Fe_slab

Finite Element Application to Slab-column Connections Reinforced with Glass Fibre-Reinforced Polymers

Qi Zhang Faculty of Engineering Memorial University of Newfoundland

April, 2004

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

Contents Introduction__________________________________________________________ 3 Modeling Slab-Column Connection _______________________________________ 5 1. Concrete constitutive model _________________________________________ 7 1.1 Solid 65 element description ______________________________________ 7 1.2 Concrete properties _____________________________________________ 8 1.3 The Consideration of Solid65 Element Input Data_____________________ 13 2. GFRP reinforcement constitutive model _______________________________ 18 2.1 Link8 Element Description _______________________________________ 18 2.2 GFRP Reinforcement Properties __________________________________ 19 2.3 Smear and Discrete Reinforcement Consideration ____________________ 20 3. Boundary Condition and Spring constitutive model ______________________ 23 3.1 Link10 Element Description ______________________________________ 23 3.2 Boundary Condition ____________________________________________ 24 4. Finite Element Discretization________________________________________ 25 5. Numerical Implementation _________________________________________ 27 6. Verification in the Elastic Stage ______________________________________ 32 Comparison of Finite Element Analysis to Test Results _______________________ 35 Finite Element Analysis Results Versus Modified Code Predictions _____________ 40 Summary and Conclusion _____________________________________________ 41 Reference__________________________________________________________ 43 Appendix: ANSYS CODE (GSHD1) ______________________________________ 45

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FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

Introduction In recent years there has been increased interest in the use of fiber-reinforced polymers (FRP) for concrete structures. As one of the new promising technologies in construction, FRP material solves the durability problem due to corrosion of steel reinforcement. Glass fiber reinforced polymer (GFRP) composites have been used for reinforcing structural members of reinforced concrete bridges. Many researchers have found that GFRP composite reinforcement instead of steel is an efficient, reliable, and cost-effective means of reinforcement in the long run.

In the structural elements, the flat slab has the large surface exposed to the outside corrosive environment, such as bridge decks, ocean oil flat slabs and parks. The flat slab system offers advantages for efficient design, the overall construction process, notably in simplifying the installation of services and the savings in construction time. However, a slab-column connection in the flat slab system is frequently subjected to the significant transverse shear forces, which can produce a punching shear failure. Punching in the vicinity of a column is a possible failure mode for reinforced concrete flat slabs. Many researchers have exerted their efforts to investigate the punching strength of slabs, due to the undesirable suddenness and catastrophic results of punching failure. However, there is no generally accepted treatment for the punching resistance of flat slabs because of the complicated dependence of shear strength on their flexural behavior as well as the fact that it is difficult to observe their internal inclined cracks.

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FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

Furthermore, most FRP elements exhibit a relatively low modulus of elasticity and lack the yield-line characteristic of traditional steel reinforcement. It is unsafe to predict the punching strength of column-slab connections in terms of directly extending or simply introducing to a material coefficient to the models and equations based on steel reinforcement. Thus, it is motivated to improve the understanding of the punching failure mechanisms and to establish a reliable method for predicting the punching strength of column-slab connections reinforced with GFRPs.

This paper, therefore, will present the application of the finite element method for the numerical modeling of punching shear failure mode using a widely accepted code, ANSYS [ANSYS, 1998]. Based on properly modeling and simulating the experiments [Rashid, 2004] carried out in the Faculty of Engineering at Memorial University of Newfoundland, the behavior of slab-column connections reinforced with GFRPs will be investigated.

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FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

Modeling Slab-Column Connection A n extensive description of previous studies on the application of the finite element method to the analysis of reinforced concrete structures and the underlying theory and the application of the finite element method to the analysis of linear and nonlinear reinforced concrete structures is presented in excellent state of-the-art reports by the American Society of Civil Engineers in 1982 [ASCE 1982]. The results from the FEA are significantly relied on the stress-strain relationship of the materials, failure criteria chosen, simulation of the crack of concrete and the interaction of the reinforcement and concrete.

Because of these complexity in short- and long-term behavior of the constituent materials, the ANSYS finite element program (ANSYS 1998), operating on a Windows 2000 system, introduces a three-dimension element Solid65 which is capable of cracking and crushing and is then combined along with models of the interaction between the two constituents to describe the behavior of the composite reinforced concrete material. Although the Solid65 can describe the reinforcing bars, this study uses an additional element, Link8, to investigate the stress along the reinforcement because it is inconvenient to collect the smear rebar data from Solid65. Due to the general experiment process in the structural lab, the edge of slab is free to lift, which is different with the simple support along all four edges according to the lines of contra flexure. Then, a spring element, Link10, along the edge, is included in this study to reflect the actual setup of Page 5 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

slab-column connection. The constructed model using ANSYS is shown in Fig.1. The detail of specimens is listed in the Table 1.

Fig.1 ANSYS numerical model representation of experimental specimens

Table 1: Details of tested slabs Compressive Tensile Steel yield Slab No.

strength

strength strength

Ec

Column

(MPa)

Efrp (MPa)

630

28400

42000

250

3.23

630

24200

42000

33

3.43

630

25700

GSHD2

34

3.51

630

GSHS1

92

5.76

GSHS2

86

5.57

depth

Spacing GFRP (mm)

ratio p%

150

240

1.18

250

150

170

1.67

42000

250

200

170

1.11

26300

42000

250

200

240

0.79

630

38600

42000

250

150

170

1.67

630

37500

42000

250

150

240

1.18

(MPa)

(MPa)

(MPa)

GS1

40

3.78

GS3

29

GSHD1

size (mm)

Average (mm)

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FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

The assumptions made in the description of material behavior are summarized below: • The concrete material is assumed to be initially isotropic elastic. • The stiffness of concrete and reinforcement is formulated separately. The results are then superimposed to obtain the element stiffness. • The smeared crack model is adopted in the description of the behavior of cracked concrete. Cracking in more than one direction is represented by a system of orthogonal cracks, and the crack direction changes with load history (rotating crack model). • The reinforcing GFRP is assumed to carry stress along its axis only and the perfect bond relationship between concrete and GFRP rebar.

1. Concrete constitutive model 1.1 Solid 65 element description Solid65, an eight-node solid element, is used to model the concrete with or without reinforcing bars. The solid element has eight nodes with three degrees of freedom at each node




translations in the nodal x, y, and z directions. The element is capable of plastic deformation, cracking in three orthogonal directions, and crushing. The geometry and node locations for this element type are shown in Fig.2.

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FE application to slab-column connections reinforced with GFRPs

Fig.2: Solid65




By Qi Zhang

3-D reinforced concrete solid (ANSYS 1998)



The eight nodal points is also along with a 2X2X2 Gaussian integration scheme. This is used for the computation of the element stiffness matrix. The element’s displacement field in terms of the nodal displacements and the shape functions can be written as:

(1)

(2)

(3)

1.2 Concrete properties Concrete is a heterogeneous material made up of cement, mortar and aggregates. Its mechanical properties scatter widely and cannot be defined easily. For the convenience of analysis and design, however, concrete is often considered a homogeneous material in the macroscopic sense. Because of the cracking of concrete in tension, crushing of concrete in Page 8 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

compression, the time-dependent effects of creep, shrinkage and temperature variation, the nonlinear response of concrete members can be observed (Fig.2). This highly nonlinear response can be roughly divided into three ranges of behavior: the uncracked elastic stage, the crack propagation and the plastic (yielding or crushing) stage.

Fig.2: Typical uniaxial compressive and tensile stress-strain curve for concrete (Bangash 1989)

Before concrete cracking takes place, the behavior of concrete could be regarded as a linear isotropic material. The stress-strain matrix [Dc] of solid65 element is defined as

(4)

The nonlinear response of concrete is mainly controlled by progressive cracking that results in Page 9 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

localized failure. After concrete cracking takes place, the critical section of the structural member is weakened, and then the stress of concrete and reinforcement will be redistributed. Different with the discrete cracks model [Ngo and Scordelis, 1967], in which crack occurs during a load cycle but the need to change the topology and the redefinition of nodal points of concrete model greatly reduce the speed of the process, Solid65 element automatically generates cracking without redefining the element mesh. These models depict the effect of many small cracks that are “smeared” across the element in a direction perpendicular to the principal tensile stress direction [Darwin, 1993].

Observing the Fig.2, the stress-strain curve for concrete is linearly elastic up to about 30 percent of the maximum compressive strength. Above this point, the crack is developed in the concrete and the stress increases gradually up to the maximum compressive strength. The stress-strain relations are modified in this stage to represent the presence of a crack in the concrete. A plane of weakness in a direction normal to the crack face and a shear transfer coefficient



t

are

introduced in the solid65 element. The shear strength reduction for those subsequent loads which induce shear across the crack surface is considered by defining the value of



t. This is

very

important to accurately predict the loading after cracking, especially when calculating the strength of concrete member dominated by shear, such as slab-column connection and shear wall. The detailed analysis of the shear transfer coefficient will be presented later.

The stress-strain relations for a material that has cracked in one direction only become: Page 10 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

(5)

where the superscript ck signifies that the stress strain relations refer to a coordinate system parallel to principal stress directions with the xck axis perpendicular to the crack face. Rt, the slope as defined in Fig.3, works with adaptive descent and diminishes to 0.0 as the solution converges.

Fig.3 Strength of Cracked Condition

ft = uniaxial tensile cracking stress Tc = multiplier for amount of tensile stress relaxation

If the crack closes, then all compressive stresses normal to the crack plane are transmitted across the crack and only a shear transfer coefficient

c

for a closed crack is introduced. Then

[Dck] can be expressed as:

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FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

(6) The transformation of [Dck] to element coordinates has the form (7) where [Tck] is the transformation matrix. The matrices and equations of the stress-strain relations for concrete that has cracked in more than one direction are detailed in the theory reference of ANSYS, 1998.

In addition to cracking, the concrete will be failed in uniaxial, biaxial, or triaxial compression. The concrete is assumed to crush in that condition. A three-dimensional failure surface for concrete is shown in Fig. 4. The most significant nonzero principal stresses are in the x and y directions, represented by the

and

yp,

respectively. Three failure surfaces are shown as projections on







xp-

yp

xp

plane. The mode of failure is a function of the sign of

direction). For example, if

xp

and

yp




zp

(principal stress in the z

are both negative (compressive) and

positive (tensile), cracking would be predicted in a direction perpendicular to




zp





zp

zp.

is slightly

However, if

is zero or slightly negative, the material is assumed to crush [ANSYS 1998].

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FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

Fig.4 Failure Surface in Principal Stress Space with Nearly Biaxial Stress

1.3 The Consideration of Solid65 Element Input Data ANSYS requires input data for material properties of Solid65 element as elastic modulus (Ec), ultimate uniaxial compressive strength ( f c' ), ultimate uniaxial tensile strength (modulus of rupture, ( f r ),

poisson’s ratio (



), density ( γ ), shear transfer coefficient for an open crack ( t),

shear transfer coefficient for an close crack ( c), compressive uniaxial stress-strain relationship for concrete. If the Solid65 element includes the representation of reinforcement, up to three rebar could be defined in Solid65 element through the real constants: reinforcement material number (Matn), volume ratio (VRn) and orientation angles (THETAn and PHIn), where n represents the up to 3 rears.

MacGrewgor (2000) defined the elastic modulus and the ultimate uniaxial tensile strength ( f r ) as follows: Page 13 of 52

FE application to slab-column connections reinforced with GFRPs

Ec = 3600 f c + 6700 '

f r = 0.6 f c

γ

By Qi Zhang

1.5

(7)

2300

'

(8)

The density ( γ ) and the poisson’s ratio (




) of concrete are considered as 2300 kg/mm3 and 0.2.

The ultimate uniaxial compressive strength ( f c' ) was measured through cylinder concrete test (Rashid, 2004).

The shear transfer coefficient for a closed crack transfer coefficient,

t

c

is widely accepted within 0.9 to 1.0. The shear

represents conditions of the crack face. The value of

t

ranges from 0.0 to

1.0, with 0.0 representing a smooth crack (complete loss of shear transfer) and 1.0 representing a rough crack (no loss of shear transfer) [ANSYS 1998]. The value of

t

used in many studies of

reinforced concrete structures, however, varied between 0.05 and 0.25 [Bangash 1989; Huyse, et al. 1994; Hemmaty 1998]. A number of preliminary analyses were attempted in this study with various values for the shear transfer coefficient within 0.125-1.0, the results are shown in the Fig.5. Although there is no significant difference of the ultimate loading with the different accompanying with the increase of

t,

t,

the ultimate loading is slightly increased after the

displacement is beyond 3mm.

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FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

250000

ultimate loading (N)

200000

Bt0.8_Bc1.0 Bt0.5_Bc1.0 Bt1.0_Bc1.0 Bt0.3_Bc0.9 Bt0.125_Bc0.9

150000

100000

50000

0 0.0

2.0

4.0

6.0

8.0

10.0

12.0

displacem ent (m m )

Fig.5.Investigation of shear transfer coefficient during the small deflection

It is possible that the ultimate loading will have larger divergence if the larger displacement takes place. GFRPs with the lower modulus of elasticity will lead to the larger deflection than the normal structure member reinforced with steel. The experiment conducted by Rashid [Rashid, 2004] proves this assumption. Thus, the ultimate loading should be sensitive to

t.

Fig. 6

simulated the load-deflection curve according to the specimen GSHS2, which proves the larger difference of ultimate loading and deflection due to different

t in

the condition of displacement

more than 15mm. Since convergence problems were encountered at low loads with t less than 0.1. Therefore, the shear transfer coefficient

t

used in this study was equal to 0.2, which gives a

good agreement with the change of stiffness of the specimen as well as the ultimate loading.


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FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

In describing the uniaxial compression stress-strain behavior of concrete many empirical formulas have been proposed. These are summarized in ASCE (1982). Fig.7 shows the simplest of the nonlinear models, the linearly elastic-perfectly plastic model, which was used by Lin and Scordelis (1975) in a study of reinforced concrete slabs and walls. Fig.7 also shows a piecewise linear model in which the nonlinear stress-strain relation is approximated by a series of straight-line segments. Although this is the most versatile model capable of representing a wide range of stress-strain curves, the softening branch of the concrete stress-strain relation is found to be related to the numerical difficulties in solution convergence during nonlinear iterations.

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FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

40

stress (MPa)

30 20 piecew ise linear model elastic-perfectly plastic model,

10 0 0

0.001

0.002

0.003

0.004

0.005

strain

Figure 7, Typical stess-strain relationship of normal concrete

Due to this unique problem in the finite element analysis of slab-column connections, two solution strategies have been applied in this study. The one is applying the piecewise linear model without defining the crush of concrete, the ultimate uniaxial compressive strength ( f c' ); the another is involved with the crush of concrete but using the elastic-perfectly plastic model. After comparing this two approach with the simple model (Fig. 11), the results in the Fig. 8 shows the advantage of the latter. The elastic-perfectly plastic model gives a good prediction of ultimate loading and displacement to the specimens.

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FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

Displacement (mm)

400 GSHD2

300 piecew ise linear model

200

elasticperfectly plastic model,

100

0 0

5

10

15

20

25

30

35

40

Ultimate loading (KN) Fig. 8 Verification of two approaches to convengent problem using the simple discrete model(GSHD2)

2. GFRP reinforcement constitutive model 2.1 Link8 Element Description Link8 element, the three-dimensional spar element is a uniaxial tension-compression element with three degrees of freedom at each node: translations in the nodal x, y, and z directions. As in a pin-jointed structure, no bending of the element is considered. The element is also capable of plastic deformation, stress stiffening, and large deflection. The geometry, node locations, and the coordinate system for this element are shown in Fig. 9.

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FE application to slab-column connections reinforced with GFRPs

Fig. 9: Link8 Element

By Qi Zhang

3-D spar (ANSYS 1998)






2.2 GFRP Reinforcement Properties GFRP reinforcement in the experimental slab-column connections was made of typical GFRP material. Through the tension test of GFRP reinforcement, the properties including elastic modulus and ultimate tensile stress in this FEM study is as Fig. 10. Elastic modulus, Es = 42,000 MPa 700

Yield stress, f y = 630 MPa

Stress (MPa)

600 500

Poissons ratio,

400

= 0.3

300 200 100 0 0

0.004

0.008

0.012

0.016

Strain

Fig. 10: the stress and strain relationship of GFRPs rebar

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FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

2.3 Smear and Discrete Reinforcement Consideration Beside the discrete reinforcement model adopted in this study, the reinforcement could be also defined using the smeared reinforcement option of the Solid65 element. The amount of reinforcement is defined by specifying a volume ratio (VRn) and the orientation angles (THETAn and PHIn) of the rears. This approach is easy to construct the reinforced concrete model, which has the uniform or simple reinforcement arrangement. In practice, the reinforcement in flat slabs is normally arranged in a uniform form in different slab strips. Therefore, it is suggested to use this smear reinforcement option when simulating the large slab model. However, the stress and location of reinforcement is difficult to obtain in the smear model. In this study, the behavior of slab-column connections was investigated by examining the stress distribution in the whole specimen. The stress of the reinforcement is critical to this investigation. The discrete reinforcement model is used in the study. On the other hand, it is motivated to construct smear reinforcement model for comparing the discrete one, then strengthen the understanding the difference between these two models and prepare for the future application.

For the comparison purpose, four simple models are constructed for predicting the load-displacement relationship in an acceptable accuracy and a low consumption of computation time. The simple model doesn’t consider the layer mesh, spring support and the column stub. In Fig. 11 there are four models. The first three are the smear reinforcement models with different thickness of the reinforcement layer, including the thickness of 20mm, 40mm and 100mm. The

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FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

last one is the discrete reinforcement model with the reinforcement spacing of 240mm. The test results of these models are plotted in the Fig.12.

It is obvious that the four models obtain the similar displacement and ultimate loading, but the smear models got a smoother curve. In the general building codes in the world, the major concerns of punching strength of slabs are the reinforcement ratio, the load area, the effective depth and compressive strength of concrete. The effect of the arrangement of reinforcement on the punching strength is ignored, which is verified in this test. Therefore, the reinforcement spacing of the simulated models is taken as 120mm and 240mm for constructing the model consistent with the mesh size and column size. This is different with the reinforcement spacing of the actual tested specimens, but the reinforcement ratio is kept same.

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FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

a).Smear Model with 20 mm thickness Rebar Layer

b).Smear Model with 100 mm thickness Rebar Layer

c).Smear Model with 40 mm thickness Rebar Layer

d).Discrete Model

Fig. 11: Smear and Discrete Model

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FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

140000 Ultimate loading (N)

120000 100000 80000


 
  

   
  

60000 40000 20000 0 0

5

10

15

20

displacement (mm) Fig. 12: Comparison of smear and discrete reinforce model

3. Boundary Condition and Spring constitutive model 3.1 Link10 Element Description LINK10 element is a three-dimensional spar element having the unique feature of a bilinear stiffness matrix resulting in a uniaxial tension-only (or compression-only) element. With the tension-only option, the stiffness is removed if the element goes into compression (simulating a slack cable or slack chain condition) [ANSYS, 1998]. The geometry, node locations, and the coordinate system for this element are shown in Fig. 13.

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FE application to slab-column connections reinforced with GFRPs

Fig. 13: Link10 Element

By Qi Zhang

Tension-Only or Compression-Only Spar (ANSYS 1998)




3.2 Boundary Condition In this study, since the effect of gravity of slab-column connection on the punching strength of specimens is little, the simulated model is constructed in the form of one quarter of the slabs due to the two axes of symmetry. Thus, the boundary condition of these two edges is defined as the symmetry of displacement, which is shown in the Fig.1.

Because the specimens are simply supported on four edges that are free to lift during the tests, the proper simulation of the boundary condition is taken into account. Link10 elements, the special nonlinear spring elements in the transverse direction are employed along four edges of labs in the numerical model, which is illustrated in Fig.1. The stiffness of spring elements is numerically set to be significant high in compression and zero in tension respectively. Furthermore, the simple support in the test is a roller support made of the steel tube and covered with 3mm-thickness rubber. Both the steel tube and rubber will be deformed during the test, the stiffness of spring would have been chosen carefully to represent the test simple support in the

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FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

condition of relatively large deflection, so that the similar load-deflection curve to the tested specimens could be obtained. However, the test setup tries to simulate the slab-column connection around the conflexure line. In spite of the careful setup, the specimen could not reflect the real deformation perfectly, while the model using ANSYS could exactly simulate this theoretical specimen. Therefore, simply simulating the test specimen spring support could not represent the real situation. In this study, the comparison between the specimens and models, including the change and development of stiffness of slabs, the crack loading and the ultimate loading are the major concern. This model can be regarded as a good reference to the real slab-column connection in the flat slab system.

4. Finite Element Discretization After constructed a model with volumes, areas, lines and key points, a finite element analysis requires meshing of the model. The model is then divided into a number of small elements, and after loading, stress and strain are calculated at integration points of these small elements (Bathe 1996). Since the FEM could approximate the real situation as close as possible depending on the selection of the mesh density, it is a significant step in finite element modeling to choose an appropriate mesh size to meet the requirement of accuracy and speed. A convergence of results is obtained when an adequate number of elements are used in a model. This is practically achieved when an increase in the mesh density has a negligible effect on the results (Adams and Askenazi 1998). Therefore, in this finite element modeling study the appropriate mesh size is Page 25 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

determined by a convergence study and the behavior of slab-column connections.

Volume elements can often be either hexahedral (brick) or tetrahedral shaped, but the solid65 element comprising of 8 nodes has no center point is recommended to be meshed in the hexahedral shape.

The ACI code (ACI, 1995) assume that the control perimeters of punching failure is located at a distance of 0.5 times the effective depth from the edge of load (column), while the British code (BS8110, 1985) considers a larger control perimeter, 1.5d. Furthermore, because the spring support is used in this study, the support failure will not occur. Therefore, the mesh size in this FE study must be small in the area around the column and the larger size can be acceptable near the edge.

On the other hand, the column stub, slab and reinforcement are required to work together in the discrete model used in this study. Then, the size of column and the spacing of reinforcement are taken into consideration of determining the mesh size. The small change of the load area (column size) gives the similar results of punching strength of slab-column connections using either of ACI code or British code. Consistent with the reinforcement spacing 120mm and 240mm determined in the above chapter, column stub size in the model is reduced from 250mm to 240mm, which only leads to 1% difference.

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FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

Ultimate loading (N)

120000 100000 80000 60000


 
   

40000 20000 0 0

5

10

15

Displacement (mm) Fig. 14: The mesh size comparison

Therefore, the mesh size is considered as 60mm or 120mm. Using the simple model described in the Fig.11 to check the difference of models meshed with the mesh global size of 60mm and 120mm and the mesh thickness size of 50mm, the results are illustrated in the Fig. 14. According to this figure, it is fair to draw a conclusion that there is no significant effect on the load-deflection relationship of the model using controlled mesh size of either the 60mm or 120mm. However, the 120mm is relatively bigger to investigate the element stress around the column because the column size is 120mm; while the model with the mesh size of 60mm is quite time consuming: 4 hours have been spent to run this simple model. In the end, the layer mesh is selected in this study, which is shown in the Fig. 1.

5. Numerical Implementation The numerical implementation of the finite element model is based on the virtual work principle or the theorem of minimum potential energy to the assemblage of discrete elements. The following Page 27 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

equilibrium equations are:

(9) The terms in Eq. 9 are derived as follows: the stiffness matrix [K] ,

(10) the nodal forces due to surface traction,

(11) the nodal forces due to body forces,

(12) the nodal forces due to initial strains,

(13) and the nodal forces due to initial stresses

(14) In Eqs. 9-14, the components of [N ] are the shape functions, {d } is the vector of node displacements, {R} is the vector of applied nodal forces, {p} is the vector of surface forces and {g} is the vector of body forces.

This is a system of simultaneous nonlinear equations, since the stiffness matrix [K ] , in general,

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FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

depends on the displacement vector {d } . The solution of this system of nonlinear equations is typically accomplished with an iterative method. The load vector {R} is subdivided into a number of sufficiently small load increments. At the finishing point of each incremental solution, the stiffness matrix of the model is adjusted to reflect nonlinear changes in structural stiffness before proceeding to the next load increment. The ANSYS program (ANSYS, 1998) uses Newton-Raphson equilibrium iterations for updating the model stiffness. Newton-Raphson equilibrium iterations provide convergence at the end of each load increment within tolerance limits. Fig. 15 shows the use of the Newton-Raphson approach in a single degree of freedom nonlinear analysis.

Prior to each solution, the Newton-Raphson approach assesses the out-of-balance load vector, which is the difference between the restoring forces (the loads corresponding to the element stresses) and the applied loads. Subsequently, the program carries out a linear solution, using the out-of-balance loads, and checks for convergence. If convergence criteria are not satisfied, the out-of-balance load vector is re-evaluated, the stiffness matrix is updated, and a new solution is attained. This iterative procedure continues until the problem converges (ANSYS, 1998).

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FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

Fig. 15, Newton-Raphson iterative solution (2 load increments) (ANSYS, 1998)



In this study, for the reinforced concrete solid elements, convergence criteria were based on force. In order to obtain fast and accurate convergence of this nonlinear analysis, “Line Search Option” and “Predictor-Corrector Option “ are set on, and the convergence tolerance limits increased to a maximum of 10 times the default tolerance limits (0.5% for force checking).

According to the test procedure in the structural lab, the displacement is applied to the specimen by a hydraulic actuator. In the other word, the actuator imposes the constant displacement to the slab, while the applied loading is adjusted according to the change of the stiffness of the specimen. Therefore, the displacement is applied to the column stub in this study for simulating the experimental process.

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FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

For the nonlinear analysis, automatic time stepping in the ANSYS program predicts and controls load step sizes. Based on the previous solution history and the physics of the models, if the convergence behavior is smooth, automatic time stepping will increase the load increment up to a selected maximum load step size. If the convergence behavior is abrupt, automatic time stepping will split the load increment until it is equal to a selected minimum load step size. Although the automatic time stepping could ensures that the time step variation is neither too aggressive nor too conservative, the amount of time step still determines the initial displacement. In this study, the reinforced concrete is cracked, which leads to dramatically reduce the specimen stiffness, subsequent to the small deflection of the slab. It is possible to fail to detect the cracking load if the initial amount of time step is relatively big. Therefore, the amount of time step is set 1000 to ensure to replicate the complete loading process during the test.

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FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

6. Verification in the Elastic Stage

Since the reinforced concrete slab-column connection can be regarded as the elastic plate before the crack forms, the finite element model, simply supported, and subjected to a centrally concentrated load, was initially investigated. These models simulated the specimen (Table 1) had the same main parameters. Based on verification of the slabs before cracking, the slabs were analyzed using plain concrete theory and finite element results were compared with hand calculations.

Fig. 16a-d show relationship of the deflection and loading of the center on the top of the slabs, as calculated by the model and by traditional hand calculations (Timoshenko, 1987). As expected, the deflection is linear with loading, and the finite element model shows excellent correlation with the hand calculation before the cracks form in the model. However, the models shown in the Fig. 16e and f are a little divergent from the hand calculations. This is because the applied uniform load in the column area is regarded as central load on the slab. When the slab is relative thin, the uniform load is well consistent with the simulated central load. However, the consistency is weakened as the depth of slab increases, which shows in Fig 16e and f.

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FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

120

90 80 70

80

Loading (KN)

Loading (KN)

100

60 40

GS1_Tested

20

ANSYS_GS1

60 50 40 30

GS3_Tested

20

ANSYS_GS3

10

0

0

0

0.2

0.4

0.6

0.8

0

0.2

0.4

160

140

140

120

Loading (KN)

120

Loading (KN)

1

Fig. 16b: Finite Elem ent Model Verification in the Elastic Stage

Fig. 16a: Finite Elem ent Model Verification in the Elastic Stage

100 80 GSHS1_Tested

40

0.8

Deflection (mm)

Deflection (mm)

60

0.6

ANSYS_GSHS1

100 80 60 GSHS2_Tested

40

A_GSHS2

20

20

0

0 0

0.2

0.4

0.6

0.8

Deflection (mm) Fig. 16e: Finite Elem ent Model Verification in the Elastic Stage

1

0

0.2

0.4

0.6

0.8

Deflection (mm) Fig. 16f: Finite Elem ent Model Verification in the Elastic Stage

Page 33 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

250

Loading (KN)

200 150 100 50

GSHD1_Tested ANSYS_GSHD1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

Deflection (mm) Fig. 16e: Finite Elem ent Model Verification in the Elastic Stage

250

Loading (KN)

200 150 100 50

GSHD2_Tested ANSYS_GSHD2

0 0

0.1

0.2

0.3

0.4

0.5

0.6

Deflection (mm) Fig. 16f: Finite Elem ent Model Verification in the Elastic Stage

Further verification of the validity of finite element models using non-linear analysis may be demonstrated by comparing the predicted response of the model with experimental results obtained from laboratory tests. Load deflection behavior obtained from the model, was compared to values for similar behavior obtained for the experimental specimen.

Page 34 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

Comparison of Finite Element Analysis to Test Results

Based on the preparation and analysis for the slab-column connections constructed by ANSYS, six finite element models were conducted for the comparison of the measured ultimate loads and the associated deflection at the center of the tested slabs. All slabs were reinforced with GFRP bars. The length and width of all the slabs were 1920 mm with the thickness of either 200 mm or 150 mm. Since the two-axis symmetry, the finite element models were only a quarter of the slabs. A concrete cover of 50 mm was used for all the slabs. Due to the brittle flexural failure of the concrete members reinforced with GFRP bars, all the slabs were initially designed to be over-reinforced, using the reinforcement ratio more than the balanced reinforcement ratio
b. This also achieved the punching failure because of high reinforcement ratio (Menetrey, 1998). The slabs of GS1 and GS3 were designed for investigating the effect of reinforcement ratio on the punching loading on the normal slab-column connection. The size effect was examined using the slabs of GSHD1 and GSHD2 with the high depth. And the application of the high strength concrete to slabs reinforced with GFRP was analyzed based on the slabs of GSHS1 and GSHS2. The details of the specimens are given in Table 1.

With the application of the three-dimensional solid element, there was reasonable agreement between the predictions by the finite element model and the measurement by the tests in Page 35 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

slab-column connections. The ratios of predicted-to-measured ultimate loads were in the range form 0.82 to 1.13; the average value was 0.98 with standard deviation of 10.8. The ratios of the associated deflections at the center of the slabs ranged from 0.68 to 0.98; and the average value was 0.85 with standard deviation of 12.4. Table 2 presents the comparison results and Fig. 17a-d show the comparison of the three-dimensional finite element model prediction to the test results.

Table 2: Comparison of finite element analyzes to test results in ultimate loading capacity and its associated deflection Compressive Slab No.

Test result

Finite Element Result Finite Element/Test

strength

Pu

Deflection

Pu

Deflection

(MPa)

(KN)

(mm)

(KN)

(mm)

GS1

40

243

41.8

246

GS3

29

236

25.1

GSHD1

33

431

GSHD2

34

GSHS1 GSHS2

Pu

Deflection

34.6

1.01

0.83

230

22.2

0.97

0.88

24.2

355

16.5

0.82

0.68

383

28.6

347

21.2

0.91

0.74

92

405

39.0

424

38.2

1.05

0.98

86

329

46.3

372

45.6

1.13

0.98

It is noted from the finite element analysis that the 40% increase of reinforcement ratio only increases around the 12% punching capacity of slab-column connections. This proves the cubic root relationship of the reinforcement ratio and punching capacity. This relationship is defined in the BS8110 code for the steel as reinforcement. According to the results, even although the GFRP rears reinforcing concrete structure have the linear elastic constitutive relationship and relatively low modulus of elasticity, different from the traditional steel reinforcement, the strength of slab-column connections reinforced with GFRP rears could be predicted by the BS8110 code if considering a term to represent the relationship of steel and GFRP rears. Page 36 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

Fig. 17a: Comparison of F.E model prediction to test result – GFRP1,3

Fig. 17b: Comparison of F.E model prediction to test result – GSHD1,2 Page 37 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

Fig. 17c: Comparison of F.E model prediction to test result – GSHS1

Fig. 17d: Comparison of F.E model prediction to test result – GSHS Page 38 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

Since the finite element model provides the considerable accurate prediction of the strength of slab-column connections, some traditional confusion for investigating two-way slabs using experimental approach could be overcome based on the finite element approach, for example, parameter study, the crack phenomenon and the form of punching cone, the failure mechanism of two-way slabs, the conflict of the punching shear prediction using major codes such as ACI code and BS8110 code.

Page 39 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

Finite Element Analysis Results Versus Modified Code Predictions

Since the two-way slab reinforced with GFRPs are a new reinforced concrete member, there have not been an accepted approach to predict the ultimate load in the building codes. EI-Ghandour (EI-Ghandour, 2003) has proposed an approach to modify the equation in the ACI code using a term referring to the reinforcement stiffness to the power of 0.33; on the other hand, he proposed to modify the equation multiplying a correction factor

. In this study, the finite element analysis results were compared with EI-Ghandour’s modified code predictions. This is given in the Table 3. It is obvious that the finite element results are consistent to the equation of the modified BS8110 code for the normal concrete strength slabs, but the equation of modified ACI code overestimates the punching resistance in the average of 29%.

Table 3: Comparison of finite element analyzes to modified codes in ultimate loading capacity and its associated deflection Slab No.

Compressive strength (MPa)

Pu

Pbs

(KN)

(KN)

Pu/Pbs

Paci (KN)

Pu/Paci

GS1

40

246

236

1.04

174

0.71

GS3

29

230

239

0.96

149

0.65

GSHD1

33

355

374

0.95

271

0.76

GSHD2

34

347

339

1.02

277

0.80

GSHS1

92

424

351

1.21

265

0.63

GSHS2

86

372

306

1.22

257

0.69

Page 40 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

Summary and Conclusion

A quarter of the full-size slab-column connections, with proper boundary conditions, were used in ANSYS for modeling to reduce computational time and computer disk space requirements.

Concrete constitutive relationship included the elastic-perfectly plastic model, crack condition and crush limit. GFRP reinforcement was defined linear elastic. Spring supports were the compression only elements.

Based on the comparison of smear reinforcement model and discrete reinforcement model, the later was used in this study because it is convenient to obtain the reinforcement information.

The layer mesh was used to obtain the enough information from critical section as well as to reduce the computation time.

For nonlinear analysis in this study, the total displacement applied to a model was divided into a number of load steps. Sufficiently small load step sizes are required, particularly at changes in behavior of the reinforced concrete connections, i.e., major cracking of concrete, and approaching failure of the reinforced concrete connections. Page 41 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

For closed cracks, the shear transfer coefficient is assumed to be 0.9, while for open cracks it should be in the suggested range of 0.05 to 0.5 to prevent numerical difficulties. In this study, a value of 0.2 was used, which resulted in accurate predictions.


The general behavior of the finite element models represented by the load-deflection plots at center show good agreement with the test data. However, the finite element models show slightly more stiffness than the test data in both the linear and nonlinear ranges.


The modified BS8110 code represents the cubic root relationship between GFRP rears and the punching strength of slab-column connections. Thus, it shows good agreement with the predictions based on finite element models.

Page 42 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

Reference Abdel Wahab EI-Ghandour1; Kypros Pilakoutas2; and Peter Waldron (2003), “Punching Shear Behavior of Fiber Reinforced Polymers Reinforced Concrete Flat Slabs: Experimental Study”, Journal of Composites for Construction, ASCE, Vol. 7, No 3, August 1, 2003, pp. 258-264 American Concrete Institute (ACI), (1995) “Building code requirements for reinforced concrete and reinforced concrete and commentary.” ACI 318-95/ACI 318R-95, Detroit. Adams, V. and Askenazi, A., (1998) “Building Better Products with Finite Element Analysis,” Santa Fe, New Mexico ASCE Task Committee on Finite Element Analysis of Reinforced Concrete Structures. (1982). State-of-the-Art Report on Finite Element Analysis of Reinforced Concrete, ASCE Special Publications. British Standards Institution (BS8110), (1985) “Code of practice for design and construction.” British Standard Institution, Part 1, London. ANSYS (1998), ANSYS User’s Manual Revision 5.5, ANSYS, Inc., Canonsburg, Pennsylvania. Bathe, K. J., (1996) Finite Element Procedures, Prentice-Hall, Inc., Upper Saddle River, New Jersey. Bangash, M. Y. H. (1989), Concrete and Concrete Structures: Numerical Modeling and Applications, Elsevier Science Publishers Ltd., London, England. Darwin, D. (1993). Reinforced Concrete. In Finite Element Analysis of Reinforced-Concrete Structures II: Proceedings of the International Workshop. New York: American Society of Civil Engineers, pp. 203-232. Ngo, D. and Scordelis, A.C. (1967). "Finite Element Analysis of Reinforced Concrete Beams," Journal of ACI, Vol. 64, No. 3, pp. 152-163.




Huyse, L., Hemmaty, Y., and Vandewalle, L. (1994), Finite Element Modeling of Fiber Reinforced Concrete Beams, Proceedings of the ANSYS Conference, Vol. 2, Pittsburgh, Pennsylvania, May 1994.



Hemmaty, Y. (1998),




Modelling of the Shear Force Transferred Between Cracks in Reinforced

Page 43 of 52

FE application to slab-column connections reinforced with GFRPs

and Fibre Reinforced Concrete Structures, Pittsburgh, Pennsylvania, August 1998.



By Qi Zhang

Proceedings of the ANSYS Conference, Vol. 1,

Lin, C.S. and Scordelis, A.C. (1975). "Nonlinear Analysis of RC Shells of General Form". Journal of Structural Division, ASCE, Vol. 101, No. ST3, pp. 523-538. MacGregor,J G,;Bartlett, F M., (2000). “Reinforced Concrete: Mechanics and Design”, Prentice-Hall Canada Inc., Scarborough, Ontario Menetrey P. (1998), “Relationships between flexural and punching failure.” ACI Structure Journal, 1998; 95(4), pp. 412-419. Rashid, M (2004), “ The Behavior of Slabs Reinforced with GFRPs”, master thesis, preparing, Faculty of Engineering, Memorial University of Newfoundland, St. John’s, Canada. S. Timoshenko and S. Woinowsky-Krieger, (1987), “Theory of Plates and Shells”. MCGraw-Hill, Inc., New York.

Page 44 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

Appendix: ANSYS CODE (GSHD1)

/PREP7 /TITLE, The FINITE ELEMENT APPLICATION TO SLAB-COLUMN CONNECTIONS REINFORCED WITH GFRPS !----------------------(1)MODEL GENERATION----------------------------------ANTYPE, STATIC ET, 1, SOLID65,,,,,2,3 !Define the Solid65 element for the concrete KEYOPT,1,1,0

!stress relaxation after crack

KEYOPT,1,5,2 KEYOPT,1,6,3 KEYOPT,1,7,1 ET,2,LINK8,

!Define the Link8 element for the reinforcement

ET,3,LINK10

!Define the Link10 element for the spring support !*

KEYOPT,3,2,0

!compression only

KEYOPT,3,3,1 MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,EX,3,,10000000 MPDATA,PRXY,3,, MPTEMP,,,,,,,, MPTEMP,1,0 MPDE,EX,3 MPDE,EY,3 MPDE,EZ,3 MPDE,NUXY,3 MPDE,NUYZ,3 MPDE,NUXZ,3 MPDE,PRXY,3 MPDE,PRYZ,3 Page 45 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

MPDE,PRXZ,3 MPDE,GXY,3 MPDE,GYZ,3 MPDE,GXZ,3 MPDATA,EX,3,,1E+007 MPDATA,PRXY,3,,0

R,1,

!Define the real constant for concrete

R,2,200,

!Define the area of reinforcement as 200mm^2

R,3,2000,0, ,

!Define size of the spring support as 2000mm^2

MP, EX,1,26700

!Define the modulus of elasticity of concrete as 26700 MPa

MP,NUXY,1,0.2

!Concrete Poisson ratio is 0.2

TB,BKIN,1,1 TBDATA,1,33,100 !Yield strength of concrete is 33MPa TB,CONCR,1,1 TBDATA,1,0.2,0.9,3.5,33 !the shear transfer coefficient of the open crack is 0.2 !the shear transfer coefficient of the close crack is 0.9 !the tensile strength of concrete is 3.5 MPa !the compressive strength of concrete is 33 MPa MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,DENS,1,,2.3e-6 !Concrete density is 2.3e-16 kg/mm^3

MP,EX,2,42000 !Define the modulus of elasticity of GFRP as 42000 MPa MP,NUXY,2,0.3

!GFRP Poisson ratio is 0.2

N,1000,960,0,-50,,,,

!Define the node for spring support

N,1001,960,60,-50,,,, N,1002,960,120,-50,,,, N,1003,960,180,-50,,,, N,1004,960,240,-50,,,,

Page 46 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

N,1005,960,300,-50,,,, N,1006,960,360,-50,,,, N,1007,960,480,-50,,,, N,1008,960,600,-50,,,, N,1009,960,720,-50,,,, N,1010,960,840,-50,,,, N,1011,960,960,-50,,,, N,1012,0,960,-50,,,, N,1013,60,960,-50,,,, N,1014,120,960,-50,,,, N,1015,180,960,-50,,,, N,1016,240,960,-50,,,, N,1017,300,960,-50,,,, N,1018,360,960,-50,,,, N,1019,480,960,-50,,,, N,1020,600,960,-50,,,, N,1021,720,960,-50,,,, N,1022,840,960,-50,,,, N,2000,960,0,-60,,,, N,2001,960,60,-60,,,, N,2002,960,120,-60,,,, N,2003,960,180,-60,,,, N,2004,960,240,-60,,,, N,2005,960,300,-60,,,, N,2006,960,360,-60,,,, N,2007,960,480,-60,,,, N,2008,960,600,-60,,,, N,2009,960,720,-60,,,, N,2010,960,840,-60,,,, N,2011,960,960,-60,,,, N,2012,0,960,-60,,,, N,2013,60,960,-60,,,, N,2014,120,960,-60,,,, N,2015,180,960,-60,,,, N,2016,240,960,-60,,,, N,2017,300,960,-60,,,, N,2018,360,960,-60,,,, N,2019,480,960,-60,,,, N,2020,600,960,-60,,,, N,2021,720,960,-60,,,, N,2022,840,960,-60,,,,

Page 47 of 52

FE application to slab-column connections reinforced with GFRPs

type,3

By Qi Zhang

!Construct the spring elements

real,3 *do,i,0,22 n1=1000+i n2=2000+i E,n1,n2 *enddo

wpstyle,10,100,0,1000,10,0,0,,5 !workplace status Bopt,Numb,off K,1,0,0,0,

!Define key points for GFRP reinforcement

K,2,960,0,, K,3,0,960,, , LSTR,1,2 LSTR, 1, 3 FLST,3,2,4,ORDE,2

!Define lines for GFRP reinforcement

FITEM,3,1 FITEM,3,-2 FLST,3,1,4,ORDE,1 FITEM,3,2 LGEN,9,P51X, , ,120, , , ,0 FLST,3,1,4,ORDE,1 FITEM,3,1 LGEN,9,P51X, , ,0,120, , ,0 FLST,2,18,4,ORDE,2 FITEM,2,1 FITEM,2,-18 LOVLAP,P51X

LATT,2,2,2, , , , ESIZE,120,0,

!Define the lines with GFRP attributes !Mesh and construct the GRFP elements

FLST,5,54,4,ORDE,49 FITEM,5,19 FITEM,5,-20 FITEM,5,22 FITEM,5,-23 FITEM,5,25

Page 48 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

FITEM,5,-27 FITEM,5,29 FITEM,5,-31 FITEM,5,34 FITEM,5,38 FITEM,5,42 FITEM,5,46 FITEM,5,49 FITEM,5,53 FITEM,5,-55 FITEM,5,57 FITEM,5,-59 FITEM,5,62 FITEM,5,66 FITEM,5,70 FITEM,5,74 FITEM,5,77 FITEM,5,80 FITEM,5,84 FITEM,5,-86 FITEM,5,91 FITEM,5,93 FITEM,5,97 FITEM,5,-98 FITEM,5,103 FITEM,5,-104 FITEM,5,109 FITEM,5,-110 FITEM,5,115 FITEM,5,-116 FITEM,5,121 FITEM,5,-122 FITEM,5,127 FITEM,5,-128 FITEM,5,133 FITEM,5,-134 FITEM,5,139 FITEM,5,-140 FITEM,5,145 FITEM,5,-146 FITEM,5,151

Page 49 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

FITEM,5,-152 FITEM,5,157 FITEM,5,-158 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y !* LESIZE,_Y1,60, , , , , , ,1 !* FLST,2,144,4,ORDE,2 FITEM,2,19 FITEM,2,-162 LMESH,P51X

wpoff,0,0,-50

!Define the volumes for concrete slab and column stub

BLC4,0,0,360,360,200 BLC4,360,0,600,360,200 BLC4,0,360,360,600,200 BLC4,360,360,600,600,200 wpoff,0,0,200 BLC4,0,0,120,120,600 FLST,2,4,6,ORDE,2 FITEM,2,1 FITEM,2,-4 VGLUE,P51X WPSTYLE,,,,,,,,0 nummrg,all numcmp,all

VATT,1,1,1,0

!Define the volumes with concrete attributes

FLST,5,4,4,ORDE,4 !Mesh and construct the concrete elements FITEM,5,5 FITEM,5,8

Page 50 of 52

FE application to slab-column connections reinforced with GFRPs

By Qi Zhang

FITEM,5,170 FITEM,5,173 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y !* LESIZE,_Y1,60, , , , , , ,1 !* FLST,5,1,4,ORDE,1 FITEM,5,159 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y !* LESIZE,_Y1, , ,4, , , , ,1 !* MSHAPE,0,3D MSHKEY,1 !* FLST,5,5,6,ORDE,2 FITEM,5,1 FITEM,5,-5 CM,_Y,VOLU VSEL, , , ,P51X CM,_Y1,VOLU CHKMSH,'VOLU' CMSEL,S,_Y !* VMESH,_Y1 !* CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 !* nummrg,all numcmp,all

Page 51 of 52

FE application to slab-column connections reinforced with GFRPs

FLST,2,6,5,ORDE,6

By Qi Zhang

!Define the symmetry boundary condition

FITEM,2,3 FITEM,2,5 FITEM,2,11 FITEM,2,13 FITEM,2,18 FITEM,2,21 DA,P51X,SYMM FLST,2,23,1,ORDE,2

!Define the spring fixed in the end

FITEM,2,24 FITEM,2,-46 !* /GO D,P51X, ,0, , , ,ALL, , , , ,

!------------------------------(2)SOLUTION-------------------------------/solu cnvtol,f,,0.05,2 !Define force as the convergence limit, 0.05 nsubst,1000,2000 !Load steps are 1000, the maximum load step are 2000 outres,all,all

!Output all the data to the result file

autots,1

!Set auto stepping on

lnsrch,1

!Set line search on

ncnv,2

!Finish but not exit if not convergent

neqit,50

!Maximum iteration in each step is 50

pred,on

!Set prediction option on

time,25

!Set time 25 second

FLST,2,1,5,ORDE,1

!Applying the displacement 25mm along -UZ

FITEM,2,10 !* /GO DA,P51X,UZ,-25 ACEL,0,-9.81, ,

!Apply the gravity of model

solve finish

Page 52 of 52