Atc-19.pdf

ATC-19 Structural Response Modification Factors by APPLIED TECHNOLOGY COUNCn.. 555 Twin Dolphin Drive, Suite 550 Redwood City, California 94065

Funded by NATIONAL SCIENCE FOUNDATION Grant No. ECE-8600721

and NATIONAL CENTER FOR EARTHQUAKE ENGINEERING RESEARCH NCEER Project No. 92-4601

PRINCIPAL INVESTIGATOR Christopher Rojahn

PROJECT CONSULTANTS Andrew Whittaker

Gary Hart

PROJECT ENGINEERING PANEL

Vitelmo Bertero Gregg Brandow . Sigmund Freeman William Hall Lawrence Reaveley* *ATC Board Representative

1995

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Preface

In 1986, the Applied Technology Council (ATC) was awarded a grant from the National Science Foundation (NSF) to evaluate structural response modification factors (R factors). R factors are used in current seismic building codes to reduce ground motions associated with design level earthquakes to design force levels. The initial objectives of the project (known as ATC-19) were to: (1) document the basis for the values assigned to R factors in model seismic codes in the United States, (2) review the role played by R factors in seismic design practice throughout the United States; (3) present state-of-knowledge on R factors; and (4) propose procedures for improving the reliability of values assigned to R. In 1991, the scope of the effort was expanded with funding from the National Center for Earthquake Engineering Research (NCEER) to address and/or document (1) how response modification factors are used for seismic design in other countries; (2) a rational means for decomposing R into key components using state-of-the-knowledge information; (3) a framework (and methods) for evaluating the key components of R; and (4) the research necessary to improve the reliability of engineered construction designed using R factors. The results from the original and expanded objectives described above are documented in this report. The primary ATC-19 project consultants, who prepared the major portions of this report, were Gary Hart and Andrew Whittaker, senior-level earthquake engineering researchers from southern and northern California, respectively. Their work was overviewed and guided by an advisory "blue-ribbon" Project En-

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gineering Panel (PEP) consisting ofVitelmo Bertero, Gregg Brandow, Sigmund Freeman, William Hall, and Lawrence Reaveley (ATC Board Representative). Nancy Sauer and Peter Mork provided editorial and publication preparation assistance. The affiliations of these individuals are provided in the Project Participants list. ATC gratefully acknowledges the valuable support and patience of the NSF Project Officer, S. C. Liu. ATC also gratefully acknowledges the valuable input of participants in the companion NCEER-funded ATC-34 Project: The late Peter Gergely (Cornell University), who served on the NCEER Research Committee and played a key role in acquiring NCEER support for this investigation; Project Director Andrew Whittaker (University of California at Berkeley); PEP members Vitelmo Bertero (University of California at Berkeley), Ian Buckle (NCEER), Sigmund Freeman (Wiss, Janney, Elstner Assoc., Inc.), Gary Hart (University of California at Los Angeles), Helmut Krawinkler (Stanford University), Ronald Mayes (Dynamic Isolation Systems), Andrew Merovich (Andrew Merovich & Assoc.), Joseph Nicoletti (URSlBlume), Guy Nordenson (Ove Arup & Partners), Masanobu Shinozuka (University of Southern California), and John Theiss (ATC Board Representative); and consultants Howard Hwang (Memphis State University), Onder Kustu (OAK Engineering), and Yi-Kwei Wen (University ofIl1inois). Christopher Rojahn ATC Executive Director & ATC-19 Principal Investigator

Preface

III

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Table of Contents Preface

iii

1. Introduction 1.1 Background 1.2 Objectives of the Report 1.3 Organization of the Report :

1 1 2 2

2. History of Response Modification Factors 2.1 Introduction 2.2 R Factor Development 2.3 Rw Factor Development 2.4 Comparison of K, R, and Rw

5 5 5 8 8

3. Use of Response Modification Factors 3.1 Introduction 3.2 R Factors in SeisnUc Building Codes 3.2.1 Europe 3.2.2 Japan 3.2.3 Mexico 3.2.4 Summary 3.3 Use of R Factor Equivalents for Bridge Design

'"

11 11 11 11 11 13 14 15

4. Components of Response Modification Factors 4.1 Introduction 4.2 Impact of the R Factor on Design 4.3 Force-Displacement Response of Buildings 4.4 Experimental Evaluation of Force-Displacement Relationships 4.5 Key Components of R 4.5.1 Strength Factor 4.5.2 Ductility Factor 4.5.3 Redundancy Factor 4.5.4 DaIllping Factor 4.6 Systematic Evaluation of R Factors 4.7 Reliability of Values for R

17 17 17 18 20 21 22 23 27 29 31 32

5. Conclusions and Recommendations 5.1 Summary and Concluding Remarks 5.2 Recommendations ;

33 33 34

Appendix A: Evaluation of Building Strength and Ductility

35

Appendix B: Glossary of Terms

49

References

51

Project Participants

53

Applied Technology Council Projects and Report Information

55

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Table of Contents

v

1. 1.1

Introduction Background

The seismic design of buildings in the United States is based on proportioning members of the seismic framing system for actions determined from a linear analysis using prescribed lateral forces. Lateral force values are prescribed at either the allowable (working) stress or the strength level. The Uniform Building Code (lCBO, 1991) prescribes forces at the al10wable stress level and the NEHRP Recommended Provisions for the Development ofSeismic Regulationsfor New Buildings, hereafter denoted as the NEHRP Provisions (BSSC, 1991) prescribes forces at the strength level. The seismic force values used in the design of buildings are calculated by dividing forces that would be associated with elastic response by a response modification factor, often symbolized asR. Response modification factors were first proposed by the Applied Technology Council (ATC) in the ATC3-06 report published in 1978. The NEHRP Provisions, first published in 1985, are based on the seismic design provisions set forth in ATC-3-06. Similar factors, modified to reflect the allowable stress design approach, were adopted in the Uniform Building Code (UBC) a decade later in 1988. The concept of a response modification factor was proposed based on the premise that well-detailed seismic framing systems could sustain large inelastic deformations without collapse (ductile behavior) and develop lateral strengths in excess of their design strength (often termed reserve strength). The R factor was assumed to represent the ratio of the forces that would develop under the specified ground motion if the framing system were to behave entirely elastically (termed hereafter as elastic design) to the prescribed design forces at the strength level (assumed equal to the significant yield level). In the UBC, gravity (dead, live, and snow) and environmental (wind, seismic) loads are prescribed at the service level. Until the recent advent of Load and Resistance Factor Design (LRFD), an ultimate strength approach that is just beginning to be used in practice, steel framing systems have typically been

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designed for service-level actions using allowable stresses. Reinforced concrete framing systems are designed for ultimate strength-level actions, which are calculated by multiplying the service-level actions by load factors. Prescribed seismic forces are calculated in the UBC by dividing the elastic spectral forces by a response modification factor (Rw ): values for Rw range between 4 and 12. In the NEHRP Provisions, loads are prescribed at the strength level. In practice, steel framing systems are designed for ultimate-level actions by using al1owable stress values multiplied by 1.7; reinforced concrete framing systems are designed at the strength level for ultimate actions. Prescribed seismic forces are calculated in the NEHRP Provisions by dividing the elastic spectral forces by a response modification factor, R. Values for R range between 1.25 and 8. The relationship between the response modification factors in the NEHRP Provisions (R) and the UBC (Rw ) is presented later in this report. When using response modification factors substantially greater than one, the designer makes a significant assumption; that is, that linear analysis tools can be used to obtain reasonable estimates of nonlinear response quantities. This assumption has recently been questioned and is discussed in detail in ATC-34 (ATC, 1995). Use of large response modification factors underlies a second common assumption of seismic design; that is, that significant nonlinear response and hence significant damage is expected if the design earthquake occurs. This assumption is of course a direct result of using design forces that are significantly less than the elastic spectral forces. The consequences of this assumption are considered in detail in this report. The R factors for the various framing systems included in the ATC-3-06 report were selected through committee consensus on the basis of (a) the general observed performance of like buildings during past earthquakes, (b) estimates of general system toughness, and (c) estimates of the amount of damping present during inelastic response. Thus, there is little technical basis for the values of R proposed in

1: Introduction

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ATC-3-06. The commentary to ATC-3-06 notes that " ... values of R must be chosen and used with judgement" and that " ... lower values must be used for structures possessing a low degree of redundancy wherein all the plastic hinges required for the formation of a mechanism may be formed essentially simultaneously and at a force level close to the specified design strength." To further underscore the uncertainties associated with the values assigned to R for different seismic framing systems, a footnote to the table listing the response modification coefficients states, "These (values for R) are based on best judgment and data available at time of writing and need to be reviewed periodically." Given the fiscal and social consequences of widespread building failure that could occur in an earthquake if poor choices for values of R are used in design, it is evident to enlightened design professionals that the values assigned to R in current seismic regulations should reflect the most current knowledge in earthquake engineering and construction practice in the United States. Nearly twenty years have passed since R factors were first introduced in the United States. In this space oftime, much has been learned about the likely performance of seismic framing systems in moderate-to-severe earthquakes, especially following the 1989 Lorna Prieta and 1994 Northridge earthquakes. This new knowledge, combined with changing public expectations of acceptable levels of earthquake-induced damage and changes in construction practice, makes 1995 an appropriate year in which to publish a formal review of response modification factors and the ways in which the factors are used (and misused) in current design practice.

1.2

Objectives of the Report

This report has several key objectives. 1. To document the basis for the values assigned to R in current seismic codes in the United States.

2. To review the role played by R factors in seismic design practice in the United States. 3. To describe how response modification factors are used for seismic design in other countries. 4. To present up-to-date information on R factors.

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5. To develop a rational means of decomposing R into key components. 6. To propose a framework (and methods) for evaluating the key components of R. 7. To recommend research necessary to improve the reliability of engineered construction designed using R factors. The primary audience for this report is licensed professional engineers familiar with both current building seismic design criteria and structural dynamics. However, the report has been written to be understandable to a broad audience, with the intent ofhaving a strong impact on the design professionals and the code-change process. The secondary audience for the report is the academic/research community.

1.3

Organization of the Report

Chapter 2 provides an historical perspective on how the values of R in use today were developed. The relationship between K factors introduced in the late 1950 s, R factors introduced in ATC-3-06, and R w factors introduced into the 1988 UBC (ICBO, 1988) is established, and the shortcomings of seismic design using R factors are enumerated. Chapter 3 discusses the use of response modification factors for the seismic design of new buildings outside the United States and for the seismic design of new bridges in the United States, to provide perspective on the conclusions drawn in this report. The factors used in three common framing systems, the European, Japanese, and Mexican codes, are compared with the corresponding values in the 1991 NEHRP Provisions (BSSC, 1991). This chapter includes some conclusions about the likely behavior ofcode-compliant buildings in the United States during severe earthquake shaking. Chapter 4 discusses the impact of R factors on the seismic design process in the United States, experimental estimates of R for two steel-braced framing systems, and proposes a new formulation for R. Unresolved issues associated with the proposed formulation for R are described, and strategies for resolving these issues are proposed. In Chapter 5, the significant issues raised in this report are summarized, and key conclusions are drawn. Recommendations for further study complete this chapter. A list of references follows Chapter 5.

1: Introduction

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Reliable values for R will likely be proposed on the basis ofthe statistical evaluation of reserve strengths and system ductility values. Reserve strength and ductility can be estimated using nonlinear static analysis. Appendix A provides an overview of nonlinear static analysis and presents the results of such an analysis of a nonductile reinforced concrete moment frame building. This analysis was performed as part of the ongoing FEMA-funded Building Seismic Safety Council (BSSC) project to develop guidelines and commentary for the seismic rehabilitation of

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buildings (ATC' s portion of this project is known as ATC-33). The results of the nonlinear static analysis presented in Appendix A are used to calculate draft strength and ductility factors. Appendix B contains a comprehensive glossary of tenns used in this report. Following Appendix B are references, a list of the individuals who have contributed to the preparation of this report, and infonnation on other available ATC reports, including companion reports and other resource documents.

1: Introduction

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2. 2.1

History of Response Modification Factors

Introduction

In 1957, a committee of the Structural Engineers Association of California (SEAOC) began development of a seismic code for California. This effort resulted in the SEAOC Recommended Lateral Force Requirements (also known as the SEAOC Blue Book) being published in 1959 (SEAOC, 1959). Commentary to the requirements was first issued in 1960. These recommendations represented the profession's state-of-the-art knowledge in the field of earthquake engineering; the seismic design requirements in the 1959 Blue Book were significantly different from previous seismic codes in the United States. For the first time the calculation of the minimum design base shear explicitly considered the structural system type. The equation given for base shear was

mended Lateral Force Requirements" are intended to provide this protection in the event of an earthquake of intensity or severity of the strongest of those which California has recorded ... The code does not assure protection against non-structural damage ... Neither does it assure protection against structural damage ..." The seismic provisions in the 1961 UBC OCBO, 1961) were adopted from the 1959 Bluebook. Seismic zonation was considered through the use of a Z factor which was set equal to 1.0 in zone 3 (the region of highest seismicity), 0.50 in zone 2, and 0.25 in zone 1. The minimum design base shear in the 1961 UBC was calculated as:

v= v=

KCW

ZKCW

(2-3)

(2-1) where all terms were defined as above.

where K was a horizontal force factor (the predecessor of R and Rw ); C was a function of the fundamental period of the building; and W was the total dead load. The K factor was assigned values of 1.33 for a bearing wall building, 0.80 for dual systems, 0.67 for moment-resisting frames, and 1.00 for framing systems not previously classified. The term C defined the shape ofthe design response spectrum, and was calculated as follows:

C = 0.05

(2-2)

3..11' where T was the fundamental period of vibration in the direction under consideration. The Blue Book was developed as a seismic design code for California alone. California was assumed to have uniform seismicity, and a seismic zone factor was not required in Equation 2-1. The intent of the Blue Book was to: "... provide minimum standards to assure public safety. Requirements contained in such codes are intended to safeguard against major structural failures and to provide protection against loss of life and personal injury... The "Recom-

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2.2

R Factor Development

The development of response modification (R) factors, first introduced in ATC-3-06 (ATC, 1978), can be traced back to the horizontal force factor (Rojahn, 1988, and Rojahn and Hart, 1988). This section summarizes the development process. The publication of ATC-3-06 defined a benchmark in seismic engineering in the United States. ATC-3-06 constituted a significant departure from previous seismic codes and embodied several new concepts that included: (a) classification of building use-group categories into seismic hazard exposure groups, (b) national seismic hazard maps, (c) tools for elastic dynamic analysis, (d) use of response modification (R) factors in lieu of K factors, (e) explicit drift limits, (f) discussion of the influence of orthogonal excitation effects, (g) materials design based on strength methods instead of allowable stress, (h) provisions for soil-structure interaction, and (i) detailed seismic design requirements for architectural, electrical, and mechanical systems and components. In regard to response modification factors, ATC-3-06 noted that:

2: History of Response Modification Factors

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... -_._ .

Aa is the effective peak. acceleration of the design ground motion (expressed as a fraction of g), R is the response modification factor, and W the total reactive weight The factor of2.5 is a dynamic amplification factor that represents the tendency for a building to amplify accelerations applied at the base.

1. . R factors were intended to reflect reductions in design force values that were justified on the basis of risk assessment, economics, and nonlinear behavior. 2. The intent was to develop R factors that could be used to reduce expected ground motions presented in the fonn of elastic response spectra to lower design levels by bringing modem structural dynamics into the design process. Figure 2.1 illustrates the use of R factors to reduce elastic spectral demands to design force levels.

Only horizontal seismic forces were considered in ATC-3-06 for two reasons. First, buildings had always been designed to withstand vertical forces greater than those produced by mean (unfactored) gravity loads, thereby providing assumed reserve capacity for vertical seismic motions, and second, because the analysis and design tools needed to account for vertical ground motion effects were not routinely available in the 1970s. Furthermore, ground motion data available at the time suggested that peak vertical motions were normally less than 2/3 of peak horizontal motions, leading to the conclusion that the responses caused by vertical motions should be less severe than those caused by horizontal motions.

Given that R was to be a response reduction factor, it was decided to place R in the denominator of the base shear equation. The end result was that R factors were inversely proportional to the K factors used in previous codes. The base shear equation for structures for which the period of vibration ofthe building T was not calculated took the following form: 2.5Aa

v= -yw

(2-4)

For structures for which the fundamental building period was calculated, the base shear equation in

In this expression, V is the seismic base shear force,

--

.....

C)

ATC 3-06 elastic response spectrum for a rock site and 5% damping

c:

0

r
CD CD

0 0 r
-...

Cii 1

Design spectrum for a special moment-resisting space frame (R =8)

0

CD Q.

UI

"tI CD

.!:!

Cii

...0E

;-----------

Z

/

0

------

--------------_ ..

Period (seconds)

Figure 2. 1: Use of R factors to reduce elastic spectral demands to the design force level. Each point on the elastic response spectrum for a rock site (top curve) is divided by R to produce the design spectrum (bottom curve) for a given structure type, in this case a special moment-resisting space frame. where R= 8.

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2: History of Response Modification Factors

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1.67)

ATC-3-06 was given as: V

=

1.2A y S Ry{).67 w

( V)

Vw ( 1.33

(2-5)

In this expression, A y is the effective peak velocityrelated acceleration, S is a soil profile coefficient, and T is the fundamental period of the building. The soil profile coefficient is used to account for soil properties that could amplify the bedrock motion; its values, as defined in ATC-3-06, range from 1.0 to 1.5. The base shear of Equation 2-4 provides an upper limit on the base shear calculated using Equation 2-5.

= 0.9

(2-6)

where Vw was the allowable-stress design lateral seismic base shear (1976 UBC) and V was the strength- design lateral seismic base shear (ATC 306). The numerical factors in Equation 2-6 accounted for differences between the allowable-stress design and strength design methods: 1.67 represented the margin of safety in allowable-stress design, 1.33 represented the pennissible increase in allowable-stress design, and 0.9 was the capacity reduction factor for flexure in strength design.

Individuals who participated in the ATC-3-06 R factor development process (ATC, 1978, page 8, Structural Design, Details, and Quality Assurance Committee) have indicated that committee members first independently developed R values for each structural system type based on their own experience. The values of R selected for inclusion in ATC-3-06 represented the consensus opinion of the experts involved in the development effort.

Using the expression for Vw as specified in the 1976 UBC (leBO, 1976), it followed that:

The first step in assigning consensus R values was to set a maximum value of R for the structure types considered to provide the best seismic perfonnance; that is, those with the highest reserve strength or ductility. This category included special moment frames and dual systems composed of reinforced concrete shear wall structures with special moment frames capable of resisting at least 25 percent of the prescribed seis.mic forces.

Substituting Z = 1= T= 1.0, Si = 1.2 in Equation 2-7:

C.W. Pinkham (personal communication), a member of the team that developed R factors, described the procedure used to calculate R for special steel moment frames. The maximum value of R was determined by equating Vw computed for allowable stress design per the 1976 UBC (equivalent to the 1974 Blue Book (SEAOC, 1974) to V computed for strength design in ATC-3-06. Implicit in this undertaking was the decision not to increase the design base shear to improve seismic performance, but rather to achieve improved seismic performance by requiring better detailing. For special steel moment frames, the maximum value of R was computed at a fundamental period equal to 1.0 second:

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ZIKCS. w(1.67) I 1.33

(1.2AS) W = 0.9Ry{).67

(2-7)

where Z was a zone factor, I was an importance factor, K was a horizontal force factor, C defined the spectral shape ( 1/( 15 ,and Si was a site coefficient.

Jh

= 1.5, Ay = 0.4, and S

(1.0)(1.0)K(0.067)(1.5) ( 1.67) 1.33

=

(2-8)

(1.2 x 0.4 x 1.2) "--~~.-:-;:~....;.. 0.9R(1.0)

resulted in (0.1256)K =

0~4

(2-9)

yielding R = 5.1

K

(2-10)

In the 1976 UBC, K was set equal to 0.67 for moment resisting frame systems. The corresponding value of R in ATC-3-06 was thus computed as: 5.1 8 R = 0.67= .

(2-11)

The response modification factor for reinforced concrete shear-wall structures with special moment frames was also assigned the maximum value of eight. Values of R for other framing systems were

2: History of Response Modification Factors

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generally assigned on the basis of Equation 2-10, then adjusted in accordance with the consensus opinion of the committee. Framing systems not considered in the 1976 UBC were assigned R values by consensus opinion of the committee.

2.3

If it is assumed that CS = 0.14 and Z = 1 in Equation 2-12, and that C = 2.75 and Z= 0.4 in Equation 2-13, it follows that

Rw Factor Development

Values for structural response modification factors for allowable-stress design (RwJ were determined by the Seismology Committee of the Structural Engineers Association of California (SEAOC) and published in the 1988 Blue Book (SEAOC, 1988). SEAOC elected to introduce Rwo rather than R, to ease the eventual transition from allowable-stress design to strength design. Similar to R, Rw is inversely proportional to K. The relationship between values of K in the 1985 UBC and values of R w in the 1988 UBC can be demonstrated as follows. The equation given in the 1985 UBC (lCBO, 1985) for calculating the design base shear at the allowable stress level (VD ) is:

VD = (ZIKC5)W

(2-12)

The parameters Z and I are used to quantify the seismic zone and the importance of the building occupancy, respectively. The parameter S is used to account for site characteristics, and C is a numerical coefficient that is a function of the fundamental period of vibration of the building and the defmed spectral shape. The maximum value of C is set equal to 0.12; the maximum value of CS is set at 0.14. K is a numerical coefficient referred to as the horizontal force factor. The 1988 Blue Book (SEAOC, 1988) and 1994 UBC (lCBO, 1994) use an alternative equation for calculating VD , namely: (2-13) where Z and I are the seismic zone and importance factors, respectively. For this example, let! = 1 in Equations 2-12 and 2-13. The factor C in Equation 213 has a maximum value of2.75 and is defmed as: C

8

where S is a site coefficient and T is the fundamental period of vibration.

= 1.255

ro·

67

(2-14)

K(0.14)C = (2.75)0.4 Rw

(2-15)

and that (2-16) Substituting Equation 2-1 o into Equation 2-16 yields the following relationship between Rw and R Rw

R = 1.54R = 7.86 5.1 .

(2-17)

.

Table 2.1 displays the values of K (/985 UBCJ and

Rw (1988 UBC) for several framing systems. Table 2.1

Relationship of K and Rw

Framing System

1985 USC

1988 USC

Rw=

Bearing wall

K = 1.33

Dual steel and concrete

K = 0.80

Rw = 10

Ductile steel and concrete

K = 0.67

Rw

Other

2.4

K

= 1.00

6

= 12

Rw=

8

Comparison of K, R. and R w

With few exceptions, the R factors tabulated in ATC3-06 are the same as those in the 1991 NEHRP Provisions. The exceptions include an increased value of R in the NEHRP Provisions for special concrete moment-resisting space frames, and the addition of R factors for concrete intermediate moment-resisting space frames. The values assigned to R w in the 1994 UBC are the same as those listed in the 1988 Blue Book. For reference, values ofR in ATC-3-06 and the 1991 NEHRP Provisions, and R w in the 1994 VBC are listed in Table 2-2, for framing systems grouped according to K value. The link between K, R, and Rw was established in the previous section. Values of the horizontal force fac-

2: History of Response Modification Factors

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tor K represented the consensus opinion of expert design professionals and academicians in the late 1950s. Despite a many-fold increase in knowledge and the advent of powerful analysis tools, no substantive review of, or changes to, response reduction factors have been made since the 1950s. Recent studies by researchers (e.g., Bertero, 1986) and design professionals, including Project ATC-34, have identified major shortcomings in the values and formulation of the response modification factors used in seismic codes in the United States. These shortcomings include the following: 1.

A single value of R for all buildings of a given framing type, irrespective of building height, plan geometry, and framing layout, cannot be jus,tified.

2. The use of the values assigned to R for some framing systems will likely not produce the desired performance in the design earthquake.

ratio cannot be used to uniformly reduce elastic spectral demands to design (inelastic) spectral demands (measured typically as base shear), R must be period-dependent. This dependence is recognized in the Eurocode and the Mexican Code (see Chapter 3 for further discussion). 4. The reserve strength (strength in excess of the design strength) of buildings designed in different seismic regions will likely vary substantially. Given that reserve strength is a key component of R (see Chapter 4), R should be dependent on either the seismic zone or some ratio of gravity loads to seismic loads. 5. Seismic design using the response modification factors listed in seismic codes and guidelines in the United States will most probably not result in a uniform level of risk for all seismic framing systems. These shortcomings and other related issues are addressed in the remainder of this report.

3. The response modification factor is intended, in part, to account for the ductility of the framing system. Recognizing that a constant ductility

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2: History of Response Modification Factors

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Table 2.2

Tabulated Values for K, Rand

Rw

Basic Structural System (K factor)

R V\TC, 1978)

Bearing Wall System (K=1.33) 1. Light Framed Walls with Shear Panels

2.

a. Plywood walls, 3 stories or less b. All other light framed walls Shear walls a. Concrete

6.5

R

rosse,

1991)

6.5 8 6

4.5

4.5

b. Masonry 3. Braced Frames Carrying Gravity Loads a. Steel b. Concrete Building Frame System (K=1.00) 1. Steel Eccentric Braced Frames (EBF) 2. Concentric Braced Frames 3. Shear Walls

3.5 4.0

3.5 4.0

a. Concrete b. Masonry Dual System (K=O.80) 1. Shear Walls a. Concrete with Special Moment Resisting Space Frame (SMRSF) b. Concrete with Concrete Intermediate Moment Resisting Space Frame (IMRSF)

6.0 6.0 6 4

7.0-8.0 7.0

10.0

5.5 4.5

5.5 4.5

8.0 8.0

8.0

8.0

12.0

6.0

9.0

c. Masonry with Concrete SMRSF d. Masonry with Concrete IMRSF 2. Steel EBF with Steel SMRSF 3. Concentric Braced Frames

6.5

6.5 5.0 7.0-8.0

8.0 7.0 12.0

a. Steel with SMRSF b. Concrete with Concrete SMRSF c. Concrete with Concrete IMRSF Moment Resisting Frame System (K=O.67) 1. Special Moment Resisting Space Frames (SMRSF) a. Steel

6.0 6.0

6.0 6.0 5.0

10.0 9.0 6.0

8.0 7.0

8.0 8.0

12.0 12.0

4.0

8.0

4.5 2.0

6.0 5.0

b. Concrete 2. Concrete Intermediate Moment Resisting Space Frames (IMRSF) 3. Ordinary Moment Resisting Space Frames a. Steel b. Concrete

10

Rw

(leBO, 1994)

4.2 2.0

2: History of Response Modification Factors

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3.

Use of Response Modification Factors

3.1 Introduction The use of response modification factors is not restricted to the seismic design of buildings in the United States. R factors, or their equivalents, are used for the seismic design of buildings in Europe, Japan, Mexico, and other countries, and for the seismic design of bridges in the United States. This chapter reviews the use of R factors in seismic building codes in Europe, Japan, and Mexico, and seismic bridge codes in the United States, in order to place seismic design practice for buildings in the United States in perspective.

3.2

R Factors in Seismic Building Codes

3.2.1 Europe The seismic design procedure in the 1988 Eurocode (CEC, 1988) is a single-level design procedure that reduces elastic spectral demands to the strength design level through the use of a period-dependent, response reduction factor (q) as follows: for T < T}

T 1 + l1(Tl~O -1)

q

=

T 1 + Tl (Tl ~O/ q - 1)

(3-1)

for T> T}

q=q

(3-2)

where T is the fundamental period ofthe building; Tl is a characteristic period of the design spectrum (lower-bound period to the constant-acceleration portion of the spectrum); 11 is a factor related to the system equivalent viscous damping ~ and equal to 1.0 for ~ equal to five percent of critical; ~O is a pseudo-acceleration spectrum amplification factor (set equal to 2.5); and q is a system behavior factor that varies as a function of material type, building strength and stiffness regularity. Values for q range between one and five for reinforced concrete framing systems. Recognizing that ductility cannot be used to

ATC-19

reduce substantially elastic force demands in the short-period range from 0 to Tl , equation 3.1 shows how q varies from q = q at T= Tl' to q=1.0 at

T=O.O. Inelastic displacement values (ds ) are estimated in the 1988 Eurocode as the product of the displacement values (de) computed using the reduced (design) seismic forces and the behavior factor q. For T less than Tl' the ratio q/q exceeds 1.0 and the inelastic displacement values exceed the elastic displacement values; for T greater than T}, the ratio q/q equals 1.0, and the inelastic displacement values equal the elastic displacement values.

3.2.2

Japan

The Japanese 1981 Building Standard Law (IAEE, 1992) includes a two-phase or two-level procedure for the seismic design of buildings. The first phase (Level I) design follows an approach similar to that adopted in the NEHRP Provisions (BSSC, 1991). Steel structures are designed at the strength level, based on allowable stress design procedures with the steel allowable stress equal to the yield stress. Strength design is used for reinforced concrete structures. The second phase (Level II) design is a direct and explicit evaluation of strength and ductility, and may be regarded as a check of whether these are sufficient for severe ground motions. Timber structures and low-rise structures satisfying rigidity, eccentricity, and detailing limitations do not require Level II design. Other structures, including all structures between 31 and 60 meters high, are subject to both Level I and Level II design. Normative practice is for the seismic framing system to be designed using the Level I procedure and for the Level I design to be checked (and modified as necessary) using the Level II procedure. Structures over 60 meters high are subject to special approval by the Ministry of Construction. In the Level I design, the seismic coefficient at each story ( Cj ) is determined as the product of four variables:

3: Use of Response Modification Factors

(3-3)

11

._----_._-_._ -

.

where Z represents the seismic zone, Rt defmes the spectral shape that varies as a function of soil type, Ai defines the vertical distribution of seismic force in the building, and Co represents the peak ground acceleration. In regions of high seismicity, Z is equal to 1.0. Except for wood structures on soft subsoil, Co is set equal to 0.2. The seismic design shear force in the i-th story (Qi ) is calculated as: (3-4) where W is the reactive weight above the i-th story. For Level I design, seismic actions are computed using unreduced seismic forces. Interstory drift is limited to 0.5 percent of the story height for the prescribed seismic forces unless it can be demonstrated that greater drift can be tolerated by the nonstructural components, in which case the drift limit can be increased to 0.8 percent of the story height. In Level II design, the engineer checks plan eccentricity, distribution oflateral stiffness, minimum code requirements (in some cases), and ultimate lateralload-carrying capacity of each story. The ultimate lateral load capacity is computed using plastic analysis and ultimate seismic demands are estimated as: (3-5)

where Qud is the lateral seismic shear for severe earthquake motions, calculated according to Equation 3-4 using Co equal to 1.0, Ds is a framing system-dependent ductility factor (less than 1.0), and Table 3.1

Fes is a measure of the regularity of the building. There is no displacement check in the Level II design.

The regularity factor (Fes ) is calculated as: (3-6)

where Fe is a measure of the plan irregularity of the building, and Fs reflects the unifonnity of the distribution of lateral stiffness over the height ofthe building. For reference, Fe and Fs range in value between 1.0 (regular) and 1.5 (most irregular). The design penalties associated with selecting a highly irregular seismic framing system are clearly evident. The ductility factor (Ds ) varies as a function ofstructural material, type of framing system, and key response parameters. Materials are identified as either steel or reinforced concrete; steel-reinforced concrete is included under the heading of reinforced concrete. Table 3-1 displays values of Ds for steel seismic framing systems from the 1981 Building Standard Law (BSL). These values range between 0.25 and 0.50. The "behavior of members" rating in the first column is based on the proportioning of the structural members. For example, members in ductile moment frames with excellent ductility have smaller width-to-thickness (or depth-to-thickness) ratios than members in ductile moment frames with/air ductility or poor ductility. Stocky bracing members in braced frames are associated with excellent ductility and slender braces are associated with/air ductility.

Coefficient Ds for Steel Framed Buildings in Japan's 1981 Building Standard law Type of Frame (1) Ductile moment frame

(2) Concentrically braced frame

(3) Frames other than (1) and (2)

A. Members with excellent ductility

0.25

0.35

0.30

B. Members with good ductility

0.30

0.40

0.35

0.35

0.45

0.40

0.40

0.50

0.45

Behavior of Members

C. Members with fair ductility D. Members with poor ductility

12

3: Use of Response Modification Factors

ATC-19

,-----_._--

-

Table 3.2

Coefficient D s for Reinforced Concrete Frame Buildings in Japan's 1981 Building Standard Law Type of Frame

Behavior of Members

(1) Ductile moment frame

(2) Shear walls

(3) Frames other than (1) and (2)

A. Members with excellent ductility

0.30

0.40

0.35

B. Members with good ductility

0.35

0.45

0.40

C. Members with fair ductility

0.40

0.50

0.45

D. Members with poor ductility

0.45

0.55

0.50

For reinforced concrete construction, values for Ds vary between 0.3 and 0.55, as shown below in Table 3.2. For steel-reinforced concrete construction (termed composite construction in the United States), values for D s are reduced from those in the table by 0.05. For a reinforced concrete ductile moment frame to be assigned excellent ductility, columns have to be designed to be flexure-critical, have a longitudinal reinforcement ratio less than 0.8 percent, and have low axial «0.35fc·) and shear «O.lfc') stresses at the fonnation of the mechanism. The limiting shear stress in beams in an excellent ductility frame is 0.15fc' . Poor ductility would be assigned to a moment frame in which the axial and shear stress values in the columns are much higher than the limits noted above, and for frames incorporating shear-critical beams or columns. For a shear wall to posses excellent ductility, the wall has to be flexure-critical and have a low shear stress «O.lfc· ) at the formation of the mechanism. The reader is referred to the Tables Cl through C4 (reinforced concrete construction), and 01 through 04 (steel construction) in the 1981 Building Standard Law for more detailed information on frame and ductility classifications.

3.2.3

Mexico

The 1987 Mexico City Building Code uses a reduction factor to reduce elastic spectral demands to a strength design level. The response reduction factor (Q' ) is period-dependent and is calculated as follows:

ATC-19

for T< TA: (3-7)

Q' = Q

(3-8)

where T is the fundamental period of the building; TA is a characteristic period of the design spectrum (lower bound period to the constant acceleration portion of the spectrum); and Q is a system behavior factor that varies as a function of material type, building strength and stiffness regularity (Gomez and GarciaRanz, 1988). Values for Q range between 1.0 and 4.0. The Mexico City Building Code recognizes that ductility cannot be used to reduce elastic force demands substantially in the short-period range from oto TA by reducing Q' from Q at period TA , to 1.0 at a period of 0.0 second. Inelastic displacement values are estimated in the Mexico City Building Code as the product of the response reduction factor Q and the displacement values computed using the reduced seismic forces. When T is less than TA , the ratio Q/ Q' exceeds 1.0 and the inelastic displacement values exceed the elastic displacement values; when T is greater than TA , the ratio Q/Q' equals 1.0, and the inelastic displacement values equal the elastic displacement values.

3: Use of Response Modification Factors

13

Table 3.3

Response Modification Factor Comparison for Rock Sites

Structural System RC Structural Wall

RC Moment Frame

Steel Moment Frame

Europe a

Period 2.0

2.5

2.5

5.5

T= 1.0 sec.

3.5

2.5

4.0

5.5

T = 0.1 sec.

2.3

3.3

2.5

8.0

T = 1.0 sec.

5.0

3.3

4.0

8.0

T = 0.1 sec.

2.5

4.0

2.5

8.0

T = 1.0 sec.

6.0

4.0

4.0

8.0

~o

equal to 2.5.

Summary

The application of response modification factors (or their equivalents) to the seismic design of buildings in Europe (1988 Eurocode), Japan (1981 Building Standard Law), and Mexico (1987 Mexico City Building Code) has been reviewed. In order to draw broad conclusions about seismic design practice in the United States from this information, consider three ductile framing systems of regular configuration, all located on rock sites: (l) a reinforced concrete structural wall, (2) a reinforced concrete moment-resisting space frame, and (3) a steel moment-resisting space frame. Assume that analysis at fundamental periods of 0.1 second and 1.0 second is sufficient for the purpose of comparison. The response modification factors determined from each code are presented in Table 3.3 together with values of R from the 1991 NEHRP Provisions. Of the seismic codes being compared, only the 1981 BSL does not use a response modification factor to reduce elastic spectral demands to a strength (first significant yield) design level. Therefore, Table 3-3 lists values for the inverse of Ds . Table 3-3 shows that the response modification factors used in the NEHRP Provisions are substantially greater than the corresponding values in the Eur
14

United States

T= 0.1 sec.

a. T 1 equal to 0.2 second, TI equal to 1.0, b. Inverse of D s ' c. TA equal to 0.2 second. 3.2.4

Mexico c

Japan b

differences in the response modification factors suggest that a building designed according to the NEHRP Provisions will likely suffer more damage in the design earthquake than similar buildings designed for the requirements of either the Eurocode or the Mexico City Building Code.

It is interesting to note that the response modification factors in the European and the Mexican codes do not account for reserve strength; that is, the factors in these two codes are intended to be a measure of ductility alone. This is in contrast to the NEHRP Provisions wherein the values assigned to R are intended to account for reserve strength and ductility. A direct comparison of the values assigned to response modification factors in the NEHRP Provisions (R) and the Japanese BSL (1/ Ds) is difficult because the factors are used differently. In the NEHRP Provisions, R is used to reduce elastic forces to the strength (first significant yielding) level for design. In the BSL, the factor is used in the Level II procedure to reduce elastic forces for comparison with the maximum strength of a framing system (often designed using the Level I procedure). This maximum strength, computed using either nonlinear static analysis or plastic analysis, may exceed the design strength at first significant yielding by upwards of 100 percent (see Chapter 4 for additional information). Assuming that the maximum strength of most framing systems in the U.S. is two to three times the design strength, and similar elastic spectral demands, and recognizing that values of R exceed those of 1/Ds by a factor of between two and three,

3: Use of Response Modification Factors

ATC-19

---------------------_

__

.

the framing systems resulting from U.S. and Japanese practice will likely be similar.

report entitled Seismic Design Guidelines for Highway Bridges. These guidelines were developed by a team of nationally recognized bridge engineering experts. The format of ATC-6 paralleled that of ATC-3. In particular, it introduced R factors to reduce elastic spectral demands to a strength design level. The ATC-6 report recommended different values of R for framing elements and connections; the values for R being smaller for connections to promote plastic hinging in the framing elements and to preclude inelastic behavior in the connections. As such, the ATC-6 design methodology for bridges differed from the ATC-3 design methodology for buildings in which one value for R was used for the entire building.

Inelastic displacement values are calculated in the NEHRP Provisions as the product of the elastic displacement values computed using the reduced seismic forces and a displacement amplification factor that is numerically smaller than the response modification factor. The calculated inelastic displacement values are thus smaller than the elastic displacement values computed using the unreduced seismic forces. A different procedure is used by the European and Mexican codes wherein inelastic displacement values are calculated as the product of the displacement values computed using the reduced seismic forces and a displacement amplification factor equal to or larger than the response reduction factor. The resulting inelastic displacement values are equal to or greater than the elastic displacement values computed using the unreduced seismic forces. The European and Mexican procedures for computing inelastic displacements are more consistent with the results of recent research (Miranda and Bertero, 1994) than the procedure adopted in the NEHRP Provisions. The reader is referred to Report ATC-34 (ATC, 1995) for additional information on the calculation of inelastic displacements.

3.3

The Caltrans Bridge Design Specification (Caltrans, 1990) makes use of a period-dependent response reduction factor, which is termed an adjustment factor for ductility and risk assessment and denoted as Z. The Z factors are used to reduce elastic spectral demands to strength-design actions, so Z performs a similar role to R. Figure 3.1 presents Caltrans Z factors as a function of period and structure/component type. The reduction in values of Z with increasing period is based in part on the increase in spectral displacements with increasing period. For slender columns, large displacements may result in significant second-order (or P-b.) effects.

Use 01 R Factor Equivalents for Bridge Design

In 1982, ATC published the ATC-6 (ATC, 1982a) 8

"~

..... N 6 .....

... ..

......

0

U

ell

II.

c

4

r...

~

~t'< ...............

.. Gl

"'r -

E III

:::l

:c c(

I - Well-confined

2

r o

o

1.0

2.0

Well-confined ductile single column bents Piers, abutment walls, and wingwalls

I - Hinge

\

ductile multi-column bents

restrainer cables (Z=1.0)

Well-reinforced concrete shear keys (Z=O.8) 3.0

Period (seconds) Figure 3.1: Caltrans Z factors.

ATC-19

3: Use of Response Modification Factors

15

,-----_.-_..-_

.

The Z factors for single- and multi-column bents are constant for periods less than 0.6 second, and decrease linearly between periods of 0.6 and 3.0 seconds. The period-dependent trends for Z in the shortperiod range are not supported by analytical studies (Miranda and Bertero, 1994). In particular, although Z tends to decrease with increasing period, strength reduction due to inelastic behavior is minimal for very stiff structures, and tends to increase with increasing period. Table 3.4 shows the values of Caltrans Z factors and ATC-6 R factors (for a period of 0.3 second) for bridges founded on rock. Caltrans defines the seismic hazard at a bridge site in terms of the maximum credible earthquake whereas AASHTO defines the seismic hazard using probabilistic techniques based on a 1O-percent probability of being exceeded in 50 years. This difference in the definition of the design earthquake is responsible for the larger response reduction factors used in the Caltrans procedure, because use of

Table 3.4

either the Caltrans or the AASHTO procedure is intended to produce columns ofa similar size (Ian Buckle, personal communication). Project ATC-32 is currently reviewing Caltrans' seismic design procedures for bridges. Improved Z factors are being developed that depend on (a) bridge importance, (b) structure-to-site period ratio, and (c) element type (column, pier, or connection). Values for Z factors for ordinary and important bridge structures are reported in Table 3.4. These improved Z factors are intended to be used with elastic analysis results that consider the stiffness degradation that will occur during a major seismic event and flexural capacities that consider probable rather than nominal material strengths. The net result of the proposed ATC-32 design procedures for ductile components is that most design quantities will often be only nominally different than those for current Caltrans designs. The ATC-32 recommendations have not been formally adopted to date by Caltrans.

Bridge Response Modification Factors ATC-3?

Frame type

Caltrans Z

ATC-6 R

6

3

Single-column bent

Ordinary Bridges

Multiple-column bent

Important Bridges :5>3 ::;3

a. Proposed, not yet adopted

16

3: Use of Response Modification Factors

ATC-19

------,--------- ------_._.---_

.

4. 4.1

Components of Response Modification Factors

Introduction

4.2

The commentary to the 1988 NEHRP Provisions (BSSC, 1988) defines the R factor as " ... an empirical response modification (reduction) factor intended to account for both damping and ductility inherent in a structural system at displacements great enough to approach the maxim um displacement ofthe system." This definition provides some insight into the developers' qualitative understanding of the seismic response of buildings and the expected behavior of a code-compliant building in the design earthquake. The components of R can be dermed in several ways, each dependent on the performance level under consideration. In this report, only the life-safety performance level is considered explicitly. Section 4.2 provides a framework for a discussion on the disaggregation of R into its primary components by discussing how R is used to link elastic and inelastic response. Section 4.3 introduces some key issues associated with describing the force-displacement response of a building (expanded on by example in Appendix A). Finally, sections 4.4 and 4.5 address the disaggregation of R into its key components.

Impact of the R Factor on Design

The key parameters influencing the response of an elastic single-degree-of-freedom (SDOF) system are indicated in Figure 4.1, which illustrates a singlestory moment frame with massless columns. The floor mass m is attached to the ground by two elastic columns (springs) of combined lateral stiffness k. Damping c is introduced by a dashpot linking the floor and the ground. The SDOF in this model is the horizontal translation of the floor with respect to the ground. The inertial force developed by the floor mass during earthquake shaking is a function of the properties of the SDOF system (m, k, and c) and the characteristics of the earthquake ground motion. For an elastic SDOF system, seismic actions and displacements can be determined using an earthquake response spectrum - the envelope of the maximum responses of SDOF oscillators to one earthquake ground motion. Response spectra vary widely in frequency content and amplitude. For reference, the pseudo-acceleration spectra (Clough and Penzien, 1993) corresponding to the £1 Centro, SCT, Sylmar, and JMA earthquake ground motions are presented in Figure 4.2. These earthquake histories were recorded during the 1940 Imperial Valley, 1985 Mexico City, 1994 Northridge, and 1995 Hyogoken-Nanbu earthquakes, respectively.

Floor mass, m

Column stiffness, kJ2

Dashpot. c

Column stiffness,

k/2

Figure 4.1: Single degree of freedom system.

ATC-19

4: Components of Response Modification Factors

17

3r------.---...,.------,.----.---,....-----.----r-----, \

EI Centro 1940 Sylmar 1994

't \ ,.\\ ,I.

2.5

' , \'J\'

:§ c .Q

2

8

1.5

~ Q) a;

lI)

a..

:

:

t

. :

:

:: •

:

I

J

I

o

~

.

... i}' t

co

1

------ \orMA 1995

.:

f

-0

.

' \: ' .

I . I ,J

~:~;.:. ~ f" \:

• • • • • • SCT 1985

~"""""""':""""""" : :

".~ \ \

\: • \: \ \.. ., ..\

I

....: ..... :;.~._

.

, ,.

'~:

.....

" ..... .', ~........ -..........

.

:

..

.. .... : '0_;-. ~ ...... _ I··:··:·~·..~.~..L..-_~i__:=======~==.=-:::.-:.~.'~-~.~.~-;-~.§-~.~OL . -. 1 1.5 2 2.5 0.5 4 o 3.5 3

..

Period (seconds) Figure 4.2: Sample elastic pseudo-acceleration spectra for 5% damping.

The impact of R on the seismic design of buildings is clearly seen by comparing the equations for the design base shear for inelastic response (Equation 24) and the base shear for elastic response (Ve): (4-1) where Se,5 is the elastic 5-percent damped pseudoacceleration response spectral ordinate calculated at the fundamental period of the building; and W is the reactive weight, equal to Mg for the simple structure depicted in Figure 4.1. Note that W in Equation 4-1 is the total reactive weight and not the reactive weight in the fundamental mode. For seismic design in the United States, the spectrum has in the past generally corresponded to an earthquake ground motion with a 10-percent probability of being exceeded in 50 years 1, which is often tenned the design earthquake. The elastic spectral ordinate in Equation 4-1 is equivalent to the term 2.5A a in Equation 2-4. Equations 21. It is anticipated that the 1997 NEHRP Recommended Provisions for the Seismic Regulation ofNew Buildings, under development by the Building Seismic Safety Council (BSSC), and the NEHRP Guidelinesfor Seismic Rehabilitation of Buildings, under development by ATC (ATC-33 project) for BSSC, will incorporate seismic hazard maps that reflect longer recurrence intervals (i.e., ground motions have a 2% probability of being exceeded in 50 years).

18

4.3

Force-Displacement Response of Buildings

A typical force-displacement relationship for a building frame is shown in Figure 4.3. This relationship describes the response of the building frame subjected to monotonically increasing displacements. For the purposes of design, this nonlinear relationship is often approximated by an idealized bilinear relationship. Two bilinear approximations are widely used and these methods are described below. Either of these methods can be used to estimate yield forces and yield displacements; the two methods will generally produce similar results for most ductile framing systems. The first approximation, developed for characterizing the load-displacement relation for reinforced concrete elements (pauley and Priestley, 1992), assumes

4: Components of Response Modification Factors

_--------_._--

.

4 and 4-1 are thus identical if the response modification factor R in Equation 2-4 is equal to 1.0. In practice, the design base shear (for inelastic response) is calculated by dividing the base shear for elastic response by the response modification factor R, a value that generally varies between 4.0 and 8.0. The substantial difference between the ordinates of elastic and design base shear spectra is clearly seen in Figure 2-1.

• ATC-19

._---_.----------------.---_._.-

-

_

.

Ductile response

->

' y - Brittle response •

f-, , 1---''

r

~ ""'--

v

Roof displacement (d)

Figure 4.3: Sample base shear force versus roof displacement relationship. Base shear force (V)

Base shear force (V)

Actual response

Vy

Vy

.75Vy

ti.y

6m

6m , 0 Isplacement (d)

Displacement (~)

(a) Paulay & Priestley

(b) Equal Energy

Figure 4.4: Bilinear approximations to a force-displacement relationship.

a priori knowledge of the yield strength (Vy) of the frame. The elastic stiffness is based on the secant stiffness of the frame calculated from the force-displacement curve at the force corresponding to O.75Vy • The determination of the elastic stiffness (K) is shown in Figure 4.4a. The second method used to approximate the forcedisplacement relation of a frame is commonly termed the equal-energy method. This method assumes that the area enclosed by the curve above the bilinear approximation is equal to the area enclosed by the curve below the bilinear approximation. This bilinear

ATC.19

approximation is illustrated in Figure 4.4b. The nonlinear relationships presented in Figure 4.4 are described by the yield force (Vy ), yield displacement (~y), maximum force (Va), displacement corresponding to a limit state (~m)' and the displacement immediately prior to failure (~u), Displacements ~m and d u are well beyond the yield displacement for ductile framing systems. The elastic stiffness (Ko) is calculated by dividing the yield force by the yield displacement. The post-yield stiffness (K 1) is commonly defmed as a fraction (a) ofthe elastic stiffness as follows:

4: Components of Response Modification Factors

19

,--------.,-----------'-_ _...

...., ..... .,._ ..

(4-2)

where all tenns in this equation are defined above. The ability of a building frame to be displaced beyond the elastic limit, while resisting significant . force and absorbing energy by inelastic behavior, is tenned ductility. Displacement ductility is defmed as the difference between t:.m and t:.y- The maximum displacement ductility is the difference between t:. u and t:.y . The displacement ductility ratio is generally defined as the ratio of t:.m to t:.Y' namely: ~~

=

t:. m t:. y

(4-3)

where t:.m is always greater than t:.y- Brittle failures are characterized by negligible ductility. Brittle failures of this type are common in older construction built before the advent of ductile detailing in the mid1960s. The force-displacement relationship for a building can be determined either experimentally or analytically. Experimental evaluation is difficult, extremely costly, and therefore rare. Pseudo-dynamic testing of full-scale buildings, and earthquake simulator testing of small- to moderate-scale models of buildings, have provided force-displacement relations for buildings of different scales. The use of earthquake simulator test data to evaluate the force-displacement response of a building is described in Section 4.4 below. Nonlinear finite element analysis software is a costeffective analytical tool used by academicians and design professionals to estimate force-displacement relationships for buildings. Nonlinear static (or pushover) analysis, somewhat routine in the larger architectural/engineering firms in Japan since the early 1980s, is now being promoted in the ATC-33 project as the preferred analysis method for seismic rehabilitation projects. For reference, a description of nonlinear static analysis, together with a sample analysis of a seven-story reinforced concrete building, is presented in Appendix A.

20

4.4

Experimental EValuation of ForceDisplacement Relationships

In the mid-1980s, data from experimental research at the University of California at Berkeley were used to develop base shear-roof displacement relationships for steel braced frames and a draft fonnulation for the response modification factor. The base shear-roof displacement relationships were established using data acquired from the testing of two code-compliant braced steel frames; one concentrically braced (Uang and Bertero, 1986) and one eccentrically braced (Whittaker et aI., 1987). The force-displacement curves were developed by plotting the roof displacement at the time corresponding to the maximum base shear force for each earthquake simulation and each model. Using these data, the Berkeley researchers proposed splitting R into three factors that account for contributions from reserve strength, ductility, and viscous damping, as follows:

(4-4) In this expression Rs is the strength factor, RIJ. is the ductility factor, and l?f. is the damping factor. Using data from the most severe earthquake simulation test, the strength factor was calculated as the maximum base shear force divided by the design base shear force at the strength level. The ductility factor was calculated as the base shear for elastic response divided by the maximum base shear force and the damping factor was set equal to 1.0. The experimentally determined values of R for the concentrically braced frame was 4.5 and that for the eccentrically braced frame was 6.0. These values were significantly less than the values of 6.0 and 8.0 adopted in the 1991 NEHRP Provisions. The experimentally determined values for the strength, ductility, and response modification factors are listed in Table 4.1. The method used to calculate values for Table 4.1

Experimental Reduction Factors for Steel Frames

System

R

Concentricallybraced

2.43

1.85

4.5

Eccentrically-braced

2.85

2.12

6.0

4: Components of Response Modification Factors

ATC-19

Elastic Spectrum (;

--en

=5%)

--tt--------

Required elastic strength = R\!Rs V d

----Lj--fl----

Maximum strength

en

ca

1---7'--,.'+

(RsVd)

0.

Design strength( V d ) ~-- Period range of

model

Period (seconds) Figure 4.5: Experimental evaluation of strength and ductility factors.

the strength and ductility factors is depicted in Figure 4.5. The reader is referred to Uang and Bertero (1986) and Whittaker et al. (1987) for additional infonnation.

4.5

Key Components of R

Much research (ATC, 1982b; Freeman, 1990; ATC, 1995) has been completed since the first fonnulation for R (Equation 4-4) was proposed. Recent studies, including those in the companion Project ATC-34, support a new fonnulation for R; that is, a fonnulation in which R is expressed as the product of three factors: (4-5) where Rs is the period-dependent strength factor, Ril is the period-dependent ductility factor, and RR is the redundancy factor. This formulation, with the exception of the redundancy factor, is similar to those proposed by the Berkeley researchers (see Section 4.4) and Freeman (1990). The Freeman formulation, which was developed independently of the Berkeley fonnulation, described the response reduction factor as the product of a strength-type factor and a ductility-type factor. The redundancy factor, developed as part of Project ATC-34, is proposed in this report for the flTSt time

ATC-19

in the literature. The function ofthis factor is to quantify the improved reliability of seismic framing systems that use multiple lines of vertical seismic framing in each principal direction of a building. A fourth factor, the viscous damping factor (R~), was considered for inclusion in the new fonnulation primarily to account for response reduction provided by supplemental viscous damping devices. Such a viscous damping factor could be used to reduce displacements in a nonlinear framing system, but cannot be used to proportionally reduce force demands, especially for highly-damped frames. Recognizing that seismic design using response modification factors will remain force-based in the near tenn, the damping factor was excluded from the new formulation. One objective of this report is to provide the reader with information regarding the key components (or factors) that influence the numerical values assigned to R in the United States. The fonnulation of R in Equation 4-5 was put forth to provide a framework for the rational evaluation of these parameters. Any evaluation of the key components of R must address the fact that the components are not independent of one another. The background infonnation and research data presented in the following subsections are intended to provide the reader with insight into the four key components (i.e., reserve strength, duc-

4: Components of Response Modification Factors

21

---------r------------.---"----------

tHity, damping, and redundancy) as well as the relationships between these four components. No relative importance should be inferred from the order in which the material is presented. The proposed formulation does not specifically address the effects of plan and vertical irregularity in framing systems. Irregularity could be addressed by reducing the response modification factor by a regularity factor similar to that prescribed for the Level II seismic design procedure in the Japanese 1981 Building Standard Law (see Section 3.2.2 for details). Significant force-based penalties (higher design base shears) for the design of irregular framing systems would both discourage the use of irregular framing and reduce the uncertainties associated with the nonlinear response of irregularly framed buildings. For additional information, the reader is referred to the ATC-34 document. 4.5.1

professionals. For example, code-mandated limits on interstory drift may require the use of member sizes in flexible (long-period) framing systems that are greater than those required for strength alone - giving rise to period-dependent strength factors for driftlimited framing systems. Also, buildings located in lower seismic zones will likely have different reserve strength values than those in higher seismic zones because the ratio of gravity loads to seismic loads will differ - resulting in zone-dependent values for the strength factor. Differences in regional construction practices and differences between actual and nominal material strength will also affect the value of the strength factor, but in less predictable ways. Osteraas and Krawinkler (1990) made some qualitative observations regarding the likely reserve strength of buildings as follows. "... Small, low-rise (buildings) with nonstructural partitions and architectural elements whose design is controlled by loading conditions other than seismic will have high (reserve strength) '" The effect of nonstructural partitions ... will decrease with increasing height, as the 'scale' of the nonstructural elements becomes small compared to that of the structural elements and as the seismic loading condition (controls the member proportions)..."

Strength Factor

The maximum lateral strength of a building will generally exceed its design strength. Merovich (unpublished) notes that: ..... In general, members are designed with capacities equal to, or in excess of their design loads. While the degree to which their capacities exceed the design requirements is a measure of the design efficiency, all properly executed designs contain some degree of overstrength or excess capacity as a consequence of the design procedure. In some instances, . geometry or other detail code provisions will dictate larger member sizes and hence greater capacities than those solely based upon conformity to stress/ strength provisions. In other instances, design provisions related to displacement parameters will produce larger member sizes than those dictated by stress/strength provisions. For members that are sized to resist significant gravity loads, a substantial percentage of the overall capacity may be available since actual loads are probably at levels far below the design value at the time of the earthquake..." The strength factor will likely depend on many parameters not immediately obvious to many design

22

A method for evaluating the reserve strength of a building follows. Sample values of Rs calculated by different researchers are also included. Evaluation of Strength Factors

Nonlinear static analysis (also termed pushover analysis) can be used to estimate the strength of a building or framing system (ATC, 1982b; Bertero, 1986; Freeman, 1990; Hwang and Shinozuka, 1994; Uang and Bertero, 1986; Whittaker et aI., 1990). The procedure used to estimate the strength of a building is straightforward, but requires the analyst to select a limiting state of response. Typical limiting responses include maximum interstory drift and maximum plastic hinge rotation. The steps in the procedure are as follows: 1.

Using nonlinear static analysis, construct the base shear-roof displacement relationship for the building.

4: Components of Response Modification Factors

ATC-19

2. At the roof displacement corresponding to the limiting state of response, calculate the base shear force Vo in the building. The reserve strength of the building is equal to the difference between the design base shear (Vd ) and Vo'

3. Calculate the strength factor using the foHowing expression: R S

= VVdo

(4-6)

This method of evaluating the strength factor was used to create the estimates of strength factors given below. Appendix A demonstrates the use of non linear static analysis to construct the base shear-roof displacement relation for a building and evaluate the strength factor for that building.

Estimates of Strength Factors The reserve strength in common seismic framing systems has been studied by a number of researchers using nonlinear static analysis. The results of some of these studies are summarized below. Freeman (1990) reported strength factors for three three-story steel moment frames, two constructed in seismic zone 4 and one in seismic zone 3. The strength factors, after modification to reflect strength design, were reported as 1.9,3.6, and 3.3, respectively. Earlier studies by Freeman (ATC, 1982b) estimated strength factors, after modification to reflect strength design, of approximately 2.8 and 4.8, for four-story and seven-story reinforced concrete moment frames, respectively. Osteraas and Krawinkler (1990) conducted a detailed study of reserve strength and ductility in three structural systems: distributed moment frames, perimeter moment frames, and concentric braced frames. The framing systems were designed assuming (a) seismic loads per UBC seismic zone 4 and soil type $2, (b) dead loads of 70 psf, (c) Jive loads of30 psf, (d) a 3bay by 5-bay building plan using 24 square foot bays, and (e) an elastic period computed using a simplified relation related to building height. Osteraas and Krawinkler reported strength factors ranging from 1.8 to 6.5 for the three framing systems. For distributed moment frames, the strength factor ranged between 6.5 in the short-period range to 2.1 at a period of 4.0 seconds. For perimeter moment frames, the strength factor ranged between 3.5 in the short-

ATC-19

period range to 1.8 at a period of 4.0 seconds, and for concentric braced frames, the strength factor ranged between 2.8 at 0.1 second to 2.2 at 0.9 second. Uang and Maarouf(l993) analyzed two buildings shaken by the 1989 Lorna Prieta earthquake: a 13story steel frame building and a six-story reinforced concrete perimeter moment frame building. The strength factors reported for these two buildings, after modification to reflect strength design, were 4.0 and 1.9, respectively. Hwang and Shinozuka (1994) studied a four-story, reinforced concrete, intennediate moment frame building located in UBC seismic zone 2. The design base shear for this building was 0.09W. The maximum lateral resistance of the building was calculated to be 0.26W, resulting in a strength factor of2.2 ifno limits are placed on the damage to the framing system. (lfthe perfonnance level selected for the design earthquake were no damage to the structural frame, the strength factor would have been approximately 1.6). The scatter in the reported values for the strength factor is significant - and too large to be of much use to the design professional community. It is clear that coordinated and systematic studies are needed to develop strength factors of sufficient reliability to be included in seismic design codes. These studies ought be conducted at the national level to effectively address the issues identified earlier in this section. 4.5.2

Ductility Factor

The seismic response parameters of displacement capacity, ductility, and ductility ratio are closely inter-related, but often confused. For example, a frame with a large displacement capacity might have smaH ductility and a small ductility ratio, and a frame with a small displacement capacity might have small ductility but a large ductility ratio. Consider the force-displacement relationships for two one-story building frames shown in Figure 4.6. The nonnalized force-displacement relationships are idealized as elastic-plastic, the yield drift ratios are assumed to be 0.2 percent (Frame A) and 1.0 percent (Frame B), and the maximum interstory drift ratios are assumed to be 1.2 percent (Frame A) and 3.0 percent (Frame B). The key seismic response parameters are listed in Table 4.2 below. The values of the response parameters are constrained by the interstory drift limit of 1.5 percent, which is consistent with the

4: Components of Response Modification Factors

23

r - Code maximum drift

Ductility of A

,.

of 1.5%

Displacement capacity of A

CI)

Ductili

f:! o LL

----

of B

...\

-------------~---

A

\.8

1.0%

2.0%

+

3.0%

Drift (% of story height)

Figure 4.6: Definition of terms for two example one-story frames. Table 4.2

Seismic Response Parameters for Two Example One-Story Frames

Response Parameter

Frame A

Frame B

Yield drift

0.2%

1.0%

Drift Capacity

1.2%

1.5%

Displacement Ductility

1.0%

0.5%

Displacement Ductility Ratio

6

1.5

drift limits set forth in the UBC. The drift values in Table 4.2 are expressed as a percentage of the story height. These data illustrate the importance of defining response parameters with respect to specific limit states. By limiting drifts to 1.5 percent, the stiffer frame (Frame A) is more ductile and has a higher ductility ratio than the more flexible frame (Frame B). However, if the drift limit state is removed, the more flexible frame has substantially more ductility (equal to 2 percent) than the stiffer frame. The ductility ratios (J.1.) can be computed at the system, story, and element levels. At the system and story levels, the ductility ratio is normally expressed in terms of the displacement ductility ratio. At the element level, ductility ratio can be expressed in terms of strain ductility ratio, curvature ductility

24

ratio, and rotation ductility ratio. For the purposes of this discussion, displacement ductility ratio at the system level is used to determine the ductility factor. The calculation of displacement ductility ratio for a building is demonstrated by example in Appendix A. It must be recognized that the ductility factor is a measure of the nonlinear response of the complete framing system and not components of the framing system, regardless of which ductility parameter is used. Assuming that reliable estimates ofdisplacement ductility are available, the next step in estimating the ductility factor is to derive a relationship between displacement ductility and the ductility factor. This step has been the subject of much research in recent years. The relationships developed by Newmark and Hall (1982), Krawinkler and Nassar (1992), and Miranda and Bertero (1994) are paraphrased below as background information for the reader. Although broad consensus has not yet been reached on the use of one of the suites of relationships outlined below, the latter two sets of relationships best fit the available data. Newmark and Hall Research

Newmark and Hall (1982) provided relationships that can be used to estimate the ductility factor (R~ for elasto-plastic SDOF systems as follows:

4: Components of Response Modification Factors

ATC-19

._-------------_._ ..•

..

8.---.---.---,..----,---,---,..-----r--...., ;.

... 6 o

U

: /: . . : J.1=4 : . j ' ';";< - ' - ' ; -'-': - ' - ' ; -'_'; _'_'; _._._

~4 :J

a 2

_._

/.

~

13

. . _ _;_._.: . :J.1=6. ;:-._.-: _._:_._ :. ..

i-'-:' "" 1...... :..

:J.1=2

.-_/

tY' OL..--.....L-.--"""----L-_--J...._ _~_ _.l..--_--..I._ _--' o 0.5 1 1.5 2 2.5 3 3.5 4 Period (seconds)

Figure 4.7: Newmark and Hall R!J. - Jl - T relationship.

R~ = [c(Jl- 1)

for frequencies above 33 Hz (periods below 0.03 second):

+ 1] lie

(4-10)

where:

RlJ. = 1.0

(4-7)

for frequencies between 2 Hz and 8 Hz (periods between 0.12 second and 0.5 second):

Ta b c(T, a) = - - + - . a 1+T T

The regression parameters a and b were obtained for different strain-hardening ratios (tenned a in Figure 4.9) as follows:

for frequencies less than 1 Hz (periods exceeding 1.0 second):

a

=0%:

a

=2%: =10%:

a (4-9) Figure 4.7 illustrates the Newmark and Hall relationships for displacement ductility ratios of2, 4 and 6. Estimates for R between 0.03 second and 0.12 second, and 0.5 se60nd and 1.0 second can be interpolated between the limiting values given by Equations 4-7,4-8, and 4-9.

Krawinkler and Nassar Research Krawinkler and Nassar (1992) developed a RI1 - J.1- T relationship for SDOF systems on ~o~k or stiff soil sites. They used the results of a statistical study based on 15 western United States ground motion records from earthquakes ranging in magnitude from 5.7 to 7.7. Developed assuming damping equal to 5 percent of critical, the Krawinkler and Nassar equation is:

ATC-19

(4-11)

a

= 1.00

b = 0.42

a

= 1.00

b = 0.37

a =1.00

b = 0.29

Note that a equal to 0% corresponds to an elastic· plastic system. The relationships between R!J. and T for displacement ductility ratios of 2, 4, and 6 are presented in Figure 4.8 for a strain-hardening ratio of 10 percent. Krawinkler and Nassar also studied the implications of extending their R!J. - J.1- T relationships to multiple-degree-of freedom (MDOF) systems. Three model types were analyzed for target ductility ratios between 2 and 8: strong column-weak beam, weak column-strong beam, and weak first story. Given that the latter two failure modes are discouraged by the model building codes and the NEHRP Provisions. only the strong column-weak beam results are reported here. The objective of this study was to develop a procedure whereby the maximum story displacement ductility ratio in a MDOF system could be limited to the corresponding ductility ratio in the

4: Components of Response Modification Factors

25

8 .--------r---,-.-----.---.- - T .- - - - r - - - r - - - , . . - - - - - - ,

.( / -':-'-' ~._.- .:-. _.~ '~-:'6'[-' -. ~. -.... 6

,g ~ u. ~

/.:

': I:

4 ../"; "":

g

o

:

_'

-:

":

:

:

:

'_ _- ';-

:

:

,

:

:

: ~ =4- :.:-

:

.

- :

_

. -

- :- -

:

II

.,

-

:" ,

':

.

:~=2

2-V

O ' - - - - " - -..........- -.......- -.........--L.-----'----I..---..I

o

1

0.5

1.5 2 2.5 Period (seconds)

3.5

3

4

Figure 4.8: Krawinkler and Nassar R~ - ~ - T relationship.

...

m

~ (J)

2 r-----..,....------.------r----~--------,

Q) (J)

co

..0

u.. oo

fg

co

1.5 ............ :

~ . ."" +---: . .. .

:

--

.......

'.

1

Q)

~ (J)

Q)

~

. .

_' 0

lIE

. .0.5 ..........................

..0

lIE

ct-

+-

-0

-

•-

CJ

0

u..

oo

0 '--

::E

0

---'

0.5

•+

~=4;a= ~=4; a= ~ = 8; a ~ = 8; a =

=

........

.........

..I-.

1

1.5

2

0% 10% 0% 10% ----l

2.5

Period (seconds) Figure 4.9: MDOF modification factors.

SDOF system. Krawinkler and Nassar concluded that the strength demands for SDOF systems must generally be increased to be applicable for MDOF frame structures. The modification factor, defined as the required base shear strength of the MDOF system divided by the inelastic strength demand of the corresponding first-mode SDOF system, limits the story ductility ratio in the MDOF system to the target ductility ratio. Modification factors for target ductility ratios of four and eight, and strain hardening ratios of o percent and 10 percent are presented in Figure 4.9. The reader is referred to Nassar and Krawinkler (1991) for additional information.

system is approximately equal to the corresponding SDOF system strength demand, suggesting that higher-mode effects need not be considered in this period range. For buildings with fundamental periods exceeding 0.75 second, higher-mode effects will necessitate an increase in the design lateral strength if target ductility ratios are to be satisfied. In general, the modification factor increases with increasing target ductility ratio and decreases with increasing strain hardening. MDOF systems without strain hardening drift more than the corresponding SDOF system, and increased lateral strength is required to limit the story ductility ratio to the target value.

For buildings with fundamental periods less than 0.75 second, the base shear demand on the MDOF

26

4: Components of Response Modification Factors

ATC-19

8.--------r---,.---...,.----,.--.....,--.------,--~

~6 o

U

rn

u.

~4

;:: (.J

::s

o

..-

2

.

: Il = 2 :

. - .:---- .::-. -

- '- -

Nassar & Krawinkler - - - Miranda - Rock .- .- .Miranda - Alluvium oL-_---1..._ _"""--_--J._ _.....L..-_----''--_--'-_ _''---_---'

o

0.5

1

1.5

2

2.5

3

3.5

4

Period (seconds) Figure 4.10: Ductility factor comparison. Miranda and Bertero Research

Miranda and Bertero (1994) summarized and reworked the RI,J. - Jl. - T relationships developed by a number of researchers including Newmark and Hall (1982), Riddell and Newmark (1979), and Krawinkler and Nassar (1992), in addition to developing general RI,J. - Jl. - T equations for rock, alluviu~, and soft soil sites. The Miranda and Bertero equatIons presented below were developed using 124 ground motions recorded on a wide range of soil conditions, and assumed five percent of critical damping. Their equation for the ductility factor is

Jl. - 1

(4-12)

RIl =--+1 <1>

where: for rock sites:



= 1+

1 _ ~e-1.5(1n(n-o.6f lOT - Jl.T 2T

for alluvium sites: = 1 + 1 _

12T - Jl.T

2:..

5T

2

e -2(ln(n- o.2 )

.for soft soil sites:

Tg

3Tg -3(1n(TIT,)-O.25)2

<1>= 1 +3T-4Te

and T is the predominant period of the ground motio~ (see Miranda and Bertero (1994) for details). A comparison of the Nassar and Krawinkler and Miranda and Bertero R Il - Jl. - T relationships for rock and alluvium sites is presented in Figure 4.10. Since

ATC-19

the differences between these relationships are relatively small, they can be ignored for engineering purposes. 4.5.3

Redundancy Factor

A redundant seismic framing system should be composed of multiple vertical lines of framing, each designed and detailed to transfer seismic-induced inertial forces to the foundation. Although redundancy is encouraged for lateral force-resisting systems designed in the United States, the trend in California in recent years has been to construct seismic framing systems composed of only a small number of vertical lines of seismic framing; that is, framing systems with minimal redundancy. This trend in California is likely a result of poor understanding by earthquake engineers of the important role played by redundancy in the response ofseismic framing systems to severe earthquake shaking. Few studies have examined the effect of redundant seismic framing in a quantitative way. However such studies have been carried out for wind framing systems by Moses (1974). In this study, it was noted that safety margins for wind framing system collapse modes depend on the sum of several strength and load variables. Therefore, the reliability of the framing system will be higher than the reliability of individual members of the framing system. Moses concluded that a partial safety factor less than or equal to one was appropriate for a redundant system, and that the required mean member strength could be reduced in such a system below that necessary for nonredundant or determinant systems. A mean strength reduction factor inversely proportional to the

4: Components of Response Modification Factors

·------l

27

._.._---_.....\

denotes a moment connection (and plastic ~J - ,__ hinge)

r

denotes an idealized pin

Mp = 100 k' Seismic framing

Seismic framing

Frame A

Frame B

Figure 4.11: Redundancy in moment frame systems.

square root of the number of independent strength terms (plastic hinges in the sway mechanism) in the redundant wind framing system was proposed. As illustrated below, similar logic can likely be applied to seismic framing systems. Consider the two framing systems with identical geometry depicted in Figure 4.11.Frame A is composed of one bay of seismic framing with each beam member capable of developing a nominal plastic moment of 200 units. Frame B is composed of two bays of seismic framing with each beam member capable of developing a nominal plastic moment of 100 units. Both limit analysis and nonlinear static analysis would assign both frames the same maxi-. mum lateral strength. However, using a methodology

similar to that proposed by Moses (1974) for wind framing systems, the ratio of the nominal moment strength (Mp ) of the beams in Frame A (eight total plastic hinges) and Frame B (sixteen total plastic hinges) should be:

MA

In

-!...B = ~=1.4 Mp

(4-13)

11./16

To achieve a similar level of reliability, the design lateral strength of Frame A should be 40 percent higher than that of Frame B. As another example, consider the two framing systems depicted in Figure 4.12. Frame C is composed

denotes an

I~ Id.a1~""

Frame C

plo

Frame D

Figure 4.12: Redundancy in shear wall systems.

28

4: Components of Response Modification Factors

r--.-------,--",----

ATC-19

of three bays of fram ing including one flexural wall capable of developing a nominal plastic moment of 1000 units. Frame D is composed of three bays of framing including two flexural walls, each capable of developing a nominal plastic moment of 500 units. Limit analysis would assign both framing systems the same lateral strength. Using the methodology of Moses (1974), the ratio of the design lateral strength of the shear walls in Frame C(one plastic hinge - at the base of the wall) and Frame D (two plastic hinges - one hinge at the base of each wall) should be 1.4 to achieve a similar level of reliability. Four lines of strength- and defonnation-compatible vertical seismic framing in each principal direction of a building have been recommended as the minimum necessary for adequate redundancy (Bertero, 1986; Whittaker et aI., 1990). It could be possible to penalize less redundant designs by requiring that higher design forces be used for such framing systems. For example, if it is assumed that four lines of strengthand deformation-compatible vertical seismic framing should form the basis of the response reduction factors in the UBC and NEHRP Provisions, redundancy could be explicitly accounted for by modifying the R factor in a manner similar to that suggested in Table 4.3. The values shown in Table 4.3 are proposed to demonstrate a likely trend, stimulate discussion among design professionals and researchers, and to promote research and study. These draft values for the redundancy factor have no technical basis and are not intended for implementation in seismic codes or guidelines. Table 4.3

Draft Redundancy factorsa

Lines of Vertical Seismic Framing

Draft Redundancy Factor

2

0.71

3

0.86

4

1.00

a. Values not intended for use in design or

codes The use of strength- and defonnation-compatib1e framing was emphasized in the previous paragraph. To illustrate the importance of setting limits on the

ATC-19

relative strength and stiffness of the lines of vertical seismic framing in a redundant framing system, consider the base shear force-roof displacement response of the reinforced concrete shear wall-steel moment frame dual system shown in Figure 4.13. This dual system was chosen because many design professionals consider the dual system to be a redundant seismic framing system. The 1991 NEHRP Provisions state that ..... the moment frame (in the dual system) shall be capable of resisting at least 25 percent of the design forces. The total shear force resistance is to be provided by the combination of the moment frame and the shear walls or braced frames in proportion to their rigidities." For the purpose of this discussion, the shear walls are assumed to be ten times stiffer than the moment frames. The design base shear values for the walls and moment frames are therefore 91 percent and 25 percent of the system design base shear (V in Figure 4.13), respectively. Assuming that the shear walls yield at a roof-drift ratio' (calculated by dividing the roof displacement by the building height) of 0.2 percent, and fail at a roof-drift ratio of 1.0 percent, it is evident from Figure 4.13 that the moment frames (also termed the back-up frames) neither contribute substantially to the force-displacement response of the building nor dissipate significant energy at a roof displacement corresponding to the displacement capacity of the shear walls. For the moment fr:unes to contricute sigr.iticantly to the response of a dual system, their stiffness and strength should be similar to that of the shear walls. The need to use elements of similar strength and stiffness applies to all lines of vertical seismic framing in a building. It is not sufficient to provide multiple lines of vertical seismic framing in a building the multiple lines of framing must be strength- and deformation-compatible to be capable of good response in a design earthquake. Seismic frames not meeting these conditions should probably not be considered redundant systems.

4.5.4

Damping Factor

Damping is the general tenn often used to characterize energy dissipation in a building frame, irrespective of whether the energy is dissipated by hysteretic behavior or by viscous damping. Damping accomplished by hysteretic behavior in a building responding in the elastic range is generally

29

4: Components of Response Modification Factors

------,----_----:---_

__

.."_. .".__.. .. ",,

"

.•.....

r

Base Shear

Dual system

,,---------f-\ #

0.91V

- Dual system "fails· at defonnation capacity of shear wall

0.5V (

Back-up moment frame

1.0% Roof Drift Index (%)

Figure 4.13: Force-displacement relationships for a dual system.

termed equivalent viscous damping and is assigned a value equal to five percent of critical. The use of five percent equivalent viscous damping is reasonable values of equivalent viscous damping ranging between five percent (steel frames) and seven percent (shear walls) (ATC, 1974), and five percent (steel frames) and eight percent (shear walls) (DOD, 1986), have been reported. However, given that such damping is probably heavily dependent on the type and arrangement of interior and exterior nonstructural elements, there is no compelling reason to reduce seismic demands on selected framing systems to reflect marginal increases of two to three percent in building damping. The damping factor as discussed in Sections 4.4 and 4.5 is intended to account for the influence of supplemental viscous damping devices on the force and displacement response of buildings. This influence has been studied by a number of researchers (Riddell and Newmark, 1979; Wu and Hanson, 1989). These studies have focused on displacement response only, and data from the Riddell and Newmark studies have been implemented in the 1994 UBC for the design of seismic isolation systems. However, current seismic design procedures using R factors are force-based procedures. The addition of viscous damping to a building frame will always serve to reduce displacements, but may increase the inertial forces if the viscous forces become substantial. This relationship can be demonstrated simply as

30

follows. Consider again the elastic SOOF system shown in Figure 4.1. The equation of motion for this frame is given as either mvt(t) + ev(t) + kv(t) = 0

or mv(t) + ev(t) + kv(t)

= -mvg(t)

(4-15)

where m, e, and k, are defined in Figure 4.1; yt(t), v(t), and v(t) are the total acceleration, relative velocity, and relative displacement of the mass, respectively; and Yg(t)is the ground acceleration. Equation 4-14 can be rewritten as: (4-16) and then simplified to read: vt(t)

= -cJv(t)-2co~v(t)

(4-17)

where co = kim and 2co~ = elm 2

In this equation, cJv(t) is the hysteretic (or spring) force per unit mass developed in the columns of the SDOF system, and 2co~v(t) is the damping force per

4: Components of Response Modification Factors

---------_.----

(4-14)

ATC-19

unit mass developed in the dashpot of the SDOF system. The solution to the equation of motion for the SDOF system is I

v(t) =

_(_1_) Iv ('t)e-~CJ){t-t)sinCOD(t - 't)d't(4-18) roD 0 g

where is the angular frequency; and roD is the damped angular frequency (Clough and Penzien, 1993). The maximum value ofv(t) is termed the spectral displacement; the spectral displacement is equal to the pseudo-displacement. From Equation 418, it is clear that an increase in damping (I;) will lead to a reduction in the displacement response. The maximum value of jit(t) is termed the spectral acceleration; the maximum value of co 2 v(t) is the pseudo-acceleration which is typically used in seismic design to represent the maximum acceleration of the mass in the SDOF system. From Equation 4-17, it can be seen that the spectral acceleration is approximately equal to the pseudo-acceleration if the damping forces (equal to cv in Equation 4-15) are small. If the damping forces are large, the contribution of the damping force to the inertial force will become significant. The foregoing discussion is limited to the elastic response of a SDOF system. The relation between spring force (= lev in the elastic range) and damping force (cv ) will be further exacerbated if the frame is designed using a large response modification factor, because the value of the spring force will be limited to the inelastic strength of the frame (= Vo in Section 4-3) whereas the damping force can continue to increase as the velocity increases. Table 4-4 lists values of the damping factor R~ for reducing displacement response only, taken from the 1994 UBC provisions for seismic isolation systems and from the work of Wu and Hanson (1989) for different levels of viscous damping. These factors should not be used to reduce hysteretic force demands unless the forces developed in the viscous elements are explicitly accounted for in the design process.

4.6

Systematic Evaluation of R Factors

Response modification factors playa key, but controversial, role in the seismic design process in the United States. No other parameter in the design base shear equation (Equations 2-4 and 2-5) impacts the design actions in a seismic framing system as does

ATC·19

r'--'

Table 4.4

Damping Factor as a Function of Viscous Damping

Viscous damping (% critical)

1994 UBC

2

0.80

5

1.00

1.00

1.20

1.19

(l?f.)

Wuand Hanson (Rc,)

7

10 12 15 20

1.39 1.50

1.56

the value assigned to R. Despite the profound influence of R on the seismic design process, and ultimately on the seismic performance of buildings in the United States, no sound technical basis exists for the values of R tabulated in seismic design codes in the United States. There is an obvious and pressing need to develop a rational technical basis for R factors if equivalent lateral force design procedures are to be retained for seismic design. Ifthe new formulation for R presented in this chapter is to be implemented in building codes in the United States, systematic and coordinated studies are required to support or modify the proposed values for the strength, ductility, redundancy, and damping factors. Research is also needed to fully characterize the interdependence of the four factors. There is no merit in replacing unsubstantiated R values with three or four times as many unsubstantiated values for strength, ductility, redundancy, and damping factors. Strength and ductility factors for most seismic framing systems will likely vary between seismic zones due to differences in the ratios of gravity loads to seismic loads. Consequently, strength and ductility factors should be evaluated for each seismic framing system in each seismic zone using standardized definitions of reserve strength and ductility. The studies conducted by Osteraas and Krawinkler (1990) for three framing systems in seismic zone 4 provide a good model for how to carry out such studies. For each seismic framing system considered, multiple plan and vertical building geometries must be analyzed in a systematic manner to provide the data necessary to quantify the R factors. The draft procedures described above for the evaluation of the different

4: Components of Response Modification Factors

31

,-----------------_.,,_.._-_

•.•....

component factors of R could be used for such coordinated studies.

have to be established through careful research.

4.7 Framing systems with less than four vertical lines of strength- and deformation-compatible seismic framing in each principal direction ofthe building, or those possessing minimal torsional redundancy, should be penalized through the use of a redundancy factor. Limits must be placed on the relative strength and stiffness of the vertical lines of seismic framing in each principal direction of a building. The numerical values assigned to the redundancy factor must be established using reliability theory. Should seismic design practice in the United States shift to displacement-based procedures rather than force-based procedures, it may be appropriate to include a damping component in the R factor. Before this can be done, values for the damping factor will

32

Rel1ab111ty of Values for R

It is of paramount importance that revised values for R and the values for the component strength, ductility, and redundancy factors be reliable, in the sense that buildings designed using these factors should meet the assumed performance level in the design earthquake. Values for the strength and ductility factors should be evaluated using a consistent methodology. It is also important that sufficient numbers and types of buildings be analyzed to permit statistical evaluation and interpretation of the responses. The values assigned to both R and its component factors should aim to provide either a uniform level of risk for all framing systems or a level of risk that is consistently less than a yet-to-be-determined threshold.

4: Components of Response Modification Factors

--.....--------,-----

ATC-19

,----------------------- ._..•-.__

.•..............

5. 5.1

Conclusions and Recommendations

Summary and Concluding Remarks

The objectives of this report were three-fold: (I) to review the role played by response modification factors (R factors) in the seismic design of buildings in the United States, (2) to document the basis of the values of R (and its working-stress equivalent Rw ) utilized in seismic codes in the United States, and (3) to propose a method for the systematic and rational determination of values of R for seismic framing systems used in the United States, including the determination of values for the key components of R. The focus of Chapter 2 was the historical development of values for Rand Rw . It was reported that values for R and Rw could be traced directly to the horizontal force factor K first introduced into seismic codes in the United States in 1959, and that the values for K represented the consensus opinion of expert structural engineers in California in the late 1950s. The values assigned to K were based on engineering judgement, and not on detailed analysis - not surprising since the requisite analytical tools were not available to the design professional community until the mid 1970s. This investigation concluded that: 1.

There is no mathematical basis for the response modification (R) factors tabulated in modern seismic codes in the United States.

2.

A single value of R for all buildings of a given framing type, irrespective of plan and vertical geometry, cannot be justified.

3. To ensure consistent levels of damage, values for R should depend on both the fundamental period of the building and the soil type on which the building is founded. 4.

The values assigned to R for a given framing systems should vary between seismic zones because the reserve strength in a framing system will probably be a function of the ratio of the gravity loads to the seismic loads. Also, detailing requirements currently vary by zone.

ATC-19

5. The values currently assigned to R for different framing system will probably not result in uniform levels of risk for all buildings. The use of response modification factors is not limited to the seismic design of buildings in the United States. A similar response reduction factor (Z) is used by Caltrans and AASHTO for the seismic design of bridges in the United States. Moreover, response modification factors or their equivalents are used in a number ofcountries for seismic design. In Chapter 3, response modification factors in use in Europe, Japan, and Mexico were studied, and the numerical values of R for three framing systems were compared with those used in the United States. The investigation showed that larger values of R are used in the United States than in Europe and Mexico. Given that the seismic performance objectives are similar in all cases, the conclusion drawn from the study was that the values of R in seismic codes in the United States may not be sufficiently conservative. Chapter 4 described the impact of R on the seismic design process, introduced force-displacement relations for buildings, summarized the experimental evaluation of R for two steel-braced framing systems, and proposed a new formulation for R. The results of an experimental program at the University ofCalifornia at Berkeley were reviewed. These results suggest that the values of R tabulated in the NEHRP Provisions for concentrically and eccentrically braced frames are not sufficiently conservative, and that further study is needed. The new formulation for the response modification factor splits R into factors related to reserve strength, ductility, and redundancy. The implications of introducing a damping factor into the new formulation were also addressed. Further, procedures for evaluating strength and ductility factors using commercially available analytical tools were proposed. The lessons to be learned from the 1994 Northridge and 1995 Kobe earthquakes are clear. First, moderate magnitude earthquakes can produce elastic seismic demands substantially greater than those assumed by seismic codes in the United States. Second, moderate

5: Conclusions and Recommendations

33

,_._-.,------------- ,-------r---.------"--.----.- .

magnitude earthquakes in major urban areas can result in huge social and economic losses. Although damage and loss of older structures was expected in these earthquakes, damage to buildings and bridges designed to modern seismic standards was not. The degree of loss in both earthquakes was too high the seismic risks can and must be mitigated. One key step in the mitigation effort is to improve the reliability of new construction in a timely manner. One obvious way to improve the reliability of new construction is to improve the reliability of the response modification factors used in the seismic codes.

5.2

Recommendations

Static lateral force analysis and design procedures are key components of routine seismic design in practice in the United States. One key step in the procedure is the calculation of design forces. These are typically calculated by dividing the elastic spectral force by a response modification factor (R). Even with the advent of powerful nonlinear analysis packages, elastic design procedures are likely to be used for much seismic analysis and design in the foreseeable future and such procedures necessarily make use of a response modification factor in one fonn or another.

34

r-'

For this reason, and recognizing that there is no sound basis for the values assigned to R in U.S. seismic codes, it is of paramount importance to establish appropriate values for R through rigorous research and study so that the intended perfonnance of the building stock can be realized with a high degree of reliability. This report has proposed a draft fonnulation for the systematic quantification of R and its component factors and has recognized the need to quantify the redundancy factor through reliability analysis. A coordinated action plan is currently being developed (ATe, 1995) to systematically evaluate R for different framing systems in all regions in the United States, and to improve the reliability of current seismic design procedures. The execution of this action plan will require significant funding from federal agencies. However, the rapid implementation of this plan will substantially reduce the exposure offederal, state, and local agencies to substantial loss in future earthquakes in the United States. Further, the knowledge gained from these studies should prove useful for the evaluation and rehabilitation of older construction.

5: Conclusions and Recommendations

ATC-19

------l.....---·---·--..,-··----················

Appendix A: Evaluation of Building Strength and Ductility A.I

Introduction

A trend in earthquake engineering practice in the United States, especially in California, is for the strength and deformation characteristics ofa building to be evaluated using nonlinear static analysis. This kind of analysis can estimate the likely maximum strength and deformation capacity of the building, identify potential weak and/or soft stories in the building, and identify poorly proportioned framing elements that give rise to excessive deformation demands at the element level. In order to evaluate the probable force-deformation response of a building, member proportions must be established or known a priori. For new construction, preliminary member proportions are generally established using the procedures set forth in a seismic code; for existing construction, member sizes are known. The purpose of this appendix is two-fold: (1) to introduce the reader to key features of nonlinear static analysis and (2) to demonstrate techniques for evaluating strength and ductility factors. Nonlinear static analysis is one of four procedures proposed by the ATC-33 project team (ATC, in progress) for the seismic rehabilitation of existing buildings - the remaining three procedures being linear static analysis, linear dynamic analysis, and nonlinear responsehistory analysis. The salient features of nonlinear static analysis are introduced below. We first provide a brief description of nonlinear analysis. This is followed by a description of a nonproprietary computer code for nonlinear analysis (DRAIN-2DX). DRAIN-2DX is then used to analyze a nonductile, seven-story reinforced concrete frame building, constructed in 1966, and to evaluate the base shear-roof displacement relationship for this building. The base shear-roof displacement relationship is analyzed in Section AA to estimate the strength and ductility factors for the subject seven-story building with the intent of demonstrating to the reader how to calculate the strength factor, the different means by

ATC-19

A.2

Nonlinear Static Analysis

A.2.1

Introduction

The two common methods of nonlinear analysis are nonlinear static analysis and nonlinear response-history analysis. For both methods of analysis, a framing system is modeled and analyzed as an assembly of elements and components. The data output from either analysis procedure includes force and deformation demands on elements and components. Nonlinear static analysis is less demanding in a computational sense than nonlinear response-history analysis, but more rigorous than linear methods of analysis. In nonlinear response-history analysis, a mathematical model of a building is subjected to digitized records of earthquake ground motions. The analysis is generally terminated at the end of the earthquake ground motion record - often after more than 2000 time steps. In nonlinear static analysis, increasing inertial forces (or displacements) are imposed on a mathematical model of a building. The analysis is terminated once a target displacement is reached - often after fewer than SO load steps. The target displacement represents a maximum building displacement during earthquake shaking; brief comments on the selection of the target displacement are provided in Section A.2.7. The remainder of Section A.2 is devoted to nonlinear static analysis. The following sections provide the reader with an introduction to the subject. Much additional information is available in the literature.

A2.2

Basics of Nonlinear Static Analysis

For the purposes of seismic analysis, a building should be modeled and analyzed as a three-dimensional assembly of elements and components. Twodimensional modeling and analysis of a building will generally be acceptable if either the torsional effects are small or the three-dimensional effects can be

AppendiX A: Evaluation of Building Strength and Ductility

.._ --

-~--------,,_

1-'---"

which to calculate the displacement ductility ratio for the subject building, and how to estimate the ductility factor.

35

accounted for separately. The mathematical model of a building should include the following: (a) all elements and components ofthe seismic and gravity framing systems, (b) nonstructural components in the building likely to possess significant stiffness and strength, and (c) elements of the foundation system (footings, piles, etc.) that are sufficiently flexible and/or weak to contribute to the response of the building. The distribution of the equivalent lateral static loads (see Section A.2.6) in the mathematical model should be adequate to capture all key dynamic effects on the seismic and gravity framing system, the nonstructural components, and the foundation. Gravity loads should be imposed on the mathematical model to reflect those loads likely to be present during earthquake shaking. The initial gravity loading conditions (QG) can be described by one of the following two equations (ATC, in progress):

reinforced concrete floor slab. Similarly, the yield strength of a wide-flange steel beam should be based on the likely yield stress rather than the nominal yield stress. Connections between framing members should be modeled unless the connection is sufficiently stiff to prevent relative deformation between the connected elements or components and the connection is stronger than the connected elements or components. A.2.4

Nonlinear Static Procedure

The nonlinear static procedure requires an a priori estimate of the target displacement. The target displacement serves as an estimate of the maximum displacement of a selected point (node) in the subject building during the design earthquake.The node associated with the center of mass at the roof level is often the target point or target node selected for comparison with the target displacement. Nonlinear static analysis is integrated into the fourstep nonlinear static procedure as follows:

QG = 1.1(QD + QL + Qs)

(A-I)

QG = O.9QD

(A-2)

Develop a two- or three-dimensional mathematical model of the building, as described in Section A.2.2.

where QD , QL , and Qs are the dead, live, and snow loads, respectively. Equations A-I and A-2 are intended to provide upper- and lower-bound estimates, respectively, on the likely gravity loads on an element or component. Other load combinations (BSSC, 1991) can also be considered.

2. Impose constant gravity loads, and then apply static lateral loads (or displacements) in patterns that approximately capture the relative inertial forces developed at locations of substantial mass.

1.

A.2.3

Modeling Elements, Components, and Connections

The mechanical characteristics (i.e., force-deformation) of each element and component of the building should be modeled in sufficient detail that their important effects on the response of the building are reasonably represented. In most instances, the mechanical characteristics estimated for the analysis will be elastic stiffness, inelastic stiffness, and yield strength. Failure modes (e.g., shear) that may occur at deformations smaller than those anticipated in the analysis should be accounted for in the element or component model. Elements and components of buildings should be modeled using actual rather than nominal geometries and mechanical properties. For example, the mechanical characteristics of a beam in a reinforced concrete frame should account for the likely presence of a

36

3. Push the structure using the load patterns of Step 2 to displacements larger than those associated with the target displacement (Le., the displacement of the target node exceeds the target displacement). 4. Estimate the forces and deformations in each element at the level of displacement corresponding to the target displacement (Step 4). The element force and deformation demands of Step 4 are then compared with the element capacities in a manner similar to that demonstrated in Section A.3. A.2.5

DRAIN Computer Code

DRAIN-2DX (Prakash et aI., 1992) is a two-dimensional, general-purpose, nonlinear, finite-element analysis program developed at the University of California at Berkeley. The modeling and analysis procedures incorporated in DRAIN-2DX are

Appendix A: Evaluation of Building Strength and Ductility

r-··---------·-'---

ATC-19

-----------_.. -._--_ ..

,----------

.,,""

'

I

summarized below. The DRAIN-2DX computer code could be used for Steps 2,3, and 4 of the nonlinear static procedure described above. Building framing systems are modeled as twodimensional (X-Y) assemblages of nonlinear elements connected at nodes. Unless a node is restrained or slaved to another node, each node has three degrees of freedom. Elements (and components) are divided into groups, although alI elements of a given type (e.g., beam-column) need not be assigned to a single group. Masses are lumped at nodes, so the nodal points should be selected to adequately captUre the inertial response ofthe building. By lumping the masses at nodes, the mass matrix is diagonal.The damping matrix can be made proportional to the element stiffness values and nodal masses. Numerous analysis types are available with DRAIN2DX, including (a) static gravity analysis for combined element and nodal loads, (b) nonlinear static analysis for nodal loads, (c) eigen analysis for the evaluation of mode shapes and periods, (d) responsespectrum analysis, (e) nonlinear dynamic analysis for ground motions defined by acceleration records, (f) nonlinear dynamic analysis for ground motions defined by displacement records, and (g) nonlinear dynamic analysis for specified initial nodal velocities (for shock analysis). The program is sufficiently flexible to allow a building (or structure) to be analyzed for several analysis segments (or types), thus facilitating sequential static and dynamic analysis. Loads are input as either patterns for static loads or as records for dynamic loads. Seven different load types are available with DRAIN-2DX, including (a) static element load patterns - typically used for gravity loads, (b) static nodal load patterns consisting of vertical, lateral. and rotational loads applied on nodes for gravity and static analysis segments, (c) ground acceleration records, (d) ground displacement records, including an allowance for multiple support excitation, and (e) response spectra. DRAIN-2DX can perform both static and dynamic analysis. In static analysis, the load is typically applied in a number of steps. The program selects load substep sizes within each step by projecting the next stiffness change (known as an event) and terminating the substep at that event. The structure stiffness is then changed at the end of each substep, and . the analysis is continued for the following substep.

ATC-19

The static analysis segment is complete once either the entire load has been applied or a target displacement value is reached. In dynamic analysis, the time step can be selected to be constant or variable. Further options for dynamic analysis include event calculations within time steps and corrections at the end of each time step to improve the energy balance or equilibrium. Second-order (or P - A) effects can be modeled in DRAIN-2DX by considering geometric stiffness for each element, and including second-order forces in the calculation ofthe resisting forces. For static analysis, the geometric stiffness is modified at each event. For dynamic analysis, the geometric stiffness can be kept constant or allowed to vary. Six element types are currently available in the DRAIN-2DX element library: (I) a truss element, Type 01; (2) a beam-column element, Type 02; (3) a connection element, Type 04; (4) a panel element, Type 06; (5) a link element, Type 09; and (6) a fiber beam-column element, Type IS. Of these six elements, the most commonly used are the truss, the beam-column and the connection elements. Some introductory remarks on these three elements folIow. The reader is referred to Prakash et al. (1992) for additional information. Truss elements transmit axial loads only and can be arbitrarily oriented in the X-Yplane. The inelastic response of these elements can be specified as either yielding in tension and elastic buckling in compression or yielding in both tension and compression. A two-component parallel model (an element consisting of elastic and inelastic components in parallel) is used to capture strain-hardening effects. Beam-column elements possess axial and flexural stiffness and can be arbitrarily oriented in the X-Y plane. Shear deformations and rigid-end offsets can be accounted for in the beam-column element. Yielding is concentrated in the plastic hinges at the element ends, and strain-hardening is approximated by a two-component parallel model. Different yield moments can be specified at the two element ends as well as for positive and negative flexure - two features necessary to model reinforced concrete columns and beams. Gravity and other static loads applied to an element can be captured by specifying end clamping or fixed-end forces. Second-order effects can be included by introducing equilibrium

AppendiX A: Evaluation of Building Strength and Ductility

37

-----------------"--_._..-

.....

correction and geometric stiffness as noted above. Three modes of deformation are available to beamcolumn elements - axial deformation, flexural rotation at element end 1, and flexural deformation at element end 2. A plastic hinge forms when the moment in the element reaches the yield moment. Inelastic axial deformations are assumed not to occur; that is, a beam-column cannot yield in axial tension or compression.

to inelastic displacements is large - for any given ground motion the ratio of elastic to inelastic displacements could range between 0.5 and 2.0. A conservative approach to calculating the target displacement, in the absence of additional information, would be to increase the target displacement by between 50 percent and 100 percent; that is, to assume that the inelastic displacement is equal to 1.5 to 2.0 times the elastic displacement.

The connection element connects two nodes with identical coordinates in theX-Yplane. This element can connect either rotational displacements of the nodes or the translational displacements ofthe nodes, and it can be specified to achieve complex inelastic behaviors. A common applica~ion for this element is the modeling of beam-column panel zones in steel frames.

Nonlinear static analysis makes use offorce-deformation relationships for beams and columns that are generally based on monotonic force-deformation analysis. This assumption will likely be adequate for buildings designed to experience less than three displacement cycles to between 80 percent and 100 percent ofthe target displacement. On the other hand, consider a building in the near-source zone with a fundamental period of 0.5 second subject to a Richer magnitude 7.5 event - this building may be subjected to 10 to 20 displacement cycles to between 80 percent and 100 percent of the target displacement. The strength and stiffness of the structural components and elements in this building will most likely degrade substantially over the course of the 10 to 20 displacement cycles. The question thus arises as to how the design professional should account for the effects ofcumulative damage. At present, there are no definitive answers for building framing systems (Reinhom, private communication). In the absence of definitive data, the design professional should reduce the monotonic deformation capacity of structural framing elements and components to indirectly account for the deleterious effects of prolonged strong ground shaking.

A.2.6

Lateral Load Profiles for Analysis

Lateral loads should be applied in patterns that both approximately capture the vertical distribution of inertial forces expected in the design earthquake and account for the horizontal distribution of inertial forces in the plane of each floor diaphragm. Load patterns that bound the plausible distributions ofinertial force should be considered for design. Two vertical distributions of inertial force commonly used for nonlinear static analysis are the distribution defined by the first-mode shape ordinates ofthe building and the constant acceleration distribution, which corresponds to the formation of a weak first story. For flexible buildings, a vertical distribution of seismic force that reflects the likely contributions of higher modes should be considered. A.2.7

Target Displacement Calculation

The method most commonly used to evaluate the target displacement is based on the assumption that elastic and inelastic displacements are equal; that is, the inelastic displacement of a SDOF oscillator with initial (elastic) period T is equal to the elastic spectral displacement calculated using period T. This assumption is based primarily on the work of Miranda and Bertero (1994) who demonstrated by exhaustive analysis that for periods greater than 0.5 second (for a rock site), mean elastic displacements were approximately equal to mean inelastic displacements. This assumption should be carefully reviewed by the design professional calculating a target displacement, because the scatter in the ratio of elastic

38

,_._--.------------

A.3

Seismic Evaluation of an Example Building

A.3.1

Description of Building

The building selected for sample analysis is a sevenstory reinforced concrete building located in Los Angeles, approximately 13 miles south ofthe epicenter of the 1971 San Fernando earthquake. This building was damaged in both the 1971 earthquake and the 1994 Northridge earthquake. This building was the subject of detailed analysis following both the 1971 earthquake (DOC, 1973) and the 1994 earthquake (Lynn, private communication). The latter analysis effort was funded by the Federal Emergency Management Agency to verify the non-

AppendiX A: Evaluation of Building Strength and Ductility

ATC-19

._--------------_._-_..---_._ _ _ ..

.

linear static analysis procedures being developed for the ATC-33 project (in progress). The analysis results presented below are an extension of the FEMA study. The results of this study are contained in a background report to ATC-33 Guidelines and Commentary for the Seismic Rehabilitation ofBuildings. The 63,000-square-foot building, designed in 1965, is approximately 62 feet by 160 feet in plan. The typical framing consists of columns on a 20-foot (transverse) by 19-foot (longitudinal) grid. Spandrel beams are located on the perimeter frames. The floor system is a reinforced concrete flat slab, 10 inches thick at the second floor, 8.5 inches thick at the third to seventh floors, and 8 inches thick at the roof. The ground floor is a four inch thick slab-on-grade, and the foundation is piled. A typical floor framing plan is presented in Figure A-I. Atypical transverse section and typical beam and column details are presented in Figure A-2. Interior columns are 18 inches square and exterior columns are 14 inches by 20 inches in plan. Spandrel beam sizes are shown in Figure A-2. The seismic framing system is composed of interior slab-column moment frames and perimeter beamcolumn moment frames. The design base shear force at the working stress level was (DOC, 1973):

v = ZKCW = 1.0xO.67xO.057xW=O.04W (A-3) The north face of the building, along column line D, has four bays of masonry infill between the ground and second floor level, all at the eastern end of the structure, between column lines 5 and 9. For simplicity, these infill walls were not included in the mathematical model described below. (Were this evaluation to be used for the purpose of seismic rehabilitation, the infill walls would have been included in the mathematical model.) The reader is referred to the Department of Commerce report (DOC, 1973, pp. 359-393) for additional information regarding the design and construction of this sample building.

A3.2

Modeling of the Building

The first question confronting the engineer charged with evaluating the building is what to include in the

ATC·19

mathematical model of the building. Although some design professionals would choose to exclude the interior slab-column framing from the mathematical model of this framing system, it is inappropriate to do so in this instance, as is demonstrated below. The second question to be answered is whether the building can be represented using two-dimensional mathematical models; that is, uncoupling the threedimensional independent framing systems along each principal axis of the building. In the sample building, the torsional response is small- especially so after the infill masonry walls are removed. As such, Lynn (private communication) modeled the three-dimensional building with two two-dimensional framing systems - one per building axis. Since the purpose of this appendix is to demonstrate the use of nonlinear static analysis, only the results of the analysis of the longitudinal framing are summarized. The reader is referred to the aforementioned background report for additional information. Two exterior frames and two interior frames were included in the mathematical model of the longitudinal framing. The mathematical model of one-half of the framing system is presented in Figure A-3. The mathematical models of the interior and exterior frames were linked together with rigid struts to simulate the assumed rigid floor diaphragm. The reactive weights assigned to the seven suspended floors are presented in Table A-I.

A3.3

Modeling of Key Elements

The mathematical model of the frames was composed of columns and beams. Beam-column joints were not included in the model. The reinforced concrete columns were modeled using their gross-section stiffness. The axial forcemoment yield surfaces were established using standard interaction curves, with capacity reduction factors (c\> ) set equal to 1. A strain-hardening ratio of five percent was assumed for all columns. The exterior reinforced concrete beams were modeled as L-beams. The slab width assumed to contribute to the strength and stiffuess of the edge beams was set equal to 30 percent of the perpendicular span (often termed 12 ), The strength and stiffness values of the interior slab-beams were calculated using a slab width equal to 60 percent of the perpendicular span. Beam and slab beam stiffness values were esti-

Appendix A: Evaluation of Building Strength and Ductillty

39

,------r---.-----,.,,----.-..---...

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·· il

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1

.

eb Figure A-1

40

... ...•

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" u

!

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.

Typical floor plan of sample building (DOC, 1973, p. 363).

Appendix A: Evaluation of Building Strength and Ductility

1--·'----.:-------··---

ATC-19

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ATC-19

~

ii

!:

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n

•:

:a 1

.." a "71 ~s ..<= ':~"1<:' "';Y<::::',.'.:-:,:.,.:: ;'(I:·I~-L( ::.<:f··' . 11 :

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Transverse section and typical details of sample building (DOC, 1973, p. 365).

Appendix A: Evaluation of Building Strength and Ductility

41

,------------------------_._._--_ _

.

r

Extenor Frame

\,

Rigid Links

\,

Interior Frame

1--.[

--<-

...., ...., ...., .....

Figure A-3

Table A-l

Mathematical model of the sample building longitudinal framing.

Sample Building Reactive Weights

Floor level

Weight 1185

7th

1350

6th

1350

5th

1350

4th

1350

3rd

1350

2nd

1548

mated as one-half of gross stiftbess; flexure yield surfaces were established using nominal material properties. A strain-hardening ratio of.five percent was assumed for all beams and slab-beams. A.3.4

Eigen Analysis Results

The modal periods and shapes of the building frame were established using the eigen solver in DRAIN2DX. The first three modal periods and the percentages of the total mass in each of these three modes are presented in Table A-2. The first three mode TableA-2

42

A.3.5

Nonlinear Static Analysis Results

The key results of three analyses of the sample building are presented below. The data from the first two analyses are presented to demonstrate differences in response resulting from the use of different load profiles; Analysis 1 uses a triangular load profile, and Analysis 2 a rectangular profile. The results ofthe third analysis are presented to demonstrate the importance of including the interior frame in the mathematical model. Analysis 3 uses a triangular load profile but considers only the response of the perimeter (exterior) frame.

(kips) Roof

shapes are shown in Figure A-4.

Dynamic Characteristics in the Longitudinal Direction, Sample Building

Mode

Perbd (sees.)

%o(TotaJ

1

1.33

84

2

0.45

11

3

0.26

3

Mass

The base shear versus roof displacement relations for Analysis 1 and Analysis 2 are presented in Figure A5. The strength of the framing system, calculated using a rectangular force profile, at a roof displacement of20 inches (2.5 percent roof drift), is approximately 10 percent larger than that calculated with a triangular profile. Using the triangular profile response data, and the equal energy method (see Section 4.3), the yield displacement was calculated to be approximately equal to 4.5 inches, and the yield force to be approximately equal to 16 percent of the reactive weight of the building. The locations of plastic hinges in the exterior frames at a roof displacement of20 inches are presented in Figure A-6 for Analysis 1 and Analysis 2. The mechanisms associated with the two force profiles are different - the triangular profile results in a sway mechanism involving the lower four stories and the rectangular profile results in concomitant mechanisms (i.e., a lower four-story sway mechanism and a fourth story sway mechanism). Although the exist-

Appendix A: Evaluation of Building Strength and Ductility

ATC-19

._-------,----------------_ ----_ .

..

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Mode 1

Figure A-4

Mode 2

Mode 3

Mode shapes of sample building in the longitudinal direction.

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ATC·19

OL.-

....l..-

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----l

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Base shear versus roof displacement relations for sample building analyses 1 and 2.

Appendix A: Evaluation of Building Strength and Ductility

43

---------_._------,.. - _-_

_..

denotes plastic hinge (a) Analysis 1

+ (

(b) Analysis 2

Figure A-6

Plastic hinge locations, roof displacement of 20 inches, sample bUilding analysis 1 and 2.

ence of two mechanisms may seem counterintuitive, it should be noted that only the hinges in the perimeter frame are shown in Figure A-6 and the forcedeformation relationship for the interior frames plays a key role in the force-deformation response ofthe building. A typical column and beam in the second story of the perimeter frame (denoted C and B in Figure A-3, respectively) were each analyzed for the purpose of demonstrating part of a typical seismic evaluation procedure. For the sample column, the maximum rotation capacity of the subject column was calculated to be 0.005

44

radian. This calculation was based on a plastic hinge length ofO.5d (8 inches) and an axial load equal to the sum of the plastic beam shear forces and dead loads above the second story. This maximum rotation of 0.005 radian was realized at roof displacement values of 12 inches and 10 inches, for Analysis 1 and Analysis 2, respectively. The maximum rotation capacity of the sample beam was estimated to be 0.03 radian, assuming a plastic hinge length ofO.5d (14 inches). This maximum beam rotation was reached at roof displacement values of 19 inches and 16 inches, for Analysis 1 and Analysis 2, respectively.

Appendix A: Evaluation of Building Strength and Ductility

r'·.-· ----.....-----·---·--",----

ATC-19

0.25.--------r-----,--,------.--

.....,

Analysis 1: All frames Analysis 3: Exterior frames only

-~ E

.~ (,)

0.2

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...............

•

0.15

'"0 ...

...

...

...

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...

...

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•

... ... ... ... ...

.

.

...

.

...

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...

f

... ... ... ... ... ... ... ... ...

~o

... (,)

m 0.1

or.

...

..

.

(/)

Q)

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CD

0.05

·' . --'.

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Figure A-7

__

...

..

--

I

...

...

...

...

...

...

...

~

'.'

10 Displacement (inches)

15

20

Base shear versus roof displacement relations for analyses 1 and 3.

This demand-capacity evaluation is simply intended to demonstrate the nonlinear static procedure. The sample evaluation is by no means sufficiently rigorous for the seismic evaluation of existing construction. In a full evaluation, all beams, columns, joints, and components should be examined closely. In this example, the sample column was assumed to be flexure-critical; this column is actually shear-critical, and it could not accommodate the shear forces associated with a plastic hinge rotation of 0.005 radian. To demonstrate the importance of considering all of the structural framing in the mathematical model, consider the base shear versus roof displacement relationships for Analysis 1 and Analysis 3 presented in Figure A-7. The data presented in this figure demonstrate that the stiffness of the exterior frames and interior frames is similar - that is, the stiffness of the interior slab-column frame approaches that ofthe exterior beam-column frame. It also shows that the strength of the interior frames and exterior frames is similar. If a designer were to ignore the stiffness and strength of the interior frames, the fundamental period of the building would be overestimated by 40% and the target displacement overestimated by a factor approaching two. Such an error in judgment might mean the difference between a ~eismic r~tr~fit involving jacketing and/or strengthemng of a limited

ATC-19

~

'...

-'.·. 5

..'-. . - - -. - . - . .. ........ .

number of columns and a seismic retrofit requiring the provision of a new seismic framing system. Further, a decision to exclude the interior frame from the analysis model could result in a flexible retrofit solution incapable of protecting the existing framing systern. A.4

Estimation of Strength, Ductility, and R Factors

A4.1

General

The calculation of strength and ductility factors is demonstrated in this section by use of a force-displacement relationship established in Section A.3 for the seven-story nonductile reinforced concrete moment frame. For the purpose ofthis discussion, the results ofAnalysis 1 (triangular load pattern) are used to derive estimates of the strength and ductility factors, and the maximum roof displacement is assumed to be eight inches. This roof displacement estimate ignores both the likelihood of shear failure in the non-ductile columns and the limited deformation capacity of the interior frame column-slab connections. These assumptions would not be valid were this an evaluation for the purpose of assessing the seismic vulnerability of the building. The displacement capacity of the frame is reduced from 12 inches

AppendiX A: Evaluation of BUilding Strength and Ductility

._---------,,-,---

45

,---------_._-_._-,.. _----_ _ ..

,.

(see Section A.3.5) to 8 inches to reflect the likely degradation in defonnation capacity resulting from multiple cycles of loading (see Section A.2.7). A.4.2

Strength Factor

The force-displacement relationship for Analysis I is reproduced in Figure A-8. The base shear force (Vo) at a roof displacement of eight inches is approximately 17 percent of the reactive weight ofthe building. The design base shear for this building at the strength level (Vd ), calculated by multiplying the working-stress design base shear (Equation A-3) by a seismic load factor of 1.40, is approximately 0.06W. Using these data, the strength factor is calculated as: Va

Rs

0.17W

= Vd = 0.06W

(A-4)

= 2.8

Note that if the interior frame was not included in the mathematical model (as was probably the case when the building was designed), the base shear force at a roof displacement of eight inches is approximately equal to 0.08W, and the resulting strength factor would be equal to 1.3. A.4.3

Ductility Factor

A bilinear approximation to the calculated force-displacement relationship is shown in Figure A-8. This approximation is based on the equal-energy method (see Section 4.3 for details) and assumes that the yield force (Vy) is equal to Vo. The maximum displacement (~m) is eight inches and the yield displacement (~y) is six inches. Using these data, the displacement ductility ratio is calculated as: ~m

J.l = ~y

8

= 6 = 1.3

(A-S)

The ductility factor can be calculated using either the Nassar and Krawinkler (1991) or Miranda and Bertero (1994) relationships. For the purpose of this discussion, the Miranda and Bertero equation (Equation 4-12) for a rock site is used to estimate the ductility factor. For a fundamental period of 1.33 seconds and a ductility ratio of 1.3, is equal to 0.76, and the ductility factor is equal to: RIJ.

J.l- 1

= '""" + 1 =

1.3 -1 0.76 + 1

= 1.4

(A-6)

Using these data and a strain-hardening ratio equal to

46

opercent (consistent with the assumptions made in constructing the bilinear approximation to the forcedisplacement relationship), the ductility factor calculated using the Nassar and Krawinkler relation (Equation 4-10) is equal to 1.3. Summazy

A.4.4

Estimates for the strength and ductility factors for the sample building have been made in the preceding two subsections for the purpose of demonstrating how the results of nonlinear static analysis can be used to evaluate key components of the response modification factor. These estimates of 2.8 for the strength factor and 1.3 for the ductility factor, are likely upper bounds on the probable values if column shear strength and joint defonnation capacity are considered. Note that if the Krawinkler and Nassar modification factor for MDOF systems is considered (see Section 4.5.2.3 and Figure 4-9 for details), the ductility factor will be reduced. The seismic framing system in the sample building is composed of a perimeter beam-column moment frame and an interior slab-column moment space frame. If the interior space frame is ignored, the building has good redundancy in the longitudinal direction, with 16 vertical bays of seismic framing (eight per face) but only marginal redundancy in the transverse direction with six vertical bays of seismic framing. If the interior space frame is included in the evaluation, and assuming that the stiffness and strength of the interior framing are similar to those of the perimeter framing, the subject building has excellent redundancy (RR equal to 1.0) in both the longitudinal and transverse directions. The response modification factor (R) could be calculated as the product of the strength-ductility, and redundancy factors. For this example, R is equal to: R

= R s R IJ. RR = 2.8 x 1.3 x 1.0 = 3.6

For the reasons cited above, this value for R is approximate only. Similar buildings constructed in the early 1960s will likely have values of R that vary between 2 and 4. Much additional study (see Chapter 5 for details) is needed before values of R can be rationally assigned to either new or existing seismic framing systems.

Appendix A: Evaluation of Building Strength and Ductility

_._---,

(A-7)

ATC-19

------....,.---_._------,.. _._---_._,

.....

,""

0.25 r - - - - - - r - - - - , - - - - - , - - - - - - - , - - Analysis 1: All frames . - - ;

0.2

Bilinear approximation .

~

Maximum base shear coefficient = 0.17

' -'

C

.~

.2

I

0.15

. .

:t:

Q)

o o

I

...

I

:

.

.

.

~ 0.1 ......... f·····:···············:···············:··············

.s:::

I

(fl

~

~

/

0.05

/

/

:

Design base shear coefficient = 0.06 ; ;

.

/ 'I

o o

L----"-------_..L-

Figure A-8

ATC-19

- - -

5

10 Displacement (inches)

.....L...-

15

.....J

20

Base shear versus roof displacement relation for analysis 1.

Appendix A: Evaluation of BUilding Strength and Ductility

0 _ _- -

47

-------.,---_._----_.__._--_._

.." ........

Appendix B: Glossary of Terms

Acceleration:

the material, element, or system level.

Effective Peak Acceleration: A coefficient representing ground motion at a period of about OJ sec. Effective Peak Velocity: A coefficient representing ground motion at a period of about 1.0 sec. Allowable-Stress Design (ASD): A method of proportioning structures such that the computed elastic stress does not exceed a specified limiting stress also known as Working-Stress Design (WSD). Base: The level at which the horizontal seismic ground motions are considered to be imparted to the building. Base Shear: Total horizontal force or shear at the base. Building: Any structure, fully or partially enclosed, used or intended for sheltering persons or property. Critical Damping: The smallest level of damping for which no oscillation occurs in free vibration response. Damping: A measure of energy dissipation. Damping in a structure is typically defined in terms of percent of critical damping. Design Action: The force in an element (axial, flexure, shear, torque) for which the element is to be designed. Design Basis Earthquake: The earthquake that produces ground motions at the site under consideration that have a 10 percent probability of being exceeded in 50 years. Diaphragm: A horizontal, or nearly horizontal, system designed to transmit seismic forces to the vertical elements of the seismic force-resisting system. Ductility: A measure of the ability of a material, element, or system to deform beyond yield. Ductility Ratio: The ratio of maximum deformation to yield deformation - only applies to deformations larger than yield. Ductility ratio can be introduced at

ATC-19

Element: One of the major parts that compose the structural system. A structural wall or a moment frame is an element. The boundary member of a structural wall, or the individual column or beam in a frame is a component of an element. Failure: The inability to satisfy a predetennined limit state. For the limit state of collapse, failure is often defined as the displacement at which the resistance (base shear) of the building falls below 80 percent of the maximum resistance. Other definitions of failure could include (a) drifts exceeding predetermined limits, and (b) plastic hinge rotations exceeding predetermined values. Flexure-critical: The failure mode of a component or element that is governed by flexural response. Frame: Braced Frame: An essentially vertical truss, or its equivalent, of the concentric or eccentric type that is provided in a building frame or dual system to resist seismic forces. Moment Frame: A frame in which members and joints are capable of resisting forces by flexure as well as along the axis of the members. Types of moment frames are: intermediate moment, ordinary moment, and special moment. Frame System: Building Frame System: A structural system with an essentially complete space frame providing support for vertical loads. Seismic force resistance is provided by structural walls or braced frames. Dual Frame System: A structural system with an essentially complete space frame providing support for vertical loads. Seismic force resistance is provided by a moment resisting frame and structural walls or braced frames. Space Frame System: A structural system composed of interconnected members other than bearing walls that is capable of supporting vertical loads and that also may provide resistance to

Appendix B: Glossary of Tenns

49

,--------_._----_.. _--_._

.

seismic forces. Load: Dead Load (QD): The gravity load due to the weight of all permanent structural and nonstructural components of a building such as floors, roofs, and the operating weight of fixed service equipment. Gravity Load (W): The total load and the applicable portions of other loads. Live Load (QrJ: The load superimposed by the use and occupancy of the buildings, not including the wind load, earthquake load, or dead load. The live load may be reduced for tributary areas as permitted by the building code administered by the regulatory agency. Load Factor: A factor by which a nominal load effect is multiplied to account for the uncertainties inherent in the determination of the load effect. Partial Safety Factor: The factor by which the element safety factor should be modified to account for its presence in the structural system, so that the overall failure probability is similar to that desired for the element. Performance Objective: A level of seismic functionality that a building owner or occupant expects of a structure. Sample performance objectives include no collapse, preservation of life safety, damage control, immediate occupancy, and fully functional. Redundancy: A measure of the number of lines of vertical seismic framing in a building. The

50

greater the number of lines of vertical seismic framing of similar strength and stiffness, the greater the redundancy (and reliability) of the seismic framing system. Reserve Strength: The difference between design strength and maximum strength. Resistance: The maximum load-carrying capacity, as defined by a limit state. Risk: Exposure to loss. Risk is defined as the probability of seismically-induced unacceptable performance. Shear-critical: The failure mode of a component or element that is governed by shear response. Story Drift Ratio: The relative displacement of two adjacent floors, divided by the story height. Story Shear: The summation of design lateral force at the level above the story under consideration. Strength Design: A method of proportioning structures based on the ultimate strength of critical sections. Uniform Risk: A term used to describe equallikelihood of loss or damage. Wall: A component, usually placed vertically, used to enclose or divide space. Wall System, Bearing: A structural system with bearing walls providing support of all or major portions of the vertical loads. Structural walls or braced frames provide seismic force resistance.

Appendix B: Glossary of Terms ...._

.. _----- •.

ATC-19

_----------------------

'---..,.------,-_.'----,

----------,-----'------_."----"-_._'--

..--.,,.

References

ATC, 1974,An Evaluation ofa Response Spectrum Approach to the Seismic Design ofBuildings, ATC-2 Report, Applied Technology Council, Redwood City, California. ATC, 1978, Tentative Provisions for the Development ofSeismic Regulationsfor Buildings, ATe3-06 Report, Applied Technology Council, Redwood City, California. ATC, 1982a, Seismic Design Guidelinesfor Highway Bridges, ATC-6 Report, Applied Technology Council, Redwood City, California. ATC, 1982b, An Investigation ofthe Correlation Between Earthquake Ground Motion and Building Performance, ATC-IO Report, Applied Technology Council, Redwood City, California. ATC, 1995, A Critical Review ofCurrent Approaches to Earthquake Resistant Design, ATC-34 Report, Applied Technology Council, Redwood City, California. ATC, in progress. Guidelines and Commentary for the Seismic Rehabilitation ofBuildings, Volumes I and II, ATC-33.03 Report, Applied Technology Council, Redwood City, California. BSSC, 1985, 1988,1991, 1994, NEHRP Recommended Provisions for the Development ofSeismic Regulationsfor New Buildings, Building Seismic Safety Council, Washington, D.C. Bertero, V.V., 1986, "Evaluation of response reduction factors recommended by ATC and SEAOC," Proceedings ofthe Third U.S. National Conference on Earthquake Engineering, Charleston, North Carolina. Caltrans, 1990, Bridge Design Specifications Manual, California Department of Transportation, Division of Structures, Sacramento, California. Clough, R.W. and J. Penzien, 1993. Dynamics of Structures, McGraw Hill, New York. CEC, 1988, Structures in Seismic Regions - Design Part 1, Eurocode No.8, Commission of the

I

DOC, 1973, San Fe mando, California Earthquake of February 9, 1971, Vol. 1, ed. L.M. Murphy, U.S. Department of Commerce, Washington, D.C. DOD, 1986, Seismic Design for Essential Buildings, TM-5-809-10-1, Departments of the Army, Navy, and Air Force, Washington, D.C. Freeman, S.A., 1990, "On the correlation of code forces to earthquake demands," Proceedings of the 4th U.S.-Japan Workshop on Improvement of Building Structural Design and Construction Practices, ATC-15-3 Report, Applied Technology Council, Redwood City, California. Gomez, R. and F. Garcia-Ranz, 1988, "Complementary technical norms for earthquake resistant design", Earthquake Spectra, EERI, 4 (3): 441460. Hwang, H. and M. Shinozuka, 1994, "Effect of large earthquakes on the design of buildings in eastern United States," Proceedings ofthe Fifth U.S. National Conference on Earthquake Engineering, Chicago, Illinois. lAEE, 1992, Earthquake Resistant Regulations, A World List, International Association for Earthquake Engineering, Tokyo, Japan. ICBO, 1961, 1976,1985,1988,1991,1994, Uniform Building Code, International Conference of Building Officials, Whittier, California. Krawinkler, H. and A.A. Nassar, 1992, "Seismic design based on ductility and cumulative damage demands and capacities," Nonlinear Seismic Analysis and Design of Reinforced Concrete Buildings, Fajfar, Krawinkler, edd, Elsevier Applied Science, New York. Miranda, E. and V.V. Bertero, 1994, "Evaluation of strength reduction factors for earthquake-resistant design," Earthquake Spectra, EERI, 10 (2): 357-379. Moses, F., 1974, "Reliability of structural systems,"

References

ATC-19

,_._-----

European Communities, Luxembourg.

--_"

-

,----_---:.._-;----------~-_

51

..__._._._--_.•... _...

Journal of the Structural Division, ASCE, 100

Proceedings ofthe 3rd US-Japan Workshop on Improvement ofStructural Design and Construction Practices, ATC-15-2 Report, Applied Technology Council, Redwood City, California.

(ST9): 1813-1820.

Nassar AA and H. Krawinkler, 1991, "Seismic Demandsfor SDOF and MDOF Systems," John A. Blume Earthquake Engineering Center, Report No. 95, Stanford University, Stanford, California. Newmark, N.M. and W.J. Hall, 1982, Earthquake Spectra and Design, EERI Monograph Series, EERI, Oakland. Osteraas, J.D. and H. Krawinkler, 1990, Strength and Ductility Considerations in Seismic Design, John A. Blume Earthquake Engineering Center, Report 90, Stanford University, California. Paulay, T. and MJ.N. Priestley, 1992, Seismic Design ofReinforced Concrete and Masonry Buildings, John Wiley and Sons, New York. Prakash, V., G.H. Powell, and F.C. Filippou, 1992, DRAIN-2DX' Base Program User Guide, Department of Civil Engineering, Report No. UCB/SEMM-92/29, University of California, Berkeley, California. Riddell, R. and N.M. Newmark, 1979, Statistical Analysis of the Response ofNonlinear Systems Subject to Earthquakes, Civil Engineering Studies, Structures Research Series, 468, University of Illinois, Urbana, Illinois. Rojahn, C., 1988a, "An investigation of structural response modification factors," Proceedings of the Ninth World Conference on Earthquake Engineering, Kyoto, Japan. Rojahn, C. and G.C. Hart, 1988b, "U.S. code focusing on R-factor ofUBC, ATC-3, and NEHRP,"

52

.----.---------_..._----

SEAOC, 1959, 1974, 1985, 1988, 1990, Recommended Lateral Force Requirements and Commentary, Seismology Committee, Structural Engineers Association of California, Sacramento, California. Uang, C.M. and V.V. Bertero, 1986, Earthquake Simulation Tests and Associated Studies ofa 0.3Scale Model ofa Six-Story Concentrically Braced Steel Structure, Earthquake Engineering Research Center, Report No. UCBIEERC-86110, University of California, Berkeley, California. Uang, C.M. and A Maarouf, 1993, "Safety and economy considerations ofUBC seismic force reduction factors," Proceedings ofthe 1993 National Earthquake Conference, Memphis, Tennessee. Whittaker, AS., C.M. Uang, and V.V. Bertero, 1987, Earthquake Simulation Tests and Associated Studies of a O.3-Scale Model ofa Six-Story Eccentrically Braced Steel Structure, Earthquake Engineering Research Center, Report No. UCBI EERC-87/02, University ofCalifornia, Berkeley, California. Whittaker, AS., C.M. Uang, and V.V. Bertero, 1990, An Experimental Study of the Behavior of Dual Steel Systems, Earthquake Engineering Research Center, Report No. UCBIEERC-88114, University of California, Berkeley, California Wu,1. and R.D. Hanson, 1989, "Study of inelastic spectra with high damping," ASCE Journal of Structural Engineering, 115 (6): 1412-1431.

References

ATC-19