrent of water through the canyon and towards Hydropolis. The failure was sudden and unexpected, and for those in the canyon witnessing the failure, there are no mobile phone services. The reservoir and dam does not have a completed emergency action plan, thus the potential impact related to failure is unknown to residents in Hydropolis. Data Provided Participants are requested to evaluate the downstream consequences as it relates to population at risk, loss of life and direct economic impact. Participants in Benchmark Theme C are provided additional data to assist in consequence assessment. This data includes Hydropolis census information in shapefile format and parcel data in shapefile format. Census Data Participants are provided with a shapefile representative of census data for the city of Hydropolis. This data includes information regarding the population and the economic activity. Table 5 and Table 2 provide attribution information for these data. Table 1: Population data field names and descriptions Field Names Total Male Munder5 M5to9 M10to14 M15to17 M18to19 M20 M21 M22to24 M25to29 M30to34 M35to39 M40to44 M45to49 M50to54 M55to59 M60to61 M62to64 M65to66 M67to69 M70to74 M75to79 M80to84 M85over
Description Total Male and Female Total Male Male: Under 5 years Male: 5 to 9 years Male: 10 to 14 years Male: 15 to 17 years Male: 18 and 19 years Male: 20 years Male: 21 years Male: 22 to 24 years Male: 25 to 29 years Male: 30 to 34 years Male: 35 to 39 years Male: 40 to 44 years Male: 45 to 49 years Male: 50 to 54 years Male: 55 to 59 years Male: 60 and 61 years Male: 62 to 64 years Male: 65 and 66 years Male: 67 to 69 years Male: 70 to 74 years Male: 75 to 79 years Male: 80 to 84 years Male: 85 years and over
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Female Funder5 F5to9 F10to14 F15to17 F18to19 F20 F21 F22to24 F25to29 F30to34 F35to39 F40to44 F45to49 F50to54 F55to59 F60to61 F62to64 F65to66 F67to69 F70to74 F75to79 F80to84 F85over
Total Female Female: Under 5 years Female: 5 to 9 years Female: 10 to 14 years Female: 15 to 17 years Female: 18 and 19 years Female: 20 years Female: 21 years Female: 22 to 24 years Female: 25 to 29 years Female: 30 to 34 years Female: 35 to 39 years Female: 40 to 44 years Female: 45 to 49 years Female: 50 to 54 years Female: 55 to 59 years Female: 60 and 61 years Female: 62 to 64 years Female: 65 and 66 years Female: 67 to 69 years Female: 70 to 74 years Female: 75 to 79 years Female: 80 to 84 years Female: 85 years and over Table 2: Economic table field names and descriptions
Field Names jobs11 jobs21 jobs22 jobs23 jobs3133 jobs42 jobs4445 jobs4849 jobs51 jobs52 jobs53 jobs54 jobs55 jobs56 jobs61 jobs62 jobs71
Description Agriculture, Forestry, Fishing and Hunting Mining, Quarrying, and Oil and Gas Extraction Utilities Construction Manufacturing Wholesale Trade Retail Trade Transportation and Warehousing Information Finance and Insurance Real Estate and Rental and Leasing Professional, Scientific, and Technical Services Management of Companies and Enterprises Administrative and Support and Waste Management and Remediation Services Educational Services Health Care and Social Assistance Arts, Entertainment, and Recreation 224
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jobs72 jobs81 jobs92
Accommodation and Food Services Other Services (except Public Administration) Public Administration
Parcel Data Benchmark participants are provided parcel data in shapefile format. This includes information regarding the zoning (e.g., residential, commercial, etc.) and the structure (e.g., stories, basement, quality). Table 3 provides a description of the zone classification provided in this data. Table 3: Parcel descriptions for zone classifications Code A-1 A-2 AP C-1 C-2 C-3 IP M-1 O-1 P P-R R-1 R-1 R-2 R-3 RA-1 RA-2 R-D R-LT RO-1 RO-20 R-T SD-LC-1 SD-RO SU-1 SU-2 SU-3 UCO WO
Description Rural Agricultural—1-Acre Minimum Zone Rural Agricultural—2-Acre Minimum Zone Airport Protection Overlay Zone Neighborhood Commercial Zone Community Commercial Zone Heavy Commercial Zone Industrial Park Zone Light Manufacturing Zone Office & Institutional Zone Parking Zone Reserve Parking Zone Residential Zone: Houses Single Family Residential Zone Residential Zone: Houses, Townhomes & Medium Density Apartments Residential Zone: Houses, Townhomes & High Density Apartments Residential and Agricultural Zone, Semi-Urban Area Residential and Agricultural Zone Residential and Related Uses Zone, Developing Area Residential Zone: Houses & Limited Townhomes Rural and Open Zone Rural and Open Agricultural Zone Residential Zone: Houses & Townhomes Limited Neighborhood Commercial Zone Residential/Office Zone Special Use Zone Special Neighborhood Zone, Redeveloping Area Special Center Zone Urban Conservation Overlay Zone Wall Overlay Zone
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Consequence Assessment Reporting Requirements All participants should summarize methods, techniques, assumptions, and key results in a paper not to exceed 15 pages. In addition, each participant is requested to generate the information described in Table 4 using results of the consequence assessment. These data focus on the consequences of failure. Table 4: Reporting requirements for consequence assessment Required Data Population at Risk
Loss of Life
Flood Severity Grid Direct Economic Impact
Data Description Participants should use the provided consequence results spreadsheet to populate the requested information. This information includes age demographic break‐downs by time to flooding and depth of flooding. A gridded dataset indicating the spatial variability of population at risk (regardless of age) is also requested. Participants should use the provided consequence results spreadsheet to populate the requested information. The loss of life estimation includes break‐downs by time from breach failure. A gridded dataset indicating the spatial variability of loss of life is requested. Gridded dataset representing the peak flood severity, classified as low, medium and high (e.g., 1, 2, 3) severity. Participants are expected to describe the assumptions used in categorizing flood severity in the submitted paper. Participants should use the provided consequence results spreadsheet to populate the requested information. This information includes break‐ downs by time from dam failure. A gridded dataset indicating the spatial variability of direct economic impact is also requested.
Consequence Metadata As described above, participants are expected to provide gridded data for some of the reporting requirements. These gridded data files must conform to the metadata of the original data provided in the benchmark. These are summarized in Table 5. Table 5: Gridded data metadata requirements Metadata Parameter Left Extent Bottom Extent Right Extent Top Extent Cellsize Columns Rows
Value 0 0 25831.81905 9930.941439 9.4760892 2726 1048
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RESULT COMPARISON OF THE PARTICIPANTS THEME C
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Results Comparison Theme C Computational Challenges in Consequence Estimation for Risk Assessment Introduction Theme C of the 12th International Benchmark Workshop on the Numerical Analysis of Large Dams involved 8 participants who submitted at least partial solutions of the Hydropolis Dam Break Case Study. This synthesis is not a complete representation of each solution. In many cases, participants conducted their own sensitivity/uncertainty analyses. For brevity, this synthesis document considered only one set of solutions from each participant. Full description of solutions can be found in the papers authored by each participant team. This synthesis of results is based on statistical methods to facilitate comparison to the greatest extent possible. The Kappa statistic is used to quantify the similarities among the participants’ study results. Kappa corresponds to a numerical rating of the degree of agreement between two raters, which in this case are the outputs (e.g., gridded data peak flood depths) of two participant models. Kappa is calculated in Equation 1. (1) where P(a) is the relative observed agreement between the two models and P(e) is the probability that the agreement is due to chance. Kappa ranges from -1 to 1. If the two models are in perfect agreement then Kappa equals 1, whereas if there is no agreement among the models except what would be expected by chance. A Kappa value less than 0 indicates more disagreement than what one can expect by chance. A commonly cited scale for interpreting Kappa values is given by Landis and Koch1 and reported in the following Table 1. The Kappa statistic quantifies the aggregate agreement between two different spatial grids, but does not provide any spatial information about where the two models differ more or less. Table 1: Kappa value interpretation as proposed by Landis Kappa
Agreement Interpretation
<0 0.01-0.20
Less than chance agreement Slight agreement
Agreement Index (used in Results Tables) Chance Slight
0.21-0.40
Fair agreement
Fair
0.41-0.60 0.61-0.80
Moderate agreement Substantial agreement
Moderate Substantial
0.81-0.99
Almost perfect agreement
Almost perfect
1
Landis J.R., and Koch G.G., The measurement of observer agreement for categorical data. 1977 Biometrics 33, 159-174 229
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Overview of Participation The benchmark case includes a diverse set of analyses and several participants only submitted results against a subset of the complete problem. Table 2 lists each of the lead authors, an ID assigned to the participant team for solution comparison, and an indication of submission completion for topics of comparison. Table 2: Summary of Participant Submissions Authors
ID
Hydraulic Solution
Davison et al. Bent et al. Mancusi et al. Williams & Buchanan Thames & Kalyanapu Altinakar et al. Saberi et al. McVan et al.
HRW LANL MANC MMC TK MUST SAB MCV
Yes Yes Yes Yes Yes Yes Yes Yes
Population at Risk Solution Yes Yes Yes Yes Yes Yes No Yes
Loss of Life Solution Yes Yes Yes Yes Yes Yes No No
Economic Solution Yes Yes Yes Yes Yes No No No
Discussion Breach formulation Participants used a range of models, including physics-based breach models using dam material information and regression equations based on previous dam failures. Figure 1 shows the breach hydrographs produced by each team. It is clear that choice of model, method, and parameters can significantly affect the timing and the magnitude of the peak discharge. In general, models using regression equations had a much earlier peak than physics-based models. The SAB and HRW hydrographs were shifted in time to better illustrate the data for comparison. The peak discharge ranges in time from under 0.5 hours to more than 18 hours and the magnitude of peak discharge ranges from near 10,000 cubic meters per second (cms) to more than 40,000 cms. SAB, MMC, LANL, MANK, and TK all have peak discharges above 20,000 cms, while MCV, MUST, and HRW have peaks below 20,000 cms. ~4 hr shift for comparison
~16 hr shift for comparison
Figure 1: discharge hydrographs 230
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Hydrodynamics Each participant used different models, which solve the same equations using different numerical schemes. The most notable difference between models appeared to be the difference in mesh or grid type used. Models using structured grids tended to have more similar results relative to peak depths than models using irregular meshes. This may be a result of interpolation between the two types of meshes for comparison. A wide variety of techniques were used by the participant teams to produce the necessary flood output data for consequence analyses. Each participant submission details the methods used. These differences can be a strong explanatory variable in the comparison of consequence results. For example, dam failure hydrodynamic simulation results are dependent on the input datasets (e.g., breach discharge, topography, roughness) and modeling\method approach. Participants used different approaches to estimate roughness coefficients. Each participant was required to interpret the roughness coefficient associated with an NLCD land use type. Because there isn’t a single accepted dataset, there were a range of coefficients used by participants. Cross Sections The participants were provided a single point location for each section, and the participants estimated the line upon which flow should be summed. Figure 2 shows the hydrographs submitted by each of the participants for all cross sections. SAB and HRW have the same time shifts as previously noted.
Figure 2: Hydrographs for all cross sections submitted by benchmark participants 231
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Peak Flood Depths The peak flood depth was defined by all participants as the peak depth at every cell regardless of the time at which it occurred. The maximum flooded area and flooded area categorized in half-meter increments of peak depths are summarized in Figure 3 and Figure 4. The total flooded area ranges from near 47 km² to just under 30 km². The Kappa statistic was applied to the gridded peak flow depths. A Kappa value was calculated for each unique combination of two models. The results of the peak flood depth Kappa analysis are summarized in Table 3. There is generally strong agreement between the models. The chance and moderate differences between MCV and SAB and other models is likely due to interpolation of the irregular mesh output from those submissions to a new mesh for comparison with other models.
Figure 3: Summary of total flooded area
Figure 4: Flood area summarized by half-meter increments of peak depth
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Table 3: Kappa statistic (model similarity) for peak flood depth HRW HRW
LANL
MANC
MMC
TK
MUST
MCV
SAB
Substantial
Substantial
Substantial
Substantial
Substantial
Moderate
Chance
Substantial
Substantial
Substantial
Substantial
Moderate
Chance
Substantial
Substantial
Substantial
Moderate
Chance
Substantial
Substantial
Moderate
Chance
Substantial
Moderate
Chance
Moderate
Chance
LANL
Substantial
MANC
Substantial
Substantial
MMC
Substantial
Substantial
Substantial
Substantial
Substantial
Substantial
Substantial
Substantial
Substantial
Substantial
Substantial
Substantial
Moderate
Moderate
Moderate
Moderate
Moderate
Moderate
Chance
Chance
Chance
Chance
Chance
Chance
TK MUST Chance
MCV Chance
SAB
Flood Wave Arrival Time There were differences among participants in defining thresholds for determining when the flood wave arrived. For example, LANL and MUST defined the flood wave arrival as the time at which a grid cell becomes wet without implying a threshold. MMC defined the flood wave arrival as the time in which the depth of water in a grid cell reaches 0.6 meters. Agreement between participants was assessed using the Kappa Statistic and are shown in Table 4. The results indicate that the level of agreement between any two models varies more for the flood wave arrival time than for the peak flood depths. It is noted again that HRW used a breach hydrograph with a much longer lag time to peak discharge than other participants. Table 4: Kappa statistic (model agreement) for flood wave arrival time HRW HRW
LANL
MANC
MMC
TK
MUST
MCV
SAB
Chance
Chance
Chance
Chance
Chance
Chance
Chance
Slight
Substantial
Slight
Fair
Substantial
Moderate
Slight
Chance
Fair
Slight
Fair
Slight
Moderate
Substantial
Substantial
Slight
Slight
Slight
Fair
Substantial
LANL
Chance
MANC
Chance
Slight
MMC
Chance
Substantial
Slight
TK
Chance
Slight
Chance
Slight
MUST
Chance
Fair
Fair
Moderate
Slight
MCV
Chance
Substantial
Slight
Substantial
Slight
Fair
SAB
Chance
Moderate
Fair
Substantial
Slight
Substantial
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Peak Unit Flow Rate Table 5 shows the level of agreement between participants for Peak Unit Flow Rate as determined by the Kappa analysis. The results show less agreement between two models for the peak unit flow rate than for peak flood depth and flood wave arrival. Table 5: Kappa statistic (model agreement) for peak unit flow rate HRW HRW LANL MANC MMC TK MUST MCV SAB
Fair Moderate Slight Slight Substantial Fair Fair
LANL
MANC
MMC
TK
MUST
MCV
SAB
Fair
Moderate Fair
Slight Substantial Fair
Slight Chance Slight Chance
Substantial Fair Moderate Fair Slight
Fair Slight Slight Slight Slight Fair
Fair Slight Slight Slight Chance Fair Slight
Fair Substantial Chance Fair Slight Slight
Fair Slight Moderate Slight Slight
Chance Fair Slight Slight
Slight Slight Chance
Fair Fair
Slight
Consequences Population The spatial distribution of population within the flooded area varied among the participants. MCV, MANC, and MUST uniformly distributed the population available in the census blocks to properly sum the affected population within partially flooded census blocks. HRW, TK, and MMC redistributed the population to the parcel data provided. MMC additionally accounted for residential population and workforce population within the parcels. Finally, LANL used imperviousness defined by developed areas in the NLCD to distribute the population from each census block. Population at Risk Population results include total PAR and PAR by age (e.g., under 14 and over 65 years). These are summarized in Figure 5 and Figure 6.
Figure 5: Summary of total, under 14-yr, and over 65-yr PAR
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Despite the wide range of peak discharges and subsequent flooded areas, the total sum of PAR is fairly consistent among participants. In addition, the participants consistently indicate that the majority of the PAR is located in flood water less than 2 meters deep. While there is consistency in total population by depth, PAR is less consistent with respect to flood arrival time. This indicates that PAR is perhaps sensitive to breach formulation. For example, HRW had nearly a 16-hour lag time for peak discharge. Therefore, the PAR in their solution is not generally at risk until more than 3 hours. This information is likely to have an impact on loss of life models that take into account warning time and evacuation plans.
Figure 6: PAR by peak depth Flood Severity Figure 1-7 shows the standard deviation among the different flood severity calculations.
Figure 7: Flood severity standard deviation There are significant differences on how the flood severity is defined across the participants. The methods used by TK and MMC were not described. HRW defined flood severity based on effects to population who are exposed (i.e., not within buildings) to the flood waters. LANL, MUST, and MANC used the U.S. Department of Homeland Security’s Dam Sector
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consequence estimation guidelines.2 These classifications are based more on structural stability and population within buildings than people directly exposed to the flood. There is complete agreement between the models on the severity assigned within the canyon (standard deviation of 0). Differences in severity classification are noted within the floodplain area, which the area where population in this case study is located. Loss of Life The total loss of life is shown in Figure 8. Although the method in which the loss of life was estimated varied between participants, the majority of estimates for loss of life was approximately 2,000 people. HRW’s loss of life estimate was the same order of magnitude as the other teams, but nearly twice as great. An explanatory factor in this difference could be the different technique used to calculate flood severity.
Figure 8: Total loss of life Economics There were differences in definition of direct economic impact. For example, LANL defined the direct economic impact as lost jobs, wages, and business outputs. LANL was the only participant to use business gross domestic product (GDP) per employee, but they did not include structural damage, which they defined as an insurable loss and not a direct economic impact. As such, LANL economic results are reported on a $/day basis, but did not estimate the duration of the impact. MAN, MMC, TK, and HRW all evaluated structural damages using the provided parcel information. The key challenge using the parcel data was assigning values to the occupancy classes. Because this information was not provided by the formulation team, each participant had to assume this information. The data used by participants ranged from available data in models, economic subject matter experts, and values reported in the literature. Figure 9 shows the total economic impact. Because of the difference in methods (e.g., GDP vs. insurable losses), and the assumption regarding asset value, the economic impact values range significantly. LANL reported $665,000/day, but the duration of the event is unknown. At the high end, MANC reported $2.6 Billion based on insurable losses.
2
Department of Homeland Security, 2011. “Estimating loss of life for dam failure scenarios.”
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Figure 9: Summary economic impact submitted by the participants
Conclusion In general, the results from each participant team are similar in terms of hydrodynamics. However, flood wave arrival times were different between teams. This is attributable to differences in the calculation of the breach hydrograph and differences in regression and physics-based formulations in addition to definition of thresholds used to quantify the arrival of the flood. The largest differences in peak flood depths were likely due to the requirement for the teams using irregular meshes to report output in a structured grid format. PAR estimates were also similar across teams. Differences are primarily due to different methods used by participants to spatially distribute population. The methods included uniform distribution of the population within the census block, uniform distribution of the population among parcels within the census blocks, and uniform distribution of the population among develop land within the census block. The most significant differences in the population estimates appeared to be the time after failure at which population is at risk. This has significance relative to flood wave warning time and potential evacuations. Finally, there were some significant differences in the economic impact. First, there were differences in interpretation of direct economic impact. This was either interpreted as business losses based on GDP or structural damage/insurable losses using parcel data. In addition, within the insurable loss estimation, there were significant differences in assessing the value of structures within parcels.
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PAPERS THEME C
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A Benchmark study on dam breach and consequence estimation using EMBREA and Life Safety Model M. Davison1, M. Hassan1, O. Gimeno1, M. van Damme1 and C. Goff1 1
HR Wallingford, Howbery Park, Wallingford, Oxfordshire, OX10 8BA, United Kingdom E-mail: [email protected]
Abstract This paper presents the modelled consequences of a hypothetical dam breach as laid out in Theme C of the 12th international Benchmark Workshop on Numerical Analysis of Dams. The EMBREA model was used to model the failure of the dam due to headcut erosion and to derive a breach hydrograph which was then used in InfoWorks ICM to model the 2D flood spreading. The results of the flood model have been used to calculate how severe the flood would be in terms of the total population at risk, loss of life using the Life Safety Model and economic impact. The paper shows that the consequences of the hypothetical flood will be severe in terms of casualties and economic damage. Introduction This paper presents the modelling of a potential breach of a hypothetical dam and the estimation of the consequences in the populated areas and economic activities downstream, within Theme C of the 12th International Benchmark Workshop on Numerical Analysis of Dams [1]. The paper is structured in three sections. Section 1 describes the dam breach model, which provides indicative predictions of a number of breach scenarios; Section 2 focuses on the flood wave propagation and Section 3 presents the human consequences and direct economic impact of the dam failure.
Dam Breach Modelling Choice of Breach Model During the last 15 years, HR Wallingford has undertaken the development of a new breach model, called EMBREA (previously known as HR BREACH). The IMPACT project [2] in 2005, Peeters et al. [3] in 2011, and Morris et al [4] in 2012 have demonstrated that EMBREA performs the best among the available breach models. On that basis, EMBREA was used to undertake the modeling for this benchmark work. Selection of Failure Modes EMBREA can simulate breach failure by overtopping and piping. For this study overtopping is considered to be the main dam failure mode, but piping was also investigated in one of the scenarios. Overtopping failure results in breach formation either due to headcut erosion or surface erosion. With headcut formation the erosion of material forms steps in the downstream face of the embankment which progressively grow in size and cut through the embankment. When the retreating headcuts reach the upstream side of the crest, the hydraulic control rapidly reduces in height and rapid failure ensues. With surface erosion, material is eroded from the embankment face and the crest area more smoothly leading to a reduced cross sectional embankment profile.
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The dominant process depends mainly on the material used in the dam construction. The hypothetical dam in this instance is a rolled earth fill structure composed of predominantly sandy clays and clayey sands with a relatively low permeability (1.9x10-6 cm/s). Based on the type of soil and the permeability provided, and using Shevnin et al. [5], its clay content is estimated to be more than 25%. In addition to the clay content, the matric suction in unsaturated soils adds to the cohesive properties of soils. The low permeability of the embankment soil indicates the high likelihood of the presence of an added cohesion due to matric suction throughout the erosion process. The overall cohesive properties make headcut erosion the most likely erosion mode during overtopping failure. The headcut erosion process is modeled in EMBREA using an erodibility coefficient (Kd), and a headcut migration coefficient (C). These parameters are usually measured on site or in a soil laboratory, but as the dam on this study is hypothetical, they were estimated based upon the soil clay content and compaction effort. To assess their uncertainty, the Monte Carlo analysis feature in EMBREA was used as described in below. A number of modeling runs were undertaken to establish the potential worst case scenarios for the hypothetical dam failure. Within these runs, consideration was given to the uncertainty and sensitivity of results to various aspects such as material erodibility, failure mode and erosion processes, as follows: Runs 1, 2 and 3. In these runs EMBREA was used to perform a Monte Carlo analysis to the overtopping failure with headcut (1000 runs with a triangular probability distribution for Kd and corresponding C values as given in Table 4). The output of the Monte Carlo run analysis was a frequency distribution of the peak breach outflow. The median value of this distribution was taken as the base run (Run 1). The corresponding values of the 75 % (Run 2) and 25 % (Run 3) exceedance probabilities were taken as the lower and upper limits respectively. Run 4. In this run, the erosion process was changed to overtopping through surface erosion rather than headcut to assess the impact of a different erosion process on the breach hydrograph. Other inputs were identical to Run 1. Run 5: In this run, a good grass protection is assumed to be present on the downstream face of the embankment to assess its impact on delaying the breach initiation and hence the wave arrival time at downstream locations. Other inputs were identical to Run 1. Run 6: In this run, the failure mode was changed to piping rather than overtopping to assess the impact of a different failure mode on the breach hydrograph. It should be noted that surface erosion is used in the model once the top of the pipe becomes unstable and fails (i.e. failure mode changes from piping to overtopping). Other inputs were identical to Run 1. Model set up This section provides a description of the model set up, including modeling boundary conditions, initial conditions, dam geometry and soil properties. The selection of such data was mainly based upon the data provided to participants in [1]. Due to the size of the reservoir, any inflow into the reservoir during the breach event has a negligible effect on the reservoir water levels, and has therefore not been included. Based on investigation of the immediate reach downstream of the dam, a low tailwater level was assumed meaning that the breach flow was not ‘drowned’. The dam geometry data is given in Table 1. Table 2 shows
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the initial conditions used in each series of runs. The dam failure for this benchmark takes place when the pool elevation is at crest elevation. Table 3 shows soil properties used in each series of runs and Table 4 shows Kd and C values for each run. The Kd values and C values were obtained from the qualitative relationships for different soil types provided in the EMBREA user manual version 1.3. Table 1: Dam geometry Crest level 272 mAD
Foundation level 211 mAD
Crest Width
Crest length
24m
360m
Upstream slope 1:3
Downstream slope 1:3
Initial breach depth (m) 0.25 NA
Initial breach width (m) 0.5 NA
Table 2: Initial conditions Run No. 1-5 6
Pipe diameter (m) NA 0.03
Pipe Level (mAD) NA 212
Initial water level (mAD) 272 272
Table 3: Soil properties Porosity Unit Wt (KN/m3) Manning’s n
0.40 19.64 0.025
Friction Angle (degrees) Cohesion (KN/m2) Plasticity Index
14 19.15 10
Table 4: Kd and C values Run No. Monte Carlo 1, 5 and 6 2 3 4
Erodibility coefficient, Kd (cm3/N-s) Triangular probability distribution with a lower value of 1, mid value of 5 and an upper value of 10 5.5 4.3 6.8 5.5
Headcut migration coefficient, C (s-2/3) Values calculated from Kd values based on EMBREA user manual version 1.3 0.0032 0.0026 0.0040 NA
Modelling results, observations and conclusions Figure 1 shows a comparison of the breach outflow results for Runs 1 to 6. Runs 1, 2 and 3 hydrographs show that Kd and C values have a significant effect on the breach peak outflow and initiation time in the case of overtopping failure due to headcut formation. Increasing them by a factor of 1.3 (Run 1) and 1.6 (Run 3) has increased the breach peak outflow by about 40 and 100% compared to run 2 respectively. Run 4 shows that changing the type of erosion process results in a significant reduction in the peak breach outflow and initiation time. The peak outflow was reduced by about 78% compared to Run 1 with less than one hour of initiation time. Run 5 shows that for the case considered, a grass protection increases the initiation time of the breach with a negligible change in the peak breach outflow compared to Run 1. Run 6 shows that changing of the failure mode on breach formation to piping
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significantly reduces the peak breach outflow and initiation time. The peak outflow was reduced by about 30 % compared to Run 1, with the shortest initiation time of all the runs.
Figure 1: Breach outflow hydrograph for Runs 1 to 6
Flood Modelling Data provided The data provided for the flood model to participants was: digital elevation model (DEM) representing pre-dam and post-dam construction gridded dataset of land use/cover The horizontal resolution of both datasets is about 9.5m. The land use/cover data follows the classification guidance and values provided in the National Land Cover Dataset (NLCD), 2006. Roughness estimation Surface roughness values for the flood simulation were estimated for each land use/cover class using the Conveyance and Afflux Estimation System (CES/AES) software [6]. The tool provides a database of roughness values extracted from various sources in the literature and given as mid, upper and lower estimates covering the range of roughness values expected within each system. This provides some measure of the uncertainty associated with the estimation of roughness by deriving an upper and a lower roughness credible scenario. The values assigned to each land use/cover class are shown in Table 5, as well as the total areas covered in the test bed region. For the developed areas, the low and high roughness values correspond to the lowest and highest values derived from the possible share between pervious/impervious areas given in the descriptions.
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Table 5: Land use/cover estimated roughness Class
ID Description
Water
11 Open Water Developed, 21 Open Space Developed, 22 Low Intensity Developed, 23 Medium Intensity Developed 24 High Intensity Barren Land 31 (Rock/Sand/Clay) 41 Deciduous Forest 42 Evergreen Forest 43 Mixed Forest 52 Shrub/Scrub Grassland/ 71 Herbaceous
Developed
Barren Forest Herbaceous Planted/ Cultivated Wetlands
Pasture/Hay 81 90 Woody Wetlands
Winter Summer Mid Low High Mid Low High 16.9% 0.010 0.010 0.010 0.010 0.010 0.010 Area
5.2%
0.029 0.025 0.034 0.029 0.025 0.034
10.7% 0.030 0.026 0.035 0.030 0.026 0.035 5.1%
0.031 0.026 0.036 0.031 0.026 0.036
0.5%
0.032 0.025 0.036 0.032 0.025 0.036
0.2% 28.4% 7.9% 0.1% 20.9%
0.022 0.073 0.251 0.142 0.073
3.5%
0.046 0.027 0.083 0.046 0.027 0.083
0.4% 0.2%
0.046 0.027 0.083 0.046 0.027 0.083 0.054 0.039 0.064 0.054 0.039 0.064
0.017 0.049 0.151 0.091 0.048
0.028 0.112 0.341 0.205 0.112
0.022 0.102 0.251 0.142 0.102
0.017 0.072 0.151 0.091 0.072
0.028 0.162 0.341 0.205 0.162
Flood model The 2D Infoworks ICM software was used to simulate spreading of the flood given by the breach hydrograph into the valley downstream of the dam. Three roughness scenarios were considered that correspond with the mid, low and high roughness values for each land use class. For each roughness scenario, the model was run for each of the six dam breach hydrographs from the breach model. The hydrographs were used as an inflow boundary condition for the flood model. This was represented using the breach width and depth corresponding to the maximum breach outflow in combination with the post-dam construction digital elevation model. Results The results of the flood model presented in this section correspond to the overtopping failure with headcut failure mechanism only, as this is the most likely failure mode for the dam. The results for other failure modes can be provided upon request. Figure 3 shows the attenuation of the breach hydrograph of the median scenario (Run 1) between the five cross sections on Figure 2. As shown, the peak flow is reduced by 10% when the flood arrives at cross section 5, which takes about 14 min from the moment of the peak breach discharge. In the scenario with 75% exceedance probability (Run 2), the peak flow attenuation is of a similar amount (9%), but the wave takes 10 min to reach cross section 5. Finally, for the scenario with 25% exceedance probability (Run 3), the peak flow hardly decreases (see Table 6). The flood extents for the three scenarios are very similar, with the flood spreading to the valley on the north between sections 3 and 4 for the median and 75% scenarios (see Figure 2). 245
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The peak flood depths for the 25%, the Median, and 75% exceedance probability scenarios are given in Table 7. Flood depths are mostly below 2m for all three scenarios, and below 4m for more than 90% of the flooded area. The spatial variability of the peak flood depths is shown in Figure 4 at 0.5m intervals for the median scenario. The higher flood depths in the city occur between cross sections 3 and 5 and also along the south end of the flood extent downstream section 5. That is also the main flowpath through the plain, with unit flow rates mainly between 4 and 10 m2/s (Figure 5), increasing to 30m2/s near section 5. The area of the city with higher unit flow rates –up to 80m2/s- is between cross sections 4 and 5. The arrival time of the first inundation relative to the time at which a breach flow initiates is shown in Figure 6. The flood reaches the populated area after 1.5h to 2h, but the majority of the city starts flooding after 2h. That is the arrival time of the small flow at the beginning of the dam breach initiation. However, the peak flow travel time in Figure 7 shows that the flood wave peak travels from the dam to the city in less than 15min, reaching the further edge in less than 90min, and hence the peak flow travel time is the travel time to consider.
Figure 2: Cross-section locations and flood extent
Figure 3: Cross section discharges median run
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Table 6: Cross section discharges Breach Cross Section 5 3 Scenario Peak flow (m /s) Time Peak flow (m3/s) Time Median 13,956 18:51 12,540 (-10.1%) 19:05 (14min) 75% Exc. Prob. 19,257 15:05 17,537 (-8.9%) 15:15 (10min) 25% Exc. Prob. 9,908 23:51 9,877 (-0.3%) 24:05 (14min)
Figure 4: Peak flood depths (m) for the median scenario Table 7: Flooded area (m2) Peak Flood Depth Range (m) 0 - 0.5 0.5 - 1 1 - 1.5 1.5 - 2 2 - 2.5 2.5 - 3 3 - 3.5 3.5 - 4 4 - 4.5 4.5 - 5 5 - 5.5 5.5 - 6 6 - 6.5 6.5 - 7 7 - 7.5 7.5 - 8 >8 TOTAL
Flooded area (m²)
25% exceedance probability 7,684,136 9,977,083 5,110,216 3,012,575 2,115,959 1,231,376 515,700 355,773 186,327 136,850 143,135 95,004 61,959 57,559 61,421 58,368 522,524 31,325,966
Median 3,842,921 8,084,897 6,912,247 3,611,785 2,523,993 2,305,789 1,875,305 1,185,490 683,978 463,618 354,426 222,336 151,037 131,372 113,772 86,025 900,657 33,449,648 247
75% exceedance probability 3,776,472 8,525,347 7,732,446 4,025,567 2,710,321 1,967,706 1,380,169 696,280 424,916 386,304 256,907 162,801 150,050 140,801 82,433 71,119 973,392 33,463,027
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Figure 5: Peak unit flow rate in m2/s for the median scenario
Figure 6: Inundation arrival time from initiation time for the median scenario
Figure 7: Peak flow travel time
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Consequence Estimation Flood severity Flood Severity has been classified according to the Defra methodology presented in technical reports [7, 8]. The method describes the flood hazard as a function of velocity, depth and the presence of debris as (1) Where, HR is the (flood) Hazard Rating, d is the Depth of flooding (m), v is the velocity of the floodwater in (m/s), and DF is the debris factor (assumed to be 0.5 for depths <0.25m, and 1.0 for depths >0.25m). Four flood hazard classes were established based on the thresholds of flood hazard rates (HR) that cause people to lose stability. Thresholds are derived from the experiments on people with different height multiplied by mass values. For this study, a 3 class categorization (low, medium and high) was required. Therefore the thresholds have been redefined as in Table 8. Figure 8 shows that the flood severity is mostly high in the flooded area. Table 8: Flood Severity classification ID
Class
1
Low
Flood Hazard Rate (HR) < 0.75
2
Medium
0.75 - 1.25
3
Significant >1.25 to high
Description Caution. Flood zone with shallow flowing water or deep standing water” Dangerous for some (i.e. children, the elderly and the infirm). Danger: Flood zone with deep or fast flowing water Dangerous for most people to all people. Danger to extreme danger: flood zone with deep fast flowing water
Figure 8: Flood severity
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People at risk Given the reporting requirements of the people at risk by time intervals and flood depths, the calculation of the number of people at risk was based on the flood extent from the hydraulic model and the population from the census data and the buildings. The average number of people in a building was calculated by dividing the total population by the number of buildings for each census zone. To calculate the number of people at risk from flooding the number of buildings in the maximum flood extent was determined from GIS and the population summed. Table 9 shows the total number of people at risk per age group and flood depth. Most of the people are at areas with flood depth below 1.5m. Table 9: Number of people at risk per age group and flood depth Peak Flood Depth Range (m) 0 - 0.5 0.5 - 1 1 - 1.5 1.5 - 2 2 - 2.5 2.5 - 3 3 - 3.5 3.5 - 4 4 - 4.5 4.5 - 5 5 - 5.5 5.5 - 6 6 - 6.5 6.5 - 7 7 - 7.5 7.5 - 8 >8 Total
Total Population At Risk 4,608 10,189 5,336 1,209 710 621 497 369 283 252 211 182 123 91 52 43 128 24,904
12-yr and Under Population at Risk 693 1,528 763 146 113 118 106 93 71 63 55 47 33 23 13 11 37 3,913
65-yr and Over Population at Risk 841 1,878 1,139 326 163 111 72 28 21 21 17 13 7 8 4 3 6 4,658
Loss of life As with the number of people at risk from flooding, the calculation of loss of life uses the same population distribution. There are a number of methods to calculate loss of life. Brown and Graham [9] is a very simple method where the population at risk is multiplied by a factor depending on the length of time between the breach and flood warning. DeKay and McClelland [10] use a simple method that uses different equations for high and low force flooding. Graham [11] expands on the simple methods of Brown and Graham [9] and DeKay and McClelland [10] using the concept of flood severity, flood warning and the understanding of the exposed population to vary the fatality rate. The UK Flood Risk to People methodology [7] calculates the loss of life based on the hazard rating, the type of people exposed and the sensitivity of the area. This is similar to the method of Graham [11] used in the US, although it is more advanced because it uses the magnitude of the flood expressed by the spatially variable hazard rating to define fatality rate rather than a global fatality rate. The sensitivity of the area is defined by the time of travel of the flood wave, type of housing, and the presence of flood warning and evacuation plans. This calculation is most appropriate for large areas for 250
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example census zones. Finally, the Life Safety Model [12] is an agent based model that determines the fate of each person that is exposed to the flood in a dynamic way through time and space. The model is designed for improving evacuation plans. The loss of life function in the model is based on the water depth and velocity. This function can be used in a static way to calculate loss of life from the maximum DV (Depth x Velocity). After consideration of all the above methodologies, the approach taken was to primarily use the loss of life function from the Life Safety Model (LSM) to provide an estimate of loss of life based on people distributed in buildings. This has been chosen ahead of the UK Risk to People (UK R2P) method because of the requirement to produce gridded output at a high resolution. The Theme C2 information [13] states that the dam fails at 11pm on Saturday night with no witnesses that are able to raise an alarm, and that there is no evacuation plan in place. It has therefore been assumed that the total population are in their homes and receive no warning for the loss of life calculations. The loss of life estimated for this pilot from the Life Safety Model approach is 4,966. That is based on conservative assumptions that a) people are on the ground, and b) take no measures to avoid the flood. The spatial variability of fatalities in Figure 9 show that the area between cross sections 3 and 5 is the most affected. All fatalities are after 3h of the breach initiation. An alternative scenario where people shelter in buildings has been performed with the Life Safety Model where loss of life occurs if the building is submerged or collapsed. Assuming that people shelter in buildings the number of fatalities is estimated to be 3,075. A more realistic estimate can be produced assuming that people take refuge in multi-storey buildings or respond to a warning and evacuate the area, for which the dynamic Life Safety Model would be appropriate. A comparison of the methods is shown in Table 10. The UK R2P method was used as a secondary check of LLOL. It is an area wide calculation which uses a simple vulnerability measure multiplied by the Hazard Rating to define the number of people at risk. The number of injuries is then estimated as 2 times the number of old and sick people as a proportion of the number at risk. The fatality rate is 2 times the Hazard Rating, and the number of fatalities is the number of injured multiplied by the fatality rate. This produces a significantly lower estimate of loss of life than the LSM because it assumes that young and healthy people can escape the flood. In this case the lack of flood warning means that the whole population should be considered at risk. The calculation has been modified so that the whole population is exposed producing an estimate of loss of life of 4,134. Table 10: Loss of life Estimation method Loss of life
Life Safety Model approach 1 4,966
Life Safety Model approach 2 (people shelter in buildings) 3,075
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UK Risk to people methodology 1,150
Modified UK Risk to people methodology 4,134
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Figure 9: Loss of life from Life Safety Model, cell size ~9.5m x 9.5m Economic damage Depth damage curves have been taken from the MCM [14] 2010 update for residential and non-residential properties that are similar to the description given in the building data provided. These curves were developed for UK properties. The damages have been converted to US$ using an exchange rate of 1.51, but no other adjustments have been made to account for differences in US and UK properties. Alternatively there is a US method that relates the damage as a percentage of the building value for given water depths [15], but this method could not be used because building values were not provided in the building data. The total economic damage for the base dam breach has been estimated as US$1,237,582,738. The spatial variability of the economic impact is shown in Figure 10, the commercial buildings having the higher damage per grid cell (~89.8m2).
Figure 10: Direct economic impact Observations It would have been preferable to distribute the number of people across the residential buildings based on knowledge of household groupings. However, the census data provided was not detailed enough. In addition the building data provided contained the overall footprint rather than the number of individual properties (in the case of flats). This meant that some census zones had large population with little or no residential buildings. For example census zone 30 has 3 large properties classified as commercial but a population of 565. To reduce 252
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uncertainty in the distribution of people within a census area, more detail is required on the demographics and number and type of buildings. The LSM can also model the behaviour of people in a flood situation as they seek to avoid the flood, thereby reducing the predicted loss of life.
Conclusions The paper presents the consequences of the breach of the hypothetical dam as described in Theme C of the 12th International Benchmark Workshop on Numerical Analysis of Dams. The dam breach analysis undertaken using EMBREA highlights the importance of choosing the correct erosion method and the impact of choosing the correct erodibility of the soil. Based on the data provided, it was estimated that the embankment fails according to headcut erosion leading to a steep hydrograph with a high peak discharge, the consequences of which are more severe than when an embankment fails due to surface erosion or piping. The effects of the different roughness scenarios considered on the flood model was negligible compared to the impact of the different hydrographs used as input, emphasizing again the need for a good dam breach model. The flood model results provide useful information for warning and evacuation planning. The model indicates that the first flow from the dam breach reaches the populated area 1.5h to 2h after the breach initiates, but the peak flow travels from the dam to the city in less than 15min, reaching the further edge in less than 90min. The total direct economic impact is estimated at more than US$1.24b. The loss of life calculations performed with the static Life Safety Model show a loss of life of 4,966, which represents nearly the 20% of the total people at risk.
References [1] 12th International Benchmark workshop on numerical analysis of dams, "THEME C: Computational Challenges in Consequence Estimation for Risk Assessment: Numerical modelling, Uncertainty Quantification, and Communication of Results. Part 1- Hydraulic Modelling and Simulation," ed: International Commission on Large Dams (ICOLD), 2013. [2] M. W. Morris, M. A. A. M. Hassan, and K. A. Vaskinn, "Conclusions and recommendations from the IMPACT Project: WP2 Breach Formation," 2004. [3] P. Peeters, T. Van Hoestenberghe, L. Vincke, and P. Visser, "SWOT analysis of breach models for common dike failure mechanisms," presented at the 34th IAHR World Congress - Balance and Uncertainty, 33rd Hydrology & Water Resources Symposium, 10th Hydraulics Conference, Brisbane Australia, 2011. [4] M. W. Morris, M.A.A.M. Hassan, T.L. Wahl , R.D. Tejral, G.J. Hanson, and D.M. [5] Temple, "Evaluation and development of physically-based embankment breach models," presented at the 2nd European Conference on Flood Risk Management, Rotterdam, The Netherlands, 2012. [6] V. Shevnin, O. Delgado-Rodríguez, A. Mousatov, and A. Ryjov, "Estimation of hydraulic conductivity on clay content in soil determined from resistivity data," Geofisica Internacional, vol. 45, 2006. [7] Defra and Environment Agency, "Reducing uncertainty in river flood conveyance, phase 2, Conveyance manual," Environment Agency, report2004. [8] Defra and Environment Agency, "R&D outputs: Flood Risk to People Phase 2.
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[9] Guidance Document," 2006. [10] Defra and Environment Agency, "Framework and guidance for Assessing and Managing Flood Risk Development, Flood Risk Assessment Guidance for New Development," Technical report2005. [11] C. A. Brown and W. J. Graham, "Assessing the Threat to Life from Dam Failure," [12] Water Resources Bulletin, vol. 24, pp. 1303-1309, 1988. [13] M. L. DeKay and G. H. McClelland, "Predicting Loss of Life in Cases of Dam Failure and Flash Flood," Risk Analysis, vol. 13, pp. 193-205, 1993. [14] W. J. Graham, "A Procedure for Estimating Loss of Life Caused by Dam Failure," ed. [15] Denver, Colorado: U.S. Department of Interior, Bureau of Reclamation, Dam Safety [16] Office, 1999. [17] BC Hydro, "Dam Safety, BC Hydro Life Safety Model: Formal model description," ed: BC Hydro,, 2004. [18] I. B. w. o. n. a. o. dams, "THEME C: Computational Challenges in Consequence Estimation for Risk Assessment: Numerical modelling, Uncertainty Quantification, and Communication of Results. Part 2- Consequence Estimation," International Commission on Large Dams (ICOLD) ed, 2013. [19] E. Penning-Rowsell, C. Johnson, S. Tunstall, S. Tapsell, J. Morris, J. Chatterton, et al., "The benefits of Flood and Coastal Risk Management: A manual of assessment techniques," ed. Middlesex: Middlesex University Press., 2005. [20] C. Scawthorn, P. Flores, N. Blais, H. Seligson, E. Tate, S. Chang, et al., "HAZUS-MH flood loss estimation methodology. II. Damage and loss assessment," Natural Hazards Review, vol. 7, pp. 72-81, 2006.
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Uncertainty in Two-Dimensional Dam-Break Flood Modeling and Consequence Analysis M.S. Altinakar1, M.Z. McGrath1, V.P. Ramalingam1, D. Shen1, Y. Seda-Sanabria2 and E.E. Matheu3 1
National Center for Computational Hydroscience and Engineering, The University of Mississippi, Brevard Hall Room 327, P.O. Box 1848, University, MS 38677-1848, USA 2 Critical Infrastructure Protection and Resilience Program, Office of Homeland Security, U.S. Army Corps of Engineers, Headquarters, Washington, DC 20314, USA 3 Critical Lifelines Branch, Sector Outreach and Programs Division, Office of Infrastructure Protection, National Protection and Programs Directorate, Washington, DC 20528, USA E-mail: [email protected]
Abstract This paper presents the preliminary results of an extensive study to evaluate and quantify uncertainty in two-dimensional numerical dam-break flood modeling and consequence analysis based on a benchmark test case. The benchmark problem consists of a hypothetical 61m-high embankment dam in a mountainous region with lightly populated urban areas located downstream. Although the benchmark problem was intended mainly for estimating the uncertainty in the evaluation of the consequences of failure of a dam near populated areas with complex demographics, infrastructure and economic activity, the present study also investigated the uncertainty in two-dimensional numerical modeling based on three control variables. DSS-WISE™ numerical model was used to calculate dam-break flood and potential loss-of-life for a total of 120 cases, which represented 40 random pairs of breach width and breach formation time, each computed with three different sets of Manning’s coefficients defined based on land use/cover type. Computed results include raster maps of maximum flood depth, maximum flood discharge per unit width and flood arrival time as well as hydrographs at 7 cross sections. Analysis of population at risk (PAR) impacted by the flood and loss-of-life were also computed for all 120 simulations. The paper presents some preliminary results based on the statistical analysis of results.
Introduction Test case consists of synthetic data specially designed for the benchmark study. A hypothetical 61m-high embankment dam located in a mountainous area impounds a reservoir having a normal storage of 38 million cubic meters. A lightly populated urban area, called Hydropolis, is located about 3.5 km downstream from the dam. The benchmark test case data for hydrodynamic simulation consists of (1) digital elevation model (DEM) of the area of interest with and without the dam; (2) dam geometric characteristics; (3) reservoir stagevolume and stage surface area curves; (4) dam and foundation material; (5) dam failure scenario and conditions ate the time of failure; and (6) gridded data of land use/cover. In addition the following data is provided for consequence estimation: (1) shapefile of the census data for Hydropolis with breakdown into various age and gender classes; (2) shapefile for twenty economic activity classes; (3) shapefile for parcel data with 28 zone classes (residential, commercial, etc.). The complete dataset and supporting documents have been published on the conference website under Theme-C and will not be repeated here. It is
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assumed that the dam fails without warning when the water surface in the reservoir is at the crest elevation and the population downstream cannot be warned.
Two-Dimensional Dam-Break Flood Simulation Model Two dimensional simulations of dam-break flood were performed using the DSS-WISE™ software developed by the National Center for Computational Hydroscience and Engineering (NCCHE), The University of Mississippi. DSS-WISE™ is an integrated platform, which couples a state-of-the-art two dimensional numerical model with GIS-based pre- and postprocessors, that are developed as an extension to ArcGIS® software developed and commercialized by ESRI. Its numerical solver – CCHE2D-FLOOD – solves the conservative form of the two-dimensional shallow water equations, given as [
]
[
(1)
]
in which, is the vector of conserved variables, directions, and . These are defined as
[
]
[
]
[
and
are flux vectors in
] [
(
)
(
)]
and
(2)
where, is the flow depth, and components of unit discharge in and directions, the bed elevation, the net source/sink mass flux per cell area per unit time, and the gravitational acceleration. The slopes of energy grade line are given by Manning’s equation √
√
(3)
with as the Manning’s coefficient of roughness. Note that and correspond to the components of the unit discharge in and directions, respectively, i.e., and . Finite volume discretization of Equation (1) over a regular Cartesian mesh provides an explicit equation for advancing the values of the conserved variables in time: (
)(
)
(
)(
)
4
In Equation (4), and define the cell size in and directions, and the time step size. The intercell fluxes are computed using the first order Harten-Lax-van Leer-Contact (HLLC) approximate Riemann solver by Toro et al. (1994), which was implemented following the methodology described by Kim et al. (2007). The details can be found in Altinakar et al. (2013a), Altinakar (2012) and Altinakar and McGrath (2012). The DSS-WISE software and its solver CCHE2D-FLOOD have been fully verified using analytical solutions and validated using laboratory and field data. Blind tests have also been undertaken in collaboration with U.S. Army Corps of Engineers (Altinakar et al., 2012).
Data for Benchmark Test Case Figure 1 shows the digital elevation model without dam. Open water surface to the East of the area of interest was converted into “NoData”. The numerical model treats NoData cells as an
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internal free outflow boundary. This prevents flood propagating over the open water surface, which would otherwise be perceived as bed topography. The dam is represented in the DEM as an idealized structure representing a wall of constant thickness. The embankment dam has a crest width of 24m and the upstream and downstream slopes of 3H:1V. Considering that the dam has a height of 61m, the width of the idealized wall was chosen as 207m, which corresponds to the average width of the embankment dam. The overtopping failure was assumed to take place when the reservoir reaches crest elevation. Thus, the initial condition models the reservoir as a stagnant water body with surface elevation equal to crest elevation. Cross Section 6
Elevation (m) 1823.5
Breach Cross Section
67.2
Idealized Dam
Open water area removed by changing elevations to No Data
Figure 1: DEM without the dam (pre-construction) with open water area removed.
Selection of Test Cases Three parameters were chosen to investigate the uncertainty of unsteady hydrodynamic simulations: (1) bottom breach width, ; (2) breach formation time, ; and (3) the set of Manning’s coefficient of roughness, . Prediction of Breach Parameters Empirical equations have been developed to estimate breach parameters based on the analysis of the data from historic dam failures (Wahl, 2004). The present study uses the empirical equations given by Froehlich (1995) to compute the breach width and breach formation time (5) (6) where is the average breach width (m), i.e. the breach width at half of the breach height (m), is a constant equal to 1.4 for overtopping failures, and is the reservoir volume at the time of failure (m3). Equations (5) and (6) were obtained based on data from past failures of zoned earthen, earthen with a clay core, and rockfill embankment dams. The dam heights ranged from 3.66m to 92.6m (with 90%<30m and 76%<15m), the volume of water in the reservoir ranged from 0.0130×106 m3 to 660.0×106 m3 (with 87%<25.0×106 m3 and 76%<15.0×106 m3) and the lake surface areas ranged from 0.045km2 to 2,165km2. The data for the benchmark test case (61m, 38×106 m3, and 1.584 km2), which is well within the range of cases used to establish Equations (5) and (6). Thus, the average breach width and breach formation time are calculated as and ̅ , respectively. Considering the
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side slopes of 1.4H:1V [6], the bottom with and top width of the trapezoidal breach cross and , respectively. section are Uncertainty of prediction of breach parameters for embankment dams was investigated by Wahl (2004) [12]. It was shown that the prediction errors using empirical equations, including Froehlich (1995) [6], exhibit a lognormal probability distribution (base 10) defined by a mean value, , and a standard deviation, . Table 1 summarizes the uncertainty estimates of the breach parameters predicted using Froehlich (1995). Using the values given in Table 1, the 95% confidence interval for the breach bottom width extends from 25m to 146m whereas the 95% confidence interval for breach formation time extends from 0.3hr to 4.9hr. Table 1: Uncertainty Estimates for Froehlich (1995) Equations (taken from Wahl, 2004). Number of Cases (log Before outlier After outlier cycles) exclusion exclusion Breach width Eq. (5) 77 75 +0.01 Breach formation time Eq. (6) 34 33 -0.22 *95% confidence interval is given around a hypothetical predicted value of 1.0 Parameter
Equation
(log cycles) ±0.195 ±0.32
95% Confidence interval* 0.40 – 2.4 0.38 – 7.3
Manning’s Coefficients Manning’s coefficients were assigned based on the given land use/cover data with 16 size classes that follow the classification guidance and values provided in the U.S. National Land Cover Dataset (NLCD). The land use/cover classes for the area of interests are shown in Figure 2. Table 2 shows the three sets of Manning’s coefficient used in the present study. It should be noted that the area of interest does not have classes 12 and 95. Figure 3 provides a visual comparison of the three sets of Manning’s coefficients for each land use/cover class.
Figure 2: Map of land use/cover classifications. Latin Hypercube Sampling for Breach Parameters Latin Hypercube Sampling (LHS) method was used to select 40 random pairs of bottom breach width, and breach formation time, . LHS is a stratified-random method which provides an efficient way of sampling variables from their distributions (Iman et al. 1981). The cumulative distribution (normal distribution of logarithms) of both variables were divided into 40 intervals of equal probability. For each variable, a value is randomly selected from every interval. The selected values are paired randomly to achieve an optimal filling of the parameter-uncertainty space defined by the 40×40 matrix of and . Design of the LHS was accomplished using the “lhsdesign” function in Matlab® software using an option that 258
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maximizes the minimum distance between the pairings. Figure 4a shows the LHS design used in the present study. Since and have lognormal distributions (base 10), the probability values, , used in LHS were converted into real values using the following expressions: Table 2: Land Use/Cover Classes and Three Sets of Corresponding Manning’s Coefficients. ( ) DSSFEMA NOAA Min. WISE(1) HAZUS(2) C-CAP(3)) Value 11 Open Water 0.0330 0.0010 0.0250 0.0010 12 Perennial Ice/Snow 0.0100 0.0100 0.0100 0.0100 21 Developed-Open Space 0.0404 0.0200 0.0350 0.0200 22 Developed- Low Intensity 0.0678 0.0500 0.1200 0.0500 23 Developed- Med. Intensity 0.0678 0.1000 0.1200 0.0678 24 Developed- High Intensity 0.0404 0.1500 0.1200 0.0404 31 Barren Land 0.0113 0.0900 0.0300 0.0113 41 Deciduous Forest 0.1000 0.1000 0.1600 0.1000 42 Evergreen Forest 0.1000 0.1100 0.1800 0.1000 43 Mixed Forest 0.1200 0.1000 0.1700 0.1000 52 Shrub/Scrub 0.0400 0.0500 0.0800 0.0400 71 Grassland/Herbaceous 0.0400 0.0340 0.0350 0.0340 81 Pasture/Hay 0.0350 0.0330 0.0500 0.0330 82 Cultivated Cropland 0.0700 0.0370 0.1000 0.0370 90 Woody Wetlands 0.1500 0.1000 0.1500 0.1000 95 Herbaceous Wetlands 0.1825 0.0450 0.0500 0.0450 (1) Altinakar et al. (2013b); (2)Luettich and Westerink (2009); (3)ARCADIS (2011) NLCD Class
Description
Mannin's Coefficient, n (m-1/3s)
DSS-WISE
FEMA HAZUS
Average Value 0.0197 0.0100 0.0318 0.0793 0.0959 0.1035 0.0438 0.1200 0.1300 0.1300 0.0567 0.0363 0.0393 0.0690 0.1333 0.0925
Max. Value 0.0330 0.0100 0.0404 0.1200 0.1200 0.1500 0.0900 0.1600 0.1800 0.1700 0.0800 0.0400 0.0500 0.1000 0.1500 0.1825
NOAA C-CAP
0.2000 0.1800 0.1600 0.1400 0.1200 0.1000 0.0800 0.0600 0.0400 0.0200 0.0000
11 12 21 22 23 24 31 41 42 43 52 71 81 82 90 95
NLCD Class
Figure 3: Comparison of three sets of Manning’s coefficients assigned to NLCD classes. ̅̅̅̅
[
]
;
̅
[
]
(7)
In the above expression, stands for the inverse of the cumulative distribution function of standard normal distribution, with mean equal to zero and standard deviation equal to 1, at the probability value of . For each variable, i.e. or , its corresponding and values were used. The selected 40 pairs are provided in Table 3. In Figure 4a, the numbers for the data points correspond to combination numbers in Table 3. Figure 4b and Figure 4c show that the randomly selected of and values display a lognormal probability distribution, as expected. Assumptions Regarding the Progression of the Breach Geometry 259
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Histogram of Generated Wb Data and Lognormal PDF Histogram of Data and Lognormal PDF
Histogram of Generated W Data and Lognormal PDF Histogram of Data band Lognormal PDF
14
16
12
14
Number Observations of Observations Numberof
Number Numberof of Observations Observations
Froehlich (1995) expression, used for estimating breach characteristics, assumes that the final breach cross section for overtopping failures is a trapezoidal with side slopes of 1.4H:1V. Figure 5a shows the mean, minimum and maximum breach sizes selected by LHS. The elevations of the cells under the footprint of the idealized dam were initially raised to the level of the crest. As soon as the simulation starts, the breach occurs by gradually bringing down these cells to the original valley elevation during the breach formation time, , as shown in Figure 5b. The cell elevations are not permitted to go below the original bed elevation.
10
8
6
4
2
12 10 8 6 4 2
0 0
50
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200
0
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0
2
Breach Bottom Width, Wb, (m)
(a)
4 6 8 Breach Formation Time (hr)
10
12
Breach Formation Time, tf, (hr)
(b)
(c)
Figure 4: (a) LHS design obtained by maximizing minimum distance between data pairs; (b) Histogram of randomly selected breach widths and the fitted lognormal distribution; (c) Histogram of randomly selected breach formation times and the fitted lognormal distribution. Table 3: List of 40 pairs of breach width and breach formation time selected using LHS. Pair No 1 2 3 4 5 6 7 8 9 10
Pair No 11 12 13 14 15 16 17 18 19 20
(hr) 2.2 1.0 0.6 5.2 2.0 0.6 1.8 0.4 3.4 0.8
(m) 38.3 130.3 116.9 68.8 58.6 83.6 80.6 37.2 66.7 177.3
Pair No 21 22 23 24 25 26 27 28 29 30
(hr) 1.4 2.1 1.6 0.6 1.7 1.3 0.5 0.2 2.5 1.7
300
300
280
290
(m) 70.6 46.2 60.7 88.0 113.1 98.5 44.7 27.4 65.7 48.9
220 200 100
150
200
250
300 350 Chainage (m)
Cross section
Dam Crest
Minimum Breach
Maximum Breach
400
450
500
(m) 88.4 53.2 49.6 78.4 92.9 63.6 31.8 73.8 47.5 36.0
(hr) 0.3 1.4 0.7 2.6 0.5 0.8 1.1 0.9 4.9 0.7
t=0
270
240
Pair No 31 32 33 34 35 36 37 38 39 40
(hr) 0.4 2.8 0.9 1.0 1.2 3.0 1.1 0.7 1.5 1.3
280
260 Elevation (m)
Elevation (m)
(m) 41.9 51.8 103.5 55.1 57.0 59.0 30.6 20.0 74.4 40.1
t = tf / 4
260 250 240
t = 2tf / 4
230
t = 3tf / 4
220
t = tf
210
200
Mean Breach
100
150
200
250
300
350
400
450
500
Chainage (m)
(a)
(b)
Figure 5: (a) Breach geometry for mean, minimum and maximum breach bottom widths; and (b) Linear progression of the breach geometry assumed in the numerical model.
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Results of Hydrodynamic Simulations A total of 120 numerical simulations were performed by combining 40 randomly chosen pairs of breach width and breach formation time with three sets of Manning’s coefficients. Each simulation provided three raster files (maximum flow depth, , flood arrival time, , and maximum specific discharge, ), and discharge hydrographs at seven cross sections (dam crest, five pre-defined cross sections along the main downstream flow, and one cross section to capture the flows towards north) recorded with a time interval of two minutes.
Average Hmax (m)
St. Dev. of Hmax (m)
Minimum Value of Hmax (m)
Maximum Value of Hmax (m)
Figure 6: Statistics of maximum flood depth,
meter 0.25 0.50 0.75 1.0 2.0 3.0 4.0 5.0 8.0 12.0 17.0 22.0 27.0 32.0 37.0 42.0 47.0 52.0 57.0 62.0
(m), based on 120 simulations.
Maps of Maximum Depth, Flood Arrival Time and Maximum Discharge per Unit Width The set of 120 raster files were used to calculate maps of ensemble average, standard deviation, and minimum and maximum values of the maximum flood depth (Figure 6), the flood arrival time (Figure 7), which is defined as the time at which the dry computational cell becomes wet, regardless of the depth, and the maximum discharge (Figure 8). In these maps, cells with a null value are not displayed. As expected, the highest flow depths and unit discharges occur in the narrow valley immediately downstream of the dam. Comparison of minimum and maximum value maps of these three variables indicate that large differences exist between individual runs. The largest standard deviations of maximum flood depth and maximum unit discharge occur in the narrow valley downstream of the dam whereas the largest standard deviation values for the arrival time are observed farthest from the dam. The area on the northern part is inundated only for certain simulations. The longest time arrival and largest standard deviations of arrival time are observed in this area (Figure 7). An understanding of the sensitivity of results to different parameters can be gained by analyzing and comparing results files of individual simulations. Figure 9 shows a comparison of the flood maps of selected pairs of breach width and breach formation time with all three sets of Manning’s coefficients. The upper right hand corner shows a plot of all 40 random pairs selected with LHS in terms of real values of and . This should be compared with the plot in probability space shown in Figure 4a. As shown, there are no simulations in the upper right corner of the plot corresponding to large breach widths with long formation times. The inspection of Figure 9 shows that the area on the north part of the DEM is inundated only for certain combinations of parameters. In space, the simulations with inundation of 261
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north part correspond to small breach formation times and large breach widths. In addition, higher Manning’s values favor inundation of the north part of the DEM. The main land use/cover types in the narrow valley are pasture/hay, woody wetlands, and evergreen forest. The n-values from NOAA C-CAP for these land use/cover types are the highest resulting in higher flow depths and, thereby spilling of the flood into northern region of the DEM for pair numbers 3, 6. 8, 18, and 31. hours
Average tarr (hr)
St. Dev. of tarr (hr)
Minimum Value of tarr (hr)
Maximum Value of tarr (hr)
Figure 7: Statistics of flood arrival time,
0.25 0.50 0.75 1 2 3 4 5 6 7 8 9 10 11 12 16 20 24 30 34
(hr), based on 120 simulations. m2/s
Average qmax (m)
St. Dev. of qmax (m)
Minimum Value of qmax (m)
Maximum Value of qmax (m)
Figure 8: Statistics of maximum discharge per unit width, simulations.
3 4 4.65 5 6 8 10 14.86 25 50 100 150 200 250 300 350 400 450 500 562
(m2/s), based on 120
Hydrographs Discharge hydrographs were recorded at seven cross sections. These include the breach cross section and five cross sections specified in the benchmark test case formulation. An additional cross section was also introduced in order to measure the flood discharge propagating to the north of the DEM under certain combinations of control parameters (see Figure 9). Due to the
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lack of space, only limited number of results will are presented in this paper. Additional details will be provided in subsequent publications. Figure 10 shows the discharge hydrographs at the breach cross section (left plot) and the relationship between peak discharge and the control parameters (middle plot) and (right plot). Runs with different n-values are distinguished with different symbols/colors. Since at the breach cross section the influence of the Manning’s coefficient is not fully felt, for a given and pair, the hydrographs obtained with different n-values coincide fairly well. The peak discharge does not seem to be highly correlated with the breach width (middle plot) but rather with the breach formation time (right plot). The peak discharge is inversely proportional with the breach formation time and all the data collapses onto a single curve. A similar plot is shown for cross section 5 in Figure 11. The peak discharges of hydrographs are attenuated. For a given and pair, the hydrographs computed with different sets of nvalues have different arrival times. The peak discharges are not correlated with the breach width (middle plot) but they still show a relatively good correlation with the breach formation time (right plot) despite the fact that the differences in the peak discharge for simulations with different n-values are larger for and pairs. DSS-WISE™ n-values
FEMA HAZUS n-values
NOAA C-CAP n-values
Figure 9: Comparison of flood depth maps for selected simulations.
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4 3.5 3 2.5 2 1.5 1 0.5 0
0
0.2
0.4
0.6
0.8
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1.2
1.4
1.6
Breach Cross Section
4
Max. cross section discharge (m3/s)
Cross section discharge (m3/s)
x 10
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time (s)
2
4.5
x 10
4 3.5 3 2.5 2 1.5 1 0.5 0 20
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4
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Max. cross section discharge (m3/s)
Breach Cross Section
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x 10
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0
1
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4
x 10
2
3
4
5
6
Breach Formation Time (s)
Figure 10: Discharge hydrographs and the peak discharge at the breach cross section (DSSWISE: black squares, FEMA HAZUS: red “plus” signs, and NOAA C-CAP: blue crosses). 3
2.5
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x 10
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Figure 11: Discharge hydrographs and the peak discharge at cross section 5 (DSS-WISE: black squares, FEMA HAZUS: red “plus” signs, and NOAA C-CAP: blue crosses).
Consequence Analysis In the present study the consequence analyses were restricted to the analysis of population at risk (by age group) to be affected by flood based on flood arrival time and peak discharges per unit width. Direct or indirect economic analyses were not performed. Population by Census Block and PAR Distribution Census block polygons for Hydropolis, colored by total population, are shown in Figure 12. Other polygon attributes include population by age and gender group and jobs in 20 different classes. The loss of life analysis in the present study was based only on total population. The map of population at risk per computational cell shown in Figure 13 was computed by assuming a uniform distribution of population in each census block. As it can be seen, there are two major urban areas (see also Figure 2). The urban area on the west is located in the narrow valley downstream of the dam and directly on the path of the flow. The urban area on the north east corner of the map is farther from the dam and located on a relatively flat area.
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Total Population 1 - 42 43 - 81 82 - 152 153 - 278 279 - 481 482 - 943 944 - 1987 1988 - 3338
Figure 12: Total population by census block (only census blocks with population are shown). PAR Analysis The number of PAR by age group for each one of the 120 simulations were counted for 7 flood arrival-time intervals and for 17 depth intervals and recorded in tables. shows the average value, standard deviation, minimum and maximum values of PAR numbers by arrival time interval. Some of the information listed in is summarized in Figure 14. The plot on the right hand side of Figure 14 shows that the range between cumulative values of minimum and maximum PAR can be significant. Final minimum and maximum cumulative PAR values correspond to 4.4% of 268% of the final average cumulative PAR.
PAR / 100m2 82
1
Figure 13: PAR distribution per computational cell (10m×10m).
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Table 4: Statistics of PAR numbers by age group and flood arrival time interval.
PAR ≥ 65
14
PAR ≤ 14
Total PAR
Age
Value Average St. Dev. Min. Max. Average St. Dev. Min. Max. Average St. Dev. Min. Max. Average St. Dev. Min. Max.
0-15min 0.186 0.548 0.043 5.110 0.035 0.147 0.006 1.286 0.129 0.031 3.507 0.021 0.030 0.007 0.317
15-30min 12.782 14.597 0.019 52.346 3.296 3.802 0.002 13.409 8.623 0.014 35.341 0.862 0.986 0.004 3.597
30-60min 38.370 28.768 8.566 144.341 7.584 3.101 2.014 19.903 25.142 5.926 93.943 5.644 7.590 0.625 30.495
60-90min 75.021 47.800 1.110 148.087 10.381 6.789 0.286 21.096 48.743 0.746 96.662 15.898 10.011 0.078 30.329
120-180min 54.556 39.032 1.210 119.794 7.987 6.049 0.157 18.265 35.977 0.853 77.542 10.592 7.275 0.200 23.987
>180min 14.201 26.624 0.087 132.010 2.037 4.041 0.010 20.306 9.376 0.061 87.380 2.789 4.983 0.016 24.324
Cumulative Total PAR as a Function of Time
80
800
70
700
60
600
Number of PAR
Number of PAR
Average PAR
90-120min 65.333 25.632 0.532 95.128 9.437 4.100 0.140 14.424 42.874 0.346 59.286 13.022 4.963 0.046 21.418
50 40 30 20
500 400 300 200
10
100
0 0-15min
15-30min PAR ≤ 14
30-60min
60-90min
14 < PAR < 65
90-120min 120-180min
PAR ≥ 65
0
>180min
0-15min
15-30min
Cumul. Total PAR
Total PAR
30-60min
60-90min
Min Cumul. Total PAR
90-120min
120-180min
>180min
Max Cumul. Total PAR
Figure 14: (Left) Average PAR by age group and by arrival time; (Right) Cumulative values of average, minimum and maximum total PAR impacted by the flood as a function of time. and Table 6 show the average value, standard deviation, minimum and maximum values of PAR numbers by flood depth interval in two separate tables. Some of the PAR information listed in Table 5 and Table 6 is summarized in Figure 14. The plot on the right hand side of Figure 14 shows that the range between cumulative values of minimum and maximum PAR can be significant. This figure shows that majority of PAR impacted by the flood (93.5%) will be subjected to a maximum flood depth of 3.5m. The range between minimum and maximum PAR values is significant. The largest differences are observed for the depth interval 0.00.5m. Minimum (98.6% PAR in less than 3.5m depth) and maximum (87.8% PAR in less than 3.5m depth) PAR both show the same tendency as the average PAR. The average, minimum and maximum values of the inundated area listed in Table 5 and Table 6 are plotted in Figure 16 as a function of flow depth. As it can be seen most of the inundated area (95.1%) will be subjected to water depths less than or equal to 5m. The largest differences between minimum and maximum values occur for the depth interval 0.0-0.5m.
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Table 5: Average and standard values of PAR numbers by age group and depth interval. Depth (m) 0.0m-0.5m 0.5m-1.0m 1.0m-1.5m 1.5m-2.0m 2.0m-2.5m 2.5m-3.0m 3.0m-3.5m 3.5m-4.0m 4.0m-4.5m 4.5m-5.0m 5.0m-5.5m 5.5m-6.0m 6.0m-6.5m 6.5m-7.0m 7.0m-7.5m 7.5m-8.0m >8.0m
Total PAR Min Max 55.53 19.25 73.38 15.41 67.36 23.75 26.36 12.31 10.30 4.18 4.65 0.87 3.23 0.47 2.57 0.67 2.24 0.83 2.18 0.85 2.02 0.90 1.81 0.95 1.63 1.08 1.41 1.09 1.17 1.06 1.03 1.13 3.55 4.92
PAR ≤ 14 Min Max 8.21 2.91 11.10 2.16 9.97 3.68 3.63 1.83 1.37 0.43 0.81 0.15 0.67 0.19 0.62 0.20 0.58 0.21 0.56 0.21 0.52 0.23 0.46 0.24 0.41 0.27 0.36 0.27 0.30 0.27 0.26 0.28 0.92 1.21
14 < PAR < 65 Min Max 37.03 48.74 44.20 16.67 6.23 2.83 2.02 1.67 1.50 1.46 1.35 1.22 1.10 0.95 0.79 0.70 2.40 -
PAR ≥ 65 Min Max 10.29 3.48 13.54 3.29 13.19 4.05 6.06 2.33 2.70 1.27 1.01 0.41 0.54 0.24 0.28 0.14 0.16 0.06 0.16 0.07 0.15 0.07 0.13 0.08 0.12 0.09 0.10 0.09 0.08 0.08 0.08 0.09 0.23 0.36
Inundated Area (m2) Min Max 7,333,416 3,206,673 9,018,780 1,693,804 8,709,363 1,270,547 5,856,884 1,471,850 3,358,918 1,008,449 2,018,934 709,981 1,453,657 533,611 1,337,561 659,157 674,860 362,878 325,181 106,113 260,150 95,178 218,419 85,614 185,084 79,717 161,414 73,249 137,548 63,610 119,563 57,089 2,226,929 550,383
Table 6: Minimum and maximum values of PAR numbers by age group and depth interval. Depth (m) 0.0m-0.5m 0.5m-1.0m 1.0m-1.5m 1.5m-2.0m 2.0m-2.5m 2.5m-3.0m 3.0m-3.5m 3.5m-4.0m 4.0m-4.5m 4.5m-5.0m 5.0m-5.5m 5.5m-6.0m 6.0m-6.5m 6.5m-7.0m 7.0m-7.5m 7.5m-8.0m >8.0m
Total PAR Min Max
38.76 49.03 12.73 7.50 3.93 2.59 1.25 0.67 0.29 0.28 0.23 0.17 0.02 0.00 0.00 0.00 0.02
134.98 102.42 92.40 45.04 19.90 7.23 4.61 3.63 4.16 4.03 3.65 3.69 4.07 4.05 4.10 4.26 20.47
PAR ≤ 14 Min Max
5.33 7.00 2.05 1.32 0.80 0.54 0.30 0.16 0.07 0.08 0.07 0.06 0.01 0.00 0.00 0.00 0.00
20.13 15.17 14.09 6.95 2.95 1.09 1.15 0.93 1.02 1.00 0.93 0.91 1.01 1.00 1.02 1.05 5.10
14 < PAR < 65 Min Max
25.84 33.25 7.80 4.98 2.82 1.84 0.85 0.46 0.20 0.19 0.16 0.11 0.01 0.00 0.00 0.00 0.01
89.84 67.26 61.01 29.01 12.39 4.38 2.59 2.08 2.78 2.70 2.42 2.45 2.71 2.70 2.72 2.84 13.92
PAR ≥ 65 Min Max
7.59 8.78 2.88 1.20 0.30 0.21 0.10 0.05 0.02 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00
25.01 19.99 17.30 9.09 4.56 1.76 0.87 0.62 0.36 0.33 0.31 0.33 0.35 0.35 0.35 0.36 1.44
Inundated Area (m2) Min Max
3,502,773 6,067,713 4,253,739 2,131,314 740,011 422,222 447,365 171,062 95,543 78,662 73,453 74,621 66,539 59,984 44,808 39,151 1,409,981
17,674,779 11,718,682 10,358,269 7,690,691 5,280,020 3,035,832 2,560,271 2,481,161 1,333,115 521,537 442,965 418,002 330,630 298,393 276,752 277,381 3,827,117
Loss of Life Analysis The loss of life is computed using the U.S. Bureau of Reclamation DSO-99-06 Procedure (Graham, 1999). In this method, the fatality rate of PAR in a given area, which in the case of a two dimensional numerical simulation can be taken as the computational cell, is computed from Table 6 based on three parameters: flood severity, warning time, and flood severity understanding. Warning time can be computed as the time difference between the time the warning issued and the arrival time of the flood. Flood severity understanding is a parameters that must be appreciated based on the characteristics of the population in the inundation area. Since in the benchmark test case it is assumed that the population cannot be warned, the flood severity understanding is no longer applicable, and Table 6 reduces to the three highlighted lines. There are no clear guidelines for estimating flood severity. In the present study, the flood severity was computed based on maximum specific discharge, , which is equal to the product of depth and flow speed. The limits of high, medium, and low flood severity
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shown in Table 7 are taken from guidelines published by the U.S. Department of Homeland Security (DHS, 2011). Average PAR
Minimum, Avarage and Maximum Total PAR
80 60
Number of PAR
Number of PAR
70 50
40 30 20 10 0
PAR ≤ 14
14 < PAR < 65
PAR ≥ 65
160 140 120 100 80 60 40 20 0
Mean Total PAR
Mean Total PAR
Minimum Total PAR
Maximum Total PAR
Figure 15: (Left) Average PAR by age group and by flood depth; (Right) Average, minimum and maximum total PAR impacted by the flood as a function of depth.
Inundated Area (m2)
Minimum, Avarage and Maximum Values of Inundated Area 20,000,000 18,000,000 16,000,000 14,000,000 12,000,000 10,000,000 8,000,000 6,000,000 4,000,000 2,000,000 0
Avg. Inundation Area
Min. Inundation Area
Max. Inundation Area
Figure 16: Average, minimum and maximum values of inundated area as a function of depth. Based on the flood severity criteria given in Table 8 and the fatality rates in Table 7, the lossof-life (LOL) maps for each simulation was established using the corresponding raster. The average value, standard deviation, and minimum and maximum values of loss of life for each computational cell were then computed from 120 loss-of-life maps. Figure 17 shows the average value, standard deviation, minimum and maximum values of estimated loss of life (LOL) based on 120 simulations. As expected, the largest loss of life numbers are observed in the western urban area located in the narrow valley. This area is closer to the dam and directly on the path of the dam-break flood. The values are much larger. The urban area on the north eastern corner of the map has relatively small values of LOL due to lower values of . It is important to note that the areas with no loss of life and with zero standard deviation are not displayed. Looking at the standard deviation map, one can observe that there are areas with zero standard deviation. These areas can be regarded as areas where the LOL has highest likelihood.
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Table 7: Fatality rates for loss-of-life estimation from dam failure (Graham, 1999). Flood Severity
Warning Time (Minutes) 15 to 60
HIGH
More than 60 15 to 60
MEDIUM
More than 60 15 to 60
LOW
More than 60
Flood Severity Understanding Not applicable Vague Precise Vague Precise Not applicable Vague Precise Vague Precise Not applicable Vague Precise Vague Precise
Fatality Rate (Fraction of PAR projected to die) Suggested Value Suggested Range 0.75 0.30 – 1.00 Apply the values shown above to the PAR values who remain in the dam failure floodplain after warnings are issued. No guidance is provided on how many people will remain in the flood plain. 0.15 0.03 – 0.35 0.04 0.01 – 0.08 0.02 0.005 – 0.04 0.03 0.005 – 0.06 0.01 0.002 – 0.02 0.01 0.0 – 0.02 0.007 0.0 – 0.015 0.002 0.0 – 0.004 0.0003 0.0 – 0.0006 0.0002 0.0 – 0.0004
Table 8: Flood severity criteria used for loss of life analysis. Flood severity LOW MEDIUM HIGH
Criteria in SI Units
Criteria in US Customary Units
# of Deaths
Minimum Value of LOL
Maximum Value of LOL
1 - 42 43 - 81 82 - 152 153 - 278 279 - 481 482 - 943 944 - 1987 1988 - 3338
St. Dev. of LOL
PAR
Average LOL
0.00 - 0.02 0.02 - 0.04 0.04 - 0.06 0.06 - 0.08 0.08 - 0.10 0.10 - 0.12 0.12 - 0.14 0.14 - 0.16 0.16 - 0.18 0.18 - 0.20 0.20 - 0.22 0.22 - 0.24 0.24 - 0.26 0.26 - 0.28 0.28 - 0.30 0.30 - 0.32 0.32 - 0.34 0.34 - 0.36 0.36 - 0.38
Figure 17: Average value, standard deviation, minimum and maximum values of estimated loss of life (LOL) based on 120 simulations.
Conclusion The uncertainty in 2D numerical modeling of dam-break flood and the resulting loss of life were studied using 120 simulations representing 40 random pairs of breach width and breach formation time calculated with three different sets of Manning’s coefficients. The results show that large variations in the extent of the inundated area, water depths, and loss-of-life occur based on the particular combination of control parameters. The paper presents the average value and standard deviation of selected computed results as well as their upper and lower bounds. Peak discharge at the breach cross section and the cross sections downstream seem to be highly correlated with the breach formation time rather than the breach width. 269
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Only part of the results could be presented. Additional findings will be presented in subsequent publications.
Acknowledgements The research for the present benchmark study and the paper was sponsored by the U.S. Army Corps of Engineers, Headquarters Office of Homeland Security. The development of the DSS-WISE™ software was funded by the Department of Homeland Security-sponsored Southeast Region Research Initiative (SERRI) managed by the Department of Energy’s Oak Ridge National Laboratory.
References [1] Altinakar, M.S., McGrath, M.Z., Yucel, O., and Ger, M. (2013a). Simulation of Supercritical Flow in an Open Channel Bend Using Cut-Cell Boundary Method. 6th Int. Perspective on Water Res. and Env. (IPWE 2013), January 7-9, 2013, Izmir Turkey. [2] Altinakar, M.S., McGrath, M.Z., Ramalingam, V.P., Strati, O. (2013b). Dam Break Flood Simulation – Doing It Faster and Simpler: DSS-WISE™, DSS-WISE™ Lite, and DSAT. 1-day Short Course, ASDSO West Regional Conf., May 8, 2013, Seattle, Washington. [3] Altinakar, M.S. (2012). Advances in Numerical Modeling and Simulation of Floods, Inundation Mapping and Consequence Analysis. Keynote Lecture presented at the 10th International Congress on Advances in Civil Engineering (ACE 2012), October 17-19, 2012, Middle East Technical University (METU), Ankara, Turkey. [4] Altinakar, M.S., McGrath, M.Z., Matheu, E.E., Ramalingam, V.P., Seda-Sanabria, Y. Breitkreutz, W., Oktay, S., Zou, J.Z., Yezierski, M. (2012): Validation of Automated Dam-Break Flood Simulation and Assessment of Computational Performance, Proc. Dam Safety 2012, September 16-20, Denver, Colorado. [5] Altinakar, M.S. and McGrath, M.Z. (2012). Parallelized Two-Dimensional Dam-Break Flood Analysis with Dynamic Data Structures. ASCE-EWRI, 2012 World Environmental & Water Resources Congress, May 20-24, 2012, Albuquerque, New Mexico. [6] ARCADIS (2011). ADCIRC Based Storm Surge Analysis of Sea Level Rise in the Galveston Bay and Jefferson County Area in Texas. Report prepared for The Nature Conservancy, Nov 28, 2011. [7] DHS (2011). Estimating Loss of Life for Dam Failure Scenarios. U.S. Department of Homeland Security, September 2011. Washington DC. [8] Froehlich, D.C. (1995a). Embankment Dam Breach Parameters Revisited. Water Res. Engineering, Proc. 1995 ASCE Conf. on Water Res. Eng., New York, pp. 887-891. [9] Graham, Wayne J. (1999). A Procedure for Estimating Loss of Life Caused by Dam Failure. DSO-99-06, U.S. Bureau of Reclamation, Denver, Colorado, September 1999. [10] Iman, R.L.; Helton, J.C.; and Campbell, J.E. (1981). An Approach to Sensitivity Analysis of Computer Models, Part 1. Introduction, Input Variable Selection and Preliminary Variable Assessment. Journal of Quality Technology, Vol. 13, No. 3, pp. 174–183. [11] Kim, D.-H., Cho, Y.-S., and Yi, Y.-K. (2007). Propagation and Run-up of Nearshore Tsunamis with HLLC Approximate Riemann Solver. Ocean Engineering, Volume 34, Issues 8-9, June 2007, pp. 1164-1173. [12] Luettich, R. and Westerink, J. Storm Surge Inundation Modeling: State of Science. Hurricane Surge Workshop, February 11, 2009, St. Petesburg, Florida [13] Toro, E. F., Spruce, M., Speares, W. (1994). Restoration of the Contact Surface in the HLL-Riemann Solver. Shock Waves, Vol. 4, No. 1, pp. 25-34. [14] Wahl, T.L. (2004). Uncertainty of Predictions of Embankment Dam Breach Parameters. Journal of Hydraulic Engineering, Vol. 130, No. 5, pp. 389-397. 270
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Risk assessment for hypothetical dam break A method for the rapid and consistent evaluation L.Mancusi1, L.Giosa2, A.Cantisani2, A.Sole2 and R.Albano2 1
Sustainable Development and Energy Resources Department, Research on Energy Systems (RSE) Spa, via R. Rubattino 54, 20134 Milano, Italy 2 School of Engineering, University of Basilicata, viale dell’Ateneo Lucano 10, 85100 Potenza, Italy E-mail: [email protected]
Abstract This paper details the technical contribution to the theme of flood risk analysis as consequence of a dam failure. According to the numerical problem proposed for the workshop, the analysis consists of the evaluation of the dam break and its consequences. The simulation includes two scenarios of dam breach: the scenario 1 that represents the case of an easy erodible dam and the scenario 2 the case of an erosion resistant dam. For each scenario, a dam failure discharge hydrograph was calculated and the subsequent flood wave and consequences have been evaluated. The methodology adopted involves, for a first part, the use of standard models for the hydraulic modelling of the dam breach and flood wave propagation. For the second part, a set of GIS scripts was written, tested and developed using the python scripting language to obtain a rapid appraisal of consequences for the population and to assess the direct economic damages for residential, commercial, and industrial buildings. Since the latter elaboration depends greatly on the type and the level of detail of available data, in this study we have been used data as generic as possible and GIS scripts that allow, for a great variety of cases, a rapid initial assessment.
Introduction Modern society considers it essential to increase the safety of the infrastructure. Risk analysis is a helpful tool for the evaluation and management of risks which can affect people, environment and human development. The purpose of this paper was to demonstrate the application of an example of quantitative risk assessment technique that consists of estimating the consequences of failure of a dam near populated areas with complex demographics, infrastructure and economic activity. The first chapter concerns the hydraulic modelling and simulation of the dam breach, the second the subsequent flood wave propagation and the last focuses on consequence estimation.
Dam failure This section includes the description of modelling of the breaching process and the subsequent discharge hydrograph for the hypothetical overtopping of the dam. Two methods have been adopted and the results are named respectively scenario 1 and 2. Scenario 1 In this case were used, a statistical method available in the literature for the dam failure peakdischarge estimation of and a physically based mathematical model for calculating the total discharge hydrograph. 271
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P. Molinaro [1] utilised 31 data sets (predominantly earthfill and some rockfill) extracted from the report of J. E. Costa [2] to develop a relationship from the peak-discharge, the height of dam and the reservoir volume at time of failure. Qmax gH
5
2
V 0.116 3 H
0.221
(1)
Qmax = peak-discharge (m3/s) g = gravity of Earth that has an approximate value of 9.81 m/s2 H = height of the dam (m) V = volume of water at breach time (m3) Applying the formula (1) for this case (H=61 m and V=30.3*106 m3) we obtain: Qmax =32800 m3/s Where:
The latter value was used to calibrate parameters of the mathematical model. The mathematical model was developed by Molinaro [3] and simulates the breach development process through an earthen dam due to overtopping. The model is developed by coupling the conservation of mass of the reservoir inflow, spillway outflow, and breach outflow with the sediment transport capacity of the quasi-steady uniform flow along an erosion-formed breach channel. The rate at which the breach is eroded is evaluated using the Engelund and Hansen [4] sediment transport relation. The dam is modeled as an isosceles triangle formed by a noncohesive material of uniform diameter D. The storage characteristics of the reservoir are described by specifying a table of volume vs. water elevation. The overtopping failure simulation starts by assigning a small initial breach whose bottom elevation must be below the reservoir water level. The first stages of erosion are along the downstream face of the dam while the breach bottom erodes vertically downward. An erosion triangular channel is gradually cut into the downstream face of the dam. The sides of the breach channel has a constant angle (α) with the vertical which is a function of the internal friction (φ) of dam’s material. The flow into the channel is determined by the broad-crested triangular weir relationship. The breach bottom is allowed to progress downward until it reaches the bottom elevation of the dam, subsequently the channel becomes trapezoidal with the sides that maintain the same slope α of the previous triangle. The following figure Figure 1 shows the sequence of the simulation of the breach formation.
Figure 1: Breach formation sequence
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The most important parameter for model calibration is the characteristic diameter D. In this case, taking into account the result of the equation 1, we have adopted the value D = 0.01 m. The discharge hydrograph obtained is shown in the Figure 3. This hydrograph was used for the first propagation of a flood wave referred to the next chapter. It is characterized by Qmax =28936 m3/s and a breach formation time of 2040 sec (0.57 hrs). Scenario 2 In addition to the above, a second method was applied for the evaluation of failure discharge hydrograph. Even in this case a regression equations was used for the estimation of dam breach parameters, and then a mathematical model was applied. Table 1 summarizes the resulting breach parameters (Wb: bottom width of the breach and tf: breach formation time) computed by several approaches available in the literature. Table 1: Breach Parameters Method MacDonald and Langridge - Monopolis (1984) Floehlich (1995a) Floehlich (2008) Von Thun and Gillette (1990)
Wb (m) 167 147 110 207
tf (hrs) 1.99 0.66 0.57 1.47
In the last column you can see that the breach formation time ranges from 0.57 to 2 hours. As in scenario 1 the resulting time corresponds with the minimum, then for this scenario the case of the maximum value of tf=2 hours was investigated. For performing the dam breach outflow hydrograph computation, HEC-RAS model was adopted. The implementation of these breach parameters in the HEC-RAS modelling system is depicted on Figure 2. The resulting discharge hydrograph compared to the previous scenario is depicted in Figure 3. As can be seen in the latter figure, the second hydrograph, having the same volume, is characterized by a peak value cut in half compared to the first, but a duration in time approximately double. For further analysis, we can consider the two hydrographs: the first as representing of a easy erodible dam and the second an erosion resistant dam.
Figure 2: HEC-RAS Dam Breach Model of scenario 2
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Dam Failure discharge hydrographs for scenario 1 and 2 35000
Discharge for scenario 1 30000
Discharge for scenario 2
Q(m3/s)
25000 20000 15000 10000 5000 0 0
20
40
60
80
100
120
Time (min)
Figure 3: Discharge hydrograph of scenario 2 compared with scenario 1
Flood simulation Hydraulic modelling has carried out using the MIKE 21 software by the DHI Water Environment Health to simulate flood wave propagation in the river and to describe the inundation on the floodplain. This software solves the shallow water equations by means of a finite difference scheme. For each scenario, the simulation has been constructed using, as upstream boundary condition, the corresponding discharge hydrograph for the hypothetical overtopping of the dam. According to Bunya et al. [5], the Manning n coefficient is spatially assigned associating the value of n with the land cover definition of 2001 from the USGS National Land Cover Data (NLCD) (Table 2), These values are selected or interpolated from standard hydraulic literature. Table 2: Manning n value for 2001 NLCD classification Lu Code 11 12 21 22 23 24 31 41
Description Open Water Perennial Ice/Snow Developed-Open Space Developed-Low Intensity Developed-Med Intensity Developed-Hight Intensity Barren Land Deciduous Forest
n Mann. (s/m1/3) 0.020 0.022 0.050 0.120 0.120 0.121 0.040 0.160
Lu Code 42 43 52 71 81 82 90 95
Description Evergreen Forest Mixed Forest Shrub/Scrub Grassland/Herbaceous Pasture/Hay Cultivated Cropland Woody Wetlands Herbaceous Wetlands
n Mann. (s/m1/3) 0.180 0.170 0.070 0.035 0.033 0.040 0.140 0.035
The results of flood modelling consist of values, for each grid cell in the study area, depth (m) and the two components of the vector unit flow rate (m2/s) for 15 minute intervals and the envelope of their maximum. Using a GIS scripts, hydrographs flow at different cross sections were extracted. Some of these are shown in Figure 4.
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Comparison of hydrographs of the two scenarios at different cross sections 35000
Scenario1 - Cross-sec.1 30000
Scenario1 - Cross-sec.5 Scenario2 - Cross-sec.1
25000 Q (m3/s)
Scenario2 - Cross-sec.5 20000 15000 10000 5000 0 0
20
40
60
80
100
120
140
160
Time (min)
Figure 4: Discharge hydrographs of the two scenarios at different cross sections Comparing the hydrographs of two scenarios in different cross sections we may note that: while having at the initial cross-section a large difference in peak flow rate, however, during the process of propagation downstream, an attenuation of the difference occurs.
Impacts assessment The potential risks associated with the failure or disruption of dams could be considerable and potentially result in significant destruction, including loss of life, massive property damage, and severe long-term consequences. The following sections contain the analysis of two categories: public safety and direct economic impact. Population at Risk and Loss of Life estimation The analyzes described in this section have been carried out by adopting the published guidelines of the report [6] that provides guidelines and recommendations for estimating loss of life resulting from dam failure or disruption. The results of flood modelling and the data from the population census are used. Geographic analyzes were carried out using Map Algebra techniques implemented in a set of scripts written, tested and developed using the python scripting language and the Open Sources GDAL libraries and NumPy Python module. To combine multiple maps in Map Algebra all data have been converted into grid format. The outputs of the hydrodynamic model have been processed to derive the information required for the analysis. Using a GIS scripts, a Flood Wave Arrival Time grid was obtained, in addition the two components of the vector unit flow rate are combined to obtain the maximum Peak Unit Flow Rate values (m2/s). These values, called parameter DV, are representative of the general level of destructiveness that would be caused by the flooding. The DV values are then categorized, as suggested in the Figure 5 extracted from guidelines, into ranges of values which define low, medium, and high severity zones.
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Figure 5: Flood severity rating criteria reported in the guidelines The vector polygons of the population census block were converted into grid format: the hypothesis assumed for the different values of the fields is that their distribution within the polygon is homogeneous. By overlaying grid maps of flood with the grid of the population is achieved as a result the map of Population at Risk (PAR). The estimate of loss of life is finally obtained by multiplying the PAR with the Fatality Rate (Fraction of people at risk projected to die). The latter was obtained by using the values of the Table 3 (from tab. 4 of guidelines) as a function of warning time and flood severity. Table 4 shows the results for the two scenarios for the hypothesis of event occurred at night (understanding=vague). In the case of the first scenario, there is a population at risk greater than 11 percent, while the estimate of the largest loss of life is almost 30 percent. The last result is mainly caused by the shorter warning time of the first scenario. It should be noted that in each case that the differences in terms of the consequences are less than the differences of the peak discharge of the two scenarios at the breach of the dam, . This result is due to the fact that the volume released from the dam is still the same.
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Table 3: Recommended Fatality Rates for Estimating Loss of Life as reported in the guidelines Flood Severity HIGHT MEDIUM
Warning Time (min) Not applicable No warning 15 to 60 More then 60 No warning
LOW
15 to 60 More then 60
Understanding Not applicable Not applicable Vague Precise Vague Precise Not applicable Vague Precise Vague Precise
Fatality Rate 0.75 0.15 0.04 0.02 0.03 0.01 0.01 0.007 0.002 0.0003 0.0002
Table 4: Population at Risk and Loss of Life estimation Time Interval (min) 0-15 15-30 30-60 60-90 90-120 120-180 >180 Total Peak Flood Depth Range (m) 0.0-0.5 0.5-1.0 1.0-1.5 1.5-2.0 2.0-2.5 2.5-3.0 3.0-3.5 3.5-4.0 4.0-4.5 4.5-5.0
14-yr and Under 65-yr and Over Total Population At Risk Population at Risk Population at Loss of Life Risk Scen. 1 Scen. 2 Scen. 1 Scen. 2 Scen. 1 Scen. 2 Scen. 1 Scen. 2 0 3 0 0 0 0 0 2 2 4 0 0 0 1 2 2 4 529 2 794 1 169 732 308 196 1 783 1 262 5 111 641 721 110 1 057 105 2 0 10 411 8 529 1 491 1 084 2 163 1 996 0 0 8 935 12 346 1 350 1 899 1 662 2 290 0 0 688 1 911 85 265 121 360 0 0 29 676 26 228 4 816 4 090 5 311 4 948 1 787 1 266 Flooded Area (m2) Scen. 1 8 308 132 6 025 869 7 854 660 5 612 806 3 711 550 2 547 251 1 746 448 1 279 417 546 680 391 332
Total Population At Risk
14-yr and Under 65-yr and Over Population at Population at Risk Risk
Scen. 2 Scen. 1 Scen. 2 Scen. 1 Scen. 2 Scen. 1 Scen. 2 4 088 784 5 854 4 825 915 706 1 017 937 7 190 976 5 683 6 770 895 1 021 987 1 237 7 858 701 8 937 7 769 1 373 1 164 1 664 1 453 5 141 466 3 768 2 879 543 391 781 676 2 970 641 1 562 1 038 197 118 404 304 1 901 885 510 380 72 60 137 98 1 453 532 274 292 43 49 68 70 711 995 209 220 37 53 49 28 392 859 125 228 35 62 8 17 310 246 153 237 43 65 11 17 277
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5.0-5.5 5.5-6.0 6.0-6.5 6.5-7.0 7.0-7.5 7.5-8.0 >8 Total
380 736 209 944 242 630 250 352 201 323 294 532 192 882 221 527 176 180 128 947 186 417 101 919 1 880 693 850 640 41 285 006 34 078 946
211 190 194 162 176 254 1 414 29 676
216 271 346 337 154 84 182 26 228
57 51 53 44 44 61 353 4 816
54 67 90 83 37 18 52 4 090
15 15 14 9 11 16 104 5 311
13 16 25 30 11 7 9 4 948
Direct Economic Impact estimation Methods and values of the parameters used in this section are drawn mostly from the report [7]. They concern the assessment of the direct economic damages for residential, commercial, and industrial buildings. The input data consist of map of land use and parcel zone map of the study area. As in the previous paragraph, for the analysis, all the data are preliminarily converted into grid format. The following assessments do not take into account Agricultural, Roads, Infrastructure and Vehicles damages. The assessment however allows the estimation of the damage to buildings and their contents, and when applied to different scenarios allows an effective comparison of the impact. The extent of damage to the buildings and its contents is estimated from the depth of flooding by the application of a depth-damage curve associated with each occupancy type. Depth damage curves demonstrate the relationship between the depth of the flood relative to the first finished floor level of buildings and the damage caused to the structures and contents. Damages are typically expressed as a percentage of depreciated building replacement value. Adopting a non-traditional approach, the adopted method models directly the content damage as a percentage of structure value rather than using a content-to structure value ratio. Not having a map of buildings, the area covered by the buildings has been derived from the land use map according to the hypothesis of Building Coverage shown in the following table. Table 5: Relationship between Land Use and Building Coverage Lu Code 11 12 21 22 23 24 31 41
Description
Building Cover. % Open Water 0% Perennial Ice/Snow 0% Developed-Open Space 10% Developed-Low Intensity 20% Developed-Med Intensity 35% Developed-Hight Intensity 50% Barren Land 0% Deciduous Forest 0%
Lu Code 42 43 52 71 81 82 90 95
Description Evergreen Forest Mixed Forest Shrub/Scrub Grassland/Herbaceous Pasture/Hay Cultivated Cropland Woody Wetlands Herbaceous Wetlands
Building Cover. % 0% 0% 0% 0% 0% 0% 0% 0%
To calculate damages, each structure must be assigned to a structure occupancy type. For each structure occupancy type an estimated replacement value and a structure depth-damage and a content depth-damage relationship must be defined. In our case, replacement values were extracted from the "Table C-3 Estimated Replacement Value", depth-damage relationship from “Table C-1 Depth Damage Curves, Defining Damages as a Percentage of Depreciated Building Value for Depth of Flooding Above Floor 278
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Height” and the height of the floor of buildings from the ground level was taken from "Table C-2 Foundation Heights" of report [7]. The following table contains the list of occupancy type categories adopted. Table 6: Occupancy types Occup. Type RES1
Description
COM
Residential One Story, No Basement Residential Two or more Stories, No Basement Commercial buildings
IND PUB FAR
Industrial buildings Public buildings Homestreads
RES2
Unit Cost ($US/sqm) 1711 3336 1528
Origin data of Depth Damage Curve USACE Generic Depth Damage Curves for residential buildings " USACE depth damage, as used in Ford (2005) [8] " " "
1528 1711 1711
The Figure 6 below shows the graph of the depth-damage curves. Contents Depth-Damage Curves as a Percentage of Depreciated Building Value for Depth of Flooding Above on the Ground Level
90%
90%
80%
80%
70%
70%
60%
60%
Damage (%)
Damage (%)
Structural Depth-Damage Curves as a Percentage of Depreciated Building Value for Depth of Flooding Above on the Ground Level
50% 40%
RES1_S RES2_S COM_S IND_S PUB_S FAR_S
30% 20% 10% 0% -1
0
1
2
3
4
RES1_C RES2_C COM_C IND_C PUB_C FAR_C
50% 40% 30% 20% 10% 0%
5
-1
0
1
2
3
4
water depth (m)
water depth (m)
Figure 6: Depth damage curves To assign at each parcel the occupancy type we chosen the values according to the Table 7. Table 7: Reclassify table: from parcel ZONINGCATE to occupancy type ZONINGCATE COMMERCIAL INDUSTRIAL / WHOLESALE / MANUFACTURING INSTITUTIONAL / GOVERNMENT OFFICE OFFICE OFFICE OPEN SPACE / RECREATION / AGRICULTURAL RESIDENTIAL RESIDENTIAL RESIDENTIAL 279
Stories any any any 1 2 3 any 1 2 3
Occupancy Type COM IND PUB RES1 RES2 RES2 FAR RES1 RES2 RES2
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RESIDENTIAL / AGRICULTURAL UTILITIES / TRANSPORTATION
any any
FAR RES1
The results of applying the method for the two scenarios are listed in the following table. Table 8: Direct Economic Impact Time Interval (min)
Direct Economic Impact ($US)
Scenario 1 Scenario 2 0-15 0 0 15-30 0 0 30-60 499 417 507 333 141 773 60-90 671 323 845 151 037 801 90-120 839 403 518 854 746 410 120-180 577 769 953 822 621 093 >180 12 807 947 64 207 745 Total 2 600 722 770 2 225 754 822 The results show that the total damage, in the case of the first scenario, are greater than 14% and that difference occurs in the first 120 minutes. Also in terms of economic loss the difference between the two scenarios are less than the differences of the peak discharge in the breach of the dam.
Conclusion In this paper we present the results of the analysis of a possible dam failure. The development of a dam break is a complex process involving numerous uncertainties: the methodology adopted in this work is a medium-scale approach type and can be used for the rapid and consistent evaluation of consequences for the population and to assess the direct economic damages for residential, commercial, and industrial buildings. Rapidity is allowed by using aggregate data: maps of land-use, population census and parcel zone. Consistency is required to ensure comparability between evaluations. For that reason the method can be used to prioritize corrective actions to achieve the greatest and quickest possible risk reduction or for identification of the most effective and better-justified measures of risk mitigation. The comparison carried out for the two scenarios is an example of use of the methodology to estimate the sensitivity of results with respect to an uncertain parameter which is the breach formation time.
Acknowledgements This work was carried out also thanks to the Research Fund for the Italian Electrical System under the Contract Agreement between RSE S.p.A.(Research for Energetic System) and the Ministry of Economic Development - General Directorate for Nuclear Energy, Renewable Energy and Energy Efficiency, stipulated on July 29, 2009 in compliance with the Decree of March 19, 2009.
References [1] Molinaro P. (1990). Metodi statistici per la stima della portata di picco defluente in una valle per la rottura di una diga in materiale sciolto. ENEL DSR-CRIS Milano. [2] Costa J. E. (1995). Floods from dam failures. U.S. Geological Survey, Open-File Report 85-560 Denver Colorado. 280
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[3] Molinaro P. (1986). Modello matematico della breccia che si sviluppa in uno sbarramento in materiale sciolto per tracimazione. ENEL DSR-CRIS Milano. [4] Engelund F., and E. Hansen (1972). A Monograph on Sediment Transport in Alluvial Streams, Teknisk Folag, Copenhagen. [5] S. Bunya, J. C. Dietrich, J. J. Westerink, B. A. Ebersole, J. M. Smith, J. H. Atkinson, R. Jensen, D. T. Resio, R. A. Luettich, C. Dawson, V. J. Cardone, A. T. Cox, M. D. Powell, H. J. Westerink, H. J. Roberts. (2010). A High-Resolution Coupled Riverine Flow, Tide, Wind, Wind Wave, and Storm Surge Model for Southern Louisiana and Mississippi. Part I: Model Development and Validation. Monthly Weather Review 138:2, 345-377. [6] Dams Sector. Estimating Loss of Life for Dam Failure Scenarios. U.S. Department of Homeland Security, Washington, D.C. , September 2011. [7] Department of Water Resources Division of Flood Management (2008). Flood Rapid Assessment Model (F-RAM) Development 2008, State of California, The Resources Agency [8] David Ford Consulting Engineers (2005). Urban flood scenario: Sacramento area levee breach scenario, report prepared for California Department of Water Resources, December 5th 2005.
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2-D hydraulic modelling of a dam break scenario O. Saberi1 , C. Dorfmann2 and G.Zenz3 1, 2
PhD student, Institute of Hydraulic Engineering and Water Resources Management, Stremayrgasse 10/2, A-8010 Graz, AUSTRIA 3 Univ. Prof. Institute of Hydraulic Engineering and Water Resources Management, Stremayrgasse 10/2, A-8010 Graz, AUSTRIA E-Mail: [email protected]
Abstract Dam failure is a catastrophic event, and study on the structure of dam is important in the field of water resource engineering. The reason is the risk to life and property below the dam structure. The dam failure occur due to many reasons, some of these reasons are related to structural problems while others are related to the hydraulic conditions. For embankment dams, main reasons are overtopping and piping through the dam body or foundation. However there are some situations that make dam failure happen suddenly, like earthquakes, landslides or wars. This study provides a two-step numerical simulation of dam failure analysis for an embankment dam which its failure happened in overtopping conditions. First step is about simulating breach of the dam and calculating breach hydrograph, maximum discharge of breach, initial time of the breach formation, and time of maximum breach discharge and other parameters of breach. BREACH GUI software was used for this step. The second part simulates flood after dam break. TELEMAC2D software was used for simulating the flood. The results include the travel time (warning time) of the flood wave to various locations in the downstream valley and the representative valley cross-sections depicting flow depth and unit flow rates. Furthermore the results of this paper can provide information to build an inundation map which can help to develop risk management analyzes.
Introduction A hypothetical embankment dam was constructed in a mountainous region. This high hazard dam sits directly above a lightly populated area which is 3.5 kilometers away from an urban environment. The primary function of this dam is flood control for heavy snowmelt and strong monsoonal weather patterns. In addition, the reservoir provides some water supply and recreational activities to nearby communities. The dam failure for this dam takes place when the pool elevation is at crest elevation. The mode of failure is assuming an overtopping failure [6]. Two primary tasks in the analysis of a potential dam failure are the prediction of the reservoir outflow hydrograph and the routing of that hydrograph through the downstream valley to determine the dam failure consequences. When populations are located close to a dam, it is important to accurately predict the breach outflow hydrograph and its timing relative to events in the failure process that could trigger the start of evacuation efforts.
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Breach Parameter Estimation Empirical Method The empirical method are used to predict breach parameters estimation and breach peak discharge and breach hydrograph estimation including geometry of breach, time to reach failure and peak flow discharge, the empirical methods are get from documented failures. The recommended empirical method for predicting dam breach parameters are MacDonald&langridge-Monopolis (1984), Froehlich (2008), Froehlich (1995 a) and Von Thun and Gillite. In this paper we calculate the dam breach parameters with Froehlich (2008) Formula. For more details we refer to the 12th International Benchmark Workshop [6]. Froehlich (2008) is depending only on the height of breach ( , the reservoir volume and the breach side-slope, also we have the failure mode factor ( ) in the Froehlich (2008) for distinguish between overtopping and piping failure. After put mentioned parameters in the Froehlich (2008) formula we have breach hydrograph in the below shape: Table 1: Results of the Froehlich (2008)
Discharge (m³*1000/s)
Froehlich (2008)
⁄ 39500
0.57
110
50000 40000 30000 20000 10000 0 0
0.2
Time (hours)
0.4
0.6
Figure 1: Outflow hydrograph from the Froehlich (2008) BREACH modelling In this project we use the National Weather Service (NWS) software Brach GUI for predicting the breach parameters and the resulting outflow hydrograph of an earthen dam [1]. BREACH model was primarily developed by Fread [9] from the National Weather Services. Since 1988 it has been used broadly. This software use a physically-based mathematical model using the principles of hydraulics, sediment transport, soil mechanics, the geometric and material properties of the dam, and the reservoir properties [1]. Furthermore it uses critical properties which are measurable from dam material descriptions. Therefore it considers a more robust model. However it is the responsibility of the engineer to determine the appropriate combination of values as the measure characteristic are normally changing with in a wide range. Available parameter of breach hydrograph The dam used in this study is a homogenous embankment dam which is a rolled earth fill structure and composed of predominantly sandy clays and clayey sands. It is located in a 284
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mountain region. The width of crest is 24 m, the length of crest is 360 m and the height of the dam is 61m. The total water volume of water behind the dam is 38,276,344 m³. The distance from this dam to the population area is 3.5 km. The dam failure will take place when the water level behind the dam is at crest elevation and the mode of failure will be over topping failure. For more details we refer to the 12th International Benchmark Workshop [6]. As base inflow we assumed a discharge equal to 100 m³/s. In Table 1 the main physical parameters of the dam are listed. Table 2: Strength parameters of soil [6] Effective cohesion (KPa) Effective friction angle Undrained strength (KPa) Shear Wave velocity (m/s) Maximum shear modulus (KPa) Saturated unit weight (kg/m3) Permeability (cm/s)
19.15 14° 43.09+0.175 152.4 46443 2002 1.9*10^-6
Calculating input parameter for Breach software For start a simulation with the BREACH software we need to calculate some parameters as input data. The most important input parameters to be provided in the BREACH software are the mean grain size D50 (mm), the initial elevation of water surface, the elevation of the top of the dam, the elevation of the bottom of the dam, the average plasticity index (PI), the critical shear stress coefficient, the slope of the upstream and downstream faces of the dam (1:3), the base inflow hydrograph, the surface area behind the dam vs. the elevation of the dam, the porosity specifications of the dam material and the breach parameter. Table 1 and the equations 1-4 according to [ ] can help with defining these input parameters: √ ⁄
(1)
where V is the shear wave velocity (m/s), shear modulus (KPa)
is the density of material (kg/m3) and G is the (2)
where is the specific gravity, of water (N/m³)
is the unit weight of soil (N/m³) and
is the unit weight
(3) where
is the specific gravity,
is the unit weight of water (N/m³) and e is the void ratio (4)
where e is the void ratio (%) and n is the porosity (%). With the parameters in Table 1 and the equations 1-4 the porosity n is calculated as 49%. D50 can be considered approximately equal to 0.066 mm [4]. The basis for this approximation comes from soils mechanics fields.
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When the diameter of Clay soil is less than 0.002 mm and the diameter of sand soil is between and 0.05 and 2.00 mm then sandy clay is a soil with 33 percent till 55 percent clay and up to 65 percent sand [5]. The average plasticity index (PI) is calculated according to the USCS plasticity chart and the relationship equations of Skempton and Henkel (1953) [3], Psterman (1959) [3] and Bjerum(1960) [3] which is equal to 42%. Furthermore PI will be used in the following formulation to calculate two more parameters required by BREACH software (CA, CB) [1]: (5) where is the critical shear stress (kPa), PI is the plasticity index (%) and CA, CB are the critical shear stress parameters. Having PI=42% and derived from Table 1, CA and CB are calculated as .
Discharge (m³*1000/s)
Output of the BREACH software Having all the above inputs into the software we can run the BREACH model to calculate the outflow hydrograph given in Figure 1. 45 40 35 30 25 20 15 10 5 0 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
Time (hours)
Figure 2: Outflow hydrograph from the breach software Furthermore the software output provides some other parameters which can be used in risk management studies. These parameters are given in Table 2: Table 3: Results of the BREACH software Maximum discharge of breach (m3/s)
41600
Time for peak out flow (hours)
4.62
Time for starting outflow (hours)
4.44
Final depth of breach (m)
61
Top width of breach at peak breach flow (m)
250
Time to reach final breach bottom elevation (hours)
4.97
Bottom width of breach at peak breach flow (m)
286
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Comparison between result of Froehlich (2008) and BREACH modelling
Discharge (m³*1000/s)
After calculating breach hydrograph from Froehlich (2008) and breach modeling in follow picture, we find out all the dam breach parameters including the failure time, the peak discharge outflow and the average of breach width of Froehlich less than BREACH software. 45000 40000 35000 30000 25000 20000 15000 10000 5000 0 4.463
BREACH software Froehlich (2008)
4.963
5.463
Time (hours)
Figure 3: merged discharge of cross sections (m3)/s 2-D hydrodynamic flood with Telemac-2D The flood from dam break can go to downstream valley and in this part we are going to determine dam failure consequences and calculating the water surface, the water depths, the unit flow rates as well as the flow rates in various cross sections. The TELEMAC-2D modelling software is used for this purpose. TELEMAC-2D is a numerical simulation software and belongs as module to the open source Telemac-Mascaret suite for the simulations of hydrodynamic flow, contaminant and sediment transport. It solves the twodimensional depth-averaged Saint Venant equations for free surface flow. A detailed description of the Telemac System is given in Hervouet’s book (2007)[7]. The main results of Telemac-2D are the water depths and the depth- averaged velocity components. The original geometry provided by the ICOLD benchmark specifications was modified by deleting a priori non flooded mesh areas in order to save computation time. Table 3 shows the resulting mesh properties: Table 4: Mesh properties Number of element: 213750 Interior elements: 210652 Edge elements: 3038 Number of nodes: 108433 Edge nodes: 3114 Interior nodes: 105319 The upstream boundary condition is located at the dam and the hydraulic boundary condition is the unsteady flow hydrograph from the BREACH software, see Figure 1. The downstream boundary condition is a free outflow.
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Results overview The results of TELEMAC-2D are stored in the binary SEARAFIN format which can be analyzed and visualized by using the free software Fudaa or BlueKenue [8]. In this project BLUE KENUE software was used. In the following the results are shown graphically for breach discharge, cross section discharge, peak flood depth, flood wave arrival time, peak unit flow rate and flooded area. Furthermore interpretation on each subject is given along with the output figure. Breach Discharge This result was calculated in the first part with BREACH GUI software (shown in Figure 1). Also from the calculated breach hydrograph we find out: With the inflow of 100 m³/s into the dam reservoir it takes 4.36 hours for the dam breach initiation. The maximum discharge takes place after 648 s after the breach initiation. The duration of flood is 2844 s. Cross-Section Discharge For the calculation of the cross section discharge the following steps are performed: First the unit flow rates are calculated (m²/s) Second the unit flow rates are integrated along the cross section width Figure 2 shows the discharges in various cross sections and the reduction of the peak flows going downstream from cross section 1 to 5. Discharge of cross sections (m3/s) 45000 40000 35000
Discharge (m3/s)
30000 cross section 1
25000
cross section 2 20000
cross section 3
15000
cross section 4 cross section 5
10000 5000 0 00:00:00 00:28:48 00:57:36 01:26:24 01:55:12 02:24:00 02:52:48 time (HR)
Figure 4: merged discharge of cross sections (m3)/s
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Peak flood depths In BLUE KENUE software we can also calculate the maximum flood depths in the project area. Figure 3 shows the results for the peak flood depths where the blue color indicates the safe area with maximum water depths less than 0.5 m and the red color indicates dangerous areas with minimum water depths of 4.5 m.
Figure 5: Peak flood depths (m) Flood wave arrival time and water depths In this part we calculated the arrival time of flood. This will help us to find out how long it takes for the flood to reach each part of downstream area. BlueKenue software visualizes the water depths separately for each chosen time interval. It means BlueKenue gives us one gridded dataset for each time intervall. Figures 4, 5 and 6 shows the water depths or flood wave arrival times for certain time intervals.
Figure 6: Water depths after 30 minutes
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Figure 7: Water depths after 2 hours and 30 minutes As we can see in Figure 4, 5 and 6 these visualizations are useful to analyze the flood wave arrival time and the locations of impact. Peak unit flow rate The unit flow rate is given in equation 6: (6)
√
where q is the unit flow rate (m²/s), U and V are the water velocity components in x and y direction (m/s), and H is water depth (m). The peak unit flow rate in every mesh node can be extracted from the time varying unit flow rate and is shown in Figure 7.
Figure 8: Peak unit flow rate (m²/s) Flooded area 290
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For calculating the flooded area we can define the total wet area by defining iso-lines. This option categorizes the flooded area for arbitrary water levels. The total flooded area is given in Table 4 and the flooded areas for different water depths with interval of 0.5 m is given in Table 5. Table 5: Total flooded area Total flooded area (m²)
47626580
Table 6: Flooded areas, intervals 0.5 m Water heights (m) >8 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5
Flooded area (m²) *10^3 4179 4290 4805 5565 6827 7910 9131 10269 11569 13349 15432 18145 22200 27109 32959 38433 41043
The results of these tables can be used in the risk management study for calculating the damage property and loss of life.
Conclusion The first part of this study describes the BREACH model for predicting the breach hydrograph of a hypothetical embankment dam. The reason for the failure of dam is assumed to be overtopping. The following results can be deduced: The maximum outflow in this project is calculated as 41558 m³/s. The water flow is transferred into the downstream direction of the dam which can cause catastrophic consequences.
The maximum discharge happens near the dam place at cross section number 1. One of the most important parameter in the risk management is the alarm time. In this
study we have calculated the break initiation time as well as the formation time. Furthermore these data can help us to determination the Property damage and loss lives. The above mentioned results used in the successive 2-D depth-averaged calculations with TELEMAC-2D. The following conclusions can be made based on the 2-D simulation results:
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The height of water and the maximum height of water which are important parameters in risk management fields are calculated to define inundation regions. These results also can be used to define warning zones according to the flooded areas.
The wave arrival time is analysed for 15 minutes intervals. This time model shows us how long it takes for the flood to reach each part of the downstream area which is useful for reducing loss of human lives in risk management studies.
Acknowledgements I would like to express my deepest appreciation to all those who provided me the possibility and help me to complete this report.
References [1] Janice Sylvestre, Developer, 2010, BREACH GUI (BREACHJ) Version 1-0-0 08-012010 [2] TELEMAC modelling system, 2D hydrodynamics, TELEMAC-2D software Version 6.0, USER MANUAL [3] A. S. Balasubramaniam, H. Cai, D. Zhu, C. Surarak and E. Y. N. Oh,Griffith School of Engineering Griffith University, Gold Coast, Australia, Settlements of Embankments in Soft Soils [4] AFTA 2005 Conference Proceedings, Placement of Riparian Forest Buffers to Improve Water Quality [5] Braja M.Das, 2008. Advance soil mechanic, third edition. This edition published 2008 [6] By Taylor & Francis 270 Madison Ave, New York, NY 10016, USA [7] 12th International Benchmark Workshop on numerical analysis of dam, 2013, Austria, Graz [8] Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method by Jean-Michel Hervouet(May 29, 2007) [9] CHC - Canadian Hydraulics Centre, National Research Council, 2010. Blue Kenue, Reference Manual, August 2010. [10] The NWS simplified dam-break flood forecasting model by Jonathan N.wetmore and Danny L.Fread (Reviesed 12/18/91) by Danny L.Fread, Janice M.Lewis, and Stephen M.Wiele
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Risk Assessment Analysis of a Hypothetical Dam Breach Using Adaptive Finite Element Methods Adaptive Hydraulic Model, ADH D. McVan1, J. Ellis2, G. Savant3 and M. Jourdan1 1Research Hydraulic Engineer, Engineer Research and Development Center, U.S. Army Corps of Engineers, Vicksburg, MS, 39180. E-mail: [email protected] 2Engineering Technician, Bowhead Science and Technology, LLC, and Onsite Contractor, Engineer Research and Development Center, U.S. Army Corps of Engineers, Vicksburg, MS 39180. E-mail: [email protected] 3
Research Water Resources Engineer, Dynamic Solutions LLC and Onsite Contractor, Engineer Research and Development Center, U.S. Army Corps of Engineers, Vicksburg, MS 39180. E-mail: [email protected]
Abstract A virtual computational test bed was developed by a committee sponsoring a session at the International Commission on Large Dam (ICOLD) 12th International Benchmark Workshop. The purpose of this virtual test bed was to serve as a platform for comparison of dam breach models and methodologies. This paper provides the results of one of the model comparisons. The ADH model, developed by the U.S. Army Engineer Research and Development Center (ERDC) was used to simulate a catastrophic dam failure and gradual dam failure. The ADH Model is described, as well as the domain, setup, and boundary conditions for the specific simulations. Results are provided as gridded and tabular data, as well as hydrographs. The gridded results include peak flood depths, flood wave arrival times, peak unit flow rates and total population at risk. Tabular results include flooded area and population as risk (including age demographics). Hydrographs are presented for both the dam failure discharge and at five down stream cross sections. Loss of life and economic impacts were not calculated.
Introduction Computational capacity, along with multi-processors computing techniques, has grown resulting in development of new models for flooding and consequence/risk assessment. These new models which are computing flood wave propagation at higher spatial resolutions can be coupled with risk assessment models which allow for more detailed analyses of the effects of these flows. This paper represents results of the numerical dam breach done using the virtual testbed developed for the International Commission on Large Dam (ICOLD) 12th International Benchmark Workshop. This virtual testbed was developed as a tool for comparison of different numerical model results and the model capabilities. The testbed includes a hypothetical dam above a hypothetical city named Hydropolis. The focus of this study was to estimate the consequences of failure of the dam. Demographics, infrastructure and dam 293
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characteristics were provided so that comparison of these models could be made which will aid in the development of simulation frameworks that can support dam risk analysis. Two model scenarios were modeled; the first model represents a catastrophic event in which the entire dam is destroyed instantaneously. The second model represents a gradual breach using Froehlich (1995a) equations.
Methodology Model Code Description ADH is a state-of-the-art code developed by the U.S. Army Engineer Research and Development Center (ERDC) to simulate both saturated and unsaturated groundwater, overland flow, three-dimensional Navier-Stokes flow, and two and three-dimensional shallow water equations, including super-critical flows and shock capturing (Berger el.al. 2010). The ADH code is parallelized and is capable of running on high performance computing systems. ADH is also an implicit code meaning its time step size is not limited by the element size as with explicit codes. These features make ADH computationally efficient for large-scale applications. ADH has been utilized to study varied phenomena such as estuarine circulation (McAlpin et al. 2009; Tate et al. 2010; Martin et al. 2010, Martin et al. 2011), riverine flow (Stockstill and Vaughan 2009; Stockstill et al. 2010), and dam breach (Savant et al. 2011). The two-dimensional shallow water (SW2) equations are used for the application presented here. This module solves the conservative form of the SW2 equations, allowing for local and global mass conservation. ADH is a temporally and spatially adaptive code. For temporal adaption, the code utilizes a variable time step such that failure to reach convergence at a given time step size does not end the simulation; rather, the time step size is reduced and the solve is attempted again (Savant et al. 2011). ADH also has the ability to allow continuous wetting and drying such that the flood front is computed and visualized accurately (Berger and Lee 2004; Savant and Berger 2011). For this model simulation, a maximum time step of 10 seconds was specified with smaller time steps being performed automatically as needed. Model Setup All hydrodynamic numerical models require certain basic input data to perform simulations. These data are bathymetric and topographic elevations for defining the model domain, inflow hydrographs or water elevations, as well as other boundary information such as rainfall and ocean water surface elevation (if the area of interest is in a tidal zone) for driving the model, as well as roughness characteristics and the ground surface and other parameterizations depending upon the objective of the modeling application. In the following sections, these domain construction data, driving data, and parameterization needs are addressed. Model Domain The bathymetric and topographic information for this workshop were provided from the ICOLD “Theme C” Formulation Team in the form of a DEM with 9.4 meter resolution. Two DEMs were provided, one representing the domain without the dam, and the other with the dam. The DEM covered an area that was approximately 250 km2. The hypothetical dam was located at coordinate 4499.66 meters and 6681.57 meters and has a crest width of 24 meters and a crest length of 360 meters. The river bed elevation at the base of the dam is 211 meters and the crest elevation is 272 meters with 3H:1V embankment slopes. The maximum storage capacity of the reservoir is approximately 38 million cubic meters.
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The ADH mesh was created using the Surface-water Modeling System (SMS) (Aquaveo 2009) meshing algorithms and consists of 13,790 triangular elements comprised of 7,138 nodes at the triangle corners. Both the catastrophic model and the gradual breach model used the same mesh. The dam was ‘built’ into the mesh at the provided coordinates specifying the width of the elements along the dam’s crest and determining the upstream and downstream length of the embankment. Element areas range from approximately 250 m2 to 25,000 m2, with the finest resolution located at the dam and along the channel and the larger areas located downstream in the lake. After the mesh was constructed, the separate DEMs were interpolated to the mesh. The DEM without the dam was interpolated to the mesh for the catastrophic model and the DEM with the dam was for the gradual breach model. Figure 1 shows the mesh at the dam site, the model domain and bathymetry. Model Boundary Conditions and Parameterization Numerical modeling of any type requires the specification of driving boundary information. For the dam breach model, this driving force is maximum volume of the reservoir at the crest elevation, 272.0 meters. This is achieved by creating initial depth conditions at each node. For the nodes representing the reservoir, the water surface elevation was entered and the initial water depth was determined from the mesh bathymetry. The rest of the mesh was
Figure 1: Model Domain. given a -0.1 meter water depth in order to simulate a dry mesh. In an attempt to prevent backup of the flow which may cause artificial inundation not created by the flooding, an evaporation boundary was applied to the downstream boundary that allows the water to be transported out of the domain. However, after the model was complete, it was observed that after the flood wave hits the boundary, the wave is reflected back into the model domain. This only occurred in the area of the lake and it was determined that it would not affected the area of interest. Bed roughness was an additional user specified parameter that defines the frictional resistance. ICOLD’s “Theme C” Formulation Team provided a gridded dataset representing the land use/land cover of the region. This data was used to identify areas of similarity and applied to the mesh as material types. Figure 2 shows the various material types used in the model. The elements and nodes associated with each material type will have the same material properties specified in the boundary conditions. Literature search describing the varying roughness coefficients for differing floodplains were used to determine the bed roughness for each material type (Arcement, G.J., Schneider, V.R., and Oregon Department 295
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of Transportation, 2005). Table 1 shows the material types used in the model along with its roughness value. Table 1: Bed Roughness Material Type Manning’s n Value Channel 0.018 Forest 0.100 Developed 0.020 Shrub/Cropland 0.050 Shrub/Forest 0.070 Lake 0.018 Reservoir 0.018 Dam Crest 0.018 Upstream Dam Face 0.018 Downstream Dam Face 0.018 Downstream Boundary 0.018 (Arcement, G.J., Schneider, V.R., ) (Oregon Department of Transportation, 2005) In the ADH numerical code, the Manning’s value is converted to an equivalent roughness height that depends on the instantaneous water depth at a particular node. The friction formulation is then derived from the logarithmic velocity profile based on open channel flow. Thus, while the user-specified frictional value (Manning’s number) is constant over the specified material type, the actual applied resistance is spatially varying according to the water depth and velocity magnitude at each node in the mesh. This results in higher frictional values for locations of shallower depths as is experienced in nature and can be extremely important when estimating flood extents.
Figure 2: Material Types For the catastrophic model, the dam does not exist in the mesh therefore the model’s initial conditions at time 0.0 simulated a wall of water at a depth of 61 meters. For the gradual breach, the dam does exist in the mesh. Equations developed by Froehlich (1995a) were used to determine the average breach width and the breach formation time and are shown below. Failure of the dam was assumed to be caused by overtopping resulting in 80-percent of the dam height removed or approximately 48.8 meters. Therefore, the maximum width of the breach was approximately 141 meters and the breach formation time was approximately 48 minutes (2940 seconds). (The height of the breach and the formation 296
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time were rounded to 49 meters and 49 minutes). To achieve the gradual breach, elevations of the nodes representing the dam were gradually lowered at specified time intervals. At the end of the breach, the elevations of the nodes representing the breach were 223 meters. (1) (2) Where:
Bave Ko Vw hb tf
= = = = =
Average Breach Width (m) Constant (1.4 for overtopping failures, 1.0 for piping) Reservoir volume at time of failure (m3) Height of the final breach (m) Breach formation time (hours)
Model Results Model Simulations The ADH model simulations for both the catastrophic and the gradual breach were performed on the SGI Altix Ice machine (Diamond) at the USACE ERDC (ERDC DSRC 2011). This machine uses an SLES (Linux) 11 operating system with a 2.8 GHz Intel Xeon Nehalem-EP Processor. The catastrophic model was run on 16 processors and simulated 6 hours 40 minutes (24,000 seconds) and required approximately 60 minutes to complete (wall time). The gradual breach was run on 12 processors and simulated 12 hours (43,200 seconds) and required approximately 2 hours to complete. Both models had output intervals of 100 seconds. In accordance to ICOLD’s “Theme C” Formulation Team, the information obtained for both the catastrophic event and the gradual breach were discharge hydrographs for the dam failure and along five cross-sections specified by the Formulation Team. The model also provided contours of the peak flood depths, peak unit flow rate, and the flood wave arrival times. Because ADH uses a mesh, the output results are also in the mesh format. In other words, each node in the mesh contains all the information for each time step. To determine the total population at risk including the age demographics as well as the flooded area, the ADH depth file were converted to a raster for each output time interval. The raster was then converted to a shapefile which contained the depth at 1 meter intervals. The Formulation Team specified 0.5 meter intervals, however due to the constraints of this methodology, only integer values could be used. The census block shapefile provided by the Formulation Team was clipped to the depth shapefile and the ratio of the clipped area compared to the entire area determined the total population potentially affected by the flood. Catastrophic Model Results As mentioned previously, the catastrophic model simulated the worst case scenario where the entire dam is removed instantaneously. The maximum discharge at the time of the breach was approximately 84,000 cms. At each cross section, the maximum discharge varied between approximately 61,000 cms and 81,000 cms. Figure 3 shows the discharge hydrographs for each location. The total volume at the reservoir was approximately 38.93 km3 and at each of the cross sections, the volume varied between 39.4 km3 and 40.9 km3. Table 2 shows the maximum discharge with the associated times of occurrence at the dam and for each of the 5 cross-sections along with the total volume for each location.
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Though ADH conserves fluid and constituent mass to machine precisions, for wet-dry problems this conservation is dependent on user specified tolerance parameters. These parameters control how accurately ADH solves the continuity and momentum equations using the Newton-Raphson iterative approach for solving a system of implicit equations. Very small tolerance parameters will drive the solution to conserve mass to machine precision but can significantly increase run time, large tolerances will significantly reduce run time but will not provide mass conservation or the correct results. A balance has to be achieved between the requirements of a particular project or simulation and whether exact mass conservation is required. For the simulations presented in this document the mass is conserved to 3-percent for catastrophic breach. ADH utilizes a diffusive wave type shock capturing scheme in wetdry areas, this scheme, for extreme wet-dry scenarios, can hide fluid mass in dry cells and is an additional, though rare, source of mass errors. All elements/nodes are included in computations at all times and no minimum ‘include in computation’ depth is used by ADH. The error shown in Table 2 shows how well ADH conserved the mass at each cross-section. Table 2: Maximum Discharge for Catastrophic Event
Location At the Dam X-Section 1 X-Section 2 X-Section 3 X-Section 4 X-Section 5
Time of Maximum Discharge (seconds) 1 26 207 293 385 546
Maximum Discharge (cms) 83,828 80,855 66,027 60,738 62,089 67,166
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Total Volume (km3) 38.93 39.37 39.47 39.69 39.65 40.91
Error 1.01 1.01 1.01 1.00 1.03
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Figure 3 Maximum Discharges for Catastrophic Event. The reservoir was completely emptied in approximately 50 minutes (3000 seconds). The flood wave was completely at the lake after 2.7 hours. As one would expect, the greatest depths occur within the channel. Maximum depths in the floodplain range between 2 and 12 meters. The unit discharge in the floodplain is less than 50 m2/s. Figures 4 and 5 show the maximum peak depth for the catastrophic model and the peak unit flow rate, respectively. The flood wave reaches the floodplain about 5 minutes after the dam is breached and reaches the downstream boundary at approximately 1.67 hours (100 minutes). Figure 6 shows the arrival time in 15 minute intervals.
Figure 4 Peak Flood Depth in meters for Catastrophic Event
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Figure 5 Peak Unit Flow Rate (m2/s) for Catastrophic Event
Figure 6 Flood Wave Arrival Times for Catastrophic Event The total flooded area at peak depths, excluding the area of the lake, was approximately 41.1 km2 and affected well over 28,000 people. The total population at risk after 180 minutes is over 26,000. These differences are the results of taking a single snap shot of the population in time. As the flood wave travels downstream, the inundated area covers a finite area. For the next time, the inundated area covers a completely different area. Sometimes these areas overlap which result in duplicating the population at risk and at other times there is a gap resulting in the population at risk not being accounted for. Therefore, when assessing loss of life and economic consequences, it is best to use the peak flood depth values. Figure 7 shows the clipped total population at risk for the peak depths. Table 3 shows the population at risk at 15 minute time intervals and includes age demographics and Table 4 shows the flooded areas at 1 meter intervals along with the total population at risk and the age demographics. For the 12-year and under age demographic, the total population at risk after 3 hours is over 4,300 and about 4,600 for the maximum flooded area. The 65-year and over age demographic has over 4,700 at risk after 3 hours and around 5,200 for the maximum flooded area. The total flooded area with depths greater than 8 meters is over 3.35 km2 which affects a total population at risk of less than 2,000. For the 12-year and under demographics, the total population at risk affected by depths greater than 8 meters is slightly over 400 and over 130 for the 65-year and over demographics.
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Table 3: Total Population at Risk for Catastrophic Event Time Intervals (minutes 15 30 45 60 75 90 105 120 135 150 165 180 Total
Total Population at Risk 3,179 12,516 9,807 629 33 19 13 12 12 11 11 11 26,253
12-yr and Under Population at Risk 815 1,936 1,449 97 7 4 2 2 2 2 2 2 4,320
65-yr and Over Population at Risk 225 2,487 1,910 132 1 1 0 0 0 0 0 0 4,759
Table 4: Flooded Area Population at Risk for Catastrophic Event Peak Flood Depth Range (meters) >/= 1 >/= 2 >/= 3 >/= 4 >/= 5 >/= 6 >/= 7 >/= 8 Total
Flooded Area excluding lake (km2) 41.09 32.37 17.55 9.73 5.94 4.56 3.82 3.35 41.09
Total Population at Risk 28,802 21,273 6,274 3,494 2,886 2,598 2,201 1,710 28,802
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12-yr and Under Population at Risk 4,641 3,429 1,207 835 710 635 535 413 4,641
65-yr and Over Population at Risk 5,208 3,994 1,044 327 220 202 167 136 5,208
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Figure 7 Total Populations at Risk for Catastrophic Event Gradual Breach Model Results Failure of the dam for the gradual breach scenario was assumed to be caused by overtopping resulting in 80-percent (49 meters) of the dam height removed. The maximum width of the breach was 141 meters and the breach formation time was 49 minutes (2940 seconds). Elevations of the nodes representing the dam were gradually lowered by 1 meter at 1 minute intervals. At the end of the breach, the elevations of the nodes representing the breach were 223 meters. The maximum discharge at the dam location was over 17,000 cms and occurred 33.33 minutes (2000 seconds) after the initial time of breach. The maximum discharges for the 5 cross-sections ranged from between 16,400 cms and 20,500 cms. Table 5 shows the maximum discharges at the dam and the 5 cross sections along with the calculated volumes of each hydrograph. Figure 8 shows the discharge hydrographs for each location. Since the lowest elevation of the final breach is 223 meters, the reservoir is assumed to retain a volume of approximately 0.36 km3. The volumes at the cross sections range from 38.5 km2 to 42.2 km2. As previously mentioned, for wet-dry problems, the conservation of mass is dependent on the tolerance parameters specified by the user which controls how accurately ADH solves the continuity and momentum equations. For the gradual breach simulation, the mass was conserved to 8-percent. The error in Table 5 shows how well ADH conserved the mass at each cross section. The maximum peak depths for the gradual breach inundated area are over 20 meters and occur in the channel. The floodplain depths range from 0.5 meters to less than 6 meters, with a few scattered pockets of slightly higher depths. The peak unit flow rates in the floodplain are less than 10 m2/s and in the channel, the peak unit flow rate is over 120 m2/s. The flood wave arrives at the floodplain about 45 minutes after the initial breach and reaches the downstream boundary after a little over 140 minutes. The flood wave was completely in the lake after 3.3 hours. Peak flood depth, peak unit flow rate and the flood arrival times are shown in figures 9 through figure 11, respectively. Table 5: Maximum Discharge for Gradual Breach Event
Location At the Dam X-Section 1 X-Section 2 X-Section 3 X-Section 4 X-Section 5
Time of Maximum Discharge (seconds) 2000 1690 2246 1029 1174 1409
Maximum Discharge (cms) 17,357 16,717 16,839 16,447 20,527 18,185
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Total Volume (km3) 38.93 38.54 39.03 40.19 40.82 42.21
Error 0.99 1.0 1.03 1.05 1.08
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Figure 8 Maximum Discharges for Gradual Breach Event.
Figure 9 Peak Flood Depth in meters for Gradual Breach Event.
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Figure 10 Peak Unit Flow Rate in square (m2/s) for Gradual Breach Event.
Figure 11 Flood Wave Arrival Times for Gradual Breach Event. The total flooded area, excluding the lake, was slightly over 20 km2. The total population at risk from this inundation is over 9,000 people. The highest population at risk occurs within the first 75 minutes of the breach and affects over 15,500 people. As mentioned previously, as the flood wave travels downstream, the single snap shots in time may over lap resulting in the duplication of the number of population at risk. This is the case for the gradual breach which results in the total population at risk at the end of 3 hours being double the total population at risk from the peak flooded area. Again, the more reasonable assessment of the total population at risk should be based on the peak depth flooded area. For the 12-year and under age demographic, the total population at risk after 3 hours is over 3,500 and half that, about 1,600 for the maximum flooded area. The 65-year and over age demographic has over 3,000 at risk after 3 hours and around 1,700 for the maximum flooded area. The maximum inundated area affected by depths greater than 8 meters is 1.99 km2. The total population affected by these depths is less than 300, with the 12-year and under population at risk slightly over 50 and under 30 for the 65-year and over demographics. Table 6: Total Population at Risk for Gradual Breach Event Time Intervals (minutes)
Total Population at Risk
12-yr and Under Population at Risk 304
65-yr and Over Population at Risk
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15 30 45 60 75 90 105 120 135 150 165 180
Total
217 2,871 2,966 4,795 4,812 1,664 800 182 82 32 30 38 18,489
32 738 760 882 728 233 114 18 6 3 3 4 3,523
29 208 223 890 1,024 395 199 62 28 8 8 10 3,084
Table 7: Flooded Area Population at Risk for Gradual Breach Event Peak Flood Depth Range (meters) >/= 1 >/= 2 >/= 3 >/= 4 >/= 5 >/= 6 >/= 7 >/= 8 Total
Flooded Area excluding lake (km2) 20.02 10.00 5.85 3.99 3.05 2.49 2.21 1.99 20.02
Total Population at Risk 9,226 4,107 2,768 2,121 1,302 326 401 272 9,226
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12-yr and Under Population at Risk 1,600 892 699 531 321 142 86 52 1,600
65-yr and Over Population at Risk 1,699 541 220 166 101 48 34 27 1,699
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Figure 12 Total Populations at Risk for Gradual Breach Event
Conclusion The results from the catastrophic breach event showed a total flooded area, excluding the lake, of 41.1 km2 which affects over 28,800 people. The total number of 12-year and under affected by the maximum inundated area is over 4.600 and the total number of 65-year and over is over 4,200. The flooded area with depths greater than 8 meters is 3.3 km 2 and affects about 1,700 individuals, over 400 are 12-year and under and over 130 are 65-year and over. The greatest number of the population at risk occurs within the first hour of the dam breach resulting in over 26,000 being affected. Three hours after the initial dam breach, the majority of the flood wave has reached the lake and no longer affects the city of Hydropolis. The gradual breach scenario assumed breaching was caused by overtopping resulting in 80percent of the dam being removed. The time to reach the maximum breach and the maximum width were determined using equations developed by Froehlich (1995a). The results showed a total flooded area excluding the lake of over 20 km2 which affects over 9,200 people. The number of 12-year and under affected by the maximum flooded area is 1,600 and the number of 65-year and over is 1,699. The flooded area with depths greater than 8 meters is 1.99 km2 affects less than 300 individuals with over 50 being 12-year and under and less than 30 being 65-year and over. The majority of the population at risk occurs within the first 90 minutes of the dam breach; however the total number is skewed because the inundated area for each time interval overlaps. These results represent the impacts of a dam breach, whether that breach is catastrophic or gradual. Though loss of life was not calculated using the described model, the population at risk was significant. Additionally, economic impacts were not considered, as this is not part of the applied model. Even without these values available for comparison, there are many data elements that will be useful in comparing the ADH model to other models presented at this workshop. The use of the virtual test bed should prove to be a valuable tool, both now and in the future, for comparison of such models.
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References [1] Aquaveo (2009). “Surface-water Modeling System Version 11.0”, Aquaveo. http://www.aquaveo.com/pdf/SMS_11.0.pdf. [2] Arcement, G.J., Schneider, V.R. “Guide for Selecting Manning’s Roughness Coefficients for Natural Channels and Flood Plains” U.S. Geological Survey Watersupply Paper 2339. http://www.fhwa.dot.gov/bridge/wsp2339.pdf [3] Berger, R.C. and Lee, L.M. (2004) “Multidimensional Numerical Modeling of Surges Over Initially Dry Land.” Coastal and Hydraulics Engineering Technical report, ERDC/CHL TR-04-10. Vicksburg, MS: U.S. Army Engineering Research and Development Center. [4] Berger, R.C., Tate, J.N., Brown, G.L., and Savant, G. (2010). “Adaptive Hydraulics: User Manual.” Vicksburg, MS: U.S. Army Engineering Research and Development Center. https://adh.usace.army.mil/. [5] Engineer Research and Development Center (ERDC) DoD Supercomputing Resource Center (DSRC), http://www.erdc.hpc.mil/docs/diamondUserGuide.html, accessed December 5, 2011. [6] Martin, S.K., Savant, G., and McVan, D.C. (2010). “Lake Borgne Surge Barrier Study.” Coastal and Hydraulics Engineering Technical Report, ERDC/CHL TR-10-10. Vicksburg, MS: U.S. Army Engineering Research and Development Center. [7] Martin, S.K., Savant, G. and McVan, D. (2011) “Two Dimensional Numerical Model of the Gulf Intracoastal Waterway Near New Orleans: Case Study”, Journal of Waterway, Port, Coastal and Ocean Engineering, doi:10.1061/(ASCE)WW.1943-5460.0000119. [8] McAlpin, T.O., Berger, R.C., and Henville, A.M. (2009). “Bush Canal Floodgate Study.” Coastal and Hydraulics Engineering Technical Report, ERDC/CHL TR-09-09. Vicksburg, MS: U.S. Army Engineering Research and Development Center. [9] Oregon Department of Transportation (2005). “Hydraulics Manual (Part 1); Chapter 8, Appendix A”. Engineering and Asset Management Unit Geo-Environmental Section. ftp://ftp.odot.state.or.us/techserv/GeoEnvironmental/Hydraulics/Hydraulics%20Manual/Chapter_08/Chapter_08_Appendix_A/ Chapter_08_Appendix_A.pdf [10] Savant, G., Berger, R.C., McAlpin, T.O., and Tate, J.N. (2011). “An Efficient Implicit Finite Element Hydrodynamic Model for Dam and Levee Breach.” Journal of Hydraulic Engineering, ASCE. Vol. 137, No. 9, pp: 1005-1018. [11] Savant, G and Berger, R.C. (2011) “Wetting and Drying Characteristics of Adaptive Hydraulics.” USACE, ERDC Technical Note, USACE/ERDC-11-DRAFT, In Progress. [12] Stockstill, R.L. and Vaughan, J.M. (2009). “Numerical Model Study of the Tuscarawas River below Dover Dam, Ohio.” Coastal and Hydraulics Engineering Technical Report, ERDC/CHL TR-09-17. Vicksburg, MS: U.S. Army Engineering Research and Development Center. [13] Stockstill, R.L., Vaughan, J.M., and Martin, S.K. (2010). “Numerical Model of the Hoosic River Flood-Control Channel, Adams, MA.” Coastal and Hydraulics Engineering Technical Report, ERDC/CHL TR-10-01. Vicksburg, MS: U.S. Army Engineering Research and Development Center. [14] Tate, J.N., Lackey, T.C., and McAlpin, T.O. (2010). “Seabrook Fish Larval Transport Study.” Coastal and Hydraulics Engineering Technical Report, ERDC/CHL TR-10-12. Vicksburg, MS: U.S. Army Engineering Research and Development Center.
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Assessment of ICOLD Benchmark Case Study using Flood2D-GPU and HEC-FIA B. A. Thames 1 and A. J. Kalyanapu 2 1
2
US Army Corps of Engineers, 801 Broadway, Nashville, TN 37203, USA Department of Civil & Environmental Engineering, Tennessee Technological University, 1020 Stadium Drive, Prescott Hall 334 Box 5015, Cookeville, TN 38505, USA E-mail: [email protected]
Abstract This paper presents a benchmark case study of a dam failure using Flood2D-Graphics Processing Unit (GPU), a two-dimensional hydraulic modeling software, coupled with the Hydrologic Engineering Center’s Flood Impact Analysis (HEC-FIA) software, a flood consequence model, to evaluate flood risk in a hypothetical test bed environment. The defined objective of this study is to present the findings prescribed by the ICOLD Theme “C” Formulation Team, which include inundation mapping, population at risk, loss of life, and direct economic impacts. The increased computational capability afforded by Flood2D-GPU allows for a much deeper analysis of the hypothetical scenario. Many studies have been conducted to evaluate the uncertainty in the estimation of breach parameters (breach width and time of failure); however, how breach parameter estimation uncertainty affects the uncertainty in consequence estimation has been rarely evaluated. A secondary objective of this study is to determine the affects of breach parameter estimation uncertainty on consequence estimation uncertainty through a comparative analysis of four breach parameter estimation (BPE) methods and a sensitivity analysis of the main breach parameters such as breach depth, breach width, time of failure, and breach side slope. Through these analyses, breach depth, typically assumed using engineering judgment, is found to be the most sensitive breach parameter. Similar breach formation times are computed for three of the BPE methods, Froehlich (1995a), Froehlich (2008), and Von Thun and Gillette, evaluated while varying breach widths are computed; however, the consequence results are similar despite the varied breach width computations highlighting the importance of breach formation time.
Introduction With the increased awareness of the infrastructure deficiencies within the United States and world-wide, dam safety and understanding the impacts of dam failures has become much more emphasized due to the large number of dams world-wide with population centers within the destructive path of dam failure discharges. Studies such as these strive to better define the hydraulics of a dam failure and the resulting flood impacts. Many US federal agencies, such as the US Army Corps of Engineers (USACE), the Department of Homeland Security (DHS), and the Bureau of Reclamation (USBR), are working diligently through research and practical analysis to better understand the mechanics of a dam failure. This case study provides insight into hydraulic and consequence modeling tools available to simulate a dam failure and evaluate possible impacts from flooding.
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Methodology Various modeling software are used to obtain the risk assessment results prescribed by the ICOLD formulation team for Theme C, which consist of an array of hydraulic and consequence-based products. To evaluate the proposed problem, a dam breach discharge hydrograph is first developed using the unsteady flow component of the Hydrologic Engineering Center’s River Analysis System (HEC-RAS) version 4.1. The dam breach discharge hydrograph is a necessary input into Flood2D-GPU, which leverages the computer’s graphics card for much improved computational capabilities of the model input [8]. Output from Flood2D-GPU, which includes inundated areas, depth grids, and flood wave arrival time grids, serve as inputs into HEC-FIA, which provides the consequence assessment for the solution to the problem statement. HEC-FIA is a single event GIS-based software that determines impacts from flooding, such as population at risk (PAR), economic damages, and loss of life (LOL) on a structure-by-structure basis [9]. Ultimately, a dam failure within the proposed hypothetical test bed is evaluated using this combination of modeling software through a comparative analysis of the four suggested BPE techniques and a sensitivity analysis of four breach parameters to evaluate the range of possible PAR, LOL, and economic damage outcomes and the sensitivity of these outcomes to breach parameter estimations. Development of Dam Failure Hydrograph (HEC-RAS) HEC-RAS is utilized to generate dam failure discharge hydrographs for input into Flood2DGPU. HEC-RAS is one-dimensional hydraulic model capable of performing unsteady flow dam failure simulations utilizing user-provided hydraulic inputs (cross-section and inline structure geometry, boundary conditions, Manning’s n-values, etc.) and breach parameters (breach opening dimensions and location, breach weir coefficients, breach formation time, etc.) [13]. The HEC-RAS model developed for this evaluation uses the information provided by the formulation team including a digital elevation model (DEM), a land use layer, and information about the dam embankment and failure scenario. Because the only purpose of the HEC-RAS model is the generation of a dam failure discharge hydrograph, the extent of the model is limited to the reach just adjacent to the proposed dam. The HEC-RAS model geometry is developed by processing the DEM using HEC-GeoRAS, a data pre- and post-processing software developed for HEC-RAS. The HEC-RAS model consists of a storage area to represent the reservoir above the dam, an inline structure representing the dam and the necessary bounding cross sections upstream and downstream of the dam. The stage-volume curve provided by the formulation team is utilized in the HECRAS model to represent the storage characteristics of the reservoir. The cross section geometry is developed directly from the DEM and imported into HEC-RAS using HECGeoRAS. The inline structure geometry is developed using a combination of information provided by the formulation team in the problem statement and the DEM. The most critical input into the HEC-RAS model is the determination of the breach geometry and parameters that include the final breach bottom width (Wb), final breach invert elevation (Elb), breach side slopes (SS), and breach formation time (tf). [16] conducted an uncertainty analysis of embankment dam breach parameters for all of the proposed BPE approaches provided by the formulation team with the exception of Froehlich (2008), which did not exist at the time of publication. This study focuses on four BPE methods: Froehlich 1995a [3], 310
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Froehlich 2008 [5], the MacDonald and Langridge-Monopolis (MLM) [10], and the Von Thun and Gillette (VTG) [15] methods, and the calculated or assumed values are presented in Table 1. [16] shows through an uncertainty analysis, that in predicting breach width, breach failure time, and peak flow through the breach, the Froehlich (1995a) technique produced the lowest mean prediction error and the narrowest width of uncertainty band for all three estimations. The only exception to this is the width of the uncertainty band for the breach width estimation which is only slightly larger than the VTG technique. Based on these results, the Froehlich (1995a) BPE technique emerges as the best compromise; however, the current study compares the four methodologies across the range of sensitivity for all methods to highlight the range of possible outcomes in terms of hydraulic and consequence results. In addition, this evaluation provides a better understanding of how breach parameter sensitivity affects consequence estimation uncertainty through a sensitivity analysis of four breach parameters, SS, Wb, Elb, and tf, for each BPE method. The sensitivity analysis approach involves computing these four breach parameters using each BPE method (Table 1) and adjusting these values to create a sensitivity range. SS, W b, and tf are increased and decreased by 25 and 50%. Because the initial assumption is that the breach height formation will extend to the toe of the dam for all BPE methods, the sensitivity range for Elb is developed by increasing the base assumption of 211m by 7.625m or one eighth of the total breach depth (hb) until the value reaches half of the total breach depth. Using the computed values from Table 1, a base condition breach failure hydrograph is developed for each BPE method using HEC-RAS serving as the basis for the comparative analysis. In addition, breach hydrographs are developed for the range of sensitivity values. Sixty eight simulations are computed to produce breach hydrographs for input into Flood2DGPU. In interest of space, the resulting breach hydrographs for the four base condition simulations are presented in Figure 1. Table 1: Computed Breach Parameter Estimates using the Four BPE Methods Parameter SS Wb hb tf
Froehlich (1995a) 1.4 62 m 61 m 0.7 hrs
Froehlich (2008) 1.0 110 m 61 m 0.6 hrs
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MLM (1984) 0.5 12 m 61 m 2.0 hrs
VTG (1990) 1.0 207 m 61 m 0.9 hrs
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50000 45000 40000 Froehlich 1995a Froehlich 2008 MLM VTG
Discharge (CMS)
35000 30000 25000 20000 15000 10000 5000 0 1
1.5
2
2.5
3
3.5
Time (hrs)
Figure 1: Computed Discharge Breach Failure Hydrographs from unsteady flow HEC-RAS Model for Base Simulations Two-Dimensional Hydraulic Modeling (Flood2D-GPU) In this study, a GPU flood model, named Flood2D-GPU, developed in NVIDIA's CUDA programming environment is used [8]. The modelling framework uses the 2D unsteady numerical flood model that solves the non-linear hyperbolic shallow water equations using a first-order accurate upwind difference scheme. These equations are developed from the Navier-Stokes equations by integrating the horizontal momentum and continuity equations over depth often referred to as the depth averaged or depth integrated shallow water equations (i.e., Saint Venant equations). The non-conservative form of the partial differential equations is [7;8;12]: h uh vh 0 ContinuityEquation(1) h uh vh t x y 0 ContinuityEquation(1) t x y u uuu uuuu vHuu gHH gS 0 Momentum Equation inxx--direction direction (2) u v u g v gS Equation in x - direction (2) g fx 0 gS fxfx 0 Momentum Momentum Equation in (2) t xtt yxx xyy xx v vvv vvv H vv gHH gS 0 Momentum Equation inyy--direction direction (3) u vuu gvv gS Equation in y - direction (3) g fy 0 gS fyfy 0 Momentum Momentum Equation in (3) t x y y t xt yx yy y where, h is the water depth, H is the water surface elevation, u is the velocity in the xdirection, v is the velocity in the y-direction, t is the time, g is the acceleration due to gravity, Sfx is the friction slope in the x-direction, and Sfy is the friction slope in the y-direction. The upwind finite difference numerical scheme is used to discretize governing equations (1-3), as it yields non-oscillatory solutions, through numerical diffusion [2;11]. A staggered grid computational stencil is used to define the computational domain with the water depth (h) in the centre of the cell and u and v velocities on the cell edges. The model requires a digital elevation model (DEM) to represent topography, Manning’s n for surface roughness representation and a flow hydrograph. The CPU environment in this module consists of the 312
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random number generator, and storage of input data and parameters and source code to calculate future time step using Courant condition, and implement hydrograph update. The GPU environment contains the computational engine that solves continuity and momentum equations, and boundary condition implementation. A 2D model is chosen because of its better representation of flood flow (especially in floodplains) simultaneous flood extent delineation and instantaneous flood velocities at all nodes in the computational domain. Flood2D-GPU is validated for accuracy and found to provide significantly reduced computational time (up to two orders of magnitude) compared to the same flood model implemented serially in a CPU-based environment [8]. Application of Flood2D-GPU in this study uses HEC-RAS dam breach hydrographs, Manning’s n map generated from the land use/cover data provided along with the digital elevation model (with dam burnt into the topography). The methodology of dam breach hydrograph is explained in previous section. To develop spatially variable Manning’s n surface map, the National Land Cover Dataset (NLCD) Land cover (LC) codes were used to estimate approximate Manning’s n values based on recommendations from [6]. Table 2 provides the LC code and its assigned Manning’s n value and Figure 2 depicts the resultant spatially variable Manning’s n map. Table 2: Estimated Manning’s values for National Land Cover Dataset NLCD LULC Class Name Open Water Developed, Open Space Developed, Low Intensity Developed, Medium Intensity Developed, High Intensity Barren Land (Rock/Sand/Clay) Deciduous Forest Evergreen Forest Shrub/Scrub Grassland/Herbaceous Pasture/Hay Cultivated Crops Woody Wetlands Emergent Herbaceous Wetlands
LULC Code 11 21 22 23 24 31 41 42 52 71 81 82 90 95
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Manning's n 0.025 0.016 0.030 0.030 0.050 0.025 0.120 0.200 0.070 0.050 0.030 0.035 0.160 0.110
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Figure 2: Land Use Land Cover Derived Manning’s Map for the case study Consequence Assessment (HEC-FIA) HEC-FIA is used to determine the consequences, such as PAR, economic damages, and LOL, using the results of the Flood2D-GPU simulations. [9] presents that “HEC-FIA is a standalone, GIS enabled model for estimating flood impacts due to flooding…that generates required economic and population data…to compute urban and agricultural economic flood damage, area inundated, number of structures inundated, population at risk, and loss of life.” Input requirements for HEC-FIA include a stream alignment, terrain grid, depth grid, arrival time grid, a structure inventory, and timing information for the purpose of evaluating a single event [9]. Based on the information provided by the formulation team, several assumptions and derivations from the provided data are made to develop the required inputs into HECFIA. ArcHydro Tools, an extension of ArcGIS Desktop, is used to create a stream centerline by processing the DEM. The depth and arrival time grids for all simulations are derived from the results of the Flood2D-GPU simulation. The depth grid required for HEC-FIA is the max depth grid; however, additional depth grids for the specified time intervals were derived for the required PAR, LOL, and economic damage results. The arrival time grid is assumed to be the time after failure at which the flood depth will reach two feet of depth. For this study, the structure inventory must include a structure name, occupancy type, damage category, structure value, and population information in order to derive the prescribed results. HEC-FIA utilizes two separate databases, a structure occupancy and global inventory table, to define the structure inventory against which all consequence estimations are made. The structure occupancy table consists of general occupancy types, such as residential, commercial, and industrial, and, for this study, is populated using occupancy types using the database that comes with Federal Emergency Management Agency’s (FEMA) HAZards United States – Multi-Hazard (HAZUS-MH) Tool [1]. The structure occupancy type table includes the content-to-structure value ratio for defining content values for each structure, links to depth-damage curves for structure and content values, and risk elevations for defining at what elevation an individual is at risk of fatality based on age. HEC-FIA only considers 314
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two age groups, those under and over the age of 65; however, to ascertain consequence results for populations under the age of 14, a separate HEC-FIA model is created for each simulation. The assumption is that anyone under the age of 14 would be accompanied by an adult and either carried vertically to safety or is mobile enough to move vertically to safety. The global inventory table consists of each structure, or parcel for this study, and specific parameters related to each structure such as depth-damage curves for the structure and contents, structure and content value, first floor elevations, and number of stories. This information is either user-defined or derived from other sources such as the DEM or through a relationship with the structure occupancy type table. For instance, the depth-damage curves for the structure and contents are adopted from the structure occupancy type table by way of the general occupancy type assigned to each structure by the user. The structure appraised values, population values for the various age groups, structure name, and number of stories is defined by the user in ArcGIS Desktop and imported into HEC-FIA. First floor elevations are derived from the DEM and increased by an assumed three feet to account for foundation height. The most time intensive process is related to the development of the global inventory. Spatially joining the parcel dataset layer with the census block layer in ArcGIS Desktop to compute the number of parcels per census block provides the ability to uniformly distribute the necessary population values for the age groups of interest, under 14, between 14 and 65, and over 65 years of age, into each parcel. The uniformly distributed population values are spatially joined back to the parcel dataset to attain an age group specific population value for each parcel. With no structure values and limited information regarding structure condition and square footage provided by the formulation team, the structure values are assumed for each occupancy type provided in the parcel dataset through collaboration with a US Army Corps of Engineers (USACE) economist to achieve reasonable values for each structure occupancy type (Table 3). Each structure is assigned a generic name based on the occupancy type. The structure name, population, structure values, and HAZUS occupancy type are compiled within ArcGIS Desktop and imported into HEC-FIA’s global inventory. Table 3: Estimated Structure Value Assumptions Zone A-1 C-1 C-2 O-1 P R-1 R-2 R-3 RA-1 R-D R-LT R-T SU-1 SU-2
Description Rural Agriculture - 1-ac Minimum Zone Neighborhood Commercial Zone Community Commercial Zone Office and Institutional Zone Parking Zone Single Family Residential Zone Residential Zone: Houses, Townhomes and Medium Density Apartments Residential Zone: Houses, Townhomes and High Density Apartments Residential and Agricultural Zone, Semi-Urban Area Residential and Related Uses Zone, Developing Area Residential Zone: Houses and Limited Townhomes Residential Zone: Houses and Townhomes Special Use Zone Special Neighborhood Zone, Redeveloping Area
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Estimated Value ($1000) 50 650 650 1000 750 125 1000 2500 75 100 750 500 1500 125
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A time window is needed for each HEC-FIA simulation, which defines a start, end, and warning issuance time for each simulation. The start time is based on the initiation of the breach. The assumption of no warning is made based on the scenario described of no cell phone service and no emergency action plan. [9] assert that warning time plays a critical role in preventing loss of life and property, and as such, is a very sensitive parameter in the HECFIA software. The sensitivity of warning time also emphasizes the importance of a robust warning system especially in situations of high risk dams just upstream of major population centers. PAR, LOL, and economic damages at specific time intervals are required results for this study. Because of the manner in which HEC-FIA treats incremental changes in PAR, LOL, and economic damages, depth grids, derived from Flood2D-GPU results, for each of these time intervals are simulated individually through HEC-FIA to develop these consequence results and post-processed to determine the PAR, LOL, and economic damages incrementally for each time interval. Because this is a labor intensive task, this time interval analysis is only conducted for the four base BPE method simulations. With all of the necessary information pre-processed and imported into HEC-FIA, two simulations, one for over and under the age of 65 and one for under the age of 14, are set up and run for each BPE method simulation, each sensitivity range, and for each required time interval totaling 184 HEC-FIA simulations. The results from these simulations are presented in the Results and Discussion section, and electronic copies of the required models, modeling outputs, GIS results, and spreadsheet results are provided supplemental to this document.
Results and Discussion Two-Dimensional Hydraulic Modelling Results In this report, the two-dimensional hydraulic modeling results using the base condition Froehlich 1995a BPE method are presented results and used in comparisons. Also, a sensitivity analysis of four breach parameters is evaluated and results are compiled and discussed. Additional results from these analyses and from other BPE methods are provided supplemental to this report. The Froehlich 1995a base simulation resulted in flood depths ranging from 0.5 m to nearly 8 m in the downstream areas of the dam. Figure 3 presents the map of the standard deviation in the peak flood depths from the multiple breach hydrographs which were derived from the sensitivity analysis of the four breach parameters. A high variation in the peak flood depths is noticed along the canyon immediately downstream of the dam. In the downstream region away from the mountainous terrain, smaller standard deviation in simulated peak flood depths is noticed that indicates less sensitivity due to the breach parameters. Thus it is likely that variability in consequence estimates including direct flood damages, LOL, PAR in the downstream region may be smaller compared to the upstream region of the floodplain. A similar trend is also observed in the maximum flood velocities for these breach hydrographs.
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Figure 3: Standard deviation map of the peak flow depths from the dam breach hydrographs developed for the four breach parameters using Froehlich 1995a BPE method Figure 4 presents the simulated flood discharge hydrographs at the five required cross-section locations. For the Froehlich 1995a BPE method, the variability of flood discharge from all the sensitivity runs are represented by the error bars. It is observed that a large variability in the magnitude of flood discharge is observed at the beginning of the dam breach. This is attributed to the influence of breach parameter sets used for generating the hydrographs. This is a clear indication that the consequence analysis, especially the LOL, PAR, direct economic damages during the initial 60 minutes of the breach, will be significantly affected by the choice of breach parameters used. With progression of time, the variability decreases at all cross-section locations.
Figure 4: Simulated flood discharge plots at the five cross-section locations for the Froehlich 1995a BPE method. Error bars indicate the variability of discharges due to various breach parameter sets. 317
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Consequence Assessment Results For the purposes of presenting results and making comparisons, the consequence results using the base condition Froehlich 1995a BPE method is considered the basis of the comparative analysis. In addition to this comparative analysis, a sensitivity analysis of four breach parameters are evaluated for each BPE method and results are compiled and discussed in this report. Additional results from these analyses are provided supplemental to this report. The Froehlich 1995a base simulation results in a PAR of 13976 persons with 2223 persons (about 16% of total PAR) under the age of 14 and 2695 persons (about 19% of total PAR) over the age of 65. HEC-FIA estimates a fatality total of 1914 persons under these simulation conditions, which is almost 14% of the total PAR. In reality, the total LOL seems to be inflated and is probably overestimated when considering the resilience and resourcefulness of a typical population. However, the assumption of no warning time appears to be the largest source of the elevated LOL totals, and a sensitivity of this assumption could provide a large range of possible outcomes for LOL. For instance, a warning just 15 minutes in advance of the breach reduces the total estimated LOL by 55% to 862 persons. A warning 1 hour in advance of the breach results in an estimated LOL of 511 persons or a 73% reduction. Direct economic damages are estimated using HEC-FIA using a depth of flooding derived from the depth grid and a first floor elevation for each structure in combination with a depthdamage curve. The total economic damages estimated by HEC-FIA are approximately $361 million dollars; however, this value is based strongly on the structure value assumptions made based on occupancy type and the content-to-structure value ratio derived from the FEMA HAZUS-MH Tool [1]. For the Froehlich 1995a base simulation, Table 4 shows incremental PAR, LOL, and direct economic impacts for specified time intervals. The PAR does not become affected by flooding until 30 minutes after the initiation of the dam breach based on flood wave propagation estimated by Flood2D-GPU and the population provided by the census block dataset. However, during the 30-60 minute time interval following the breach a large percentage of the PAR becomes affected and the greatest LOL occurs during this period. The greatest percentage of the PAR actually becomes affected in the 120-180 minute time interval due to an area of low population in the center of the damage reach. Interestingly, the LOL during the 120-180 minute time interval is not substantial even with the larger PAR due to lower flood depths allowing individuals to retreat vertically to safety within each structure. The decreased flood depths also explain why the direct economic damages are also greatest in the 30-60 minute time interval as opposed to the 120-180 minute time interval. Table 4: Incremental Time Interval Results for the Froehlich 1995a Base Simulation Time Interval (min)
Total PAR
0-15 15-30 30-60 60-90 90-120 120-180 > 180 Total
0 0 2915 512 1410 7638 1501 13976
14-yr and Under PAR 0 0 755 122 168 956 222 2223
65-yr and Over PAR 0 0 210 31 347 1835 273 2695
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LOL
Direct Economic Impact ($US)
0 0 1549 329 5 7 24 1914
$0.00 $0.00 $128,971,979.93 $77,632,731.34 $42,877,015.97 $85,161,297.09 $26,714,802.91 $361,357,827.24
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The majority of the flooding by area occurs at the lower depths of flooding (0 to 1.5m) with a majority of the PAR contained within these same lower depths of flooding. Table 5 illustrates the distribution of PAR and flooded area across the various depth ranges. Table 5: Incremental Depth Range Results of PAR and Flooded Area Peak Flood Depth Range (m) 0 - 0.5 0.5 - 1 1 - 1.5 1.5 - 2 2 - 2.5 2.5 - 3 3 - 3.5 3.5 - 4 4 - 4.5 4.5 - 5 5 - 5.5 5.5 - 6 6 - 6.5 6.5 - 7 7 - 7.5 7.5 - 8 >8 Total
Flooded Area (m²)
Total PAR
14-yr and Under PAR
65-yr and Over PAR
3498463 6433454 5310192 3706161 2874378 2379332 1600888 756803 427789 337724 372655 378761 252956 247030 200874 198091 2305519 31281068
2786 4993 2056 887 424 262 107 68 69 115 217 249 155 282 228 245 827 13976
447 664 267 89 49 35 29 20 19 32 59 71 43 78 61 61 199 2223
464 1079 488 286 126 71 8 4 5 8 15 17 11 22 16 12 62 2695
Supplemental results for the various simulations are provided electronically. When initially choosing a BPE method, the question of which technique provides the most accurate results is raised driving the need for a comparative and sensitivity analysis. The goal of this evaluation is to understand how the uncertainty of these methods and the sensitivity of the critical breach parameters affect the results of a consequence analysis. Table 6 summarizes the consequence results of both the comparative and sensitivity analysis. For the comparative analysis, the results from the Froehlich 1995a, Froehlich 2008, and VTG base simulations are extremely similar while the results of the MLM base simulation are much lower than the other three simulations, which is expected based on the dam failure hydrographs computed using HEC-RAS. The mean results for all parameters is typically only slightly less than the base simulation results. In addition, the base simulation results are typically much closer to the higher end of the range of values from the sensitivity analysis. A closer look at the results of the breach 319
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invert sensitivity analysis shows that consequence results are very sensitive to the breach invert assumption. These breach invert simulations are skewing the range of consequence results on the lower end of the range while also lowering the mean of the sensitivity simulations. The large values for standard deviation for all consequence results can also be attributed to the breach invert simulation runs. For the Froehlich 1995a sensitivity analysis, removing the breach invert sensitivity simulations from the statistical calculations produces a much tighter range for all consequence results with a 66% reduction in total PAR standard deviation, a 68% reduction in LOL standard deviation and an 80% reduction in direct economic impacts standard deviation, which indicates that these reductions in standard deviation are also due to the sensitivity of the breach invert parameter. In terms of sensitivity, breach invert is, above all, the most sensitive breach parameter; however, when comparing the breach hydrograph and consequence results of the various BPE methods, time of formation also drives the shape and peak of the breach hydrograph, which is directly related to flood consequence results. In fact, the difference in breach hydrograph peak and shape between the MLM and other three BPE methods, which is quite significant, is primarily due to the longer time of formation computed for MLM. In interest of space, many of the results discussed are not presented in this document; however, the results are provided supplemental to this report electronically.
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Table 6: HEC-FIA Base Simulation Results for Comparison and Sensitivity Simulation Statistics
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Conclusions This paper presents a solution to the problem statement using Flood2D-GPU coupled with HEC-FIA offering possible hydraulic and consequence outcomes under the hypothetical scenario provided. The comparative analysis of various BPE methods provides recognition of the similarities and differences between the methods. The sensitivity analysis of the breach parameters demonstrates the importance of specific parameters by illustrating the effects of these parameters on not only the hydraulic outputs but also the consequence results. These analyses combined begin the conversation into utilizing the most appropriate BPE method and focusing in on which breach parameters are the most critical to approximate the most accurate hydraulic and consequence results in a study of this nature. The results of the comparative analysis suggests that similar consequence results can be expected when using the Froehlich 1995a, Froehlich 2008, and VTG breach parameter estimation methods but, without additional case studies using this same approach, relative replication of these results is not substantiated. However, all of these approaches are based on straightforward regression relationships and should produce similar results. The MLM BPE method is similar in many aspects to the other three BPE methods with the exception of the inclusion of a volume of material eroded term. Additional case studies could provide insight into the differences of results experienced during this study. In general, the results of the sensitivity analysis follow the expected trends. As breach depth, width, and side slopes increase and the time of breach formation decreases, the breach discharge hydrograph shape becomes more compressed resulting in higher peak discharges, which ultimately results in more significant flood impacts and consequences. Several conclusions can be drawn from these analyses. Final breach invert is very difficult to estimate and is an assumed value not calculated by any of the BPE methods. Unfortunately, the results of this analysis show that the breach hydrograph and consequence results are extremely sensitive to the breach invert assumption. When conducting a risk-based assessment of a dam failure, special care should be taken to ensure that the best estimation of breach invert is assumed based on the sensitivity of breach hydrograph and consequence outcomes to the estimation of this parameter. Ultimately, additional research could provide a better approach to estimating this parameter drastically reducing the uncertainty in these types of analyses. When comparing the breach parameter estimations of SS, ElB, Wb, and tf, using the three BPE methods that produced similar consequence results, with the results of the sensitivity simulations, the breach formation time proves to be a critical parameter in the development of breach discharge hydrographs. Even with a Wb varying between 49 and 146 m, the resulting breach hydrograph shape and peaks are similar due to the similarity in formation time, which ranges from 0.6 to 0.9 hours.
References (arranged in alphabetical order) [1] Department of Homeland Security, Federal Emergency Management Agency (2007), Multi-Hazard Loss Estimation Methodology Flood Model HAZUS-MH MR3 Technical Manual, Washington, D. C. [2] Ferziger, J.H., and Peric, M. (2002). “Computational methods for fluid dynamics, 3rd edition” Springer, New York. 322
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[3] Froehlich, D. C. (1995a). “Embankment dam breach parameters revisited.” Water Resources Engineering, Proceedings of the 1995 American Society of Civil Engineers Conference on Water Resources Engineering, New York, 887–891. [4] Froehlich, D. C. (1995b). “Peak outflow from breached embankment dam.” Journal of Water Resources Planning and Management, American Society of Civil Engineers, 121(1), 90–97. [5] Froehlich, D. (2008). “Embankment Dam Breach Parameters and Their Uncertainties.” Journal of Hydraulic Engineering, 134(12), 1708–1721. [6] Hossain, AKM Azad, Yafei Jia, and Xiabo Chao. (2009). "Estimation of Manning's roughness coefficient distribution for hydrodynamic model using remotely sensed land cover features." Geoinformatics, 2009 17th International Conference. [7] Judi, D. R., Burian, S. J., and McPherson, T. N. (2010). “Two-dimensional fast-response flood modeling: Desktop parallel computing and domain tracking.” Journal of Computing in Civil Engineering, in press. [8] Kalyanapu A. J., Shankar S., Stephens A., Judi D. R., Burian S. (2011). “Assessment of GPU computational enhancement to a 2D flood model” Journal of Environmental Modeling and Software, 26, 1009-1016. [9] Lehman, W. and, Needham, J. (2009). “Consequence estimation for dam failures.” Hydrologic Engineering Center, US Army Corps of Engineers. [10] MacDonald, T. C., and Langridge-Monopolis, J. (1984). “Breaching characteristics of dam failures.” Journal of Hydraulic Engineering, 110(5), 567–586. [11] Patankar, S.V. (1980). Numerical heat transfer and fluid flow, Taylor and Francis. [12] Tingsanchali, T. and Rattanapitikon, W. (1999). “2-D modeling of dambreak wave propagation on initially dry bed.” International Journal of Science and Technology, 4(3), 28-37. [13] United States Army Corps of Engineers. (2010). HEC-RAS River Analysis System Users Manual, Hydrologic Engineering Center, US Army Corps of Engineers. [14] United States Army Corps of Engineers. (2012). HEC-FIA Flood Impact Analysis - Users Manual, Hydrologic Engineering Center, US Army Corps of Engineers. [15] Von Thun, J. L., and Gillette, D. R. (1990). “Guidance on breach parameters.” Internal Memorandum, U.S. Dept. of the Interior, Bureau of Reclamation, Denver, 17. [16] Wahl, T. L. (2004). “Uncertainty of Predictions of Embankment Dam Breach Parameters.” Journal of Hydraulic Engineering, 130(5), 389-397.
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Dam Failure and Consequence Assessment with Standardized USACE MMC Procedures International Commission on Large Dams 12th International Benchmark Workshop on Numerical Analysis of Dams (Theme C: Computational Challenges in Consequence Estimation for Risk Assessment) D. Williams1 and K. Buchanan2 U.S. Army Corps of Engineers, Tulsa, Oklahoma, USA U.S. Army Corps of Engineers, Huntington, West Virginia, USA 1
2
Email: [email protected]
Abstract The U.S. Army Corps of Engineers’ (USACE) Modeling, Mapping and Consequences (MMC) Production Center has submitted an entry for the International Commission on Large Dams (ICOLD) 12th International Benchmark Workshop on Numerical Analysis of Dams (Theme C: Computational Challenges in Consequence Estimation for Risk Assessment). As part of this exercise, standard MMC processes were used to develop hydraulic modeling of a dam break inundation as well as the associated consequences. Modeling techniques included one- and two- dimensional numerical models, use of geographic information systems, and economic and life loss estimation. Breach parameters were developed from standard regression equations. Based on the rules and regulations that USACE has established for the estimation of risk, economic consequences and life loss modeling results from this breach analysis corresponded to a high hazard dam. This analysis demonstrates the standardized MMC process for consistently evaluating all USACE dams while providing the flexibility to analyze projects that range from small and simple to large and extremely complex.
Introduction The Modeling, Mapping and Consequences (MMC) Production Center supports the U.S. Army Corps of Engineers (USACE) Institute for Water Resources (IWR) Risk Management Center (RMC) and the HQ USACE Office of Homeland Security, Critical Infrastructure Protection and Resilience (CIPR) program by analyzing the potential consequences of dam and levee infrastructure failures. The mission of the MMC includes production of hydraulic models, consequences estimates, and inundation maps for USACE dams and levees in support of these programs. All products are developed in accordance with MMC standard operating procedures. The intent of MMC products is to support a risk-based assessment, prioritization, and management framework for USACE CIPR, Dam Safety, and Levee Safety programs. Analysis of the risk associated with dam and levee failure events allows decision makers to prioritize investments to reduce the risk to life and property. MMC dam safety analysis is typically based on five failure and five non-failure dam breach scenarios needed for CIPR and Dam Safety program evaluations. Breach parameters for earthen dams (e.g. average breach width, breach formation time, and side slopes) are calculated using the following empirical equations: MacDonald and Langridge – Monopolis (1984) [1], Von Thun and Gillette (1990) [2], Froehlich (1995) [3], and Froehlich (2008) [4]. Results from all of these methods are compared, and results from a single empirical equation 325
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is selected for the analysis. Parameters from multiple equations are not considered to be interchangeable. Hydraulic modeling is performed using either HEC-RAS [5] (one-dimensional) or FLO-2D [6] (two-dimensional) unsteady flow analysis. The use of FLO-2D in the MMC process is generally restricted to areas where the one dimensional flow assumption utilized in HEC-RAS is not appropriate; such as coastal plains, alluvial fans, flat dense urban areas, and other large flat unconfined floodplains. Both models provide unsteady flow simulation. Steady flow models are not permitted within the MMC process. Following the development of the hydraulic model, the output from the dam failure analysis is processed using ArcGIS [7]. Inundation depth grids and inundation boundary shapefiles are generated, and these files are used in conjunction with structure and population data from tax parcels, point shapefiles, or the Federal Emergency Management Agency (FEMA) HAZUS [8] dataset to develop a HEC-FIA (Flood Impact Analysis) [9] model. The structure information is compared with the extent and timing of the flooding to estimate damages and life loss. This information (in the form of a consequence assessment report), along with an atlas of the downstream inundation, is provided by the MMC to USACE district offices for emergency action planning.
Project Description The MMC is participating in the International Commission on Large Dams (ICOLD) 12th International Benchmark Workshop on Numerical Analysis of Dams (Theme C: Computational Challenges in Consequence Estimation for Risk Assessment). Participants in Theme C are free to select the type and sophistication of modeling used to develop flood inundation areas and consequences estimates. When possible, the standard MMC process [10] was followed in the development of the dam break modeling and consequence estimation for this project. The numerical problem proposed for the workshop consists of estimating the consequences of failure of a dam in a mountain valley above the fictitious city of Hydropolis. The hypothetical dam, which was constructed for flood control, is 3.5 kilometers upstream from the city. Sandy clays and clayey sands are the predominant material within the rolled earth fill dam. With a 61-meter vertical distance from toe to crest, the structure has a high head differential when the reservoir is completely filled. The corresponding volume of the reservoir at the dam crest elevation is 38,276,000 cubic meters [11] (Figure 1).
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Figure 1: Storage-elevation curve for the reservoir. This exercise examines the downstream consequences of a catastrophic failure of the dam resulting from overtopping. Expected outcomes from this study include gridded data (peak flood depth, flood wave arrival time, peak unit flow rate, population at risk, loss of life, flood severity, direct economic impact), tabular data (flooded area, population at risk, loss of life, direct economic impact), and hydrographs (dam failure discharge, cross-sectional hydrographs). All of these products except for the peak unit flow rate are standard deliverables in a MMC dam breach analysis.
Model Development Modeling for this exercise was developed using both one- and two-dimensional techniques that are consistent with MMC standard operating procedures. Initially, only a HEC-RAS hydraulic model was planned for this study. This model computes flood wave parameters at cross-sectional nodes along a one-dimensional channel. Inspection of the terrain grid provided for this study revealed that the use of HEC-RAS for the entire downstream reach would be inappropriate since Hydropolis is located within a relatively flat plain below the base of the mountain range. In this reach, it was determined that a two-dimensional model would be more appropriate; therefore, FLO-2D was also used. Existing terrain downstream from the dam was provided by the Theme C formulation team. This dataset was provided in a gridded format with a cell width of 9.476 meters. No projection was assigned to the terrain dataset, but all geospatial data provided by the formulation team was in the same unknown coordinate system. 327
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The terrain dataset was used to extract cross sections in ArcGIS 10 (HEC-GeoRAS [12]) for use in a HEC-RAS model. The spatial extent of this model was restricted to 200 meters below the dam (modeled as an inline structure) as it was used to generate a dam breach hydrograph for a FLO-2D inflow boundary condition (Figure 2). The HEC-RAS model was run as an unsteady flow simulation with a computation interval of 5 seconds. A breach hydrograph was written to file with a 1-minute output interval, which was sufficient for defining the dam breach hydrograph.
Figure 2: HEC-RAS model geometry for breach hydrograph determination. Parameters from the MacDonald and Langridge – Monopolis empirical breach equation were not specifically considered in this analysis since it is considered to be an envelope equation, although parameters from the other equations were checked to ensure that they fell within the envelope range. The Froehlich (1995), Froehlich (2008), and Von Thun and Gillette equations were carried forward as unique scenarios in the HEC-RAS model. Since each of the three methods estimated a short breach formation time, a computational time step of 5 seconds was used in the HEC-RAS model in order to capture the peak discharge. Following the initial model development with the Froehlich (1995), Froehlich (2008), and Von Thun and Gillette breach parameters, the Froehlich (1995) equation was carried forward as the preferred method. Froehlich (1995) analyzed 63 earthen, zoned earthen, earthen with a core wall, and rockfill dam failures [3]. Data from these events was used to develop regression equations for average breach width ̅ (meters) and breach formation time tf (hours): ̅
(1) (2)
For this exercise, K0 = 1.4 (overtopping failure). The reservoir volume Vw (38,276,344 m3) and breach height hb (61 meters) were provided by the organizers of the benchmark workshop. This method provided a short breach development time coupled with a relatively wide breach width (Table 1): Table 1: Breach parameters for hypothetical dam Formation Time (hours)
Breach Equation 328
Bottom Width (m)
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MacDonald and Langridge-Monopolis Froehlich (1995) Froehlich (2008) Von Thun and Gillette
1.8 0.7 0.6 1.5
5 62 39 177
A breach hydrograph generated in HEC-RAS was then used as the inflow boundary condition in a FLO-2D model (Figure 3). The model assumed a static pool elevation of 272 meters with no inflow. Overtopping of the dam crest was the prescribed mode of failure. Unlike the onedimensional flow model (HEC-RAS), FLO-2D computes flow across a network of grid cells with eight possible directions of flow into or out of each cell. This type of flow distribution makes FLO-2D well-suited for analysis of the flood wave downstream from the dam.
Figure 3: Breach hydrograph resulting from Froehlich (1995) parameters Faster computation time of the FLO-2D model required a grid-cell width larger than 9.476 meters. By choosing an integer multiple of the terrain grid-cell size, conversion between the two could be simplified when resampling. Therefore, a computational grid-cell dimension of 94.76 meters (square) was selected. Elevation values were interpolated for the larger computational grid dimensions when imported into FLO-2D. Surface roughness values were developed by converting a National Land Cover Database (NLCD) shapefile [13] to corresponding Manning’s ‘n’ values [14]. The NLCD shapefile was provided by the Theme C formulation team. Once a new surface roughness shapefile was created, the values were interpolated for the computational grid dimensions when imported into FLO-2D. The FLO-2D model was run with a simulation time of 12 hours and an output interval of 5 minutes. Since the downstream reach is both short and steep, the simulation time was more than sufficient for routing of the breach hydrograph through the entire study area. An output 329
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interval of 5 minutes was necessary to capture the peak of the flood wave at downstream locations. A depth grid, flood wave arrival time grid, and flood severity grid were generated by using MAPPER [6], a post-processing tool for FLO-2D models. Peak unit flow rate is not a FLO2D output. This variable was therefore determined by taking the maximum flow rate for each grid cell and dividing by the computational grid-cell width. Cross-sectional hydrographs, which were required at five locations downstream from the dam, were developed by setting those locations as outflow nodes in sequential FLO-2D model runs. The time-series discharge for each cross-sectional outflow grid cell was then summed to construct a representative hydrograph for each key location. This was necessary because FLO-2D does not generate time-series discharge for grid cells within the computational domain if no channel element is used; it does write this output to file for outflow nodes, however. Hydraulic modeling results were then used for the determination of consequences based on a structure inventory. A structure inventory was developed using three datasets provided for the workshop [15]. These included a parcel shapefile with zoning information, a census block shapefile with population and employment information, and a land use grid. Typically, the economic and life loss analysis in a MMC study includes the utilization of a FEMA HAZUS dataset supplemented by imagery, land use, or parcels. Since this was not provided for the benchmark exercise, non-standard methods were used to estimate consequences. The following assumptions were required for the creation of a structure inventory with population values in HEC-FIA:
Since the dam fails at 11:00pm on a Saturday, all residential populations are in their homes. Due to the day and time, only 10% of the workers in certain job types are working. These included the following jobs: Mining, Quarrying, and Oil and Gas Extraction; Utilities; Manufacturing; Transportation and Warehousing; Administrative and Support and Waste Management; Health Care and Social Assistance; Arts, Entertainment, and Recreation; and Accommodation and Food Services. All workers come from census blocks outside the inundation area (not enough data to model movements from one census block to another). All workers are under the age of 65. All data was up to date and accurate (parcels, blocks, and land use).
Once the structure inventory and hydraulic data for the dam breach had been developed, they were used to develop the HEC-FIA model. Depth grids and arrival time grids were developed from FLO-2D output and imported into the model. The arrival time grids represented the time at which the water depth at each cell reached 0.6 meters (approx. 2 feet). Impact areas were created by reclassifying the arrival time grid into arrival time zones required for reporting results (0-15min, 15-30min, etc.). A polygon was created from that file. Some features had to be manually merged to create a unified impact area polygon for the HEC-FIA model. The structure inventory was created from the parcel file with populations and values developed in the pre-modeling phase. Structure placement was at the centroid of each parcel and was done within HEC-FIA. For estimates of necessary evacuation time for each structure, a hazard boundary shapefile was created representing the boundary where the inundation becomes 0.6 meters (2 feet) or less, which is considered a safe zone. Evacuation was modeled as a path from the structure to the nearest point on this boundary at an average speed of 16 kilometers per hour. 330
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The warning time used for the loss of life calculation was set at 30 minutes after the breach initiation. The problem statement specified that the failure was sudden and unexpected, and that witnesses within the canyon had no mobile phone service. A warning would not be issued until after the water reached the urban area at the end of the canyon, approximately 20 minutes after the failure. The warning curve used was the HEC-FIA default curve for the United States Emergency Alert System, and the mobilization curve was the default HEC-FIA curve included in the program. The HEC-FIA program evaluates damages with depth-damage curves based on occupancy type. Population distribution is also performed by occupancy type. To determine occupancy types, a cross walk table was created that assigned each parcel zone description to a specific occupancy type typically used in a HEC-FIA structure inventory. Populations and jobs were provided for each census block and broken down by category. To facilitate the distribution to individual parcels, the populations were combined into the following categories by census blocks:
Resident Population under age 14 Resident Population under age 65 Resident Population over age 65 Working Population (10% of the workers from industries potentially in operation at 11pm on a Saturday)
In order to determine the amount of population for each structure, each occupancy type was assigned a number of residential households or working households. Then for each census block, the residential populations were divided by the total number of residential households in the block and the working populations were divided by the total number of working households in the block. The resulting residential and working household sizes per census block were then used to create the populations for each parcel based on the number and type of households assigned to the occupancy type of the parcel. For example, if the residential household size in a census block was 2.5, then a single family residential structure in that block would get 2.5 people while a five to nine unit multiple family dwelling in the same census block would get 7 households, or 17.5 people. Table 2 shows the final occupancy types used and the number of assigned households. In the beginning there were more occupancy types included, but the ones listed below represent all of the types included in the final inventory after being clipped to the inundation boundary. The methodology, including the number of households assigned to each occupancy type, is representative of a typical HAZUS based structure inventory used by the MMC. Table 2: Population distribution by occupancy type Occupancy Type COM1 COM4 GOV1 RES1-1SNB RES1-2SNB RES1-3SNB
Occupancy Type Description Average Retail Average Prof/Tech Services Average government services Single Family, 1 Story no Basement Single Family, 2 Story no Basement Single Family, 3 Story no Basement 331
Residential Households 1 1 1
Working Households 1 1 -
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RES3AI RES3CI RES3DI
Multi Family Dwelling, Duplex Multi Family Dwelling, 5-9 Units Multi Family Dwelling, 10-19 Units
2 7 14.5
-
No data was provided on structure values in the parcel dataset. A standard USACE dam failure consequence estimate will often use the National Structure Inventory (NSI) developed by the USACE Hydrologic Engineering Center and based on HAZUS and Census data. The team decided to use average values from the NSI for each occupancy type to determine structure values. The NSI data is broken down by county, so a national average would have required significant effort. Instead of using a national average, the team reviewed data on housing values from the 2011 American Community Survey and determined that the state of Maine had the closest housing value to the national average. Data from the NSI for each county in Maine was combined and average values for each occupancy type were calculated. Vehicle and content values were also assigned (Table 3). Table 3: Average values by occupancy type Occupancy Type COM1 COM4 GOV1 RES11SNB RES12SNB RES13SNB RES3AI RES3CI RES3DI
Occupancy Type Description Average Retail Average Prof/Tech Services Average government services Single Family, 1 Story no Basement Single Family, 2 Story no Basement Single Family, 3 Story no Basement Multi Family Dwelling, Duplex Multi Family Dwelling, 5-9 Units Multi Family Dwelling, 10-19 Units
Average Structure Value $545,888
Average Content Value
Average Vehicle Value
$545,888
$21,819
$550,027
$550,027
$22,707
$481,998
$481,998
$27,020
$122,061
$61,030
$14,825
$121,957
$60,528
$14,766
$127,135
$63,568
$15,175
$206,625
$103,312
$19,795
$753,610
$376,805
$26,195
$1,288,765
$644,383
$29,931
Since the parcel shapefile appeared only to have information about zoning and not specific information about structures, the team assumed that not every parcel had a structure on it. Also, some parcels had “unknown” zone information so those parcels were removed from the final inventory. Information in the census blocks and the land use file also needed to be evaluated to inform the structure inventory development. One particular issue involved a large number of parcels in the middle section of the study area. These parcels appeared to be subdivided into mostly single family home lots. However, the census block data showed no residential population in those areas. Figure 4 shows census blocks labeled with total residential population with the parcels overlain on top in blue. The red census blocks have zero population.
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Figure 4: Comparison of parcels and census block populations The land cover file showed no development in the area in question except for the roads themselves; no development is shown to indicate structures. Figure 5 shows development as red or pink. The roads in the center area are listed as Developed-Open Space and everything other than roads is listed mostly as grassland or shrub (brown and pale green colors).
Figure 5: Land use downstream of the dam This data can be interpreted in several ways. For example, it may be a situation where a subdivision has been planned for future development, and the land has been parceled off and a 333
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road system developed but no houses have been built and occupied. Alternatively, it could be a scenario where the parcel data was recently updated while the land cover and census data are several years old and do not reflect the recent development. With no way to ground truth the data through aerial imagery or site visits the team chose to adopt the first interpretation, assuming that all data was current and structures had not yet been built in that area. The large group of central parcels was removed from the final inventory by deleting all residential parcels in census blocks with no residential population. In addition, all parcels more than 100 meters outside the dam failure inundation area were deleted. This minimized the parcels to only those necessary for the HEC-FIA structure inventory.
Results and Discussion The hydrograph that was computed from Froehlich (1995) breach parameters had a peak discharge of 30,917 m3/sec at the dam site. The time to peak discharge occurred 31 minutes after breach initiation began. Nearly all of the reservoir volume was evacuated within 90 minutes of the breach initiation. Discharge immediately downstream from the dam remained confined to the mountain valley. This part of the reach, in which the flood wave was limited to a maximum width of 400 meters, continues downstream for approximately 3,000 meters. Minimal attenuation of the flood wave occurred through the valley, which was verified by developing cross sections downstream to the base of the higher terrain (Figure 6). At the lower end of the valley, the peak discharge of the cross-sectional hydrograph had only decreased to 26,474 m3/sec.
Figure 6: Breach hydrographs at selected locations downstream from the dam 334
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When the flood wave reached the more highly populated area below the base of the mountain, a change in the slope and layout of the surrounding terrain resulted in a more pronounced attenuation and lateral dispersion of the inundated area. This behavior of the flood wave is consistent with the transition from a steep mountain valley to a relatively flat, unconfined plain such as an alluvial fan or coastal area. This attenuation was highlighted by peak unit flow rates, which ranged from 300 m2/sec immediately downstream from the dam to less than 1 m2/sec for most of the area in the broader flood plain. Peak depths of the flood wave ranged from 24.5 meters immediately downstream from the dam to less than 1 meter at the downstream computational boundary (Figure 7). Much of the populated area fell within the 1- to 3- meter range, although some structures in the lower part of the mountain valley were inundated by depths as high as 12 meters. Arrival times for the flood wave were only 30 minutes for the lower part of the mountain valley. The population residing within the broader flood plain below the valley fell within an arrival time zone of 1 to 1.5 hours. The most distant reach of the modeled area coincided with an arrival time of 4 hours.
Figure 7: Peak inundation depths (meters) below the dam Initial results from the HEC-FIA model are displayed below in Tables 5 and 6 (by arrival time and by depth of flooding) as required by the benchmark reporting requirements. Table 4: Consequences by arrival time Time Interval (min) 0-15 15-30 30-60 60-90 90-120 120-180 > 180 Total
Total Population At Risk 3,951 7,612 12,672 24,235
14-yr and Under Population at Risk 1,002 933 1,952 3,887
65-yr and Over Population at Risk 282 1,855 2,166 4,303
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Loss of Life 1,844 34 11 1,889
Direct Economic Impact ($US) $142,504,368 $266,968,912 $248,115,360 $657,588,640
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Table 5: Consequences by depth Peak Flood Depth Range (m)
Flooded Area (m²)
Total Population At Risk
14-yr and Under Population at Risk
65-yr and Over Population at Risk
0 - 0.5 0.5 - 1 1 - 1.5 1.5 - 2 2 - 2.5 2.5 - 3 3 - 3.5 3.5 - 4 4 - 4.5 4.5 - 5 5 - 5.5 5.5 - 6 6 - 6.5 6.5 - 7 7 - 7.5 7.5 - 8 >8 Total
3,699,606 11,646,576 23,248,253 12,625,355 6,151,044 2,990,216 2,128,172 1,670,211 1,068,576 511,839 377,144 448,981 233,470 152,654 287,348 296,328 1,562,455 69,098,227
2,749 4,249 9,162 3,183 1,353 540 153 129 173 133 259 299 250 136 286 502 679 24,235
399 660 1,401 446 160 85 13 28 49 32 71 78 67 33 69 131 167 3,887
511 687 1,626 716 373 122 55 19 15 9 17 27 15 9 19 43 43 4,303
Conclusion This exercise demonstrates the practices and procedures that the MMC uses to analyze the large inventory of dams that are owned and operated by USACE. Most of these dams are larger and more complex than the hypothetical dam that was analyzed in this project. Some include complex reaches that extend for hundreds of miles downstream with travel times that last for days or weeks. Elements of each of these dams and their downstream reaches are unique, and it is important to emphasize that the sophistication of MMC models, while incorporating a standardized process, varies by project. Since the reach in this exercise was both steep and short, flood wave travel times to the downstream boundary of the model were very quick. As a result of these conditions, comparisons of the inundation areas resulting from the Froehlich (1995), Froehlich (2008), and Von Thun and Gillette equations were very similar. Modeling with dam breach parameters from all three equations resulted in a flood wave with a steep peak and fast travel time through the confined mountain valley below the dam followed by lateral attenuation of the flood wave in the broad plain below the base of the mountain range. Breach parameters from Froehlich (1995) were carried forward in the modeling used for consequences analysis. Estimation of economic damages and life loss was challenging because FEMA HAZUS data, a standard input for MMC HEC-FIA models, was not available for the benchmark exercise. Therefore, average values were used, which may or may not be representative of Hydropolis. Total damages of $657M and a life loss of nearly 2,000 persons were estimated for this dam breach.
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Based on the rules and regulations that USACE has established for the estimation of risk, these results correspond to a high hazard dam. The procedures that were used in developing the hydraulic modeling and consequences estimation for the benchmark exercise were consistent with USACE MMC processes. These guidelines, while providing consistency in the analysis of all USACE owned and operated dams, also provide enough flexibility to analyze projects that range from small and simple to large and extremely complex.
Acknowledgments Funding for this project was provided by the U.S. Army Corps of Engineers, Office of Homeland Security. Internal review of the numerical modeling and consequences estimation results was provided by the U.S. Army Corps of Engineers, Hydrologic Engineering Center.
References [1] MacDonald, T.C., and Langridge-Monopolis, J. (1984). Breaching characteristics of dam failures. Journal of Hydraulic Engineering, 110(5): 567-586. [2] Von Thun, J.L. and Gillette, D.R. (1990). Guidance on breach parameters. Unpublished internal memorandum, U.S. Bureau of Reclamation, Denver, CO, March 13, 1990. [3] Froehlich, D.C. (1995). Peak outflow from breached embankment dam. Journal of Water Resources Planning and Management, 121(1): 90-97. [4] Froehlich, D.C. (2008). Embankment dam breach parameters and their uncertainties. Journal of Hydraulic Engineering, 134(12): 1708-1721. [5] U.S. Army Corps of Engineers (2010). HEC-RAS River Analysis System (Version 4.1) [Computer software]. Davis, CA: Hydrologic Engineering Center. [6] O’Brien, J.D. (2009). FLO-2D (Version 2009) [Computer software]. Nutrioso, AZ: FLO Engineering. [7] ESRI (2011). ArcGIS Desktop (Release 10) [Computer software]. Redlands, CA: Environmental Systems Research Institute. [8] Federal Emergency Management Agency (2012). HAZUS-MH (Version 2.1) [DVD]. Washington, DC: FEMA. [9] U.S. Army Corps of Engineers (2012). HEC-FIA Flood Impact Analysis (Version 2.2) [Computer software]. Davis, CA: Hydrologic Engineering Center. [10] U.S. Army Corps of Engineers (2013). MMC standard operating procedures for dams. Vicksburg, MS: Modeling, Mapping and Consequences Production Center. [11] International Commission on Large Dams (2013). Part 1 – Hydraulic Modelling and Simulation. Graz, Austria: 12th International Benchmark Workshop on Numerical Analysis of Dams (Theme C). [12] U.S. Army Corps of Engineers (2011). HEC-GeoRAS Extension for ArcGIS 10 (Version 10.0) [Computer software]. Davis, CA: Hydrologic Engineering Center. [13] Homer, C.G., Huang, C.C., Yang, L., Wylie, B., and Coan, M (2004). Development of a 2001 national landcover database for the United States. Photogrammetric Engineering and Remote Sensing, 70(7): 829-840. [14] Mattocks, C., and Forbes, C. (2008). A real-time, event-triggered storm surge forecasting system for the state of North Carolina. Ocean Modelling, 25: 95-119. [15] International Commission on Large Dams (2013). Part 2 – Consequence Estimation. Graz, Austria: 12th International Benchmark Workshop on Numerical Analysis of Dams (Theme C).
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OPEN THEME
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Behavior of an arch dam under the influence of creep, AAR and opening of the dam/foundation contact E. Robbe1 1
Hydro Engineering Center of EDF,Savoie-technolac, 73373 Le Bourget du Lac, FRANCE E-mail: [email protected]
Abstract This paper summarizes the behavior analysis of a thin, 37m high, arch dam built in 1956. At first, the behavior of this dam was quite difficult to understand: irreversible displacements occur during the years following the first filling, and then, from 1960 until now, others displacements were observed and difficult to read considering usual behavior of arch dam. In order to understand it and to evaluate the stress level of the concrete, finite-element analyzes are leading. First linear, then non-linear analysis taking increasingly into account dam/foundation contact opening, swelling, creep and damage are used to answer the problem and understand the historic behavior of this dam
Introduction The subject of this paper is an arch dam with a particularly complex behavior observed. The purpose of the study is to understand this behavior and to evaluate the level of stress of the dam. Increasingly complex finite-element analyzes are used (from linear to non-linear) and calibrated on the observed displacements of the dam. Stress levels are also compared to crack patterns observed on the structure. In this presentation, geological conditions and safety evaluation of the gravity abutment will not be discussed.
Presentation of the dam
Figure 1: view of the dam
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The studied dam is a thin arch, with a gravity abutment on the right side. 37 m high above its foundation made of gneiss, the dam is divided into 6 cantilevers (about 17m wide) for the arch part and 2 cantilevers for the right abutment. Figure 1 shows the design of the dam: the thickness goes from 2 m on the crest to 5.4 m at the bottom, with a sloping upstream face in the lower part only. The valley is quite wide (L/H about 3) and clearly asymmetrical. The dam has been built from 1954 to 1956. In 1961, works has been done in order to grout 2 parallels faults discovered on the left abutment and to build an apron to protect the foundation in case of overflowing.
Observed behavior This part describes the behavior of the dam monitored with points on the downstream face of the dam from the first filling in 1956 until 2001. These points are represented on Figure 2. From 2001 until now, pendulums are used in order to follow the dam’s behavior. A statistic analysis of the monitoring data is used in order to evaluate the part of the hydrostatic and the seasonal effects on the dam’s displacements. Once theses reversible effects evaluate, the irreversible effects can be estimate. Regarding these data, the behavior can be separated in 2 phases: During the 5 years after the first filling of the dam (1956-1960) 40 years from 1961 until 2001 Behavior after first filling The analysis of the radial displacements of the dam shows the following behavior: At the crest, the central part moves downstream while the top of the sides cantilevers move upstream, All the points at the central cantilever records downstream displacements. Long-term behavior The radial displacements of the monitored points located on the downstream face of the dam are represented on the Figure 2 for approximately 40 years. Theses displacements can be described the following way: The crest of the dam moves upstream continuously (0.3 mm/year for the central cantilever) The lower part of the arch goes first downstream during 10 years then upstream after 10 years of stabilization. The velocity of the downstream displacement is close to the one observed on the crest. A first global interpretation of these observations is that the concrete is under the influence of swelling, which can explains the upstream displacements of the crest. For the lower part, creep is more important during the first 10 years, that explains the first downstream displacements, and decreases gradually and therefore swelling become leading.
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Figure 2: irreversible radial displacements (from 1956 to 2001)
Linear analysis A finite-element linear analysis of the dam is realized with Code_Aster [1]. The physical properties for concrete and rock are chosen by comparison between the monitored data and the results of the FE analysis. The calibration is done considering hydrostatic load and thermal load in order to represent correctly the behavior of the dam in winter and summer for example.
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Figure 3: Mesh of the dam and foundation The linear analysis shows that the particular design leads to an unconventional behavior of the dam: the lower part works as cantilever while arch-effects appear in the upper part. That leads to high vertical tensile stress at the heel of the dam (close to 8 MPa in winter) and in the middle of the downstream face where seepage are already observed on a lift joint. Because of theses high tensile stresses, the FE is upgraded in order to take into account the opening of the dam/foundation contact and the lift joint, and to evaluate their consequences.
Non-linear analysis including dam/foundation contact The Figure 4 presents the joint elements introduced in the mesh in order to simulate the nonlinear behavior: dam/foundation contact under the 6 cantilevers, dam/foundation contact under the right gravity abutment, horizontal lift joint at the change of inertia of the dam, contraction joints. Joint elements open under normal tensile stress but share behavior is still elastic-linear whatever the opening-state of the joint. Except for vertical contraction joints, uplift is taking into account. In particular, spread of the uplift under the dam with joint opening is used.
Figure 4: mesh of the joint elements 344
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Initial shape With joint element
Elastic behaviour
Figure 5: comparison of the deformed shapes at the crest (elastic, with joint elements) The introduction of the joint elements modifies the deformation at the crest of the dam (Figure 5) Theses irreversible displacements are equivalent to the ones observed during the first years after the first filling of the dam and presented earlier. The analysis shows that the dam/foundation contact is open at the heel of the dam, particularly in winter (5 mm). That leads to an increase of the uplift under the central cantilever. This is confirmed by a recently installed piezometer which records the almost full uplift during winter.
Non-linear analysis including AAR and creep If the previous section evaluates the behavior of the dam after few years, the recorded displacements during the following 40th years need to be taken in account in order to estimate correctly the stress state of the dam. Considering the displacements recorded, the behavior of the dam is under the influence of swelling and creep and therefore, the model used have to be able to represent correctly these phenomena. In this purpose, the model developed by Grimal [2] for AAR on concrete dam is particularly adequate. This visco-elasto-plastic orthotropic damage model includes chemical pressure induced by AAR and takes into account the influence of creep on the behavior. It has been used to evaluate the behavior of Chambon's dam [2] and showed the model’s capability to reproduce displacements with acceptable accuracy. The main developments brought by this model concern interactions between AAR pressure and long term strain (creep) on the one hand, and the swelling anisotropy induced by oriented cracking on the other hand. Here, this model is coupled to the joint elements presented earlier. The calibration of the model is realized the following way: first, the kinetic of the chemical pressure induced by AAR and the volume of AAR gel are chosen in order to fit the displacement monitored on the crest of the dam then, the parameters for creep are chosen in order to fit displacements of the lower part of the dam The result of this calibration is presented on Figure 6.
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Figure 6: irreversible radial displacement estimated and calculated Considering the complexity of the dam's behavior, the calibration is considered acceptable despite the fact that the central cantilever's behavior isn't perfectly represented. However, the model is able to represent the change of direction of the lower part of the dam (displacements downstream then upstream) while the crest is moving continuously upstream. The influence of the long term behavior of the dam is analyzed regarding the evolution of the dam/foundation contact opening (Table 1) and of the stress state of the concrete (Figure 7). For the first point, the model shows a closure of the dam/foundation contact, particularly in summer while an opening is still likely possible (3.3 mm). That can explain why the full uplift is recorded only in winter under the dam. The analysis of the stress state of the dam shows an important increase of the compression stress close to the left abutment: from 5 to 10 MPa between 1960 and today. On the right side, the gravity abutment brings more flexibility, which leads to lower stress values. In-situ stress measurements with flat jack are under consideration in order to confirm the stress values calculated. Table 1: evolution of the opening of the dam/foundation contact with swelling
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Opening of the dam/foundation contact (mm)
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Figure 7: main arch stress on the upstream face in summer: without (1961) and with swelling (2011)
Conclusion At the beginning of this study, the dam's behavior was difficult to understand: multiple mechanisms were involved and therefore, the problem could not be solved with basic interpretations. As often, model increasingly complex allow to separate each phenomena (opening of the dam/foundation contact, swelling, creep) and to understand their influences on the general behavior of the dam. The last model is used to evaluate the impact of the longterm behavior on the stress state of the dam and to decide what need to be done to assure the safety of the structure.
References [1] Code_Aster software, www.code-aster.org [2] Grimal E., Bourdarot E (2013). Modeling AAR on concrete dam. ICOLD 81th Annual Meeting Symposium, Seattle, 2013, pp. 99-109.
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Need for transient thermal models, with daily inputs, to explain the displacements of arch gravity dams I. Escuder-Bueno1, D. Galán2 and A. Serrano3 1
Instituto de Ingeniería del Agua y Medio Ambiente. Universitat Politècnica de València (Spain) http://www.ipresas.upv.es 2
División de Seguridad de Presas. Canal de Isabel II Gestión (Spain) http://www.gestioncanal.es 3
iPresas, SPIN-OFF UPV. web: http://www.ipresas.com E-mail: [email protected]
Abstract La Aceña Dam, with 66 meters of maximum height, belongs to the typology of concrete gravity dams, and is operated as part of the water supply infrastructures of a Spanish major city. Many instrumentation records were available but among them, the ones provided by four direct pendulums outstood by its quality and consistency. The range of the values of the movements registered by those pendulums, of almost 4 cm and of totally elastic nature (showing no irreversible movements), set some interpretation challenges. The apparent incapacity of the models to reproduce the observed behaviour was used as a starting point for the diagnosis of the main sources of uncertainty: the nature of the foundations and the state of the joints, among others. These aspects have constituted the developmental axis of a series of works that have led to a number of effective and efficient actions on the dam during the last years. However, an updated transient thermal model with daily inputs on external and internal measured temperatures, coupled with a mechanical model has shown an spectacular improvement in the explanation of the displacements recorded by the instrumentation.
Software and constitutive models The first numerical model of global simulation of the stress-deformation behaviour of the dam of la Aceña was done as part of the works “First review and general safety analysis”, carried out by the consultancy firm OFITECO in 2005 for CANAL DE ISABEL II, owner of the dam. The model was done with SAP2000NL and intended, among others, to capture some existing radial movements that have been registered by the four direct pendulums and that were higher that expected. Based on the results of these works, Francisco Blazquez Prieto and Ignacio Escuder presented the case as one of the problems to be tackled by independent calculation teams during the “Ninth International Benchmark Workshop on Numerical Analysis of Dams”, organized by the Dam Calculation Committee of the International Commission on Large Dams (ICOLD) in Saint Petersburg, Russia, in June 2007. According to Escuder and Blazquez (2007), despite the great diversity of software (SAP 2000NL, CANT-SD, COQ-EF, SOFiSTiK, MERLIN, DIANA y ANSYS) and thermal models (three of which were transient), none of the seven teams that worked on the records of the radial movements tracked by the pendulums from January 1999 till September 2001, was able to predict the magnitude of the recorded displacements when using realistic values of stiffness of the dam body and foundations. 349
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In 2012, a re-evaluation of the model was done, which is presented in the present paper. The distinctive feature and more promising fact of this re-evaluation when adopting different software and a new behaviour model, was the fact having a much longer list of data (now running until 2011) that included daily temperature measurements (whereas at the time of the Benchmark Workshop, the experts had to work on monthly measurements given that daily measurements were missing in 1999 and an important part of 2000). With regard to thermal modelling, it is worth mentioning that the 2005 model, though done with a SAP 2000NL Licence, a very appropriate code for structures in general, could not guarantee to make the most of the data, mainly due to the limitations of the thermal module of the package (stationary nature). Upon consideration of this aspect, it was decided to develop the numerical models with a different software package, FLAC (Itasca, 2004), which would allow a transient analysis of temperature transmission and thus provide the best-adjusted results realistically attainable. FLAC (Itasca, 1994) is a finite differences code (explicit scheme) that allows the simulation of the behaviour of soils, rocks, concrete structures, etc. as well the interaction among these elements. The program is based on a scheme of Lagrangian calculation in which each element behaves accordingly to a specifically prescribed stress-displacement relation, as a response to the applied forces and the existing constrains in its boundaries. FLAC is doted with an internal programming language (FISH) that allows the definition of each calculation organization (i.e., complex construction sequences) and calculations of very different nature (i.e., for each constitutive relation). The main potential of the base code is the capacity of the software to model tensional states through the use of pre-set constitutive models and others purposely defined by the user. For la Aceña case, a mechanical elastic and linear model was adopted and coupled with a transient thermal model. The stiffness parameters that define the mechanic model are Young’s Modulus (E) and Poisson’s Ratio (), which are somehow equivalent and that can express the straindisplacement relations accordingly to the Volumetric Deformation Coefficient (K) and Shear Deformation Coefficient (G). With regard to the thermal module of FLAC 3D, it must be noticed it incorporates diffusion models as well as advection ones. The simulation of the first type of model, used for the case of la Aceña, allows the analysis of heat transmission through the materials along time and the development of thermally induced stresses and displacements. The process of heat diffusion that the program solves through the optional thermal module is defined by the following differential equation:
2T 2T 2T T k· 2 2 2 q v z y t x
·C·
(1)
Where ρ is the density of the medium, C the specific heat, k is the conductivity and qv is the source of hydration heat. In the development of the model, all involved parameters (density, conductivity and specific heat) have been assumed as being constant and independent of the temperature. Indeed, since the variations of concrete properties with temperature are very small, they can be neglected without concern. With respect to the boundary conditions, the external parameters of the dam are considered as convective surfaces, that is, as behaving accordingly to the following relation: 350
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qn h·T Te
(2)
Where qn is the component of normal flux pointing towards the outside of the considered boundary expressed in W/m2, h is the convective heat transfer coefficient expressed in W/m2ºC, T is the temperature of the boundary surface and Te is the temperature of the surrounding fluid, with both temperatures expressed in ºC. In the coupled thermal-mechanical model, the solution of the thermal stresses requires a reformulation of the incremental relations of stress-displacements that FLAC 3D achieves through a subtraction of the part of movement due to the change of temperature from the total incremental deformation. Finally, the laws of movement defined in a continuous medium are transformed through the former approximation into a system of equations that are applied to the nodes that represent the structure. These equations correspond to the laws of Newton applied to these discretized nodes and result in a system of ordinary differential equations that can be solved numerically from the explicit method of finite differences with respect to time. Summing up, it can be stated that the code FLAC gathers the necessary conditions to tackle successfully the study of La Aceña dam behaviour, taking advantage of the existence of thermal and movements records of a level of detail and quality very unusual in such structures.
Simulation methodology The present analysis simulates the stress-deformation behaviour of the dam of La Aceña in the period of 1999-2011, which is the one for which movement data were available through pendulum readings. In particular, two radial measurements were available (crest and horizontal gallery) from 4 pendulums (pendulums from 1 to 4). The mechanical properties used in the model are the same employed in former studies of the dam and are shown hereafter (Table 1): Table 1: Dam and concrete mechanical properties Young’s Modulus (dam) Poisson’s Ratio (dam) Density (dam) Young’s Modulus ( (foundation) Poisson’s Ratio (foundation) Density (foundation)
20 · 109 Pa 0.2 2405.7 kg/m³ 10 · 109 Pa 0.2 2242.6 kg/m³
The main difference between the current analysis and those carried out in the past is that better data are now available. In particular: 1. Daily records of the external temperatures in the dam since October 2000. Before, only average monthly temperatures were available. 2. Daily data from the three thermometers located in the concrete since January 2008. Prior to this date there was no data about dam concrete temperatures. 3. Data on the reservoir water temperature (year 2011, monthly periodicity).
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Higher periodicity (daily) of pendulums readings since July 2004 thanks to the installation of automatic readers. In order to understand the effect the different hypotheses have on the results, several models of the dam have been done, through the combination of the following options: Geometry: 2d and 3d Foundation: With and without foundation Temperatures: simplified hypothesis and improved hypothesis Since 2008 there exists a register of the temperatures measured in three points inside the concrete that have helped to improve the original thermal hypothesis in the terms indicated hereafter, and that correspond to what has been named “improved hypothesis”: The temperature of the concrete in contact with the air is equal to the ambiance temperature plus the effect of solar radiation (∆T = 10 °C). The temperature of concrete in contact with the water has been assumed to be constant and equal to 5° C except for the superior layer that acts as a transition element between the ambiance temperature and water. Specific heat: 3.5 W/m K, thermal conductivity: 650 J/kg K, convection coefficient: 16 W/m² K and linear thermal expansion coefficient: 10-5 Figure 1 shows the global geometry of the model and Figure 2 the location of the pendulums whose movements, measured in the crest, were used to check the goodness of the adjustment. Figure 3 represents the whole of the external variables (water level and external average temperature) as well as the pendulums measurements.
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nsulting Group, Inc.
odel Perspective 1 2012 Rotation: X: 40.000 Y: 0.000 Z: 30.000 Mag.: 1 Ang.: 22.500
shown
Group, Inc. USA
Figure 1: Complete geometrical model
Figure 2: Location of the four direct pendulums
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Figure 3: External variables (level in the reservoir and average ambiance temperature) and movements measured by the pendulums
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Summary of the results and conclusions The main conclusions that can be extracted from the current analysis of the stress-deformation behaviour of the dam of La Aceña through the simulation methodology described along the former sections, established in order to make the most of the whole set of records of thermal monitoring and movements, are: The elaborated models reproduce in a reasonably satisfactory way the thermal behavior of the structure as well as the movements observed in the dam. The fact of having carried out a transient thermal simulation for each day, capturing the impact of the direct sun exposure on the downstream side of the dam and estimating with accuracy the water temperature have been key elements to the goodness of the obtained numerical model. Figure 4 shows a summary of the average errors incurred by the different models. As it can be observed, the best model is the so called “reference one” (3D, with foundation and the hypothesis of improved temperature).
Promedio de errores absolutos
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Figure 4: Average errors in mm for each model (shown for each pendulum, from P1 in darker red to P4, in a lighter grade.) It can be stated that thanks to the recently developed numerical models a new tool is at hand to interpret the strain-deformation behaviour of the dam of La Aceña. This tool can complement the auscultations that are currently being done on the dam and that have provided an excellent level of information about both the thermal and movement behaviour. Also, the numerical tool has permitted the satisfactory and consistent reproduction of the stressdeformation behaviour auscultated until the end of 2011 and can act in the future as a way of contrasting the impact of new actions planned on the dam as well as a form of detecting potential behaviour change trends. Figure 5 shows the comparison between the reference model (with the hypothesis of improved temperature) and the model with the original hypothesis of temperature.
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Figure 5: Extent of the improvement of the thermal model (reference model in red, nonupdated thermal model in blue)
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Acknowledgements Some of the calculation routines have been developed within the following Research Project: “Integration of the components of anthropic risk in the systems of dams and reservoirs global safety management” (Incorporación de los componentes de riesgo antrópico a los sistemas de gestión integral de seguridad de dams y embalses BIA2010-17852), funded by the Spanish Ministry of Science and Innovation and with funds FEDER.
References [1] I. Escuder and F.Blazquez, Analysis of the Elastic Behaviour of an Arch Dam, Hydropower and Dams, Issue 5, (2007)
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The rehabilitation of Beauregard Dam: the contribution of the numerical modeling A. Frigerio1 and G. Mazzà1 1
Ricerca sul Sistema Energetico – RSE S.p.A., via R. Rubattino 54, Milan, ITALY [email protected]
Abstract The paper describes the case-history of the Beauregard dam (Italy), a concrete arch-gravity structure 132 m high built between 1951 and 1960 on the Dora di Valgrisenche River. The design reservoir volume was about 70 mil. m3. The geological and geotechnical investigations carried out since the dam construction and deepened in the last decade have underlined that a Deep-Seated Gravitational Slope Deformation (DSGSD) is located on the left slope. Since the first fillings of the reservoir, the interaction between the DSGSD and the dam was recognized to have relevant implications on the dam structural safety. For that reason, the Italian Dam Authorities in 1969 prescribed a limitation of the reservoir level with a corresponding reduction of the its volume to 6.8 million m3, about 1/10 of the initial design volume. The studies, which include a detailed analysis and thorough interpretation of the monitoring data over a time span of more than 50 years, have allowed to gain insights into the understanding of the DSGSD behavior and its interaction with the dam. To interpret the already experienced effects of the bank movements against the dam and to forecast the future possible trends, the numerical modeling activities have played a key role. The solution chosen to guarantee a long-term safety operation foresees the demolition of the upper part of the dam in order to drastically reduce the cracking pattern of the dam body caused by the compression of the vault for the DSGSD movement.
Introduction The Beauregard dam, located in the Aosta Valley, Italy (Figure 1) was completed in 1958. Operated by the Italian Electric Energy Company (ENEL) up to July 2001, the entire scheme was then acquired by CVA (Compagnia Valdostana delle Acque).
Figure 1: Location of the Beauregard dam on the Italian territory and aerial view of the dam
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The 132m high arch-gravity dam has a crest length of 408m, it is 45.6m thick at the foundation level and 5m thick at the crest level. With a maximum operating design level of 1770m asl, the total reservoir volume was estimated in 70 mill. m3. The filling of the reservoir was undertaken in stages between 1958 and 1968. Since the dam construction, the monitoring system installed on the left slope abutment showed the presence of a clear relationship between the reservoir level and the rate of movement of the Deep Seated Gravitational Slope Deformation (DSGSD) located on this left slope (Figures 2, 3, and 4).
Figure 2: Displacement vectors measured from surface targets
Figure 3: Horizontal displacement distribution along the plumb-lines installed in the lower portion of the slope close to the dam
Figure 4: Zones of maximum shear strain rate: overall slope (safety factor = 1.401.45)
Coherently with the left bank movement, the dam was observed to deflect upstream, due to the trust of the slope against the vault, having as a consequence the appearance of cracks on the downstream face (Figure 5). As a consequence, in 1969 the operational reservoir level was lowered down to 1710m asl, corresponding to a reservoir volume of 6.8 mill. m3, as enforced by the Italian Dam Authorities.
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Figure 5: The dam was observed to deflect upstream with cracks appearing on the downstream face and openings of some upstream vertical joints The continuous monitoring of the DSGSD and of the dam, carried out by ENEL, first, and by CVA, in recent years, has allowed to operate the hydroelectric power scheme under closely controlled conditions. Moreover, some works carried out to control the superficial water flows due to melting snow and rain, in order to limit their filtration into the sliding body, have allowed to reduce the yearly speed of the slope from 1-2 cm to few mm. However, the interaction between the DSGSD and the dam has been recognized to have relevant effects on the long-term dam behavior as the cracking pattern was continuously progressing in time. Figure 6 shows the results of an on-site investigation carried out with a tomographic system that put into evidence the areas of the dam where cracking is particularly important, i.e. along the peripheral joint and close to the downstream dam toe.
Figure 6: P-wave velocity tomogram: (left) downstream face; (right) main cross section The strong interaction between the DSGSD and the dam has been recognized to have relevant possible implications from the civil protection as well as energy production points of view, and posed important territorial and environmental issues. For this reason the Owner decided to start a further study, making also reference to the support offered by numerical modeling, with the aim to find the most suitable and long-term solution of the problem.
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The contribution of the numerical modelling Synthetically, the main aims of the numerical model were: The interpretation of the dam behavior experienced since its first fillings, The calibration of the mechanical parameters of the dam-rock system in terms of comparison between the observed dam behavior and model results, The forecast of the future dam behavior at short-middle term adopting the calibrated model, The support to the designer for the analysis of possible technical solutions to guarantee the long term operation of the dam. The numerical simulations of the dam were carried out using the FEM code ABAQUS/Standard [1]. A three-dimensional finite element model of the concrete arch dam including a proper portion of the surrounding rock mass foundation was generated in order to carry out numerical analyses (Figure 7). In the rock mass foundation the active portion of the DSGSD has been taken into account as well as the shear surface, modeled in terms of an interface whose behavior is described by a frictional law. The nodes where the DSGSD movements were imposed are outlined with a red point in Figure 7.
Figure 7: Finite Element mesh of the dam and the surrounding rock mass A linear elastic isotropic constitutive law was assumed for the rock mass foundation whose mechanical parameters were defined on the basis of the results provided by the more recent geotechnical studies. The Concrete Damaged Plasticity (CDP) constitutive law, available in ABAQUS library, was adopted to describe the concrete behavior (Figure 8). The model assumes that the uniaxial tensile and compressive responses of concrete are characterized by damaged plasticity. Under uniaxial tension the stress-strain response is linear-elastic until the failure value is attained; beyond this value a strain softening relationship follows. In compression the softening behavior occurs after an initial stress hardening. On the basis of the results of the indirect tensile tests, two different sets of material parameters were assigned to the concrete of the dam body and the "pulvino".
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Figure 8: The Concrete Damage Plasticity constitutive law (Fenves, 1998) has been assumed for concrete and pulvino of the dam Several preliminary analyses were carried out to identify the numerical model (calibration phase) in order to reproduce the real dam behavior dealing with the interaction between the structure and the slope sliding. The movement of the left slope abutment was imposed by appropriate displacement boundary conditions applied at the nodes marked in red (as explained above, Figure 7) and the hydrostatic load was firstly considered taking into account the variation relevant to the first fillings; later, a constant reservoir level at 1710 m asl was assumed, according to the present operational conditions. The identification process was mainly based on: The measured displacements along the dam crest (Figure 9); The monitoring points of two plumb lines, located inside the dam body in the main cross section and in the fourth block towards the left abutment respectively; The crack pattern of the structure as determined by tomographic investigations (Figure 10, to be compared with Figures 5 and 6). As shown in Figures 14-15, the identified numerical model exhibits a good agreement with the deformation of the dam crest as well as the.
Figure 9: Deformation of the dam crest: (left) based on monitoring; (right) based on computations
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Figure 10: Localisation of damage computed with the 3D numerical model: (left) on the dowstream face of the dam; (right) on the main cross section Afterwards, the identified numerical model was used to predict the future behavior of the structure considering a continuous increase of the landslide movement according to the trend foreseen from the past behavior. The main purpose of the analyses was to assess if the dam might undergo to local or global instability phenomena such as snap-back. The curve nr. 1 in Figure 11 allows to exclude snap-back instability for the overall structure because beyond the peak value the reaction force on the main cross section of the dam decreases gradually as the landslide movement increases. Although the measured displacement towards the upstream direction of the middle point of the dam crest (about 0.21 m) is close to the peak value of the reaction force curve, it has to be bear in mind that the structure is subjected to an imposed deformation rate due to the landslide movement. For this reason the structure will be able to follow the softening branch of the reaction force curve avoiding sudden failure when the peak value will be attained.
Figure 11: Reaction force on the main cross section of the dam vs the upstream-downstream displacement of the middle point of the crest dam
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The dam rehabilitation Different possible long-term solutions were considered by the designer (Studio Marcello, Milan) who was in charge of the rehabilitation program. Numerical modeling was widely adopted in order to analyze the different solutions which main aim was the reduction of the trust effect of rock sliding against the dam. Among the different solutions, the two more widely analyzed were those shown in Fig. 12.
Figure 12: Two of the possible long-term solutions adopted for the rehabilitation of Beauregard dam The partial demolition solution (right picture of Figure 12) was the one finally decided by the rehabilitation designer. The demolition of the upper part of the dam will lower the crest elevation from 1772m to 1720m asl. The estimated volume of the demolished concrete will be about 150,000 m3. The demolished material will be used to fill the two volumes located at the upstream and downstream dam toes (Figure 13). Additional works have been carried out in the frame of the rehabilitation interventions, among which the improvement of the hydraulic scheme with the construction of a gated spillway and a discharge tunnel.
Figure 13: Cross section of the dam: final configuration after the demolition
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The methodology chosen for the demolition is blasting. The demolition started in the Spring of 2013 and will end in 2014. Figures 14 and Figure 15 show some phases of the dam demolition that is in progress. Just to give some figures about each demolition step, the case shown in Figure 15 is relevant to an explosion carried out with about 850 ton of explosive located in 120 boreholes with a diameter of about 8 cm and a length of 6m. The demolished concrete shown in the picture regards a volume of about 6m high x 7m thick x 50m wide.
Figure 14: Different sequences of the dam demolition: the dam before the starting of works (up-rigth); the preparation of one explosion step (up-left and down)
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Figure 15: Different sequences of the dam demolition
Conclusion The case history of Beauregard dam is an emblematic case that has posed to the engineers in charge of operation and safety very critical choices, considering that the construction of new dams, and structures in general, represents their main mission. In the Beauregard case it has been deemed necessary to proceed, as sometimes happen for the human life, with a sort of a limb amputation in order the save the dam life. The decision has been difficult and it has been necessary to investigate very deeply and for long time before all possible aspects affecting the system, deciding which could have been the most reasonable solution from the engineering point of view. The choice has been taken considering the different facets of the problem: safety, mainly, environmental protection, energy production. Environmental as well as economic and financial aspects are not in the aim of the present paper and could be widely shown in a future paper. Analogously, also some other technical aspects relevant to the civil works that are under construction at the downstream toe and facing to check the presence of possible leakages, will be discussed in further papers.
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The present paper has the aim to put into evidence the support that numerical modeling can give to decision makers in a very complex problem like the one proposed by Beauregard dam.
Acknowledgements The Authors wish to express their gratitude to Lorenzo Artaz and Morena Colli of CVA who has given the permission to present the present work.
References [1] Hibbitt, et al, ABAQUS/Standard Theory Manual, Release 6.6, 2005. [2] G. Barla et al., The Beauregard dam (Italy) and the deep-seated gravitational deformation on the left slope. Hydropower 2006, Kunming, China, October 2006.
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Earthquake Assessment of Slab and Buttress Dams H. B. Smith1,2 and L. Lia2 1
2
Norplan AS, Oslo, Norway Department of hydraulic and environmental engineering, NTNU, Trondheim, Norway E-mail: [email protected]
Abstract This paper outlines work on the examination of the dynamic properties and seismic safety of slab and buttress dams [1]. The work has been carried out with a linear elastic numerical model established in the Finite Element Method program Abaqus. One single dam section in a typical Norwegian slab and buttress dam with heights of 12 and 25 meters has been modeled in regards to varying reservoir water level and lateral bracing. Abaqus has been used for the frequency analysis and the dynamic time-history analysis. The natural vibration modes of the dam represent movements in separate directions. Through a seismic event the greatest response will be represented by the buttress deflection in direction along the dam axis (axial direction). The resistance towards earthquakes will depend significantly on individual stability of elements rather than the global stability. In particular, the tensile stresses occurring in the buttress when deflecting in the axial direction are found to be a potential failure mechanism. Providing lateral bracing by struts positively influences the response of the slab and buttress dam in seismic events. When lateral bracing is provided, the ability of the dam to transfer the inertial forces to the abutment is important.
Introduction The American-Norwegian engineer Nils Ambursen developed the slab and buttress dam, the Ambursen dam, in the early 20th century. Several of these dams were built until the 1970s, especially in North America and Norway. Figure 1 presents the Håen Dam, a typical Norwegian slab and buttress dam with a structural height of 12 meters. [2] [3]
Figure 1: The Håen Dam, view of downstream side with insulating wall The reinforced concrete structure consists of the buttresses and an upstream inclined slab. The typical section and dimensions of Norwegian slab and buttress dams are presented in Figure 2. Insulating walls and lateral bracing, such as struts, are not always present. The upstream
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slab is often constructed as continuous over two or three sections, with joints where the moment is zero. The resulting moment diagrams are as presented in Figure 3. Earthquake safety regulations were introduced for Norwegian dams in 2010. Due to the recently updated regulations and the low number of slab and buttress dams in earthquake prone regions, there is uncertainty regarding the dynamic assessment of existing slab and buttress dams in Norway. The slab and buttress dam is proven to be highly efficient for static loading. Based on the experiences from registered dam breaches, earthquake incidents and upgrades, the slab and buttress dam needs to be checked for earthquake loads in both horizontal directions. Structural slenderness of buttresses and slabs makes the structure particularly vulnerable to axial loading. Load distribution and interaction between the various elements could cause the strength and stability of individual elements to be more critical than the global stability. When dynamic loads are considered, several possible failure mechanisms are identified.
Figure 2: Typical cross section [4]
Figure 3: Moment distribution in continuous slabs [4]
Finite Element Model Based on a literature review, existing computational methods and site inspections of slab and buttress dams, a linear elastic numerical model was established in the Finite Element Method (FEM) program Abaqus. The importance of various structural details in a seismic context has been analysed and utilised for the modeling of a single dam section in typical Norwegian slab and buttress dams with heights of 12 and 25 meters. Two models have been used for the analysis regarding height variation, reservoir water level and presence of lateral bracing. Dynamic properties, global response and impact on the structural elements have been analysed through eigenfrequency analysis and dynamic response-history analysis. For convenience the results presented in this paper focus on the base case of a 25 meters high dam section. The dam section was modeled, as presented in Figure 4, as a single buttress and continuous slab with symmetrical
Figure 4: Illustration of the model and generated mesh 370
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boundary conditions in the slab’s axial direction at the mid-span. The dimensions of the modeled section are:
5 meters c/c between buttresses Crest width of 2.5 meters 5V:4H inclination of the upstream slab 5V:1H downstream inclination of the buttress Both the slab and buttress thickness increases 20 mm per meter from t = 300 mm, at the dam crest
The model was partitioned in order to provide nodes in the locations of all lateral bracing. The generated mesh gave a total of 2676 shell elements and 2863 nodes. The foundation rock is expected to be of good quality and is therefore idealised with no mass and with great stiffness compared to the slender slab and buttresses. The slab and buttress dams are normally designed without transmission of reinforcement to the foundation or between the structural elements. The joint between slab and buttress is designed with a shear key to avoid displacement in the axial direction. As these joints will allow some rotation, all connections are modeled as hinged. Lateral Bracing When present, struts and insulating walls will provide lateral bracing for the buttresses in addition to the upstream slab. Insulation walls are introduced to reduce the temperature gradient through the slab. With typical thickness of t = 120 mm lightly reinforced, low concrete quality, the actual contribution of the insulation wall in a seismic event is not included. Only the effect of lateral bracing on the dynamic properties of the slab and buttress dams’ by struts is therefore investigated. Struts are normally simply supported between the buttresses, i.e. only compression forces are transferred Figure 5: Selected placement of through the struts. The struts are modeled by massless struts elastic springs that ensure the stiffening effect in the axial direction. Working both in compression and tension, the springs were modeled only on one side of the buttress in the chosen locations presented on Figure 5. Loads The applied static loading includes gravity and hydrostatic water pressure. Uplift, sediment loads and downstream water pressure are neglected, in addition to temperature loads. Seismic loads are applied according to Eurocode 8 [5] regulations. This gave a maximum peak ground acceleration for mainland Norway of 1.6 m/s2 horizontally and 1.0 m/s2 vertically. The applied acceleration time histories, presented in Figure 6, are developed for ground type A and adjusted for Norwegian geological and seismic conditions. The acceleration time histories are applied along the foundation in all three directions simultaneously. The acceleration time history in the axial direction is also applied along the edges of the slab. 371
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Because of the upstream inclination, the horizontal hydrodynamic pressure is simulated according to Zangar’s theory of added mass [7]. The overlaying masses of water are assumed to be directly accelerated by the dams’ vertical accelerations [8]. The applied hydrodynamic added mass was decomposed from the horizontal and vertical contributions to work perpendicular to the upstream slab, and applied as inertia point masses at different levels, according to the presented distribution from Figure 7. Because of the flat slab, no hydrodynamic interaction was taken into account in the axial direction.
a) Applied in the upstream-downstream direction
b) Applied in the axial direction
c) Applied in the vertical direction
Figure 6: Utilized acceleration time histories [6]
Figure 7: Distribution of applied hydrodynamic added mass
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Material Properties, Strengths and Prescribed Damping The concrete is modeled with elastic properties corresponding to a B25 concrete, described in Eurocode 2 [9]. The struts are modeled as elastic springs, with corresponding stiffness k = EA/L, acting both in tension and compression. The concrete strength was determined in accordance with Eurocode 2 [9] regulations, adjusted for a 50 % increased dynamic tensile strength, according to Raphael [10]. The struts’ capacity was determined by calculations of buckling and compression capacity. Viscous Rayleigh damping, embedded in the concrete material properties and the hydrodynamic point masses, are used to model all damping in the system, including damwater-foundation interaction and frictional dissipation. The additional stiffness proportional damping from the struts was not taken into account. Because the model represents the structure’s elastic behaviour, a damping ratio of two percent was considered appropriate. It should be noted that an increased damping ratio would reduce the systems response.
Results Frequency Analysis The frequency analysis described the structures’ natural vibration modes with movements in separate directions. In all analysed cases, the first mode is governing for the axial direction. Figure 8 describes the vibration mode shape and variation of period for oscillations in the slab and buttress dam section with water at crest level for different section heights and with or without lateral bracing. a)
b)
c)
Figure 8: The first vibration mode a) Mode shape without lateral bracing b) Mode shape with struts present c) Variation of the vibration period The first mode describing movement in the upstream-downstream and vertical direction was also governing in these directions for all cases. Figure 9 describes the mode shape and variation of period for the slab and buttress dam section with or without presence of water at crest level, for different section heights without lateral bracing.
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a)
b)
c)
Figure 9: The first vibration mode describing movement in the upstream-downstream direction a) Mode shape without water b) Mode shape with water at crest level c) Variation of the vibration period Dynamic Response-History Analysis Through the seismic event, the models were never exposed to compressive stresses close to the concrete design capacity. The maximum principal stresses are therefore presented. It is assumed that cracking of concrete will occur when the design dynamic tensile strength fctd,dyn = 1,5 MPa, is exceeded, marked by gray in the graphic presentations. The typical deflected shape and maximum principal stress state during a seismic event are presented in Figure 10, for the 25 meters high dam section with water at crest level and with or without lateral bracing in the form of struts. a) Without lateral bracing
b) With lateral bracing
Figure 10: Typical deflected shape during seismic event, deformation scale factor of 100
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The maximum principal stresses on each side of the buttress with water at crest level, are presented on Figure 11 without lateral bracing and on Figure 12 with struts.
Figure 11: Maximal principal stresses on both sides of the buttress without lateral bracing during a seismic event
Figure 12: Maximal principal stresses on both side of the buttress with lateral bracing during a seismic event. The time-history plot of transferred inertia forces to the highest loaded strut in the dam section of 25 meters height is presented in Figure 13. Representing both the struts on each side of the buttress, the plot describes the absolute value of the forces through the given elastic spring.
Figure 13: Time-history plot of the highest loaded strut in the 25 meters high dam section.
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Discussion Dynamic Properties The distinct differences in the structures vibration modes in different directions, combined with the variation of the vibration period, leads to the following observations: Dam height
Water level Lateral bracing
An increase in the section height results in a greater mass while the dam stiffness decreases both in the axial and the upstreamdownstream direction. Thus, a greater vibration period in all directions is obtained. At higher water levels, the additional hydrodynamic mass increases the vibration period in the upstream-downstream direction. Increased stiffness in the axial direction due to lateral bracing reduces the vibration period in the same direction.
Seismic Safety Dam safety requirements for slab and buttress dams include overturning and sliding stability in addition to the required structural strength for all cross sections. The dam safety should be considered in a seismic event and retain its integrity during the earthquake. The seismic safety has been evaluated based on global stability and occurring stress levels in the buttress. The struts are additionally considered. As the upstream slab is designed for tensile stresses, it is not further considered. Global Stability The maximum fluctuations of the reaction forces acting in different directions on the dam’s sections would, in combination with one another, constitute a danger to the system's global stability. Nevertheless, the maximum fluctuations occur at various times throughout the event and are only applicable in very short time periods. The system will not have time to react, and there will be no danger of collapse even if the theoretical factor of safety against sliding or overturning in a given time goes below 1.0. The consequences may be minor deformations if exceedance of capacity is repeated. The theoretical safety factor against sliding through the seismic event for the 25 meters high dam section with water at crest level is presented in Figure 14.
Figure 14: Variation in theoretical safety factor against sliding for a 25 meters high dam section with water at crest level.
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Also considering the rest capacity when assuming a frictional factor of tan = 1, the risk of sliding is not the critical case. Nevertheless, the theoretical safety factor against sliding for slab and buttress dams is reduced as the dam height increases. The upstream inclination keeps the dam stable against overturning throughout the seismic event, as tensile stresses never occur along the buttress foundation. Stresses in the Buttresses A lower section height reduces both the response and stress level of the buttress. The presence of lateral bracing prevents the buttress deflection out of its own plane, and largely reduces the stresses. When struts are present, the buttress is bent around its connection points, and the maximum stresses are found in those locations. The occurring tensile stresses in the numerical model have to be seen in relation with the presence of reinforcement. Before 1955 it was common to construct the slab and buttress dams without reinforcement in the buttresses [11]. In addition to tensile stresses, construction joints would endanger the buttress stability in such a case. For dams constructed later than 1955, a minimum amount of reinforcement will always be present. Even though tensile stresses can cause cracking of the concrete, the strength and ductile nature of reinforcement will nevertheless increase the safety level compared to an unreinforced buttress. Struts The impact on the struts from a single section must be seen in relation to the struts compression capacity, found to be dimensioning over the buckling capacity. The transferred inertia forces are accumulated towards the abutments through the struts from neighbouring sections. When beams are not continuous over the dam length, unfavourable moments can be introduced into the buttresses. Other Potential Failure Mechanisms The following potential failure mechanisms should also be taken into consideration; the connections between elements, shear capacity and punching shear capacity of the buttresses, stress accumulation around cutouts, strength of the foundation and bearing capacity of the struts. Uncertainties – Sources of Error Because only one single dam section has been modeled, several assumptions have been required. Among others, the influence of adjacent sections, especially of different heights, is of importance. As the results show a non-linear response and because reinforcement is not taken into account, the response and maximum occurring tensile stresses in the numerical model can only be indicative.
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Conclusions The results demonstrate how the dam vibration modes represent movements in separate directions. An increased dam section height results in an increased response because of the greater oscillating mass. Meanwhile, a raised water level in the reservoir increases the period for oscillation in the upstream-downstream direction. Through a seismic event, the greatest response will be represented by the buttress deflection in the axial direction. The resistance towards earthquakes will be mainly dependent on the individual stability of elements rather than the global stability. In particular, the tensile stresses occurring in the buttress, as a result of deflection in the axial direction, is found to be a potential failure mechanism. Providing lateral bracing by struts positively influences the response of the slab and buttress dam in seismic events. For the Norwegian slab and buttress dams in particular, the contribution of the existing insulating walls is considered to be small but positive. When lateral bracing is provided, the ability to transfer the inertial forces to the abutment is important. In addition to the assessment of simplified analytical methods, it is recommended that further numerical studies should focus on the slab and buttress dams’ nonlinear response, on the importance of global dam geometry, and verification of the numerical input parameters by physical vibration tests of an existing slab and buttress dam.
References [1] Smith. Earthquake Assessment of Slab and Buttress Dams, Master thesis at the Department of Hydraulic and Environmental Engineering, NTNU, Trondheim, Norway, 2013. [2] Various authors. Ambursen dams - Selected articles from the Ambursen Hydraulic Construction Company. R. M. Rudolph, Approximate Date 1912. [3] NORUT Narvik AS. Innovativ forvaltning av betongdammer, 2009. In Norwegian. [4] Guttormsen. TVM4165 - Vannkraftverk og vassdragsteknikk. Tapir akademiske forlag, 2006. Compendium of the Department of hydraulic and environmental engineering, NTNU. In Norwegian. [5] Standards Norway. Eurocode 8: Design of structures for earthquake resistance - Part 1: General rules, seismic actions and rules for buildings, 2004. NS-EN 19981:2004+NA:2008. [6] Kaynia. Communicated by. NTNU and NGI, Oslo, Norway. [7] Zangar. Hydrodynamic pressures on dams due to horizontal earthquake effects. Engineering monographs, No. 11, 1952. [8] Scott et al. Dam Safety Risk Analysis, Best Practices Training Manual. U.S. Department of the Interior, Bureau of Reclamation in cooperation with the U.S. Army Corps of Engineers, 2011. Version 2.2. [9] Standards Norway. Eurocode 2: Concrete structure - Part 1-1: General rules and rules for buildings, 2004. NS-EN 1992-1-1:2004+NA:2008. [10] Raphael. Tensile strength of concrete. Journal of the American Concrete Institute, 2:158 – 165, 1984. [11] Kleivan, Kummeneje and Lyngra. Concrete in Hydropower Structures. Hydropower Development. Norwegian Institute of Technology, Division of Hydraulic Engineering, 1994. 378
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Solution of dam-fluid interaction using ADAD-IZIIS software V. Mircevska1, M. Garevski1, I. Bulajic2 and S. Schnabl3 1
Institute of Earthquake Engineering and Engineering Seismology, univ. “Ss. Cyril and Methodius”, Box 101, 1000 Skopje, R.Macedonia 2 Mining institute, Batajnicki put 2, 11000 Belgrade, Serbia 3 Faculty of Civil and Geodetic Engineering, University of Ljubljana, 1000 Ljubljana, Slovenia E-mail: [email protected]
Abstract Fluid dam interaction has a remarkable impact on the dynamic response of dams and could play an important role in assessment of their dynamic stability. This is particularly emphasized when dams are subjected to strong seismic excitations. The phenomenon has been for the first time physically explained and mathematically solved by Westargaard. The very first dynamic analysis based on application of “Added Mass Concept”; underestimated the random earthquake nature in assessment of the hydrodynamic effects. In the recent years, various BEM-FEM and FEM-FEM techniques have been developed to account for many significant parameters that influence the accuracy of calculated hydrodynamic effects. This paper presents a BEM-FEM orientated solution based on the use of the matrix of hydrodynamic influence as a very effective tool for analyses of extensive domains of fluiddam-foundation rock systems for two major reasons: the computation time is far more effective than that in direct or iterative coupling methods and stability of the solution. The presented analyses are based on the use of genuine software originally written for static and dynamic analysis and design of arch dams.
Introduction The ADAD-ver.3 computer program [1], originally written for static and dynamic analysis and design of arch dams, is under development for the last several years. It implements an analytical procedure for the three-dimensional dynamic analysis of arch dams including the effects of dam-water interaction (water incompressibility), soil-structure interaction and the nonlinear behavior of the of contraction joint manifested by partial joint opening and closing as well as tangential displacement. The process of generation of mathematical model runs parallel and interactively with the process of design of the particular dam. The program gives an option for computer design of the dam body [2,3], whose embedment is in accordance with topology of the terrain. Program offers automatic pre-processing for generation of finite element mesh of dam and part of the foundation mass to account for the dam-foundation interaction phenomenon, as well as effective way of generation of boundary element mesh, sufficiently accurate in following the topology of the terrain to account for the fluid structure interaction. The program use sensitivity search analysis to detect the “most adequate” location of the truncation surface [4], where non-reflecting truncation boundary conditions should be imposed. The truncation surface should be located in a way to define the required completeness of the wave field where expansion of P compressive and dilatational waves takes place followed by scattering and radiational effects. Its further displacement away from
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the dam should have a negligible impact on the calculated magnitude of the hydrodynamic effects. The conducted analyses are based on an original and simple FEM–BEM fluid–structure interaction solution embedded in the ADAD–IZIIS software. This solution eliminates the difficulties of “direct” and “iterative” coupling methods by analyzing independently the two physically coupled sub-domains. The interaction effect is obtained in an uncoupled way computing the matrix of hydrodynamic influence by applying the concept of virtual work of “unit accelerations”. The suggested method does not belong to direct coupling or to iterative coupling methods, yet with its computational steps it offers a two-way coupling by transferring the fluid forces to the structure and the structural accelerations back to the fluid. The paper presents BEM-FEM oriented solution of the fluid-dam interaction along with the boundary element discretization of the reservoir domain. The dam was subjected to El Centro earthquake excitation with duration of 7 sec., scaled to the pick acceleration of 0.3g. The dam properties are the following: Dam height H=130m; Young’s modulus E = 31.5 GPa; mass density ρ = 2450 kg/m3; Poisson’s ratio ν = 0.2; the acoustic wave velocity in water c = 1440 m/s.
Numerical Model of Arch Dam and Fluid Domain The process of generation of the discrete mathematical model of the arch dam and the fluid domain runs parallel with the process of design of the dam body. This process starts by digitization of topographic data of the terrain and the shape of the main central cantilever, Figure 1. The developed pre-processing within the ADAD–IZIIS software enables each topographic isoline to be mathematically presented by a set of equations of a second order, i.e., curves passing through three neighboring digitalized points on it. Each isoline is stored in the computer by means of a certain number of polynomials. The program gives an option for computer design of the dam body [2,3], whose embedment is in accordance with topology of the terrain. Arch dam body can be modeled in a form of few circular segments as well as in the form of a parabola. During modeling, it is possible to observe the shape of the arches at all elevations along with the corresponding tables containing their geometrical parameters and to observe their mutual position in order to control overlapping. Figure 2, presents the mathematical model of the dam generated automatically, using adopted shapes of the arcs at all selected elevations in accordance with the topology of terrain. The model is formed by 199 substructures. The substructures are automatically digitized into a certain number of finite elements that are not presented in the figure. The model contains 6294 finite elements and 11170 external nodes. The bend of the system is 3111. The model posses 2013 contact elements that are involved at the connections between substructure’s blocks. The contact elements are generated automatically and the contact element mesh refinement is in accordance of the model analyst request. ADAD-IZIIS software offers a very efficient and accurate modeling of the 3D fluid boundaries according to the downstream topology of the canyon terrain. The boundary elements at the extrados of the arch dam are directly extracted from the general FE model while the boundary element mesh that represents the boundaries of the reservoir, i.e., both banks, the reservoir bottom, the water mirror and the reservoir end, very accurately follows the shape of the topographic isolines. Generation is simply by giving the number of planes that intersect the terrain along with their distances from the uppermost point of the crown cantilever. In case of highly irregular and twisted terrain, more section planes and a more refined mesh should be used in order to model the complexity of the terrain in the most accurate way, which is undoubtedly easily feasible and with shorter computational time if BEM technique is engaged. However, for the concrete configuration of the terrain and detected “most adequate” location of the truncation surface situated at the downstream 380
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distance of 210m from the dam, the number of boundary elements used in the model for accurate modeling of the 3D fluid boundaries is 1600, figure 3.
Figure 1: a) Topology of the terrain b) Shape of the central cantilever
Figure 2: Mathematical model of a dam (substructures and construction joints)
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Figure 3: Boundary element mesh of the fluid domain Fluid-structure interaction is affected by the irregularity of the terrain in the near surrounding of the dam-fluid interface. The topology of the terrain dictates the “most adequate” location of the truncation surface where non-reflecting truncation boundary conditions should be imposed. The detection of the “most adequate” location of the truncation surface is an important task in development of a reservoir model due to the fact that the HDP intensities on the dam-fluid interface are sensitive to the extent and type of waves generated by the boundaries. The truncation surface should be located in a way that its further displacement away from the dam has a negligible impact on the calculated HDP intensities. The program offers sensitivity search analysis to detect the most adequate location of the truncation surface. For the presented BE model in figure 3., the elapsed CPU time for performing such analysis is approximately 15 min. The numerical procedure is based on the conventional BEM. Laplace differential equation that governs the incompressible and inviscid fluid motion is used. The procedure is conducted over a rigid dam-canyon-walls assemble. It follows a horizontal acceleration of 1g applied in downstream direction. The acoustic elastic P waves were generated as a result of the vibration of the considered upstream dam face and the rigid canyon’s walls. The expansion of generated waves and the way of their propagation as compressive or dilatation waves depends not only on the specified boundary conditions but on the shape of the reservoir boundaries in respect to the direction of the seismic excitation. Three different types of truncation boundary conditions were considered: a) stationary type of truncated boundary conditions, i.e., perpendicular acceleration at all the points on the truncation surface is set to zero; b) hydrodynamic pressure at all points on the truncation surface set to zero; and c) non-reflecting boundary condition. Sixteen different location of the truncation surface were considered and analyzed. The curve associated with the TBC that allows dissipation of the outgoing waves shows mostly decreasing trend until reaching the meeting point of the curves of TBC type (a) and TBC type (b), at distance of L=180 m (L=1.6-1.7Hw) away from the dam. Further on, this curve remains almost horizontal, indicating unaltered value of HDP. This means that it is irrelevant whether TS is positioned at a greater distance than L=180m, since the effect of the amount and type of generated waves is negligible with further increase of the model length. However, this effect is remarkable along length L<180m, wherefore placing TS closer to the dam means overestimation of the HDP. In accordance with the applied direction of ground acceleration and due to the irregular configuration of the terrain in the vicinity of the dam, the right bank generates mostly dilatation waves while the left bank generates compressive waves. For the concrete topological conditions, the considered depth of impounded water, according the results presented in [4], the “most adequate” location of the truncation is selected at a distance of L=1.9Hw=210m.
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Figure 4: Variation of the normalized hydrodynamic pressure magnitudes as a function of the considered 16 locations of the truncated surface and different truncation boundary conditions (selected node at the bottom and at the middle of the crown cantilever)
BEM-FEM Solution of Fluid-Dam Interaction Fluid–structure interaction is the interaction of a moveable and/or deformable structure that is immersed in a fluid and/or contains a fluid. A model that captures such an interaction must use two-way coupling model, where the fluid motion affects the structure’s motion and the structure’s motion affects the fluid’s motion. Coupling should provide compatible link of both media which means equilibrium and correct transition of the physical variables at the interface. There exist various algorithms for coupling the merits of both BEM and FEM numerical methods, direct [5-8] and iterative coupling methods [9-12]. ADAD-IZIIS software [1] is based on BEM-FEM oriented solution of the coupled structure and the incompressible and inviscid fluid. The solution of the coupled system is accomplished by calculating in advance the matrix of hydrodynamic influence utilizing the concept of virtual work of “unit accelerations”. This matrix is stored in the system and recalled in any time step of the dynamic response of the dam. Hence the solution of the coupled systems is actually separated and mutually independent. Hydrodynamic forces are obtained as a product of the matrix of hydrodynamic influence and the vector of manifested total accelerations along the normal at any interface node. The interaction effect at the fluid–solid interface is enforced by adding the matrix of hydrodynamic forces to the classic equation of dynamic motion of the dam, eq. (3). The governing equation for solving the small amplitude irrotational motion of the impounded incompressible and inviscid fluid is governed by the three-dimensional Laplace’s equation as follows: 2W 2W 2W 2 2 0 x 2 y z
(1)
where W(x,y,z) is a function of the potential in the fluid domain. The equation (1) has to be amended by the specified “essential” and “natural” type of boundary conditions that exists at the boundaries of the analyzed fluid domain. Applying BEM technique, Brebbia [13], the discretization of boundary surfaces is by an assemble of eight nodded quadratic “Serendipity” type of boundary elements as follows:
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(2)
where: i=1,NBE; NBE is a number of nodes in the boundary ele ment model. The differential equation of motion of a discrete system written in an incremental form for the "i"-th time increment is the following: ̂ ̈
̂ ̇
̂
̂
̂
(3)
where: ˆ FHD is a vector of hydrodynamic force increment and ˆ Pi is a vector of seismic i force increment. In eq. (3), the vector of hydrodynamic force increment eq, 4 is calculated by use of previously defined matrix of hydrodynamic influence and directly added to the vector of seismic force increment.
ˆ FHD Wij anjtot i
i, j 1, NPT
(4)
where: Wij is a matrix of hydrodynamic influence; anjtot is the absolute acceleration along the boundary element normal; NPT is the total number of nodes at dam-fluid, bottom-fluid and banks-fluid interfaces, where the natural type of boundary condition exists Figure 5, presents the time history response of relative displacement velocity and acceleration for the selected node at the dam crest, top of the crown cantilever, where the extremes of the response occurred. Obviously, fluid structure interaction based on BEM-FEM numerical solution modifies the extreme of the response acceleration at the dam crest by 38% in respect to the dam responce with empty reservoir. Additionally calculated are hydrodunamic effects by use Westergaard added mass concept. It gives lower modification of the dam responce, i.e, the exterem of the relative responce acceleration at the dam crest is modified by 31% Nothe that, the time of extreme occurance is not coinside. Westergaard added mass concept do not give recogition to the impact of the dam flexibility on the amount of generated energy in the fluid domain and therefore on the intensity of the manifested FSI effects. The flexibility property of the dam and the influence of the reservoir domain alter the behavior of the fluid significantly and consequently the coupled system has a stronger response. The figure 6 shows the izolines of distribution of the principal stress G3 that acts along the arches with and without included hydrodynamic effects, whereat hydrodynamic effects are calculated according to added mass method and coupled BE-FE method. The stress extreme is increased by 15% if added mass method is used and 49% if coupled BE-FE method is used. Despite the influence of the terrain irregularities on the amount of energy transferred to the fluid domain, this effect so far, has not been analyzed in detail by exsisting software regardless whether boundary element method (BEM) or finite element method (FEM) is used for fluid discretization. ADAD-IZZIS software gives an opption for taking into account the influence of terrain irregularities on the magnitide of calculated hydrodymanic effecs. Figure 7., presents a snapshot of hydrodynamic pressure distribution over the interface, at time T=4.95sec. It is obtained under the assumptions that the topology of the canyon has a regular shape as indicated in the drawing.
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Figure 5. Modification of the dam response at the dam crest, relative displacement in (m) ; relative velocity in (m/sec) and relative acceleration in (m/sec2). FSI effects defined by use of BEM-FEM solution and Westergaard added mass method
a)
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b)
c) Figure 6: Distribution of principal stress G3 at the extrados face at t=2.42 s a) empty reservoir b) FSI using BEM-FEM regular terrain c) FSI using added mass method
Figure 7: Snapshot of hydrodynamic pressure distribution at time T=4.95 sec 386
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Conclusion ADAD-IZIIS software is based on BEM-FEM oriented solution of the coupled structure and incompressible and inviscid fluid domain. The software offers process of generation of the mathematical model that runs parallel and interactively with the process of design of the arch dam body. Automatically are generated the finite element mesh of the dam and part of the foundation mass for accounting the phenomena of dam-foundation interaction, and boundary element mesh that presents the boundaries of the fluid domain for accounting the fluid structure interaction. The solution of the coupled system is accomplished by use of matrix of hydrodynamic influence utilizing the concept of virtual work of “unit accelerations”. Comparison of the calculated hydrodynamic effects using both BEM-FEM solution and added mass method is made. The added mass method provides acceptable results only in the range of Westergaard restricted hypothesis. Since it neglects the dam flexibility and water compressibility and does not require any discretization of the reservoir domain wherever these features have an impact on the magnitude of hydrodynamic effects there will be discrepancy of the obtained resultants.
References [1] Mircevska V., Bickovski V. (2008). ADAD-IZIIS software: Analysis and Design of Arch Dams, IZIIS- univ. “Ss. Cyril and Methodius”, User’s Manual. [2] Houard L., Boogs (1997). Guide for Design of Arch Dams. A Water Resources, Technical Publication, Engineering Monograph No 36., 1-17. [3] Wiliam, P., Greager, and Joel D., Justin (1984). Engineering for Dams, John Wiley & Sons [4] Mircevska V., Bickovski V., Aleksov I., and Hristovski V. (2013). Influence of irregular canyon shape on location of truncation surface. Engineering Analysis with Boundary Elements, Vol. 37, pp. 624–636. [5] Estorff von O., Prabucki M.J. (1880). Dynamic response in the time domain by coupled boundary and finite elements. Computational Mechanics Vol. 6, pp. 35–46. [6] Pavlatos G.D. Beskos D.E. (1994). Dynamic elastoplastic analysis by BEM/FEM. Engineering Analysis with Boundary Elements Vol. 14, pp. 51–63 [7] Czygan O., Estorff von O. (2002). Fluid–structure interaction by coupling BEM and nonlinear FEM. Engineering Analysis with Boundary Elements Vol. 26, pp. 773–779. [8] Yu G. Mansur, W.J., Carrer J.A.M., Lie S.T. (2001). A more stable scheme for BEM/FEM coupling applied to two-dimensional elastodynamics. Computers & Structures Vol. 79, pp. 811–823. [9] Lin C-C, Lawton EC, Caliendo JA, Anderson LR.(1996). An iterative finite elementboundary element algorithm. Computers & Structures Vol. 39(5), pp. 899–909. [10] Feng YT, Owen DRJ. (1996). Iterative solution of coupled FE/BE discretization for platefoundation interaction problems. International Journal for Numerical Methods in Engineering Vol. 39(11), pp. 889–901. [11] Soares Jr D, von Estorff O, Mansur WJ. (2004). Iterative coupling of BEM and FEM for nonlinear dynamic analyses. Computational Mechanics Vol. 34, pp. 67–73. [12] Estorff von O., Hagen C. (2006). Iterative coupling of FEM and BEM in 3D transient elastodynamics. Engineering Analysis with Boundary Elements Vol. 30, pp. 611–622 [13] Brebbia C.A. (1978).The boundary element method for engineers, Pentech Press
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Influence of Surface Roughness on Sliding Stability Tests and numerical modeling Ø. Eltervaag1, G. Sas2 and L. Lia3 1
Department of Structural Engineering, Faculty of Engineering Science and Technology, Trondheim, Norway 2 Northern Research Institute, Narvik, Norway 3 Department of Hydraulic and Environmental Engineering, Faculty of Engineering Science and Technology, Trondheim, Norway E-mail: [email protected]
Abstract Lightweight concrete dams slide when the shear capacity of one or more sliding planes in the dam’s structure or foundation is exceeded. Several laboratory shear tests were carried out on concrete rock samples. The two materials were mated trough teeth-sawed interfaces with different inclination profile. This paper presents the results of the numerical modeling of those tests. The results of the shear tests were compared to the predictions of the model used in the Norwegian guidelines. It has been found that the model used in the guidelines do not predict the shear capacity accurately. Through finite element analyses a better representation of the tests has been achieved, especially regarding the influence of roughness.
Introduction In Norway, the stability of a dam is reconsidered every 15 to 20 years, depending on the consequence class for the given dam. In the past 40 - 50 years, since the large hydropower development époque in Norway, the level for required safety has increased. Consequently some dams constructed during that period no longer are considered to have the sufficient safety and expensive upgrading is needed.
Figure 1: Profiles of test samples [1]
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Two of the samples with geometry 1 and 4 were casted with bond between the rock and concrete. The remaining ten samples had no bond. During casting of concrete a plastic film was attached on the rock surface of those ten samples to prevent bonding.
Theory According to the Norwegian regulations [2] the sliding stability is assessed using the shear friction method, and is expressed by a factor of safety (FS). For a horizontal sliding plane this factor is found from the following equation. FS
c A V tan
(1)
H
In equation (1) c is the cohesive parameter, A is the shearing plane area, V and H are the vertical and horizontal forces acting on the plane respectively. The frictional factor of the plane surfaces is expressed by tan (ϕ). Thus, the factor of safety is calculated from the averaged normal stress and friction factor of the sliding plane. The shear friction method is based on the Mohr-Coulomb criterion [3] for describing the shear capacity of the sliding plane. This criterion states that the shear capacity, τ, is linearly dependent on the applied normal stress, σ, trough a material specific frictional parameter tan(ϕ) plus a cohesive parameter, c. This can be expressed as follows:
c tan
(2)
In the 1960’s it was recognized that the failure envelope for rock mass (rocks with joints and faults) was curved. One of the major contributions to this understanding was when Patton [4] derived a bi-linear failure criterion from experiments with “saw-toothed” rock specimens shown in Figure 2. Patton observed that sliding occurred at lower levels of normal stress than what was needed to cut off the saw-toothed geometry. This failure mechanism can be described on the form: tan(b i) (3) Were ϕb is the material specific friction angle, simply denoted ϕ in the Mohr-Coulomb relation, equation (2), and i is the angle of the asperities, called the asperity inclination or dilation angle. When the normal stress exceeded a certain value the saw teeth were cut off at their base. Patton [4] explained this as a change of governing failure mode, from a sliding failure along the material interface, to a failure in the material itself. The shear capacity regarding this material failure is described by:
cx tan(r )
(4)
Where cx is the bond of the failing material, and ϕr is the residual friction angle of the failure plane. The shear envelope obtained from these expressions is shown in Figure 2 .
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Figure 2: Patton's bi-linear failure criterion [4] According to Patton [4] the bi-linear failure envelope illustrates that there are two possible failure mechanisms for the rock specimens studied. The first mechanism is sliding over the asperities (saw-teeth) and occurs at low normal stresses. The second mechanism is shearing through the asperities and occurs at relatively high normal stresses. These different failure modes have been further described by Johansson [5]. Johansson [5] developed a conceptual model to describe sliding failure of one idealized quadratic asperity. Three failure modes were identified; sliding along the side of the asperity facing the load, shear-failure along the base of the asperity, and tensile failure in the rock-base of the asperity. A sketch of an idealized asperity is shown in Figure 3.
Figure 3: Principle sketch of an idealized 2D asperity [5] To describe sliding along the loaded face of the asperity (the left side in Figure 3) Johansson [5] uses Patton’s formulation, eq. (4), for shear capacity for low normal stresses (transformed from stresses to forces).
T N tan(b i)
(5)
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The equation for shear failure along the base of the material is derived based on the MohrCoulomb criteria, eq. (2) and Patton’s equation for high normal stress, eq. (4).
T ci Lsp 2 N tan(i )
(6)
For a tensile failure to occur in the rock-base, the vertical tensile stresses must exceed the tensile capacity of the rock. To calculate the average tensile stress in the rock Johansson [5] assesses the moment-equilibrium about point O in Figure 3.
T
(3 ci 4 ti ) Lasp 2
(7)
2 tan(i)
Where σti is the tensile strength and σci is the compressive strength of the rock.
Method For this work the finite element software ATENA-Science developed by Cervenka Consulting has been used [6]. The geometry of tested samples is modeled as a 2D plane stress problem, left side of Figure 5. To avoid numerical instability when applying the loads directly on concrete, the surrounding steel shear box was modeled also, right side and medallion in Figure 4. In Atena the boundary conditions were applied as fixed lines along the outside of the lower steel frame, the loading was introduced by a vertical line load along the top of the steel frame, and horizontal displacement along the left edge of the upper steel frame. Monitoring points were added at the left edge of the upper steel frame and at the right edge of the concrete material to enable load/displacement curves.
Figure 4: 2D geometrical model of a test sample with ten degree asperity angle ATENA uses interface elements to model contact between two parts of a model. To model the behavior of the interface elements ATENA uses the Penalty Method [6]. The physical properties of these interface elements are governed trough the Mohr-Coulomb criterion presented in eq. (1). To obtain input values of the interface properties the results from the tests are compared to the failure criteria described above. However, these classic formulations present large errors compared to the test results. Reliable input parameters were obtained by applying a hybrid formulation where the shear capacity is determined from the actual failure 392
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mechanism for the tests; sliding over or shearing off the asperities, eqs. (5) or (6). This hybrid formulation for shear capacity of the samples can be expressed as follows, V
c A N tan(b i) cos i(1 tan i tan b )
(8)
where i is the residual inclination angle of the asperities, which may differ from the original inclination angle if the asperities are sheared off. The basic friction angle is material constant and therefore equal for all the samples. Thus, the basic friction angle can be found from a sample without asperities, and is determined to be 34.4˚. The cohesive parameter is found by inserting the known values from the results of each of the tests into eq. (8). These values are listed in Table 1. The physical explanation of this parameter is believed to be wear of the interface as cut-off of micro asperities (micro roughness) as shown in eq. (6). The cut-off shows a decreasing effect as the samples slide against each other. To account for this a softening behavior of the cohesive parameter is introduced. In ATENA the effect of the asperities (macro roughness) is included trough the geometry of the interface. In ATENA the loading is applied incrementally. First the normal pressure was applied as a vertical line-load. The line-load for the different tests has been found by distributing the applied experimental vertical force presented in [1] over the length of the samples (240mm). The applied line-loads are presented in Table 1. Table 1: Interface and loading input parameters Sample 1.1 1.2 1.4 2.1 2.2 2.3 3.1 3.2 3.3 4.2
Test results [1] i [˚] δHmax [mm] 40 15.23 40 15.44 40 26.34 20 24.99 20 21.98 20 20.07 10 33.16 10 32.94 10 33.23 0 33.23
N [kN] 27.52 46.91 68.49 27.57 47.77 68.36 24.82 45.98 67.65 67.23
i [˚] 0 0 0 20 20 20 10 10 10 0
Input used in ATENA δH “Cohesion” Q [kN/m] [mm] [kPa] 20 1245 114 20 1720 195 20 2371 285 20 282 115 20 781 199 20 1552 285 20 309 103 20 382 192 20 536 282 20 62 280
Then the horizontal force is applied trough prescribed deformation of ten millimeter at the left edge of the upper steel frame, see Figure 4. To achieve plane horizontal displacement (avoid rotation) of the upper part master-slave boundary conditions are introduced along the upper edge in this interval. Interval three and four are identical to interval two, except that the master-slave boundary conditions are removed. The standard incremental and iterative Newton–Raphson method for material nonlinear structural analysis was used in the numerical simulations, based on the finite element method. The specimens were modeled with a mesh of 8-node serendipity plane stress finite elements. A Gaussian integration scheme with 2 x 2 integration points was used for all the concrete elements, [6,7].The mesh is shown in Figure 5.
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Figure 5: Mesh of test 2.2 Both the concrete and rock materials are modeled using a fracture-plastic model that combines constitutive sub-models for tensile and compressive behavior, as presented in the ATENA user manual [6], see Figure 6. This fracture model employs the Rankine failure criterion and exponential softening, with the hardening/ softening plasticity component based on the Menétrey– William failure surface [6]. The concrete post-cracking tensile behavior was simulated by a softening function in combination with the crack band theory [6].
ef c ef
ft
d
fct
E0
c
t
0
eq
unloading
wc = 5.14
Gf
loading
U
Gf fct
f
ef c
wc
Figure 6: Constitutive model for concrete (left) and softening function (right) [6] The decision for modeling the rock as concrete is based on the behavior of the actual rock samples during compressive standardized tests. It was observed that this behavior is more in line with the available material models for concrete than rock. The steel shear box was modeled as a linear elastic material. Theoretical background of the above mentioned constitutive models are given in the ATENA Theory manual [7]. Input parameters for the rock and concrete material are given in Table 2.
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Table 2: Material parameters for concrete material models Material Young’s Modulus [GPa] Poisson’s Ratio Tensile strength [MPa] Compressive Failure Stress [MPa] Fracture Energy [MN/m] Plastic strain Onset of crushing [MPa]
Concrete 37 0.2 4.1 58 1.03x10 -4 -0.00147 -8.61
Rock 100 0.2 2.317 280 1.05x 10-5 -0.000296 -1.54
Critical compressive displ. [m]
-0.0005
-0.0005
23
23
3
Density [kN/m ]
In Table 1it is seen that the geometry of tests 1.1 to 1.4 is modeled with zero inclination angle. The reason for this is that these tests showed a shearing failure mode, where the asperities were cut off, therefore there were no sliding between asperities but material failure. A total shearing failure mode has been hard to obtain from the numerical models. The main reason for this is that the analyses crash after the material starts to crack due to instability problems with zero or negative Jacobian for the stiffness matrix. This error message is an indication of an ill-conditioned system of equations [8]. Perhaps a more refined mesh with more integration points could have been used. Due to low computational capacities this was not possible at the time when this work was carried out. In Figure 7 the concrete material failure is shown. The blue field in the left part indicates that the tensile strength of the material is reached.
Figure 7: Material failure for test 1.4. Due to the numerical instability problems only the peak shear capacity was obtained from these analyses. However, this capacity is governed by the concrete material parameters alone, not by the interface parameters. In the test report [1] only the compressive strength was given for the rock and concrete materials. Thus, the tensile strength and onset of crushing have been generated automatically by ATENA. To overcome these problems the tests in series 1 have been modeled using the residual inclination angle of the asperities (zero) according to eq. (8).
Results The results from the numerical analyses have been monitored by recording the horizontal displacement and reaction-force along the loaded edge of the model. The load-displacement curves from the analyses (labeled ATENA) are presented in the same diagrams as the curves from the actual tests (labeled LTU after the lab where the tests were conducted). In total, eleven of the twelve shear tests were analyzed. Due to space limitation in Figure 8 below, the graphical results are shown for selected analyses only. Full description of all results are presented in [9].
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Figure 8 Load displacement diagrams showing tests and numerical results In general the procedure of assuming cut-off of micro roughness as cohesion gives a good representation of the peak shear capacity of the actual tests. When coupled with a softening behavior of the cohesive parameter, the capacity along the failure development and the residual shear capacity are represented sufficiently accurate. Since the input parameters are found from each specific test the finite element models should produce results that match both the peak and residual shear capacity from the tests exact. However, studying the results, it becomes evident that some errors still occur. The difference between the peak shear capacity from the laboratory tests and the numerical analyses are listed in Table 3.
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Table 3: Error in peak shear capacity from the numerical models. Sample
Vmax[kN] test
Vpeak [kN] ATENA
1.1 1.2 1.4 2.1 2.2 2.3 3.1 3.2 3.3 4.1H 4.2
90.53 131.19 183.42 50.74 100.59 162.83 35.38 58.70 85.43 240.00 49.56
89.91 125.80 173.21 50.81 100.57 162.08 34.65 57.56 83.65 225.75 49.52
[kN] 0.62 5.39 10.41 0.07 0.02 0.75 0.73 1.14 1.78 14.25 0.04
Error [%] 1 4 6 0.1 0.02 0.5 2. 2 2 6 0.1
The error-% is calculated with respect to the actual value from the tests [1]. The average error is approximately 2% with a maximum of 6% for two analyses. Compared to the results from the Mohr-Coulomb and Patton criteria in Table 4 this is a significant improvement. Table 4: Error in peak shear capacity from hand calculations. From tests [1] Sample I [˚]
N [kN]
V [kN
Mohr-Coulomb V [kN] Error [%]
Patton V [kN]
1.1 1.2 1.3 2.1 2.2 2.3 3.1 3.2 3.3
27.52 46.91 68.49 27.57 47.77 68.36 24.82 45.98 67.65
90.53 131.19 183.42 50.74 100.59 162.83 35.38 58.70 85.43
361.60 380.99 402.57 20.32 35.21 50.39 18.29 33.89 49.86
113.73 193.86 283.04 41.19 71.89 102.88 26.06 48.28 71.04
40 40 40 20 20 20 10 10 10
332 190 119 59 65 69 48 42 42
Error [%] 26 48 54 18 28 37 26 18 17
Although the errors in Table 3 are not large, their cause needs to be addressed. One possible source is the inaccurate recording of the residual capacity in the tests. Due to the fact that when concrete was cracking the analysis was not always stable, the residual capacity only describes the capacity as long as the interface asperities are intact. This explains why the residual capacity is not well represented for test series 2 where some deformation of the asperities was registered in the tests. For test series 1 this source of error is avoided as the interface is modeled with the inclination of the residual sliding plane from the actual tests. It is hard to determine why stable results after material failure in the concrete was not obtained, but it is believed that a fine mesh and more refined material models would improve this. Not all the required input parameters for the material models were obtained from the physical tests, thus some default values were used in the modeling. With lacking parameters, a full parametrical study would be needed to refine the materials. The results from the analyses show that the actual shear capacity of the tests are better represented through numerical modeling than the formulations available for hand calculations. Especially the opportunity to represent the softening behavior after the peak shear capacity is reached, is a large benefit.
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Conclusion Through finite element modeling, more sophisticated analyses regarding stability towards sliding might be obtained. This allows more sustainable design of structures subjected to sliding instability. However, it must be noted that due to the limited amount of samples analyzed and the scale effects encumbered with shear capacity, further work is needed to utilize the potential of finite element modeling of stability towards sliding for full scale dams. A full parametric study is needed to improve the method.
Acknowledgements Acknowledgements are given to the professor Kjell H. Holthe at the Norwegian University of Engineering Science and Technology and Dr. Dobromil Pryl at Cervenka Consulting. Acknowledgements are also given to EnergiNorge for financial support.
References [1] Liahagen, S., (2012) Stabilitet av betongdammer - Ruhetens påvirkning på skjærkapasiteten mellom betong og berg. Master’s Thesis. NTNU,IVM, Trondheim, unpuplished. [2] NVE, Norwegian Water Resources and Energy Directorate, (2005), Retningslinje for betongdammer. [3] Coulomb, C. A. (1776). Essai sur une application des regles des maximis et minimis a quelquels problemesde statique relatifs, a la architecture. Mem. Acad. Roy. Div. Sav., vol. 7, pp. 343–387. [4] Patton, F.D (1966). Multiple modes of shear failure in rock. 1st ISRM Congress, September 25 – October 1, 1966, Lisbon, Portugal. LNEC [5] Johansson, F., (2009). Shear Strength of Unfilled and Rough Rock Joints in Sliding Stability Analyses of Concrete Dams. Ph.D. Thesis. Royal Institute of Technology (KTH), Stockholm. [6] Cervenka, V. Cervenka, J. Zednek, J. Pryl, D. (2013) ATENA program Documentation, Part 8: User’s Manual for ATENA_GiD Interface. http://www.cervenka.cz/assets/files/atena-pdf/ATENA-Science-GiD_Users_Manual.pdf [7] Cervenka, V., Jendele, L., Cervenka. J., (2012) ATENA Program Documentation, Part 1: Theory. http://www.cervenka.cz/assets/files/atena-pdf/ATENA_Theory.pdf [8] Pryl, D., Cervenka, V. (2013) ATENA Program Documentation, Part 11: Troubleshooting manual. http://www.cervenka.cz/assets/files/atena-pdf/ATENATroubleshooting.pdf [9] Eltervaag, Ø. (2013) Sliding stability of Lightweight Concrete Dams – Development of numerical models. Master’s Thesis. NTNU, IKT, Trondheim. Unpublished.
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Seismic Analysis of a Concrete Arch Dam Considering Concrete Heat Generation Damage Effects P. Dakoulas1 1
University of Thessaly, Volos, GREECE E-Mail: [email protected]
Abstract This study evaluates the seismic performance and safety of Tavropos arch dam (Greece) considering the dynamic canyon-dam-water interaction. Tavropos dam has a height of 83 m and crest length of 220 m. The numerical model of the arch consists of 17 concrete cantilevers separated by vertical joints, whereas the foundation-abutment model represents a large region of the canyon. The interface behavior at the vertical joints and the horizontal base joint is modeled in three different ways: as bonded surfaces, as surfaces in contact with frictional strength, and as a combination of bonded and contact surfaces. To account for possible prior damage, the study simulated the thermo-mechanical phenomena that took place during dam construction, related to heat generation due to concrete hydration. The seismic analysis is performed against a MCE having a magnitude of 7.5 and peak ground acceleration of 0.47g. The seismic evaluation is based on an extensive parametric investigation of the effects of various key factors, including concrete strength, joint interface behavior, water level, hydrodynamic pressures, ambient temperature, canyon rock flexibility, spillway geometry, earthquake excitation characteristics, etc. It is concluded that the expected overall stability and performance of Tavropos concrete arch dam is satisfactory.
Introduction Lake Plastiras was created after construction of an arch dam (Fig. 1) in Tavropos river, a tributary of Acheloos river in Northern Greece [4]. The double-curvature Tavropos arch dam has a height of 83 m and crest length of 220 m. Construction started in 1955 and was completed in 1959, during which the dam was impounded. Today, Lake Plastiras provides drinking water to the city of Karditsa and nearby villages, and irrigation to the region near Larissa. Its electric power is 130 MW, whereas the lake is an attractive tourist destination with more than 120,000 tourists annually.
Figure 1: Tavropos Arch Dam in Lake Plastiras, Greece 399
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The objective of this work is to conduct a seismic performance and safety evaluation of the data, considering the dynamic canyon-dam-water interaction and current data about the local seismicity based on recent research [11][3][8][9][10].
Figure 2: Finite element discretization of (a) concrete arch dam geometry consisting of 17 parts and (b) Part No.10 geometry at the middle section
Figure 3: Finite element discretization of the canyon geometry (a) model A and (b) model B consisting of full quadratic finite elements for the entire canyon region.
Numerical Model Dam geometry The study is based on a detailed numerical model of the dam and canyon using the code ABAQUS [1]. Fig. 2a illustrates the numerical discretization of the arch dam body. The dam body consists of 17 parts, having a width equal to 12 m, except of the central part, which has width equal to 16 m. The 17 parts are connected with vertical joints. Fig. 2b shows Part 10, located at the mid-section of the dam. As all parts near the central region of the arch, Part 10 consists of 4 concrete blocks: A, B, C and D. Blocks B, C and D at the lower upstream side were designed as to increase the stability of the dam in the case of a potential strong earthquake during the phase of construction. They are all attached to main block A, but there are horizontal joints between blocks B and C, as well as between blocks C and D to reduce seismic tensile stresses. To improve numerical efficiency, only 2 out of the 4 actual horizontal joints of each part were discretized, without any loss of accuracy. All dam parts are
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discretized using solid hexahedral elements C3D8, having 8 nodes and 8 integration points. A total number of about 4400 elements were used for the modeling of the arch dam geometry. Canyon geometry The discretized canyon geometry has length of 600 m, width of 600 m and depth of 250 m. It consists of two parts: (a) the first part discretizes the irregular geometry near the foundation of the dam. It is discretized by using quadratic, reduced integration hexahedral elements C3D20R, each having 20 nodes, whereas the irregular rock surface and dam foundation excavation, are discretized using quadratic modified tetrahedral elements C3D10M, which are robust, accurate and suitable for frictional contact behavior; (b) for the rest of the canyon, two different models have been utilized: (1) model A (Fig. 3a) consists of hexahedral solid elements C3D8, having 8 nodes (2) model B (Fig. 3b) consists of quadratic hexahedral elements C3D20R, having 20 nodes. About 30500 and 27000 elements were used for canyon models A and B, respectively. Damage plasticity model for concrete The plastic-damage constitutive model for cyclic loading by Lee and Fenves [6] is used here for modeling the behavior of concrete. The model takes into account the effects of strain softening, distinguishing between the damage variables for tension and compression. It incorporates a degradation mechanism that represents the effects of damage on the elastic stiffness and the recovery of stiffness after crack closure. Three values of compressive strength have been adopted, based on large cubic specimens (20cm x 20cm x 20cm) obtained during construction and tested after 365 days. The use of large size specimens is important to avoid the negative effect that the large diameter aggregates may have on strength prediction. The three values of compressive strength are: (A) the mean value fc =46 MPa (B) the mean value minus three standard deviations, fc 3 ≈42 MPa and (C) f c =32 MPa, i.e. an unlikely, much lower value to account for possible additional uncertainties. For f c =46 MPa, the stress-strain behavior during uniaxial loading-unloading-reloading of a cube of concrete, subjected separately to compression and tension tests is given in Figure 4. For the cyclic tension tests, the softening behavior and the effect of the accumulated damage are taken into account, whereas for the cyclic compression, these effects are ignored in the present study. Table 1: Concrete Properties Property Density, ρ [kg/m3] Compressive strength, fc [MPa] Dynamic tensile strength, ftd [MPa] Static Young’s modulus, E [GPa] Dynamic Young’s modulus Ed, [GPa] Poisson’s ratio, ν Coefficient of thermal expansion, a [°C-1] Specific heat, c [J/(kg °C)] Conductivity, k [W/(m °C)] Heat transfer coefficient [W/(m2 °C)] Hydration heat, q(t) [kcal/kgcement] (t : in days)
Case A 2350 46 6.4 31 42 0.15 10-5 879 2.5 16
Case B 2350 42 6.0 31 42 0.15 10-5 879 2.5 16
Case C 2350 32 5.3 31 42 0.15 10-5 879 2.5 16
q(t ) t /(0.0653 0.0175t )
Joint behaviour The behavior at the vertical joints and the horizontal joint at the dam base is modeled in three different ways: as bonded surfaces, as surfaces in contact with frictional strength, and as a 401
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combination of bonded and frictional surfaces. The bonded joints are considered to have the tensile strength of concrete of the adjacent blocks. The surfaces with only frictional strength, have a coefficient of friction equal to 0.5 for vertical joints and 0.8 for the horizontal joints.
Figure 4: Damage plasticity concrete model: stress-strain relation during loading and unloading in uniaxial (a) tension and (b) compression. Canyon rock properties The canyon rock material is fine-layered limestone containing sparse intermediate layers of radiolarites. During the static and dynamic analyses of the dam-canyon system, the canyon rock is considered to behave as a linearly elastic material. Table 2 summarizes the canyon rock properties. Two scenarios are examined for the dynamic analyses, namely, case R1 and case R2, having shear wave velocities equal to Vs = 2800 and 2000 m/s, respectively. Table 2: Canyon Rock Properties Property Density, ρ [kg/m3] Static Young’s modulus, E [GPa] Dynamic Young’s modulus, Ed [GPa] Poisson’s ratio, ν Shear wave velocity, Vs [m/s] Hysteretic damping, ξr
Case R1 2450 16 48 0.25 2800 0.03
Case R2 2450 16 25 0.25 2000 0.03
Effect of heat generation during construction It is of interest to examine concrete behavior immediately after construction, especially for high performance concrete, as cracking may develop due to deformation caused by heat that is generated during cement hydration reactions [7][2]. Potential cracking generated during or immediately after construction of various dam segments might have developed weak areas, which could affect its seismic performance during earthquakes. To examine the effects of premature cracking, a numerical simulation of the dam construction process is conducted, modeling the cement-hydration heat generation, radiation, convection, conduction and ambient temperature variation for a period of about 24 months. Fig. 5 plots the distribution of the temperature in a section of the dam after 200 days, whereas Fig. 6 plots the temperature variation with time at points A, B, C, D and E, as well as the ambient temperature. The maximum temperature occurs in the middle point (C) of the dam wall and has a value of 50.5 C at 200 days from the construction start. Examination of the major principal plastic strain caused by the transient temperature differences within the dam showed that the concretehydration heat did not cause any significant cracking within the main body of the dam. Any 402
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such cracking within the dam body is practically either zero or negligible. Some thermal cracking may have developed within the upstream aseismic concrete blocks (see Fig. 2b), but it has absolutely no effect on the performance and stability of the dam in its current state.
Figure 5: Distribution of temperature within the dam body at 200 days since construction start
Figure 6: Variation of temperature at points A, B, C, D and E during construction versus time
Seismic analysis Dynamic characteristics of the dam
The fundamental natural frequency of the dam for full reservoir and canyon rock with shear wave Vs =2800 m/s is 3.97 Hz, whereas for Vs =2000 m/s, it reduces to 3.81 Hz. Earthquake excitation The seismicity of the region has been studied by Panagiotopoulos and Papazachos [10]. For the present study, a Maximum Credible Earthquake of magnitude M 7.5, with an epicentral distance from the dam site about R = 15 km, is considered. The Maximum Probable Earthquake is based on a magnitude of M 7.0 and minimum distance R =15 km. The seismic excitation considered consists of three different acceleration records. The first two have been recorded at rock sites (Lucerne and Pacoima Dam records). The third record is synthetic and has response spectra that match the Eurocode 8 design spectra for rock sites. All records have been base-line corrected and scaled to a peak horizontal outcrop-rock acceleration equal to 0.47g and peak vertical outcrop-rock acceleration equal to 0.30g. Fig. 7 plots the horizontal and vertical acceleration time histories of the synthetic record. Fig. 8 plots the acceleration response spectra for the horizontal component and the Eurocode spectra for rock site. The input excitation at the canyon base is computed through de-convolution of the selected 403
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motions. To be able to account for the radiated seismic energy, appropriate dashpots are placed at the canyon boundary and the excitation is imposed in terms of seismic stresses.
Figure 7: Synthetic excitation at outcrop rock (a) horiz. acceleration (b) vertical acceleration
Figure 8: Synthetic horizontal acceleration spectra and Eurocode spectra for rock sites Hydrodynamic pressures The hydrodynamic pressures acting on the upstream side of the dam have a significant effect on the earthquake behavior of the dam. They are considered here by using the added-mass formulation proposed by Zangar [11], which was implemented in code ABAQUS [1]. The amount of mass b at a node is equal to:
z 1 z z z b h Cm (2 ) (2 ) w A 2 h h h h
(1)
where h = depth of the reservoir, Cm = coefficient based on the angle of upstream surface to the vertical, z = depth of node below water surface, w = water density and A = area around node contributing to hydrodynamic forces.
Results and discussion The objective of the parametric analysis is to investigate the effect of various factors on the seismic performance of the Tavropos Arch Dam. More specifically, the parametric study examines effect of concrete strength variation, seasonal temperature variation, reservoir water level, canyon rock flexibility, joint behavior, hydrodynamic pressures, spillway geometry, and earthquake excitation characteristics. Due to lack of space, only representative results of the most likely scenario are presented in this article, whereas additional results will be published elsewhere. The results presented below are based on a concrete compressive strength of 46
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MPa, dynamic tensile strength of 6.4 MPa, critical summer temperature variation, maximum reservoir water level (792 m), and canyon rock S-wave velocity 2800 m/s. Both models A and B (shown in Fig. 3) are utilized for the analysis. First, the overall response and the stability of the dam are considered. Relative displacements Results from two scenarios are presented here for the evaluation of the response and stability of the dam. The first scenario assumes that the vertical joints and the base joint behave as bonded surfaces. Figs. 9a and 9b plot the relative u/d displacement at mid-crest of the dam subjected to the synthetic record for bonded joints, using the two canyon models A and B, respectively (Fig. 3). The computed relative displacement by the two models is rather similar, with the peak value about 4.5 cm. There is no residual relative displacement at the end of shaking. The peak relative displacement for the other two excitations (not shown here) are about 4 cm. Fig. 10a plots the distribution of the u/d relative displacement for bonded joints using model A at a moment of peak response (t = 6 s). The second scenario is quite conservative as it assumes that the joints have zero tensile strength and behave as surfaces in contact having only frictional strength. Fig. 10b plots the residual relative u/d displacement for the case of un-bonded frictional joints, allowing opening and sliding during shaking. The maximum relative displacement in this case is less than 5 cm, which is considered as safe with regard to the stability of the dam for this very conservative scenario.
Figure 9: Time history of relative u/d displacement at mid-crest assuming bonded joints obtained from (a) model A and (b) model B
Figure 10: (a) Peak relative u/d displacement at t = 6 s assuming bonded joints and (b) Residual relative u/d displacement at the end of shaking assuming un-bonded frictional joints. Stresses and strains Fig. 11 shows the distribution of major principal stress, 1 , at the moment of maximum response of the dam subjected to the Synthetic record based on Model A (Fig. 3). As shown in the figure, high tensile stresses develop at the upper middle part of the arch dam, reaching in some elements the concrete tensile strength for a very small time increment (e.g. 0.01s) and causing some limited plastic tensile strain. These tensile stresses occur when the upper part of the dam has a relative displacement towards upstream with respect to its base. In this case, the 405
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combined effect of dam and water inertia (negative hydrodynamic pressure) cause the upper part of the dam to deform towards upstream. This upstream relative displacement becomes maximum at t=5.32 s for the Synthetic record (see point A in Figure 9a). Such deformation causes an extension of the arch dam along its length, which is maximum at the upper-middle area of the arch, resulting to relatively high, short-duration tensile hoop stresses. It is noted that the tensile stresses become maximum at the upstream surface of the central crest area of the arch, whereas they are quite smaller on the downstream side. Figs. 12a and 12b plot the major principal stress time history at mid-crest, evaluated at the upstream surface element using the two models A and B, respectively. The results show that 1 reaches the tensile strength momentarily in Fig. 12a, but it remains below the tensile strength in Fig. 12b.
Figure 11: Distribution of major principal stress 1 at a moment of maximum tension of the dam subjected to the Synthetic record excitation (time t = 5.32 s)
Figure 12: (a) Major principal stress and (b) minor principal stress time histories at midcrest (upstream surface element)
Figure 13: Accumulated major principal plastic strains at the end of shaking
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Despite the large tensile stresses that may develop momentarily during shaking, the accumulated major principal plastic strain 1p at the end of shaking is very limited. Fig. 13 plots the distribution of 1p at the end of shaking for the Synthetic record based on model A. Its maximum value is only about 5 10-5, located at the upstream face of the mid-crest (concrete part 10). In reality, it is expected that the value of the tensile plastic strain 1p will be even smaller than the values shown in Fig. 13. This is because the actual tensile strength at the joints is expected to be about 0.7 f t , where f t is the tensile strength of the dam concrete. Thus, in areas of higher tensile stress concentration, as at the mid-crest area, any cracks most likely will occur at the weakest points, i.e. the vertical joints. As shown in other analyses not presented here, if the entire vertical joints rupture before shaking (i.e. the joint tensile strength is zero), the tensile stresses developing within the mid-crest area of the arch dam are significantly reduced, due to the instantaneous, local opening of the vertical joints. Therefore, the accumulated major principal plastic strains near the mid-crest area are much smaller. Note that the results in Figs. 11, 12a and 13 are based on model A. By using the more refined model B, it is shown that the tensile stresses developing in the mid-crest area are slightly smaller and do not exceed the tensile strength. Thus, no plastic strains develop in the dam body for the presented case of concrete with compressive strength equal to the mean value f c =46 MPa. Figure 14 plots the distribution of the minor principal stress 3 (maximum compression) at a moment of maximum response, corresponding to time t = 6 s (point B in Figure 9a). This occurs when the dam base and canyon moves upstream, while the dam is pushed by the concrete and water inertia (positive hydrodynamic stresses) towards downstream, thereby increasing the compressive stresses in the mid-crest area of the arch. The maximum value of the compressive stresses at the mid-crest area during shaking is less than 12 MPa, i.e. the compressive stresses are very small compared to the compressive strength of concrete in uniaxial load ( f c =46 MPa). In this case, the minimum available factor of safety against compressive failure FSc fc /(1 3 ) during shaking is larger than 2.5. Thus, no plastic deformations occur due to compressive stresses. The compressive stresses for the other two records are smaller than those obtained for the Synthetic record.
Figure 14: Distribution of minor principal stress 3 at a moment of maximum compression of the dam subjected to the Synthetic record excitation (time t = 6.00 s)
Conclusions Evaluation of the results of all parametric studies leads to the general conclusion that, for the maximum earthquake intensity anticipated, the dam is safe against any instability and its overall performance is satisfactory. More specifically, the main conclusions are the following:
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1. The concrete-hydration heat generated during construction did not cause any significant cracking within the main body of Tavropos dam. 2. The maximum seismic relative displacement of the dam for the case of grout-bonded joints is small (about 3 to 4.5 cm). There are no permanent relative displacements at the end of shaking. 3. For the extreme scenario in which the vertical joints and the joint at the base are already ruptured before the earthquake, the relative displacement of the dam at the end of shaking is limited to values less than 5 cm. Thus, even for this extreme scenario, the stability of the 17 cantilever dam parts is fully assured. 4. Based on the results of all analyses, the maximum compressive stresses developing during seismic shaking range between -10 MPa to -16 MPa and do not cause any plastic strains. 5. The magnitude of tensile stresses within the concrete parts depends on the behavior of the vertical joints. The results of the parametric analysis show that the tensile plastic strains (or cracking) developing within the dam body are very small. Therefore, the performance of the dam with respect to the development of tensile cracking during shaking is satisfactory. 6. The summer environmental temperature condition is the most critical one, as it results to higher tensile plastic strains within the dam body. 7. Among the four reservoir water elevations investigated, the most critical one is the maximum water level (792 m), which results to relatively higher tensile plastic strains within the dam body. 8. The stiffer rock, having a wave velocity Vs =2800 m/s, yielded consistently larger values of tensile plastic strains compared to those obtained for Vs =2000 m/s.
Acknowledgment The financial support by the Public Power Corporation is gratefully acknowledged.
References [1] ABAQUS (2012). Users’ Manual, Simulia, Pawtucket, Rhode Island. [2] Azenha, M., Faria, R., Ferreira, D. (2009), Identification of early-age concrete temperatures and strains: Monitoring and numerical simulation, Cement & Concrete Composites, 31, 369–378. [3] Chopra, A. (2008). Earthquake analysis of Arch Dams: Factors to be considered, 14th World Conference on Earthquake Engineering, October, 2008, Beijing, China. [4] Dakoulas, P. Seismic analysis of Tavropos Arch Dam, Research Report to Public Power Corporation, University of Thessaly, Volos, Greece [5] Escuder, I.B. and Blazquez, F.P. (2007). Lessons learned in the analysis of the elastic behavior of an arch-gravity dam performed by seven independent engineering teams, 9th Bench. Work. Num. Analysis of Dams, ICOLD, St. Petersburg, Russia, June 22-24. [6] Lee J., and Fenves, G.L (1998). A plastic-damage concrete model for earthquake analysis of dams, Journal of Earthq. Eng. & Struct. Dynamics, 27: 937-596. [7] Meghella, M. and Frigerio, A. (2009). Theme A: Initial strain and stress development in a thin arch dam considering realistic construction sequence, 10th Benchmark Workshop on the Numerical Analysis of Dams, ICOLD, Paris, France, Sept. 16-19. [8] Mills-Bria, B., Nush, L., and Chopra, A. (2008). Current Methodology at the Bureau of Reclamation for the Nonlinear Analysis of Arch Dams Using Explicit FE Techniques, 14th World Conference on Earthquake Engineering, October, 2008, Beijing, China.
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[9] Mojtahedi, S. and Fenves, G. (2000). Effect of contraction joint opening on Pacoima Dam in the 1994 Northridge earthquake, California Strong-Motion Instrumentation Program Data Utilization Report, CSMIP/00-05 (OSMS 00-07), Sacramento, Calif. [10] Panagiotopoulos, D. and Papazachos, K.B. (2008). Active tectonics of Thessaly and seismicity of Karditsa, 1st Conference on Development, Karditsa, Greece. [11] US Bureau of Reclamation (2006). State-of-Practice for the Nonlinear Analysis of Concrete Dams at the Bureau of Reclamation, USBR Report, Colorado, USA.
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Earthquake safety assessment of arch dams based on nonlinear dynamic analyses S. Malla1 1
Axpo Power AG, Parkstrasse 23, CH-5401 Baden, SWITZERLAND E-mail: [email protected]
Abstract For the earthquake safety assessment of two major arch dams in Switzerland, various linear and nonlinear dynamic analyses were performed. For these analyses, a safety evaluation earthquake (SEE) with a return period of 10,000 years was considered in each case. Even though the input motion has a PGA of less than 0.2 g, the crest region would be subjected to amplified accelerations significantly larger than 1.0 g. Moreover, high tensile stresses occurring in the crest region are likely to exceed the tensile strength of the vertical contraction joints and the horizontal lift joints. Consequently, these joints could open during the earthquake shaking, possibly resulting in an upper portion of a dam block becoming fully detached from the rest of the dam. The dynamic analysis of possibly detached portions in the crest region showed that they could undergo some sliding and rocking motions. However, they would remain stable during and after the earthquake. Based on the results of the nonlinear dynamic analyses, it is concluded that the earthquake loading could cause some limited damages, but an uncontrolled release of reservoir water is unlikely to occur. Therefore, the investigated dams satisfy the safety requirements for the SEE.
Introduction A program of systematic assessment of earthquake safety of all significant dams in Switzerland is scheduled to be completed by the end of the current year 2013. A document containing state-of-the-practice guidelines for this investigation program was issued by the Swiss Federal Office of Energy in 2003. The primary goal of this program is to ensure the safety of the downstream population against loss of life and property damage (Panduri et al., 2012). For this purpose, it must be shown that the safety evaluation earthquake (SEE) would not cause a dam failure resulting in an uncontrolled release of reservoir water. Depending on the height and the reservoir volume, a dam is categorized as class 1, 2 or 3 and it has to be checked for the SEE ground motion with a return period of 10,000, 5,000 or 1,000 years, respectively. At the time of the design of most of the existing dams, the earthquake action was usually considered as a pseudo-static loading and typically a horizontal seismic coefficient of 0.1 g was assumed. This is clearly inadequate even in regions with relatively low seismicity. In practically all major arch dams, the crest region would experience horizontal accelerations significantly exceeding 1.0 g in the event of the SEE, even when the ground motion at the rock level has a horizontal peak ground acceleration (PGA) of the order of 0.20 g only. In a linear elastic analysis, the earthquake shaking would typically cause high horizontal stresses in the arch direction in the crest region of the dam. As the tensile strength of the vertical contraction joints is usually quite low, these joints are likely to open during the earthquake shaking. Once the vertical joints are open, the dam blocks would behave as vertical cantilevers, leading to high vertical tensile stresses. Since the tensile strength of 411
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horizontal lift joints is usually substantially lower than that of monolithic mass concrete, a horizontal crack may form, causing the upper portion of a dam block to become fully detached from the rest of the dam. Such a detached portion would be subjected to rather high accelerations and it could thus undergo sliding and rocking motions during the earthquake shaking. As long as a possibly detached portion remains stable during and after the earthquake, an uncontrolled release of reservoir water could be ruled out and the safety requirements for the SEE would be satisfied. To investigate the dynamic stability of possibly detached upper portions in the crest region and to analyze the behavior of vertical contraction joints in an arch dam during an earthquake, nonlinear dynamic analyses have to be carried out. In this paper, two illustrative examples are presented to show how earthquake safety of arch dams can be assessed on the basis of results of 2D and 3D nonlinear dynamic analyses.
Methodology for simplified dynamic stability analysis A simplified analysis of the seismic stability of possibly detached concrete blocks in the central upper portion of an arch dam can be performed in the following steps (Malla and Wieland, 2006; Wieland and Malla, 2012): (i) Linear elastic dynamic time history analysis of a three-dimensional (3D) finite element (FE) model of dam-reservoir-foundation system; (ii) Selection of detached concrete blocks for dynamic stability analysis based on envelopes of principal dynamic tensile stresses and absolute accelerations; (iii) Obtaining time histories of radial and vertical components of absolute acceleration at the base of each detached concrete block from the results of analysis step (i); (iv) Setting up 2D FE model of each detached concrete block using contact elements to simulate the cracked lift joint (gap elements may be employed to prevent the concrete block from moving beyond the downstream face of the dam in view of the geometrical constraints in an arch dam); (v) Nonlinear dynamic analysis of rocking-sliding response of each detached concrete block subjected to input base acceleration obtained in step (iii); (vi) Evaluation of the maximum sliding movement and the maximum crack opening displacements at the upstream and downstream edges of the base of each detached concrete block for at least three different earthquake ground motions; and (vii) Assessment of the dynamic stability of detached concrete blocks based on results of step (vi). The seismic safety of the Roggiasca and Gigerwald arch dams in Switzerland was checked by employing this simplified procedure. The main results of this analysis are presented and discussed in the following sections. Furthermore, nonlinear 3D FE analysis of the Gigerwald arch dam was performed to investigate contraction joint openings during the earthquake. All the dynamic calculations were performed with the help of the general-purpose FE software ADINA (ADINA R & D, 2008). For the dynamic analyses, the three components of earthquake excitation were simulated by artificially-generated spectrum-compatible acceleration time histories. The hydrodynamic pressure was modeled as added masses acting perpendicular to the upstream face of the dam.
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Earthquake safety assessment of Roggiasca arch dam The 68 m high Roggiasca arch dam was completed and first impounded in 1965. With a crest thickness of 2.5 m and a maximum thickness at the base of 7.5 m only, the dam has a relatively high Lombardi slenderness coefficient of 21. At the location of this dam, the 10,000-year SEE has a horizontal PGA of 0.17 g. The linear seismic response of the Roggiasca dam was analyzed using a 3D FE model, which comprised the dam, the foundation rock, the sediment deposit on the upstream side and the soil fill on the downstream side. The main results of this 3D analysis performed for 3 different ground excitations (designated as earthquakes 1, 2 and 3) are listed in Table 1. The largest accelerations and dynamic stresses were obtained at the middle and also at the two quarter points of the dam crest. The results showed that the upstream sediment deposit and the downstream soil fill would not play a significant role in the dynamic behavior of the dam. Table 1: Main results of 3D linear elastic earthquake analysis of Roggiasca dam subjected to SEE ground motion under full reservoir condition Dynamic response (envelope) Relative crest displacement (mm) Along-stream direction Across-stream (left-right) direction Vertical direction Absolute crest acceleration (g) Along-stream direction Across-stream (left-right) direction Vertical direction Principal tensile stress (MPa) Principal compressive stress (MPa)
Earthquake 1
Earthquake 2
Earthquake 3
16.2 8.4 2.0
15.9 7.9 2.1
18.7 8.2 2.1
1.45 0.62 0.41
1.34 0.64 0.39
1.27 0.72 0.35
6.1
5.6
6.6
-6.3
-6.0
-6.2
The magnitude of the largest dynamic tensile and compressive stresses in the crest region of the dam is about 6 MPa (see Table 1 and Figure 1). Even higher elastic stresses are computed at the upstream and downstream edges of the dam-rock interface due to the stress singularities at these reentrant corners.
Figure 1: Principal stress vectors due to earthquake 1 (without static loads) at time t = 7.19 s when the highest tensile stress occurs under full reservoir condition 413
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The largest compressive stress in the dam body under the combination of the static and dynamic loads is about -10 MPa, which is not a problem for the dam concrete with a static compressive strength of 46 MPa. In spite of the compressive stresses due to the static loads (self-weight and hydrostatic pressure), relatively high horizontal tensile stresses of up to about 4 MPa oriented in the arch direction still occur in the crest region during the earthquake shaking. If the thermal stresses in winter and the effect of the ongoing chemical expansion of the dam concrete would also be considered, the tensile stresses would be even higher. In order to investigate the stability of possibly detached concrete blocks during the earthquake excitation, simplified 2D FE models shown in Figure 2 were employed. As the Roggiasca arch dam is rather thin, any detached block in the crest region would be quite slender.
Figure 2: 2D FE models of 7 m and 13 m high detached blocks in crest region of Roggiasca arch dam (gap elements prevent any movement beyond the downstream face) In a dynamic analysis involving rigid body motions, it is considered prudent to use only the stiffness-proportional part of the Rayleigh damping model, as the mass-proportional part corresponds to external viscous dampers connected to the nodes of the model (Hall, 2006). Three different Rayleigh damping models were considered in the stability analysis (see Table 2). Model A corresponds to that used in the 3D linear dynamic analysis. In model B, only the stiffness-proportional part is kept. In model C, the stiffness-proportional part is further substantially reduced, which is very conservative. Table 2: Rayleigh damping models used for 2D dynamic stability analysis of detached cantilever blocks in crest region of Roggiasca dam Rayleigh damping model
Parameter -1
Parameter (s)
A
(s ) 2.20
B
0.00
0.00124
C
0.00
0.00020
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The results of the dynamic stability analysis listed in Table 3 show that the detached portions in the crest region have a tendency to undergo only rocking with virtually no sliding, a behavior explained by the slender form. Even the rocking motion is quite small and causes dynamic crack openings of only a few millimeters at the base of the detached portion. Table 3: Main results of 2D dynamic stability analysis of detached portions in crest region of Roggiasca dam subjected to SEE ground motion under full reservoir condition Rayleigh damping model
Earthquake
Max. horiz. crest displacement (mm)
Max. sliding displacement (mm)
(a) 13 m high detached block above crack at level 942 m a.s.l. A 1 -4 0.0
Max. crack opening, u/s face (mm)
Max. crack opening, d/s face (mm)
0.0
0.7
B
1
-5
0.0
0.0
0.8
C
1
-6
0.0
0.0
1.1
B
2
-11
0.0
0.0
2.2
C
2
-14
0.0
0.0
3.0
B
3
-11
0.0
0.0
2.2
C
3
-17
-0.2
0.0
3.8
(b) 7 m high detached block above crack at level 948 m a.s.l. A
1
-15
-0.1
0.0
5.6
B
1
-16
-0.1
0.0
6.1
C
1
-79
-0.2
0.1
30.7
B
2
-22
-0.1
0.0
8.5
C
2
-128
-0.4
0.0
49.8
B
3
-25
-0.1
0.0
9.7
C
3
-171
-0.7
0.1
66.7
The results of the dynamic stability analysis show that the detached portions remain dynamically stable even when subjected to peak horizontal accelerations about twice as large as the pseudo-static overturning acceleration. This behavior can be explained by the fact that acceleration spikes would have a relatively high frequency of about 5 Hz corresponding to the dominant natural frequency of the dam. Hence, such a spike would exceed the pseudo-static overturning acceleration only for a very short duration of less than one-tenth of a second, which is too short to produce any significant block rotation. A review of literature on dynamic overturning of rigid blocks also confirms that acceleration peaks would have to be many times larger than the pseudo-static overturning acceleration for a block with dimensions of the order of a few meters to be toppled by a dynamic base excitation at such a frequency, as illustrated in Figure 3 (Shi et al., 1996; Zhang and Makris, 2001).
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Figure 3: Boundaries between stable and unstable regions of dynamic rocking motion based on analytical solutions for half and full sinusoidal base excitations (Shi et al., 1996, Zhang und Makris, 2001) Note: A ground excitation with an amplitude of 1.5 g and a frequency of 5 Hz corresponds to the red square (p/p = 22.4 und ap/g = 3.2) and the red triangle (p/p = 29.6 und ap/g = 4.5) in the case of, respectively, the 7 m and 13 m high detached portions.
Earthquake safety assessment of Gigerwald arch dam The 147 m high Gigerwald dam was completed and first impounded in 1976. The dam thickness varies from 7 m at the crest to about 22 m at the base. The main results of the linear elastic dynamic analysis of a 3D FE model of this doublecurvature arch dam for 3 different earthquakes are listed in Table 4. The crest region would be subjected to amplified horizontal accelerations as high as about 2.0 g during the 10,000-year SEE with a horizontal PGA of 0.19 g. Table 4:Main results of 3D linear elastic earthquake analysis of Gigerwald dam subjected to SEE ground motion under full reservoir condition Dynamic response (envelope) Relative crest displacement (mm) Along-stream direction Across-stream (left-right) direction Vertical direction Absolute crest acceleration (g) Along-stream direction Across-stream (left-right) direction Vertical direction Principal tensile stress (MPa) Principal compressive stress (MPa)
Earthquake 1
Earthquake 2
Earthquake 3
48.0 15.3 6.6
45.6 16.0 7.4
44.5 18.0 6.9
1.83 0.84 0.75
2.04 0.86 0.78
1.98 0.80 0.71
8.7
9.8
8.7
-10.3
-9.5
-9.6
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The earthquake shaking produces dynamic tensile stresses of nearly 10 MPa in the central crest region of the dam. Even after the combination with the compressive stresses due to the hydrostatic water load, high horizontal tensile stresses of up to about 6 MPa would still remain in the arch direction. In reality, such tensile stresses cannot develop due to the presence of the vertical contraction joints, whose tensile strength is normally quite low. The next step was to perform the dynamic analysis of a 3D FE model in which all 23 vertical contraction joints were simulated as frictional contact surfaces and the dam concrete was assumed to be linear elastic (uncracked). This analysis was performed for 3 different input motions. Figures 4 to 6 depict some results obtained for earthquake 2. This analysis showed that the seismic shaking would cause relative sliding displacements of nearly 1 cm between the adjacent blocks and the contraction joints would open by maximum about 4 mm. The largest compressive stress in the dam obtained in the nonlinear analysis approaches nearly -19 MPa (excluding the corner singularity at the dam-rock interface), which is not a problem for the dam concrete. In comparison, the maximum compressive stress in the case of the linear analysis is about -16 MPa. In spite of the joint displacements, the dynamic displacements and accelerations of the dam computed in the nonlinear analysis do not deviate significantly from those in the corresponding linear analysis. The main difference lies in the absence of any significant horizontal tensile stresses in the arch direction in the central crest region in the case of the nonlinear analysis owing to the presence of the contraction joints. During the brief openings of the contraction joints, the upper portion of a dam block acts temporarily as a cantilever, due to which relatively high transitory vertical tensile stresses exceeding 6 MPa appear on the downstream face of the dam, as depicted in Figure 6. Hence, horizontal cracks are likely to form, especially at the lift joints, possibly resulting in the detachment of the uppermost portion of a central block from the rest of the dam body.
Figure 4: Time histories of opening and sliding displacements of vertical contraction joint 11/12 at crest level due to earthquake 2 under full reservoir condition 417
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Figure 5:Maximum opening and sliding displacements of vertical contraction joints at crest level due to earthquake 2 under full reservoir condition
Figure 6: Stresses due to cantilever behavior during opening of vertical contraction joints The safety of possibly detached portions in a central dam block subjected to the SEE shaking was assessed using simplified 2D models (see Figure 7). This analysis showed that an 8 m high detached portion could slide by up to about 50 cm towards the reservoir during the SEE and the rocking motion would result in crack opening displacements of up to about 7 cm. However, the detached block would remain stable during and after the earthquake and the earthquake damage would not lead to an uncontrolled release of water.
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Figure 7: 2D FE model of 8 m high detached portion of a central block of Gigerwald arch dam (gap elements prevent any movement beyond the downstream face) For a less conservative dynamic stability assessment, a nonlinear analysis was also performed using a 3D model in which the uppermost 8 m of the central dam block was assumed to be detached from the rest of the dam along the vertical contraction joints at the sides and an assumed horizontal crack along a lift joint. As shown in Figure 8, the 3D analysis showed that such a detached block would slide by up to 16 cm towards the reservoir, which is only about one-third of the result obtained in the corresponding more conservative 2D analysis. The substantially smaller sliding displacement in the 3D analysis can be attributed mainly to the additional frictional resistance at the vertical contraction joint on each side, an effect that could not be taken into account in the simplified 2D approach. The maximum crack opening also decreased to about 3 cm in the 3D analysis.
Figure 8: Sliding displacement of 8 m high detached concrete block in central cantilever of Gigerwald arch dam subjected to earthquake 2 under full reservoir condition
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Conclusions The results of the earthquake safety assessment of the Roggiasca and Gigerwald arch dams can be summed up as follows: 1. The ground excitation during the 10,000-year SEE produces relatively high accelerations and tensile stresses in the central upper portion of a large arch dam. 2. The tensile strength of the vertical contraction joints and the horizontal lift joints is usually significantly lower than that of the monolithic concrete. Thus, these joints may open during the earthquake shaking, possibly resulting in the upper portion of a dam block becoming fully detached from the rest of the dam. 3. Any detached uppermost part of a dam block in the investigated dams may be subjected to some sliding and rocking motions, but it will remain stable during and after the earthquake. 4. The earthquake loading could cause limited damages in the analyzed dams, but it will not lead to an uncontrolled release of reservoir water. Hence, the investigated dams satisfy the safety requirements for the SEE.
References [1] ADINA R & D (2008). ADINA User Interface, Command Reference Manual, Vol. I: ADINA Solids & Structures Model Definition, Report ARD 08-2, Watertown, Massachusetts, USA. [2] Hall, J.F. (2006). Problems encountered from the use (or misuse) of Rayleigh damping. Earthquake Engineering and Structural Dynamics. Vol. 35, pp. 525-545. [3] Malla, S. and Wieland, M. (2006). Dynamic stability of detached concrete blocks in arch dam subjected to strong ground shaking. Proceedings of the 1st European Conference on Earthquake Engineering and Seismology (ECEES), Geneva, Switzerland, 3-8 September 2006. [4] Panduri, R., Droz, P., Malla, S., Radogna, R., Wieland, M. and Darbre, G.R. (2012). Ongoing seismic safety assessment of Swiss dams. Proceedings of the 24th International Congress on Large Dams, ICOLD, Kyoto, Japan, 6-8 June 2012. [5] Shi, B., Anooshehpoor, A., Zeng, Y. and Brune, J.N. (1996). Rocking and overturning of precariously balanced rocks by earthquake. Bulletin of the Seismological Society of America, Vol. 86, No. 5., pp. 1364-1371. [6] Wieland, M. and Malla, S. (2012). A simple method for seismic stability analysis of detached concrete blocks and estimation of contraction joint openings in large arch dams. Hydropower and Dams, Vol. 19, Issue 3, pp. 113-119. [7] Zhang, J. and Makris, N. (2001). Rocking response of free-standing blocks under cycloidal pulses. Journal of Engineering Mechanics, ASCE, Vol. 127, No. 5, pp. 473-483.
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proceedings 12th International
BENCHMARK WORKSHOP
ON NUMERICAL ANALYSIS OF DAMS, 2nd - 4th October, 2013, Graz – AUSTRIA Edited by
Gerald Zenz and Markus Goldgruber
Heft / Volume 39
proceedings ICOLD 12th benchmark workshop
I C O L D
Austrian National Committee on Large Dams Stremayrgasse 10/II, A-8010 Graz, AUSTRIA, Phone: ++43/316/8861, [email protected], www.atcold.at
Published by the Austrian National Committee on Large Dams with the support of the Austrian Reservoir Commission (Staubeckenkommission)
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