simplified performance based earthquake engineering

Department of Civil and Environmental Engineering

Stanford University

Report No.

The John A. Blume Earthquake Engineering Center was established to promote research and education in earthquake engineering. Through its activities our understanding of earthquakes and their effects on mankind’s facilities and structures is improving. The Center conducts research, provides instruction, publishes reports and articles, conducts seminar and conferences, and provides financial support for students. The Center is named for Dr. John A. Blume, a well-known consulting engineer and Stanford alumnus. Address: The John A. Blume Earthquake Engineering Center Department of Civil and Environmental Engineering Stanford University Stanford CA 94305-4020 (650) 723-4150 (650) 725-9755 (fax) [email protected] http://blume.stanford.edu

©2009 The John A. Blume Earthquake Engineering Center

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©Copyright by Farzin Zareian 2006

All Rights Reserved

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ABSTRACT

This study proposes a simplified approach towards Performance-Based Earthquake Engineering

(PBEE) and is limited to Performance-Based Design (PBD) and Performance-Based Assessment

(PBA). The proposed simple PBD procedure incorporates three domains: Hazard Domain,

Structural System Domain, and Loss Domain in which the focus is on the mean values of the

ground motion intensity, building response, and losses. This simple PBD procedure helps

engineers make decisions on global structural parameters and the choice of an effective structural

system and material based on several performance targets that are defined upfront. This

conceptual design should be followed by a detailed performance assessment phase in which the

building performance is evaluated considering all sources of uncertainty and final design

decisions are made.

The ease of the simple PBD approach and its semi-graphical presentation (denoted as

Design Decision Support System, DDSS) will provide engineers with an insight about the

contribution of different building subsystems to the building total loss. A byproduct of this

simple PBD process is that it can be used as simple PBA process by only changing the flow of

information from ground motion hazard to loss estimation. Special consideration is given to

design for collapse prevention. We show a simple way for incorporating different sources of

uncertainty (aleatory and epistemic) in the proposed simple PBD approach for collapse

prevention. The effectiveness of the proposed PBD and PBA process is illustrated through a

simple example.

The DDSS proposed in this study is supported with a database of structural response

parameters obtained for a wide-range of combination in structural parameters of frame and wall

structural systems. This research suggests a new way for describing structural component

parameters by which the component backbone curve and cyclic deterioration parameter are

defined based on the inelastic characteristics of the component. Using the aforementioned

database of structural response parameters, sensitivity of several building response parameters to

variations of structural properties was studied. Closed-form equations for addressing the collapse

potential of a building based on corresponding structural parameters were developed that may

assist the proposed simplified PBEE.

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ACKNOWLEDGMENTS

This work was supported primarily by the Earthquake Engineering Research Centers Program of

the National Science Foundation through the Pacific Earthquake Engineering Research Center

(PEER). Any opinions, findings, and conclusion or recommendations expressed in this material

are those of the author(s) and do not necessarily reflect those of the National Science Foundation.

This document was originally published as the Ph.D. dissertation of the first author under the

supervision of the second author. The authors would like to thank Professors Allin Cornell,

Gregory G. Deierlein, and Eduardo Miranda, for providing their constructive feedback and

comments on this manuscript.

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CONTENTS

1 INTRODUCTION ..................................................................................................................1

1.1 Objective and scope ........................................................................................................1

1.2 Objective and scope ........................................................................................................2

2 PERFORMANCE-BASED ASSESSMENT AND PERFORMANCE-BASED

DESIGN...................................................................................................................................5

2.1 Introduction.....................................................................................................................5

2.2 Performance-based assessment methodology.................................................................5

•Hazard Analysis 6 •Structural Analysis 7 •Damage Analysis 7 •Loss Analysis 8

2.3 Conceptual performance-based design methodology .....................................................9

2.3.1 Hazard Domain..................................................................................................11

2.3.2 Loss Domain......................................................................................................13

2.3.3 Structural System Domain.................................................................................14

2.3.4 Framework for a Design Decision Support System ..........................................15

2.3.5 Approximation in Design Decision Support System ........................................19

2.4 Simplified performance-based assessment methodology .............................................20

3 LOSS DOMAIN....................................................................................................................27

3.1 Introduction...................................................................................................................27

3.2 Background in earthquake loss estimation and overview of the loss domain

considered in this study.................................................................................................27

3.3 Definition of subsystems...............................................................................................29

3.4 Methodology for development of mean loss curves for subsystems tems....................30

3.4.1 Mathematical approach for development of mean loss curves for

subsystems ........................................................................................................30

3.4.2 Component level mean loss functions...............................................................33

3.4.3 Probability of being in a damage state for a component ...................................36

3.5 Samples of subsystem mean loss curves.......................................................................37

3.5.1 Story-level subsystems and their mean loss curves...........................................38

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3.5.2 Building-level subsystems and their mean loss curves .....................................39

3.6 Building loss at collapse................................................................................................40

3.7 Summary .......................................................................................................................41

4 STRUCTURAL SYSTEM DOMAIN: MODELING, PARAMETER

SELECTION, AND ANALYSIS .........................................................................................45

4.1 Introduction...................................................................................................................45

4.2 Structural systems and behavior....................................................................................47

4.2.1 Moment-resisting frames behavior under seismic loads ...................................47

4.2.2 Shear walls behavior under seismic loads.........................................................49

4.3 Structural Component Model........................................................................................50

4.3.1 Backbone curve .................................................................................................51

4.3.2 Hysteretic Model ...............................................................................................55

4.3.3 Cyclic deterioration model ................................................................................56

4.4 Generic moment-resisting frames and range of structural parameters..........................58

4.4.1 Geometry and number of stories of generic moment-resisting frames .............59

4.4.2 Fundamental Period of generic moment-resisting frame structures..................59

4.4.3 Viscous damping of generic moment-resisting frame structures ......................60

4.4.4 Variation of stiffness and strength along the height of generic moment-

resisting frame structures ..................................................................................60

4.4.5 Base shear strength of generic moment-resisting frame structures ...................61

4.4.6 Variation of column to beam strength ratio.......................................................62

4.4.7 Plastic hinge rotation capacity θp of generic moment-resisting frames

components .......................................................................................................62

4.4.8 Post-capping rotation capacity ratio θpc/θp of generic moment-resisting

frames components............................................................................................63

4.4.9 Capping strength ratio Mc/My of generic moment-resisting frames

components .......................................................................................................64

4.4.10 Cyclic deterioration parameter of generic moment-resisting frames

components .......................................................................................................64

4.5 Generic shear walls and range of structural parameters................................................64

4.5.1 Geometry and number of stories of generic shear walls ...................................65

4.5.2 Fundamental Period of generic shear walls.......................................................65

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4.5.3 Viscous damping of generic shear walls ...........................................................66

4.5.4 Bending strength of generic shear walls............................................................66

4.5.5 Variation of stiffness and strength along the height of generic shear

walls ..................................................................................................................66

4.5.6 Plastic hinge rotation capacity θp of generic shear walls components ..............67

4.5.7 Post-capping rotation capacity ratio θpc/θp of generic shear walls

components .......................................................................................................67

4.5.8 Capping strength ratio Mc/My of generic shear walls components ...................68

4.5.9 Cyclic deterioration parameter λ of generic shear walls components...............68

4.6 Development of database of structural response parameters ........................................69

4.6.1 Base case generic structural systems.................................................................69

4.6.2 Variation in global modeling variables of base case generic structures............70

4.6.3 Variation in component variables of base case generic structures ....................71

4.6.4 Incremental Dynamic Analysis and determination of structural response

parameters .........................................................................................................72

5 STRUCTURAL SYSTEM DOMAIN: RESPONSE OF STRUCTURAL

SYSTEMS CONDITIONED ON NO-COLLAPSE...........................................................89

5.1 Introduction...................................................................................................................89

5.2 Statistical evaluation of EDP|IM...................................................................................90

5.3 Sensitivity of EDPs related to nonstructural losses to variation of structural

parameters in generic structures....................................................................................92

5.3.1 Sensitivity of story-level and building-level drift-related EDPs to

variation of structural parameters for a case study generic moment-

resisting frame structure....................................................................................93

5.3.2 Sensitivity of story-level and building-level acceleration -related EDPs

to variation of structural parameters for a case study generic moment-

resisting frame...................................................................................................97

5.3.3 Sensitivity of story-level and building-level drift-related EDPs to

structural parameters variation for a case study generic shear wall

structure.............................................................................................................98

5.3.4 Sensitivity of story-level and building-level acceleration-related EDPs

to structural parameters variation for a case study generic shear wall............100

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5.4 Summary .....................................................................................................................101

6 STRUCTURAL SYSTEM DOMAIN: ASSESSMENT OF PROBABILITY OF

COLLAPSE AND DESGIN FOR COLLAPSE SAFETY..............................................129

6.1 Objective and scope ....................................................................................................129

6.2 Collapse fragility curves .............................................................................................130

6.3 Aleatory and epistemic uncertainties in probability of collapse .................................131

6.4 Design for tolerable probability of collapse at discrete hazard levels and MAF

of collapse ...................................................................................................................134

6.5 Sensitivity of collapse fragility curves to structural parameters variation in

generic structures ........................................................................................................136

6.5.1 Sensitivity of collapse fragility curves to structural parameters variation

in case study generic moment-resisting frame ................................................137

Effect of post-capping rotation capacity ratio θpc/θp 139 Effect of cyclic deterioration parameter λ 139 Effect of variation of stiffness and strength along the height 140 Effect of column to beam strength ratio 140 Effect of P-Delta 141

6.5.2 Sensitivity of collapse fragility curves to structural parameters variation

in case study generic shear wall ......................................................................141

Effect of plastic hinge rotation capacity θp 142 Effect of post-capping rotation capacity ratio θpc/θp 143 Effect of cyclic deterioration parameter λ 143 Effect of reduction of bending strength along the height 144 Effect of P-Delta 144

6.6 Sensitivity of collapse fragility curves to ground motion ε ........................................145

6.7 Development of closed-form equations for estimation of median and dispersion

of collapse fragility curves of generic structures ........................................................146

6.7.1 Development of closed-form equation for estimation of median of

collapse capacity of generic moment-resisting frame structure......................147

6.7.2 Development of closed-form equation for estimation of median of

collapse capacity of generic shear wall structures ..........................................149

6.8 Summary .....................................................................................................................150

7 IMPLEMENTATION OF PROPOSED DESIGN DECISION SUPPORT

SYSTEM..............................................................................................................................185

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7.1 Introduction.................................................................................................................185

7.2 Implementation of DDSS for conceptual design using building-level

subsystems ..................................................................................................................186

7.2.1 Information to be provided in the Hazard Domain. ........................................186

7.2.2 Information to be provided in the Loss Domain. ............................................187

7.2.3 Information to be provided in the Structural System Domain. .......................188

7.2.4 Implementation of DDSS using building-level subsystems............................189

7.3 Implementation of DDSS for conceptual design using story-level subsystems .........197

7.4 Incorporating the effect of epistemic uncertainty in design for collapse safety .........201

7.5 Concluding remarks ....................................................................................................203

8 SUMMARY AND CONCLUSIONS.................................................................................223

8.1 Development of a framework for Simplified PBEE ...................................................223

8.2 Introduction of Subsystem concept.............................................................................226

8.3 Introduction of a new method for describing structural components behavior

(monotonic and cyclic)................................................................................................226

8.4 Development of a comprehensive database of structural response parameters

for combinations of structural systems parameters.....................................................227

8.5 Assessment of sensitivity of structural response parameters to variation of

structural parameters ...................................................................................................227

8.6 Concluding remarks and suggestions for future work ................................................229

APPENDIX A VARIATION OF STIFFNESS AND STRENGTH ALONG THE

HEIGHT OF GENERIC MOMENT-RESISTING FRAMES....................241

A.1 Relation between stiffness of beams and columns in moment-resisting frame

structures .....................................................................................................................241

A.2 Variation of stiffness along the height of moment-resisting frame structures ............242

A.3 Variation of strength along the height of moment-resisting frame structures ............245

APPENDIX B PROPERTIES OF BASE CASE GENERIC FRAMES AND

WALLS.............................................................................................................253

B.1 Properties of the base case generic moment-resisting frame models..........................253

B.2 Properties of the base case generic shear wall models................................................254

APPENDIX C LINEAR MULTIVARIATE REGRESSION ANALYSIS ..........................281

APPENDIX D MOELING PLASTIC HINGES AND STRUCTURAL DAMPING..........285

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D.1 Introduction.................................................................................................................285

D.2 Rayleigh Damping ......................................................................................................285

D.3 Implementation of Rayleigh Damping in DRAIN-2DX and State of the

Problem .......................................................................................................................286

D.4 Solution to the Damping Problem...............................................................................287

D.5 Development of an Equivalent Element Model for Beam Element............................289

D.5.1 Development of an equivalent element model for beam element (two-

end-spring) ......................................................................................................289

D.5.2 Development of an equivalent element model for beam element (one-

end-spring) ......................................................................................................291

APPENDIX E LIST OF EDPS FOR GENERIC STRUCTURAL SYSTEMS...................295

E.1 List of EDPs for generic moment resisting frames .....................................................295

E.2 List of EDPs for generic shear walls...........................................................................300

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LIST OF FIGURES

Fig. 2.1 Performance-Based Assessment methodology layout.................................................... 22

Fig. 2.2 Layout of Design Decision Support System (DDSS)..................................................... 22

Fig. 2.3 DDSS for acceptable monetary loss at discrete hazard level. ........................................ 23

Fig. 2.4 DDSS for tolerable life loss at discrete hazard level. ..................................................... 23

Fig. 2.5 DDSS for acceptable mean annual monetary loss.......................................................... 24

Fig. 2.6 DDSS for tolerable mean annual frequency of collapse................................................. 24

Fig. 2.7 Simplified PBA for estimation of monetary loss at discrete hazard. level..................... 25

Fig. 3.1 Damage fragility curves for partitions (information obtained from Taghavi &

Miranda 2003) ................................................................................................................ 42

Fig. 3.2 Probability of being in different damage states for generic non-structural drift

sensitive components in Van Nuys Hotel Building (information obtained from

Taghavi & Miranda 2003) .............................................................................................. 42

Fig. 3.3 Replacement/repair cost of partition wall for different damage states (2001

dollars) (information obtained from Taghavi & Miranda 2003) .................................... 43

Fig. 3.4 Mean loss curve for partition wall (2001 dollars) .......................................................... 43

Fig. 3.5 Generic form of mean monetary loss curve for story-level non-structural drift

sensitive subsystem ........................................................................................................ 44

Fig. 3.6 Generic form of mean monetary loss curve for building-level structural

subsystem ....................................................................................................................... 44

Fig. 4.1 Modes of deformation in structures (after Miranda, 1999) ............................................ 73

Fig. 4.2 Moment-resisting frames and corresponding structural model: (a) geometry, (b)

structural model .............................................................................................................. 73

Fig. 4.3 Modes of failure in reinforced concrete shear walls: (a) shear wall loading

during a seismic event, (b) failure due to yielding of flexural reinforcement, (c)

failure due to diagonal tension, (d) failure due to sliding shear, and (e) failure

due to shear/flexural yielding (after Paulay and Priestley, 1992) .................................. 74

Fig. 4.4 Cantilever shear wall and corresponding structural model: (a) geometry, (b)

elastic and inelastic deformations, and (c) structural model .......................................... 74

Fig. 4.5 Component back-bone curve and its parameters: (a) old definitions, (b) new

definitions ....................................................................................................................... 75

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Fig. 4.6 Sensitivity of yield rotation and plastic hinge rotation to (M/V)/h ................................ 76

Fig. 4.7 Definition of θp and its approximation to plastic rotation. ............................................. 76

Fig. 4.8 Peak-Oriented hysteretic model used in this study......................................................... 77

Fig. 4.9 Peak-Oriented hysteresis model with 4 modes of cyclic deterioration (After

Ibarra & Krawinkler 2005)............................................................................................. 77

Fig. 4.10 Peak-Oriented hysteresis model with 4 individual modes of cyclic deterioration

(After Ibarra & Krawinkler 2005) .................................................................................. 78

Fig. 4.11 Relation between number of stories and period of generic moment-resisting

frames (Data obtained from Goel and Chopra, 1997) .................................................... 79

Fig. 4.12 Variation of stiffness along the height of generic moment-resisting frame (N =

8, αt = 0.15) .................................................................................................................... 79

Fig. 4.13 Effect of variation of stiffness along the height of generic moment-resisting

frame (N = 8, αt = 0.15) on first mode shape ................................................................. 80

Fig. 4.14 Yield base shear coefficients γ = Vy /W for generic moment-resisting frames.............. 80

Fig. 4.15 Schematic representation of three variations of column strength in generic

moment-resisting frames (SCB = 1.2-1.2, 2.4-1.2, 2.4-2.4)........................................... 81

Fig. 4.16 Variation of plastic hinge rotation capacity from column test results and the

range used in generic moment-resisting frames (data from Haselton et. al., 2006) ....... 82

Fig. 4.17 Variation of post-capping rotation capacity ratio from column test results and

the range used in generic moment-resisting frames (data from Haselton et. al.,

2006)............................................................................................................................... 82

Fig. 4.18 Variation of capping strength ratio from column test results and the average

value used in generic moment-resisting frame components (data from Haselton

et. al., 2006) .................................................................................................................... 83

Fig. 4.19 Variation of cyclic deterioration parameter from column test results (data from

Haselton et. al., 2006)..................................................................................................... 83

Fig. 4.20 Modeling of generic shear walls used in this study: (a) shear wall global model,

(b) shear wall component model, (c) inelastic spring in the shear wall

component model, and (d) elastic element in shear wall component model.................. 84

Fig. 4.21 Relation between number of stories and period of shear wall structures (Data

obtained from Goel and Chopra, 1997).......................................................................... 85

Fig. 4.22 Shear wall bending strength and yield base shear ......................................................... 85

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Fig. 4.23 Yield base shear coefficients γ = Vy /W for generic shear walls.................................... 86

Fig. 4.24 Variation of plastic hinge rotation capacity from shear wall test results and the

range used in generic shear walls (data from Fardis and Biskinis, 2003) ...................... 86

Fig. 4.25 Sensitivity of pushover curve to post-capping rotation capacity ratio for a 9-

story generic shear wall structure, T = 0.9 sec. .............................................................. 87

Fig. 4.26 Sensitivity of median of collapse capacity to post-capping rotation capacity

ratio for 9-story generic shear wall structure, T = 0.9 sec. ............................................. 87

Fig. 4.27 Map of variation to the base case structural system due to three component

parameters θp, θpc/θp, and λ ............................................................................................ 88

Fig. 4.28 Typical IDA curves for an 8-story generic moment-resisting frame subjected to

the LMSR-N record set .................................................................................................. 88

Fig. 5.1 Incremental Dynamic Analysis and corresponding mean and median value of the

EDP............................................................................................................................... 103

Fig. 5.2 Difference between mean and median value of EDP conditioned on no-collapse

and the mean and median value of EDP for all data .................................................... 103

Fig. 5.3 Pushover curves for case study moment resisting frame with variation in

structural parameters .................................................................................................... 104

Fig. 5.4 Effects of γ on drift demands of case study moment-resisting frame............................ 105

Fig. 5.5 Effects of SCB factor on drift demands of case study moment-resisting frame ........... 106

Fig. 5.6 Effects of Stff.& Str. parameters on drift demands of case study moment-

resisting frame .............................................................................................................. 107

Fig. 5.7 Effects of component θp on drift demands of case study moment-resisting frame ....... 108

Fig. 5.8 Effects of component θpc/θp on drift demands of case study moment-resisting

frame............................................................................................................................. 109

Fig. 5.9 Effects of component cyclic deterioration parameter λ on drift demands of case

study moment-resisting frame ...................................................................................... 110

Fig. 5.10 Effects of γ on acceleration demands of case study moment-resisting frame ............. 111

Fig. 5.11 Effects of SCB factor on acceleration demands of case study moment-resisting

frame............................................................................................................................. 112

Fig. 5.12 Effects of Stiff. & Str. Parameters on acceleration demands of case study

moment-resisting frame................................................................................................ 113

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Fig. 5.13 Effects of component θp on acceleration demands of case study moment-

resisting frame .............................................................................................................. 114

Fig. 5.14 Effects of component θpc/θp on acceleration demands of case study moment-

resisting frame .............................................................................................................. 115

Fig. 5.15 Effects of component cyclic deterioration parameter λ on acceleration demands

of case study moment-resisting frame.......................................................................... 116

Fig. 5.16 Pushover curves for case study shear wall with variation in structural

parameters..................................................................................................................... 117

Fig. 5.17 Effects of γ on drift demands of case study shear wall................................................ 118

Fig. 5.18 Effects of strength distribution on drift demands of case study shear wall ................. 119

Fig. 5.19 Effects of component θp on drift demands of case study shear wall ........................... 120

Fig. 5.20 Effects of component θpc/θp on drift demands of case study shear wall ..................... 121


study shear wall ............................................................................................................ 122

Fig. 5.22 Effects of γ on acceleration demands of case study shear wall ................................... 123

Fig. 5.23 Effects of strength distribution on acceleration demands of case study shear

wall ............................................................................................................................... 124

Fig. 5.24 Effects of component θp on acceleration demands of case study shear wall............... 125

Fig. 5.25 Effects of component θpc/θp on acceleration demands of case study shear wall ......... 126

Fig. 5.26 Effects of component cyclic deterioration parameter λ on acceleration demands

of case study shear wall ................................................................................................ 127

Fig. 6.1 Obtaining collapse fragility curve with Incremental Dynamic Analysis: a)

obtaining data point, b) collapse fragility curve........................................................... 152

Fig. 6.2 Uncertainty and collapse fragility curve........................................................................ 153

Fig. 6.3 Standard Gaussian variate KY for different confidence levels ...................................... 153

Fig. 6.4 Sensitivity of collapse fragility curve parameters to plastic hinge rotation

capacity θp and base shear coefficient γ in case study generic moment-resisting

frame: a&b) sensitivity of ηc, c) sensitivity of βRC, d) Pushover curve ....................... 154

Fig. 6.5 Sensitivity of collapse fragility curve parameters to post-capping plastic hinge

rotation capacity θpc/θp and base shear coefficient γ in case study generic

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moment-resisting frame: a&b) sensitivity of ηc, c) sensitivity of βRC, d)

Pushover curve ............................................................................................................. 155

Fig. 6.6 Sensitivity of collapse fragility curve parameters to cyclic deterioration

parameter λ and base shear coefficient γ in case study generic moment-resisting

frame: a&b) sensitivity of ηc, c) sensitivity of βRC, d) Pushover curve ....................... 156

Fig. 6.7 Sensitivity of collapse fragility curve parameters to variaton of stiffness and

strength along the height and base shear coefficient γ in case study generic

moment-resisting frame: a&b) sensitivity of ηc, c) sensitivity of βRC, d)

Pushover curve ............................................................................................................. 157

Fig. 6.8 Sensitivity of collapse fragility curve parameters to column to beam strength

ratio and base shear coefficient γ in case study generic moment-resisting frame:

a&b) sensitivity of ηc, c) sensitivity of βRC, d) Pushover curve ................................... 158

Fig. 6.9 Sensitivity of collapse fragility curve parameters to P-Delta effects and base

shear coefficient γ in case study generic moment-resisting frame: a&b)

sensitivity of ηc, c) sensitivity of βRC, d) Pushover curve ............................................ 159

Fig. 6.10 Sensitivity of collapse fragility curve parameters to plastic hinge rotation

capacity and base shear coefficient γ in case study generic shear wall: a&b)


Fig. 6.11 Sensitivity of collapse fragility curve parameters to post-capping rotation

capacity ratio and base shear coefficient γ in case study generic shear wall: a&b)


Fig. 6.12 Sensitivity of collapse fragility curve parameters to cyclic deterioration

parameter and base shear coefficient γ in case study generic shear wall: a&b)


Fig. 6.13 Sensitivity of collapse fragility curve parameters to reduction of bending

strength along the height and base shear coefficient γ in case study generic shear

wall: a&b) sensitivity of ηc, c) sensitivity of βRC, d) Pushover curve.......................... 163

Fig. 6.14 Sensitivity of collapse fragility curve parameters to P-Delta effects and base

shear coefficient γ in case study generic shear wall: a&b) sensitivity of ηc, c)

sensitivity of βRC, d) Pushover curve............................................................................ 164

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Fig. 6.15 Sensitivity of median of collapse capacity to ε in case study moment-resisting

frame: a) ε from BJF, and b) ε from AS97................................................................... 165

Fig. 6.16 Sensitivity of median of collapse capacity to ε in case study shear wall: a) ε

from BJF, and b) ε from AS97 ..................................................................................... 166

Fig. 6.17 Discrimination map of data for multivariate regression analysis of median

collapse capacity in generic moment-resisting frames ................................................. 167

Fig. 6.18 Base factor b0,MRF for estimation of median of collapse capacity in moment-

resisting frames............................................................................................................. 168

Fig. 6.19 Estimation error (epistemic) in estimation of median of collapse capacity in

moment-resisting frames .............................................................................................. 169

Fig. 6.20 Scatter plots for median of collapse capacity in moment-resisting frames: (a) N

= 4 & T1 = 0.4sec., (b) N = 4 & T1 = 0.6sec., (c) N = 4 & T1 = 0.8sec., (d) N = 8

& T1 = 0.8sec., (e) N = 8 & T1 = 1.2sec., (f) N = 8 & T1 = 1.6sec............................... 170

Fig. 6.21 Scatter plots for median of collapse capacity in shear walls: (a) N = 12 & T1 =

1.2sec., (b) N = 12 & T1 = 1.8sec., (c) N = 12 & T1 = 2.4sec., (d) N = 16 & T1 =

1.6sec., (e) N = 16 & T1 = 2.4sec., (f) N = 16 & T1 = 3.2sec....................................... 171

Fig. 6.22 Ratio of computed median collapse capacity to estimated median collapse

capacity for variation in structural parameters of moment-resisting frames: (a)

N, (b) T1, (c) γ, (d) θp, (e) θpc/θp, (f) λ .......................................................................... 172

Fig. 6.23 Comparison between the computed and estimated value for median of collapse

capacity for moment-resisting frames as a function of θp: (a) N = 4 T1 = 0.4, (b)

N = 4 T1 = 0.6, (c) N = 4 T1 = 0.8, (d) N = 16 T1 = 1.6, (e) N = 16 T1 = 2.4, (f) N

= 16 T1 = 3.2................................................................................................................. 173


capacity for moment-resisting frames as a function of θpc/θp: (a) N = 4 T1 = 0.4,

(b) N = 4 T1 = 0.6, (c) N = 4 T1 = 0.8, (d) N = 16 T1 = 1.6, (e) N = 16 T1 = 2.4,

(f) N = 16 T1 = 3.2 ........................................................................................................ 174


capacity for moment-resisting frames as a function of λ: (a) N = 4 T1 = 0.4, (b)

N = 4 T1 = 0.6, (c) N = 4 T1 = 0.8, (d) N = 16 T1 = 1.6, (e) N = 16 T1 = 2.4, (f) N

= 16 T1 = 3.2................................................................................................................. 175

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collapse capacity in generic shear wall ........................................................................ 176

Fig. 6.27 Base factor b0,SW for estimation of median of collapse capacity in shear walls .......... 177


shear walls .................................................................................................................... 178



0.4sec., (e) N = 8 & T1 = 0.6sec., (f) N = 8 & T1 = 0.8sec........................................... 179



1.6sec., (e) N = 16 & T1 = 2.4sec., (f) N = 16 & T1 = 3.2sec....................................... 180

Fig. 6.31 Ratio of computed median collapse capacity to estimated median collapse

capacity for variation in structural parameters of shear walls: (a) N, (b) T1, (c) γ,

(d) θp, (e) θpc/θp, (f) λ ................................................................................................... 181


capacity for shear walls as a function of θp: (a) N = 4 T1 = 0.2, (b) N = 4 T1 =

0.3, (c) N = 4 T1 = 0.4, (d) N = 16 T1 = 0.8, (e) N = 16 T1 = 1.2, (f) N = 16 T1 =

1.6 ................................................................................................................................. 182


capacity for shear walls as a function of θpc/θp: (a) N = 4 T1 = 0.2, (b) N = 4 T1 =

0.3, (c) N = 4 T1 = 0.4, (d) N = 16 T1 = 0.8, (e) N = 16 T1 = 1.2, (f) N = 16 T1 =

1.6 ................................................................................................................................. 183


capacity for shear walls as a function of λ: (a) N = 4 T1 = 0.2, (b) N = 4 T1 = 0.3,

(c) N = 4 T1 = 0.4, (d) N = 16 T1 = 0.8, (e) N = 16 T1 = 1.2, (f) N = 16 T1 = 1.6........ 184

Fig. 7.1 Mean hazard curves for T1 = 0.4, 0.8, 1.6 for the location of the design example........ 206

Fig. 7.2 Mean $loss curves for building-level subsystems: (a) NSDSS, (b) NSASS................. 206

Fig. 7.3 Sample of mean $loss curves for building-level SS subsystems: (a) moment-

resisting frame, (b) shear wall ...................................................................................... 207

xx

Fig. 7.4 Example of DDSS implementation in conceptual design with performance

targets at discrete hazard levels, using building-level subsystems and moment-

resisting frame structural systems GF1, GF2, GF3, and GF4 ...................................... 208


targets at discrete hazard levels, using building-level subsystems and shear wall

structural systems GW1, GW2, and GW3.................................................................... 209


targets at discrete hazard levels, using building-level subsystems and comparing

“best” shear wall (GW2) and “best” moment-resisting frame(GF2) alternatives ........ 210

Fig. 7.7 Expected value of $loss given IM for “best” moment-resisting frame (GF2) and

“best” shear wall(GW2) alternatives ............................................................................ 211

Fig. 7.8 MAF of $loss for “best” moent-resisting frame (GF2) and “best” shear wall

(GW2) alternatives ....................................................................................................... 211

Fig. 7.9a Example of DDSS implementation in conceptual design with performance

targets at discrete hazard levels, using story-level subsystems and moment-

resisting frame structural systems GF1, GF2, GF3, and GF4, (STORY 1) ................. 212

Fig. 7.9b Example of DDSS implementation in conceptual design with performance



Fig. 7.9c Example of DDSS implementation in conceptual design with performance




targets at discrete hazard levels, using story-level subsystems and shear wall

structural systems GW1, GW2, and GW4, (STORY 1)............................................... 215







xxi


targets at discrete hazard levels, using story-level subsystems and comparing

“best” shear wall (GW3) and “best” moment resisting frame (GF2) alternatives

(STORY 1) ................................................................................................................... 218




(STORY 4) ................................................................................................................... 219




(STORY 8) ................................................................................................................... 220

Fig. 7.12 Implementation of DDSS for collapse safety, incorporating epistemic and

aleatory uncertainties for design alternative GF2......................................................... 221


aleatory uncertainties for design alternative GW3 ....................................................... 221

Fig. 7.14 Effect of confidence level on probability of collapse at 2/50 hazard level for

design alternatives GF2 and GW3................................................................................ 222

Fig. 7.15 Effect of confidence level on MAF of collapse for design alternatives GF2 and

GW3 ............................................................................................................................. 222

Fig. A.1 Modes of deformation in structures (after Miranda 1999). .......................................... 248

Fig. A.2 Variation along the height of ρ in SAC structures (pre-Northridge design) ................ 249

Fig. A.3 Variation along the height of beam stiffness ratio in SAC structures (pre-

Northridge design)........................................................................................................ 249

Fig. A.4 Variation along the height of column stiffness ratio in SAC structures (pre-

Northridge design)........................................................................................................ 250

Fig. A.5 Schematic representation of three variations along the height of beam moment of

inertia (“Shear”, “Unif”, “Int.”).................................................................................... 250

Fig. A.6 Effect of different ρavg and variation along the height of beam moment of inertia

on first mode period of a 9-story moment-resisting frame structure............................ 251

Fig. A.7 Effect of different variation along the height of beam and column moment of

inertia on first mode period of a 9-story moment-resisting frame................................ 251

xxii

Fig. A.8 Schematic representation of three variations of column strength in generic

moment-resisting frames (SCB = 1.2-1.2, 2.4-1.2, 2.4-2.4)......................................... 252

Fig. C.1 α Critical values for F Distribution with n1 and n2 degrees of freedom , where

( )1 2 1 2, , ;n n n nP F f α α≥ = . (a) α = 0.05, (b) α = 0.01................................................................. 284

Fig. D.1 Beam element and equivalent model consisting of an elastic beam element with

springs at both ends ...................................................................................................... 292

Fig. D.2 Stiffness matrix of the elastic beam element ............................................................... 292

Fig. D.3 Stiffness matrix of the original beam element............................................................. 293

Fig. D.4 Stiffness coefficients for elastic beam in the equivalent model, springs at both

ends of elastic element.................................................................................................. 293


spring at one end........................................................................................................... 294

Fig. D.6 Stiffness coefficients for elastic beam in the equivalent model, spring at end i of

elastic element .............................................................................................................. 294

-xxiii-

LIST OF TABLES

Table A.1: Average of ρ for SAC structures ...............................................................................248

Table B.1 Modal properties of generic moment-resisting frames: N = 4, T1 = var., Stiff.

& Str. = Shear ...........................................................................................................255


& Str. = Int................................................................................................................255

Table B.3 Modal properties of generic moment-resisting frame: N = 4, T1 = var., Stiff. &

Str. = Unif. ................................................................................................................255


& Str. = Shear ...........................................................................................................256


& Str. = Int................................................................................................................256


Str. = Unif. ................................................................................................................256


& Str. = Shear ...........................................................................................................257


& Str. = Int................................................................................................................257


& Str. = Unif. ............................................................................................................258

Table B.10 Modal properties of generic moment-resisting frames: N = 16, T1 = var.,

Stiff. & Str. = Shear ..................................................................................................258


Stiff. & Str. = Int.......................................................................................................259


Stiff. & Str. = Unif. ...................................................................................................259

Table B.13 Weight, stiffness and strength properties of generic moment-resisting frames:

N = 4, T1 = 0.4, Stiff. & Str. = Shear. ......................................................................260


N = 4, T1 = 0.4, Stiff. & Str. = Int. ...........................................................................260

xxiv


N = 4, T1 = 0.4, Stiff. & Str. = Unif. ........................................................................260




N = 4, T1 = 0.6, Stiff. & Str. = Int. ...........................................................................261


N = 4, T1 = 0.6, Stiff. & Str. = Unif. ........................................................................261




N = 4, T1 = 0.8, Stiff. & Str. = Int. ...........................................................................262


N = 4, T1 = 0.8, Stiff. & Str. = Unif. ........................................................................262




N = 8, T1 = 0.8, Stiff. & Str. = Int. ...........................................................................263


N = 8, T1 = 0.8, Stiff. & Str. = Unif. ........................................................................264




N = 8, T1 = 1.2, Stiff. & Str. = Int. ...........................................................................265


N = 8, T1 = 1.2, Stiff. & Str. = Unif. ........................................................................265




N = 8, T1 = 1.6, Stiff. & Str. = Int. ...........................................................................266


N = 8, T1 = 1.6, Stiff. & Str. = Unif. ........................................................................267

xxv


N = 12, T1 = 1.2, Stiff. & Str. = Shear. ....................................................................267


N = 12, T1 = 1.2, Stiff. & Str. = Int. .........................................................................268


N = 12, T1 = 1.2, Stiff. & Str. = Unif. ......................................................................268




N = 12, T1 = 1.8, Stiff. & Str. = Int. .........................................................................269


N = 12, T1 = 1.8, Stiff. & Str. = Unif. ......................................................................270




N = 12, T1 = 2.4, Stiff. & Str. = Int. .........................................................................271


N = 12, T1 = 2.4, Stiff. & Str. = Unif. ......................................................................271




N = 16, T1 = 1.6, Stiff. & Str. = Int. .........................................................................272


N = 16, T1 = 1.6, Stiff. & Str. = Unif. ......................................................................273




N = 16, T1 = 2.4, Stiff. & Str. = Int. .........................................................................274


N = 16, T1 = 2.4, Stiff. & Str. = Unif. ......................................................................274



xxvi


N = 16, T1 = 3.2, Stiff. & Str. = Int. .........................................................................275


N = 16, T1 = 3.2, Stiff. & Str. = Unif. ......................................................................276

Table B.49 Modal properties of generic shear wall: N = 4, T1 = var..........................................276

Table B.50 Modal properties of generic shear wall: N = 8, T1 = var..........................................276

Table B.51 Modal properties of generic shear wall: N = 12, T1 = var........................................277

Table B.52 Modal properties of generic shear wall: N = 16, T1 = var........................................277

Table B.53 Generic shear wall weight and strength properties: N = 4, T1 = var., Str. =

var. ............................................................................................................................277


var. ............................................................................................................................278


var. ............................................................................................................................278


var. ............................................................................................................................279

-1-

1 INTRODUCTION

1.1 Objective and scope

Performance-Based Earthquake Engineering (PBEE) provides the knowledge needed to address

cost-effective management of resources for the design and construction of man-made facilities in

the context of seismic hazard. As stated by Krawinkler and Miranda (2004) “Performance-Based

Earthquake Engineering (PBEE) implies design, evaluation, construction, monitoring the

function, and maintenance of engineered facilities whose performance under common and

extreme loads respond to the diverse needs and objectives of owners-users and society”.

Development of methodologies for achieving the goals of PBEE has been the focus of research

programs in many countries and research centers around the globe.

The Pacific Earthquake Engineering Research (PEER) Center is one of the leading

research centers involved in the development of PBEE knowledge. PEER has developed a

detailed methodology for the seismic Performance-Based Assessment (PBA) of buildings,

bridges, and other engineered facilities. This methodology is capable of predicting building

performance in a probabilistic format. The primary motivation for the study summarized in this

dissertation was to develop a methodology for Performance-Based Design (PBD). The goal was

do develop an easy-to-use approach for PBD that can be used by engineers in their day-to-day

design practice and by code writers for the development of performance-based seismic design

codes.

In the process of development of the PBD methodology, several auxiliary targets were

identified and addressed in this research. The major objectives of this research are as follows:

• Development of a simplified PBD methodology that can be used for a conceptual design

of structural systems.

2

• Development of a simplified PBA methodology that can provide a rapid assessment of

the performance of a building.

• Identification and quantification of structural parameters that permit a description of

component behavior at all levels of performance.

• Development of a database of structural response parameters for a wide range of

structural system and component parameters.


The general methodology for PBA and PBD is discussed in Chapter 2. In this chapter we first

describe the detailed PBA methodology developed in PEER. We will show the potential of PBD

in conceptual design and the benefits provided by this approach to engineers in the process of

making informed decisions on basic structural parameters. The graphical interface developed for

conceptual PBD (denoted as Design Decision Support System, DDSS) is introduced in this

chapter. It is demonstrated how conceptual PBD and simplified PBA can be accomplished by

using this interface. The DDSS comprises the following three domains: Hazard Domain, Loss

Domain, and Structural System Domain. Each domain deals with mean values of random

variables that describe the hazard, loss, and structural response. The use of mean values is the

basis of the simplified PBEE approach proposed in this research.

The DDSS proposed in Chapter 2 relies heavily on information provided in the three

aforementioned domains. The design decision making in the DDSS is accomplished in the

Structural System Domain in which several design alternatives are checked for appropriateness

according to specified performance objectives. The supporting information on design alternatives

is developed through extensive nonlinear dynamic analyses and stored in a database that is

described in Chapter 4.

Important aspects of the Hazard Domain are discussed in Chapter 2. These include

selection of the ground motions intensity measure (scalar or vector) and of a set of ground

motion records that are representative for the location of the building.

Chapter 3 discusses the Loss Domain. This domain describes the relation between

structural response parameters and losses in building components and subsystems. The concept

of a “subsystem” is introduced to address losses in a subset of building components. We present

a general approach for obtaining such relations in the form of mathematical equations.

3

Chapters 4, 5, and 6 of this dissertation discuss the Structural System Domain. The DDSS

relies heavily on the information provided in this domain for design decision making. In Chapter

4 the structural systems and structural parameters used for developing the database of structural

response parameters are discussed in detail. A new approach for describing the structural

component backbone curve and cyclic deterioration parameters is presented in this chapter.

Chapter 5 illustrates relations between the ground motions intensity and structural

response parameters and presents examples of the sensitivity of response parameters to variations

in system and component parameters.

Chapter 6 is about structural collapse. In this chapter we first show how we quantify the

collapse potential of a building through collapse fragility functions and how these functions are

computed. We present methodologies for incorporating different sources of uncertainty in the

process of obtaining the collapse potential of a building. Sensitivity of the collapse potential of a

building to variations of structural parameters is studied and closed-form equations are provided

to help predicting this potential.

Chapter 7 shows the implementation of the DDSS described in Chapter 2 using the

information provided in Chapters 2 to 6 for conceptual design of a building. We show the

effectiveness of the DDSS for conducting a conceptual PBD based on multiple performance

objectives. Special attention is given to design for collapse safety.

Chapter 8 provides a summary of major contributions and conclusions made in this

research. Areas for future work are identified, which could be used as departure points for future

research. Several appendices are included at the end of this dissertation that summarizes

auxiliary information used in conducting this research.

5

2 PERFORMANCE-BASED ASSESSMENT AND PERFORMANCE-BASED DESIGN

2.1 Introduction

In the context of this research, Performance-Based Earthquake Engineering (PBEE) implies

design and assessment of a building whose performance complies with objectives expressed by

stakeholders (owner, user, and society). Implementation of PBEE in quantitative evaluation of

the performance of a given building (either an existing building or a completed design of a new

building) is denoted here as Performance-Based Assessment (PBA). PBA provides stakeholders

with information about the building (usually expressed in probabilistic terms) that facilitates

informed decision making for risk management. Performance-Based Design (PBD) is another

implementation of PBEE that incorporates desired performance in the design of a new building.

In this chapter we first focus on the rigorous PBA methodology developed by researchers

of the Pacific Earthquake Engineering Research (PEER) Center. This PBA methodology is

capable of predicting building performance in a probabilistic format. In the second part of this

chapter we illustrate the potential of PBD in the conceptual design process. We show that using

the proposed conceptual PBD methodology, engineers can make informed decisions on basic

structural parameters (e.g., structural system and its stiffness, strength and deformation capacity)

that fulfill targeted performance objectives. In the last part of this chapter, a simplified PBA

approach is proposed. This approach enables its users to perform PBA faster and simpler than the

rigorous PBA methodology, at the expense of providing only mean values of performance rather

than a full probabilistic description of performance.

2.2 Performance-based assessment methodology

6

The PBA methodology developed in the PEER Center (Cornell and Krawinkler, 2000;

Krawinkler, 2002; Krawinkler and Miranda, 2004; Moehle and Deierlein, 2004) involves four

types of random variables and incorporates four consecutive stages as shown in Figure 2.1.

These four variables are denoted as: IM, EDP, DM, and DV. The IM (Intensity Measure)

describes the intensity of a ground motion (e.g., spectral acceleration associated with 5%

damping at the first mode period of the building, Sa(T1)); an EDP (Engineering Demand

Parameter) is a structural response parameter (e.g., maximum roof drift); a DM (Damage

Measure) quantifies the level of damage to a building component (e.g., punching shear cracks in

a column-slab connection), and a DV (Decision Variable) describes the performance of the

building (e.g., monetary loss). The analysis processes associated with these stages are described

in the following paragraphs.

• Hazard Analysis

In Hazard Analysis the frequency with which the intensity of a ground motion is

exceeded is calculated. The main output of Hazard Analysis is a seismic hazard curve

that shows the relation between an IM and its annual frequency of exceedance (i.e.,

λ(IM)). The IM could be a scalar (e.g., Sa(T1)) or a vector (e.g., a combination of Sa(T1)

and peak ground acceleration, PGA). Traditionally, Sa(T1) (i.e., scalar IM) has been used

as IM for its simplicity and easiness of computational work. Recently, the use of vector

IMs has shown some advantages in describing ground motion characteristics (Baker and

Cornell, 2005), especially in the case of near fault ground motions (Alavi and

Krawinkler, 2004).

Hazard Analysis is performed deterministically or probabilistically. In Deterministic

Seismic Hazard Analysis (DSHA), the ground motion hazard is evaluated based on a

particular seismic scenario (Kramer, 1996). Probabilistic Seismic Hazard Analysis

(PSHA), first proposed by Cornell (1968), has become the preferred tool for seismic

hazard assessment. It incorporates uncertainties in size, location, and occurrence rate of

earthquakes in the estimation of seismic hazard. The outcome of a PSHA is expressed in

terms of the Mean Annual Frequency (MAF) of exceedance of IM (i.e., λ(IM)) and is

represented by the mean seismic hazard Curve. For instance, if Sa(T1) is used as IM, the

mean hazard curve obtained from PSHA shows the relation between Sa(T1) and λ(Sa(T1))

7

(the MAF in Sa(T1) is exceeded). In this research, mean hazard curves obtained from

PSHA are used.

• Structural Analysis

Given the information provided by PSHA (i.e., seismic hazard and representative ground

motions at the location of the building), and an analytical model of the building, a vector

of structural response parameters (EDPs) is obtained in the Structural Analysis stage.

These EDPs should include all relevant building responses that correlate well with

damage in structural components, non-structural components, and content of the building.

EDPs obtained in this stage are used to relate component damage measures (i.e. DMs) to

IMs. The relationships between IM and EDPs can be obtained through multi-intensity

inelastic response analyses of the building model that incorporates the structural system,

non structural systems, and geotechnical aspects of the building (i.e., Incremental

Dynamic Analysis IDA; Cornell and Vamvatsikos, 2002). The output of the Structural

Analysis stage is a probabilistic assessment of building response (i.e., EDPs) at different

hazard levels: P [ edp ≥EDP | im = IM] (i.e., probability that the variable edp exceeds a

certain value of EDP given that the variable im is equal to a value of IM). For instance,

the maximum interstory drift ratio (IDR) is an EDP that relates well to non-structural

component losses. By performing IDAs and using Sa(T1) as IM, we obtain the relation

between IDR at various stories and Sa(T1) in the form of P[idr ≥ IDR | sa(T1) = Sa(T1)]. A

detailed discussion of the Structural Analysis stage is provided in Chapters 5 and 6 of this

dissertation.

• Damage Analysis

In this stage, EDPs obtained in the Structural Analysis stage are related to damage

measures in building components. Building components are usually categorized into

three types, i.e., structural, non-structural, and content. For each component a variable,

defined as the Damage Measure (DM), describes the level of damage experienced in an

earthquake. The art of Damage Analysis is to firstly identify damage states in building

components (i.e., DMs), and secondly to obtain relationships between EDPs and DMs in

8

the form of P(dm = DM | edp = EDP) (i.e., probability of being in damage state DM, given

that the variable edp is equal to the value of EDP). DMs are defined as a function of level

of damage that trigger different repairs or replacement actions of building components

due to the damage induced by earthquakes. Usually, the relation between EDP and DM is

obtained in the form of a fragility function, which describes the probability that a

component reaches, or exceeds, a damage state given the value of EDP (i.e.

P(dm≥DM | edp = EDP)). The useful form of DM-EDP relations is

P(dm = DM | edp = EDP) and is obtained by subtracting the probabilities of exceeding two

subsequent damage states given the value of EDP.

• Loss Analysis

In this stage, losses (i.e., DVs) due to damages to building components are estimated.

DVs are divided into three categories of losses, which are monetary loss, downtime loss,

and life loss. DVs, in contrast with DMs, which are defined at component level, are

defined at the system and/or building level (e.g., total repair cost, total downtime, and

total number of casualties). The probabilistic representation of such DVs could be in

terms of a scenario-based probability of exceedance of a certain value, or the mean

annual frequency of exceedance value. An in depth description of Loss Analysis is

provided in Chapter 3 of this dissertation.

Using the aforementioned four stages, the process of executing the PBA methodology

can be completed. The outcome of the PBA methodology is a probabilistic representation of

DVs. Two different probabilistic representations of DV are common: scenario-based realization,

and MAF-based realization. The steps for obtaining these realizations are illustrated in Figure 2.1

and are elaborated below.

In the scenario-based realization of DV, the probability of DV exceeding a certain value

given the value of IM is estimated, P[dv ≥ DV | im = IM] (i.e., probability that variable dv

exceeds a given value of DV given that variable im is equal to the value of IM). This probability

is obtained in accordance with the total probability as follows:

( ) ( ) ( ) ( )

| | | |all all

EDPs DMs

G DV IM G DV DM dG DM EDP dG EDP IM= ∫ ∫ (2.1)

9

In Equation 2.1, G(DV|IM) is the scenario-based realization of the DV and the G functions

represent complementary cumulative distribution functions (e.g., G(DV|IM) is identical to P[dv ≥

DV | im = IM]). For instance, in order to obtain the total monetary loss for a building at a certain

hazard level, this equation is completed as follows: intensity measure(s), IM(s), (e.g., Sa(T1)), are

obtained from a Hazard Analysis; relevant engineering demand parameters, EDPs, (e.g., story

drifts) are predicted from Structural Analyses for given values of IM(s) (and representative ground

motions); component damage states are developed from repair strategies, DM fragility curves are

developed for each component through Damage Analysis; and predictions of DVs (e.g.., total

amount of Dollar loss in the building) are made through a Loss Analysis.

In the MAF-based realization of DV (i.e., MAF of DV ≡ λ(DV)), λ(DV) is obtained by

integrating G(DV|IM) obtained with Equation 2.1 over all hazard levels as shown in Equation

2.2:

( ) ( ) ( )

|allIMs

DV G DV IM d IMλ λ= ∫ (2.2)

The PBA methodology formulation (i.e., Equation 2.1 and Equation 2.2) has a

probabilistic format that enables us to quantify the propagation of uncertainties in estimation of

key variables from IM to DV. Uncertainties have different sources but can be categorized in the

two main groups: aleatory and epistemic. Aleatory uncertainty in estimation of a variable is

rooted in the random nature of that variable. Epistemic uncertainty in estimation of a variable is

rooted in the limited information that we have for estimating of that variable. Theses

uncertainties affect the characteristics of the complimentary cumulative distribution functions (G

functions), and the MAF of seismic hazard in Equation 2.1 and Equation 2.2. A detailed

discussion about uncertainties is provided in Chapters 5 and 6 of this dissertation.

The PBA methodology that is summarized in this section has been applied in detail to a

reinforced-concrete building located in Van Nuys, California (Krawinkler, ed., 2005). This

document is a good example of the level of data gathering and computational effort that is need

for completing this process.

2.3 Conceptual performance-based design methodology

The PBA methodology, as described in the previous section, is comprehensive and general, but it

is currently impractical for most cases in design practice. In the PBA methodology the building

10

and its structural components and configuration have to be known in order to complete the

Structural Analysis stage and estimate EDPs. In PBD, the building is yet to be designed. One can

perform an iterative assessment that starts with a judgmental conceptual design and it is refined

in each iteration of the PBA methodology. This is not a desirable PBD process because targeted

performance objectives are not considered explicitly when conceptual design is performed. Also,

the PBA methodology has a probabilistic basis that incorporates the effect of various aleatory

and epistemic uncertainties in the evaluation of the basic random variables (i.e., IM, EDP, DM,

and DV). This requires much data gathering and computational effort and obscures

understanding of the building’s behavior, which is an essential component of a good conceptual

structural design.

The conceptual PBD methodology proposed in this investigation is based on concepts

that incorporate performance objectives up front in the design decision process. It focuses on

mean values of basic random variables (i.e., IM, EDP, DM, and DV). This approach enables the

designer to focus on important global behavior aspects without having to deal with rigorous

mathematical formulations. Such a design could/should be followed by a rigorous or simplified

PBA methodology for verification and refinement, and for incorporation of uncertainty

evaluation and uncertainty propagation. The conceptual PBD methodology enables the designer

to choose between structural systems (e.g., moment-resisting frame and shear wall), structural

system materials (e.g., reinforced-concrete and steel), and global structural system parameters

(e.g., strength, stiffness, plastic deformation capacity, etc.), based on pre-selected global

performance targets and based on economic and life safety considerations. The proposed

conceptual PBD methodology facilitates design decision making, and for this reason from here

on is referred to as design decision support system, DDSS.

In concept, the flow of information in the DDSS is opposite to that of the PBA

methodology. In the DDSS, limits for relevant EDPs (e.g., story drifts, floor accelerations, etc.)

are specified, given that performance objectives are expressed in terms of targeted DV values.

These relevant EDPs are used for decision making on structural system, structural system

material, and structural system parameters. There is no single design parameter that satisfies all

performance objectives simultaneously, and trade-offs have to be made based on multiple

performance objectives. For instance, damage to non-structural components is usually controlled

by interstory drift limitations and is often educed by increasing the stiffness of the structure. On

the other hand, damage to building contents is controlled by floor accelerations limitations and is

11

reduced by reducing the stiffness and/or strength of the structure, which is in conflict with the

non-structural component damage limitations. This simple example shows that different

performance objectives may impose conflicting limitations on decisions on structural system,

structural system material, and structural system parameters. The DDSS enables the designer to

easily compare the trade-offs between different design alternatives that satisfy competing

performance objectives and make informed decisions based on life-cycle considerations rather

than on up-front construction costs alone.

The DDSS comprises three domains; the Hazard Domain, the Structural System Domain,

and the Loss Domain, as illustrated in Figure 2.2. The Hazard Domain provides information on

the intensity of the ground motions, the Loss Domain provides information on performance

objectives, and the Structural System Domain contains all design alternatives worthy of

exploring. The objective is to search for the best combination of structural system, structural

system material, and structural system parameters that fulfill all the performance objectives in

the most effective manner. As stated, in order to maintain focus on the global behavior of the

building, the expected (mean) values of random variables are used in the DDSS, and uncertainty

evaluation and uncertainty propagation can be incorporated in the subsequent performance

assessment phase if so desired.

2.3.1 Hazard Domain

The Hazard Domain contains the relation between the ground motion intensity IM and its mean

annual frequency of exceedance λ (i.e., hazard curve) and associated representative ground

motion records. The selected ground motion records link the Hazard Domain to the Structural

System Domain as they are used in order to obtain the IM-EDP relationships and collapse

fragility curves. In the following selection, several aspects of choice of IM, and procedures for

selecting ground motions are discussed.

Selection of intensity measure

Ground motion intensity measure IM is a scalar parameter or a vector of parameters that

represent the intensity of the ground motion. Usually, and for estimation of building response

values (i.e., EDPs), spectral acceleration at the fist mode period of the structural system Sa(T1) is

12

considered as the ground motion intensity. Several researchers have introduced other scalar IMs

(Tothong, 2006; Hutchinson et. al., 2004; Luco and Cornell, 2001; Cordova et. al, 2000) and

vector IMs (Baker and Cornell, 2005; Bazzurro and Cornell, 2000; Shome and Cornell, 1999).

The appropriateness of the selected IM is measured by its efficiency and sufficiency (Luco ,

2002). An IM is considered to be efficient, if the predicted EDP using this IM has small

variability (i.e., small dispersion). An IM is considered to be sufficient if the predicted EDP

using this IM is independent of magnitude M, source-to-site distance R, and any other site-

specific characteristic of the ground motion that affects the building response (e.g., near-fault

effects, soft-soil effects, etc.). In other words, An IM is sufficient when is capable of carrying

enough information about the ground motion.

Recently, Baker and Cornell (2005) proposed a vector IM which consists of Sa(T1) and a

parameter denoted as ε which represents a measure of dispersion in estimation of Sa(T1) at the

corresponding hazard level. They have shown that not considering ε in the process of obtaining

IM-EDP curves using only Sa(T1) leads to conservative estimation of building response

parameters at high probability hazard levels (see Chapter 6).

An essential characteristic of an IM is that it should be feasible to find the relation

between seismic hazard and the selected IM for the location of the structure. This relation is the

link between the seismicity of the structure’s location and the IM. Such relationship is obtained

by performing Probabilistic Seismic Hazard Analysis (Cornell, 1968) as discussed in Section 2.2.

Ground motion selection

Ground motions are selected for performing nonlinear response history analyses that lead to

development of IM-EDP relationships. These ground motions should be representative of the

seismicity of the structure’s location and of the hazard level at which design or assessment is

performed for. Careful selection of ground motions can increase the efficiency and sufficiency of

selected IM. The current state-of-practice in ground motion selection is based on M and R values

that represent the seismicity of the structures location (Stewart et. al. 2001; Bommer and

Acevedo, 2004). Baker and Cornell (2006) proposed a new method for selecting ground motions,

which is based on ε, M, and R. A comparison between this method and other methods for

selecting ground motions is presented in Baker and Cornell (2006) in which it was concluded

13

that the ground motions selected using ε, M, and R eliminate biases and reduce dispersion in

estimation of structural response parameters.

An important aspect of ground motion selection is the potential impact of special ground

motions (e.g., near-fault records, soft soil records). Near fault motions, especially those

influenced by forward directivity can be characterized by an impulsive motion which exposes

structures to a high input energy at the beginning of the record (Alavi and Krawinkler, 2001).

Due to this characteristic, Sa(T1) is not a proper IM for near-fault ground motions. As shown by

Krawinkler and Alavi (2001), near fault ground motions can be characterized by the pulse shape,

pulse period, and pulse amplitude. Tothong (2006) shows that using the nonlinear spectral

displacement (Sdi) as the IM improves the assessment of near-fault ground motions on structural

response.

2.3.2 Loss Domain

The Loss Domain contains the relations between induced losses and associated structural

response parameters. Obtaining these relations involves consideration of two sets of information:

1) relationship between structural response parameters and damage states in building

components, and 2) relation between building component damage states and the component loss.

For a given value of building response parameter, the value of loss is obtained by integrating the

associated losses in building components weighted by the probability of being in different

damage states conditioned on the value of building response parameter.

The Loss Domain is divided into two sub-domains, one containing losses conditioned on

collapse not occurring (i.e., NC sub-domain), and the other containing losses conditioned on

collapse (i.e., C sub-domain). Both of these sub-domains contribute to three categories of losses:

monetary loss, downtime loss, and life loss.

The NC sub-domain of the Loss Domain is partitioned into “subsystems”. A “subsystem”

is a collection of components in a building whose aggregated loss is well-represented by a single

EDP. A building could be divided into subsystems based on functional use (e.g., structural

subsystem, non-structural subsystem, content subsystem) or according to the EDP that correlates

well with the subsystem’s loss (e.g., interstory drift, floor acceleration). The strategy for dividing

the building into subsystems depends on factors such as availability of data on relation between

14

the subsystem loss and its corresponding EDP, spatial distribution of valuable components in the

building, and many others. The objective is to obtain a relationship between each subsystem loss

and its corresponding EDP so that the latter can be used by the engineer to guide design

decisions. A detailed discussion of subsystems and the process for computing their loss-EDP

(i.e., E(DV | EDP & N C )) relationships is provided in Chapter 3.

Collapse may be a major contributor to losses in a building, especially at low probability

hazard levels (i.e., large return periods, like. those associated with a 2% probability of

exceedance in 50 years, [2/50 hazard level]). Collapse could be due to dynamic instability caused

by large lateral displacements or triggered by the loss of vertical load carrying capacity of one or

several columns. As will be discussed in Chapter 6, in this study, only collapse due to dynamic

instability is considered. The effect of building collapse is quantified in the C sub-domain of the

Loss Domain. In this sub-domain, the building is not divided into subsystems, and the loss due to

collapse is set equal to the total building loss.

2.3.3 Structural System Domain

The Structural System Domain contains the information about EDPs and the probability of

collapse for selected design alternatives. It provides the link between the Hazard Domain and the

Loss Domain. In general, the Structural System Domain is the domain in which decisions are

made on a suitable structural system, structural system materials, and structural system

parameters. The effect of such decisions at discrete hazard levels (i.e., link to Hazard Domain) is

observed in the Loss Domain (i.e., link to Loss Domain). In order to complete the link between

the sub-domains of the Loss Domain and the Hazard Domain, the Structural System Domain is

also divided into a NC sub-domain (No-Collapse sub-domain of the Structural System Domain)

and a C sub-domain (Collapse sub-domain of the Structural System Domain).

The NC sub-domain includes information about relations between building subsystem

EDPs and the IM. For each subsystem, mean (expected) IM-EDP curves for various design

alternatives are presented. The IM is the intensity measure employed in the Hazard Domain (e.g.,

Sa(T1)) and the EDP is the one that correlates well with the loss in a specific subsystem (e.g.,

average of maximum interstory drift ratios for the non-structural drift sensitive subsystem, or

maximum floor acceleration at floor i for acceleration sensitive subsystem). Mean IM-EDP

curves (i.e., E(EDP | IM & N C )) are obtained by subjecting structural system alternatives with

15

specific properties to sets of ground motions representative of specific IM values, as provided in

the Hazard Domain. If it can be assumed that the frequency content of the ground motions is

insensitive to magnitude and distance within the IM range of primary interest, then Incremental

Dynamic Analyses (IDAs) (Cornell and Vamvatsikos, 2002) can be used to obtain mean IM-

EDP curves. In this research, such curves have been developed and stored in a database for many

EDPs (enlisted in Appendix E) for a wide range of moment-resisting frames and shear walls. The

range of parameter variation for development of this database is discussed in Chapter 4 and

properties of such system are presented in Appendix B. In order to implement the proposed

DDSS, it is necessary to have such mean IM-EDP curves available for the range of design

alternatives.

The C sub-domain of the Structural System Domain includes collapse fragility curves,

which show the probability of collapse as a function of the intensity measure (i.e., P(C | IM)).

Such curves are obtained by increasing the IM in an IDA until the slope of the IM-EDP curve

approaches zero, which indicates dynamic instability. A collapse fragility curve is the

Cumulative Distribution Function (CDF) of the IM values at which dynamic instability occurs

for a set of representative ground motion records. In this context, using component hysteresis

models that account for cyclic and monotonic deterioration of strength and stiffness is important

and has been the subject of recent research on collapse capacity of structural systems by Ibarra

and Krawinkler (2005). Detailed information about components deterioration models are

presented in Chapter 4. Methods for obtaining IM-EDP and collapse fragility curves, and

uncertainties involved in this process are presented in Chapter 5 and Chapter 6 of this

dissertation.

2.3.4 Framework for a Design Decision Support System

In the previous sections, we introduced the domains that comprise the conceptual PBD process.

In this section we show how design decision making is performed in conceptual PBD. As stated,

the main focus of conceptual PBD is to provide the designer with an effective combination of a

structural system, structural system materials, and structural system parameters based on

performance objective that are defined at discrete hazard levels. Such a process is illustrated

schematically in Figure 2.3 for a case in which the performance objective is to limit monetary

loss (denoted as $loss) at a certain hazard level to an acceptable value (e.g., acceptable $loss at

16

50/50 hazard level). As stated previously, losses may occur in several subsystems that are

sensitive to different EDPs. Losses in different subsystems could be assessed simultaneously, or

the focus could be placed on the one subsystem that contributes most to the value of the building,

and other subsystem losses could be evaluated subsequently. For simplicity we assume here that

the building consists of one subsystem.

The lower left portion of Figure 2.3 shows the mean $loss-EDP curve for the single

subsystem conditioned on no-collapse occurs, E($loss | EDP & N C ), and the lower right portion

shows the expected value of $loss conditioned on collapse occurs, E($loss | C). The upper left

portion of Figure 2.3 shows the mean hazard curve for the building location, for a period

corresponding to a set of design alternatives to be evaluated. The upper central portion shows

mean IM-EDP curves for several design alternatives, and the upper right portion shows collapse

fragility curves for the same design alternatives. The process starts at the lower left portion of

Figure 2.3 where E($loss | EDP & N C ) is shown. The designer enters this graph with a value of

acceptable $loss and obtains the associated EDP on the mean loss curve. The designer then

enters the hazard curve with the hazard level at which the $loss is acceptable and obtains the

associated IM. The intersection of a horizontal line at this IM value and a vertical line at the

previously obtained EDP value in the Structural System domain, which contains mean IM-EDP

curves, can be viewed as a “design target” point♠. All design alternatives, represented by

individual mean IM-EDP curves, that intersect the IM line to the left of the design target point

are “feasible” solutions (i.e., the associated expected $loss is smaller than the target acceptable

loss).

This process continues to the C sub-domain of the structural system domain where the

designer finds the probability of collapse, P(C | IM), for a design alternative at the hazard level of

interest, and by continuing vertically, the loss associated with collapse. The total expected $loss

of each design alternative, at the hazard level of interest, can then be expressed by summation of

weighted losses in each sub-domain:

( ) ( ) ( )$ | $ | &NC (NC| ) $ | &C (C| )E loss IM E loss IM P IM E loss IM P IM= × + × (2.3)

In Equation 2.3, E($loss | IM) is the total expected loss of a design alternative at intensity level

IM. P(NC | IM) is the probability of no-collapse conditioned on the value of IM and is equal to 1-

P(C | IM). E($loss | IM & C ) is the total maximum $loss of the building in case of collapse. In a

♠ This is an approximation that is discussed in Section 2.3.5

17

general case, in which the building is divided into several subsystems, the total expected loss of a

design alternative (i.e., E($loss | IM)) is obtained by summing the expected subsystem losses for

the NC case, as is shown in Equation 2.4.

( ) ( ) ( )

$ | $ | &NC (NC| ) $ | &C (C| )all

subsystems

E loss IM E loss IM P IM E loss IM P IM⎛ ⎞⎜ ⎟= × + ×⎜ ⎟⎜ ⎟⎝ ⎠

∑ (2.4)

These losses can be evaluated for various design alternatives, and when combined with

associated construction costs (an issue that is not addressed in this work), informed design

decisions can be made that should lead to an efficient structural design.

As stated by Rosenblueth (1979), optimum amount to be spent on the design process

together with the all the cost items in the building can be obtained by minimizing the summation

of costs and losses. Using this approach along with the information provided by Equation 2.4 can

be used to obtain the optimized design alternative among various design alternatives.

The conceptual PBD process for design decisions making based on tolerable life loss and

acceptable downtime loss is basically the same as the process illustrated above for decision

making based on acceptable monetary loss. The corresponding total expected life loss at a

specific hazard level and total expected downtime loss at a specific hazard level is expressed as

shown in Equation 2.5 and Equation 2.6, respectively. In Equation 2.5 the random variable life

loss is denoted as loss, and in Equation 2.6 the random variable downtime loss is denoted as

loss.

( ) ( ) ( )

| | &NC (NC| ) | &C (C| )all

subsystems


∑ (2.5)

( ) ( ) ( )

| | &NC (NC| ) | &C (C| )all

subsystems


∑ (2.6)

The completion of the conceptual PBD process for tolerable life loss and acceptable

downtime loss in the form presented by Equations 2.5 and 2.6 is a challenge that is not addressed

in this work. Research in the area of downtime modeling is in progress (Comerio, 2005). In the

area of life loss modeling, quantification of casualties in both the collapse and no-collapse

regimes is also a matter of ongoing research (Kano et al. 2006; Horie, 2006; Yeo and Cornell,

2003). In this work we use probability of collapse as a surrogate for life loss. We assume that

collapse will lead to an unacceptable loss of life. Figure 2.4 shows concepts of a DDSS for life

DDDD DD

DD

18

safety assessment. The mean loss curves in the collapse sub-domain and no-collapse sub-domain

of the Loss Domain in Figure 2.4 are plotted with dashed lines showing that for the time being,

information about these sub-domains are not available. The process for conceptual PBD for life

safety starts at the upper right portion of Figure 2.4. The collapse fragility curve is entered with a

value of tolerable probability of collapse. The intersection of the line denoting the IM value at

the specified hazard level with the line denoting the tolerable probability of collapse divides the

design alternatives into a feasible and an unfeasible solution space.

In most codes and guidelines, it is assumed that adequate collapse safety is provided by

limiting the maximum story drift at the design earthquake level to a specific value (e.g., a drift

limit of 0.02 at the 10/50 hazard level). As shown by Ibarra and Krawinkler (2005), the

dispersion in estimation of EDPs such as maximum story drift in a near-collapse regime of a

building is very large due to significant deterioration in stiffness and strength of the structural

components, which makes deformation quantities such as drift an unreliable parameter for

assessment of the probability of collapse. By using deterioration models that do account for

important aspects of deterioration it has become possible to trace the response of structures to

collapse (Ibarra et. al., 2005; Song and Pincheira, 2002) and estimate the probability of collapse

as a function of an IM rather than an EDP such as drift. The dispersion of this measure of

collapse capacity is much smaller than the dispersion of the other method (Ibarra and

Krawinkler, 2005).

The conceptual PBD process, as implemented in the proposed DDSS, can be taken to the

next level and address performance objectives defined independent from the hazard level as an

average of total loss. For each design alternative, an IM versus expected DV curve can be

obtained by determining the expected total loss for a number of hazard levels as discussed in the

previous paragraphs. Then, the expected annual loss can be computed by numerical integration

of the IM versus expected DV curve over the hazard curve. Such an approach is illustrated in

Figure 2.5 for a targeted acceptable average annual $loss. Two design alternatives are considered

in this figure. Each solid dot and solid rectangle in Figure 2.5 shows the expected total loss of the

two design alternatives at the corresponding hazard level. By connecting the solid dots and

rectangles, the IM versus expected DV curve of the two design alternatives is obtained, and by

integrating each of these curves over the hazard curve (i.e., Equation 2.7), the expected annual

$loss of each of the design alternatives is calculated.

19

( ) ( ) ( )$ $ | IMIM

E loss E loss im d imλ= ∫ (2.7)

The same approach can be used to obtain the expected annual life loss and expected

annual downtime, using Equation 2.8 and 2.9.

( ) ( ) ( ) | IMIM


( ) ( ) ( ) | IMIM


We use the mean annual frequency of collapse as a surrogate for the mean annual number

of casualties. This quantity is estimated by integrating the collapse fragility curve of each design

alternative over the hazard curve as shown schematically in Figure 2.6 and expressed

mathematically in Equation 2.10.

( ) ( )|C IMIM

P C im d imλ λ= ∫ (2.10)

The Design Decision Support System described in this section is exercised in detail in

Chapter 7. The proposed DDSS provides the user with the ability to graphically inspect the trade-

offs between different design alternatives and to make informed design decisions that

simultaneously consider performance objectives at various hazard levels.

2.3.5 Approximation in Design Decision Support System

In the previous section, we illustrated how design decision making is performed in conceptual

PBD. In terms of monetary losses, we approximated E($loss | IM& N C ) by evaluating

E($loss | EDP & N C ) in the Loss Domain at the expected value of EDP associated with IM and

conditioned on NC as shown in Equation 2.11.

( ) ( )( )| ,NC

$ | ,NC $ | &NCEDP E EDP IM

E loss IM E loss EDP=

≈ (2.11)

In other words, the expected value of function Y = g(x) is approximated as:

( ) ( ) ( )( ) ( )E Y E g x g E x= ≈ (2.12)

This approximation needs some elaboration. If we write the Taylor series expansion of

g(x) about the expected value of x:

DD DD

20

( ) ( ) ( ) ( ) ( ) ( )2 2

2

( )( ) ( )

( ) ( )2dg x x E x d g x

g x g E x x E xx E x x E xdx dx

−= + − + +

= = (2.13)

The expected value of g(x) is approximated as:

( ) ( ) ( )22

21( ) ( )

( )2 x

d g xE g x g E x

x E xdxσ

⎡ ⎤≈ + ⎢ ⎥=⎣ ⎦

(2.14)

Comparing Equation 2.14 with Equation 2.12 (which is an equivalent to Equation 2.11)

we conclude that the methodology presented by DDSS is accurate if the mean loss curves in the

Loss Domain are linear (no curvature or in other words the second derivative of the mean loss

curve is zero). If mean loss curves in the Loss Domain are not linear, using the DDSS as

described in Section 2.3.4 has an approximation along with it. This approximation depends on

the level of curvature of the mean loss curve and the dispersion in EDP.

2.4 Simplified performance-based assessment methodology

In Section 2.2 of this chapter we briefly explained the rigorous approach for PBA and remarked

that completion of this process for a given building needs a very large amount of data gathering

and computational work (Krawinkler, ed., 2005). Such a comprehensive approach to PBA may

not be necessary for most ordinary structures. With a reversal in the flow of information in the

three-domain approach presented for the DDSS, a simplified PBA process can be created. Given

the building, it location and characteristics, the mean information in the Hazard Domain,

Structural System Domain, and Loss Domain can be generated. Using the simplified PBA

process, the user can estimate the expected value of DV at a discrete hazard level (in the form of

E(Loss | IM)) and/or independent from the hazard level (in the form of E(Loss )) of the building.

This approach is illustrated in the schematic example shown in Figure 2.7, in which we

demonstrate how to estimate the expected value of monetary loss of a given structure at a

specific hazard level (i.e., E($loss | IM)).

Based on the location and characteristics of the building (fundamental period), the mean

hazard curve is obtained and drawn in the Hazard Domain. Then a building model is created and

mean IM-EDP curves and the collapse fragility curve are obtained by performing structural

analysis. These curves are drawn in the Structural System Domain as shown in upper central

portion of Figure 2.7. Dividing the building into several subsystems follows the same logic as

was discussed in the conceptual PBD process. In Figure 2.7, for simplicity, we have assumed

21

that the building consists of a single subsystem. Mean loss curves (E($loss | EDP&NC)) are

obtained for the structure based on the building characteristics and drawn in the Loss Domain.

The process for assessing the expected value of $loss at hazard level IM follows the path from

IM to EDP to E($loss | EDP&NC), and also the path from IM to collapse fragility curve to

E($loss | C).

The total expected $loss at the hazard level associated with IM is found using Equation

2.3 or 2.4, as applicable. Similarly, life loss and downtime loss are obtained using the same

approach but with corresponding information in the three domain, and by using Equations 2.5

and 2.6, respectively. MAF of losses can be obtained by using Equations 2.7, 2.8, and 2.9 for

three loss categories of monetary loss, life loss, and downtime loss.

The advantage of this PBA approach is its simplicity and the ability to observe, in the

graphical format presented here, the contributions of various subsystems to the total loss.

Perhaps of primary importance is the ability to estimate expected annual losses based on the

integration scheme illustrated in Figure 2.5.

22

Hazard Analysis

Structural Analysis

Damage Analysis

Loss Analysis

( )IM IMλ →

IM EDP→

EDP DM→

DM DV→

( )|G EDP IM

( )|G DM EDP

( ) ( ) ( )

| | |allDM

G DV DM dG DM EDP G DV EDP=∫

( )|G DV DM

( ) ( ) ( )

| | |allEDP

G DV EDP dG EDP IM G DV IM=∫

( )IMλ

( ) ( ) ( )

|allIM

G DV IM d IM DVλ λ=∫PBA at mean annual

frequency level

PBA at discrete hazard level

Hazard Analysis

Structural Analysis

Damage Analysis

Loss Analysis

( )IM IMλ →

IM EDP→

EDP DM→

DM DV→

( )|G EDP IM

( )|G DM EDP

( ) ( ) ( )

| | |allDM

G DV DM dG DM EDP G DV EDP=∫

( )|G DV DM

( ) ( ) ( )

| | |allEDP

G DV EDP dG EDP IM G DV IM=∫

( )IMλ

( ) ( ) ( )

|allIM

G DV IM d IM DVλ λ=∫PBA at mean annual

frequency level

PBA at discrete hazard level

Fig. 2.1 Performance-Based Assessment methodology layout.

( )( | ) ( | & ) ( | ) | ( | )E Loss IM E Loss IM NC P NC IM E Loss C P C IM= × + ×

Hazard Domain

( )| &E EDP IM NC

Collapse

( | )P C IM

Loss Domain

( )| &E Loss EDP NC ( )|E Loss C

Structural System DomainNo Collapse

CollapseNo Collapse

-Monetary Loss = loss$ -Downtime Loss = loss -Life Loss = lossD

Mean Hazard Curve(s) for Design

Alternatives

Mean IM-EDP Curves for Design Alternatives

Mean Loss Curves Mean Loss due to Collapse

λ(IM)

Collapse Fragility Curves for Design Alternatives

( )( | ) ( | & ) ( | ) | ( | )E Loss IM E Loss IM NC P NC IM E Loss C P C IM= × + ×

Hazard Domain

( )| &E EDP IM NC

Collapse

( | )P C IM

Loss Domain

( )| &E Loss EDP NC ( )|E Loss C

Structural System DomainNo Collapse

CollapseNo Collapse

-Monetary Loss = loss$-Monetary Loss = loss$ -Downtime Loss = loss-Downtime Loss = loss -Life Loss = lossDD

Mean Hazard Curve(s) for Design

Alternatives

Mean IM-EDP Curves for Design Alternatives

Mean Loss Curves Mean Loss due to Collapse

λ(IM)

Collapse Fragility Curves for Design Alternatives

Fig. 2.2 Layout of Design Decision Support System (DDSS).

23

( )($ | ) ($ | & ) ( | ) $ | ( | )E loss IM E loss IM NC P NC IM E loss C P C IM= × + ×

E D P( )I Mλ

E($

loss

| ED

P &

NC)

S tructural S ystem D om ainH azard D om ain

Loss D om ain

I M I M

( | )P C I M

E D P

M ean H azard C urve (s ) for

D es ign A ltern atives

M ean IM -E D P C u rvesfor D esign Alternatives

Co llapse Fragility C u rvesfo r D esign Alte rnatives

M ean $Lo ss C urve(s) (N o C ollapse) M ean $Loss V alue (C ollap se)

( )$ |E lo ss C

E D P( )I Mλ

E($

loss

| ED

P &

NC)


Loss D om ain

I M I M

( | )P C I M

E D P

M ean H azard C urve (s ) for

D es ign A ltern atives

M ean IM -E D P C u rvesfor D esign Alternatives

Co llapse Fragility C u rvesfo r D esign Alte rnatives

M ean $Lo ss C urve(s) (N o C ollapse) M ean $Loss V alue (C ollap se)

( )$ |E lo ss C

Fig. 2.3 DDSS for acceptable monetary loss at discrete hazard level.

( )( | ) ( | & ) ( | ) | ( | )E loss IM E loss IM NC P NC IM E loss C P C IM= × + ×DDD ( )( | ) ( | & ) ( | ) | ( | )E loss IM E loss IM NC P NC IM E loss C P C IM= × + ×DDDDDD

E D P( )I Mλ


Loss D om ain

I M I M

( | )P C I M

E D P

E(

loss

| E

DP

& N

C)DD

( )|E lo ss CD( )|E lo ss CD

M ean IM -EDP Curvesfor Design Alternatives

DM ean Loss Curve(s) (No Collapse)DM ean DDM ean Loss Curve(s) (No Collapse) D Loss (Collapse)M ean D Loss (Collapse)DD Loss (Collapse)M ean

Collapse Fragility Curvesfor Design Alternatives

M ean Hazard Curve(s) for

Design Alternatives

Fig. 2.4 DDSS for tolerable life loss at discrete hazard level.

24

Hazard Domain Loss Domain

( )IMλ

IM

Hazard Curves forDesign Alternatives

Mean $loss for Discrete IMsfor Design Alternatives

Design for Mean Annual Frequency of $loss

($ | )E lo ss I M

( ) ( )$ $ | ( )IMIME loss E loss im d imλ= ∫


( )IMλ

IM




($ | )E lo ss I M


Design for Mean Annual $loss


( )IMλ

IM




($ | )E lo ss I M



( )IMλ

IM




($ | )E lo ss I M


Design for Mean Annual $loss

Fig. 2.5 DDSS for acceptable mean annual monetary loss.

( | ) ( )C IMIMP C im d imλ λ= ∫

Hazard Domain Structural System Domain

( )IMλ

I M

( | )P C I M



Design for Mean Annual Frequency of Collapse

( | ) ( )C IMIMP C im d imλ λ= ∫

Hazard Domain Structural System Domain

( )IMλ

I M

( | )P C I M



Design for Mean Annual Frequency of Collapse

Fig. 2.6 DDSS for tolerable mean annual frequency of collapse.

25


Simplified PBA for $loss at Discrete Hazard Levels

E D P( )I Mλ

E($

loss

| ED

P &

NC

)

Structural System DomainHazard Domain

Loss Domain

I M IM

( | )P C I M

E D P

Mean Hazard Curve(s) for

the building location

Mean IM-EDP Curvesfor the given building

Collapse Fragility Curvesfor the given building

Mean $Loss Curve(s) (No Collapse) Mean $Loss Value (Collapse)

( )$ |E loss C



E D P( )I Mλ

E($

loss

| ED

P &

NC

)


Loss Domain

I M IM

( | )P C I M

E D P






( )$ |E loss C



E D P( )I Mλ

E($

loss

| ED

P &

NC

)


Loss Domain

I M IM

( | )P C I M

E D P






( )$ |E loss C

Fig. 2.7 Simplified PBA for estimation of monetary loss at discrete hazard. level

-27-

3 LOSS DOMAIN

3.1 Introduction

Implementation of the Design Decision Support System (DDSS) outlined in Chapter 2 and

illustrated in Chapter 7 requires the availability of mean loss curves (i.e., relationships between

expected (mean) value of building response and expected [mean] value of loss) for various

collections of building components (i.e., subsystems). This dissertation does not address the

development of mean loss curves in detail, primarily because of the lack of data needed for

quantification (Aslani and Miranda, 2005). It provides only basic comments on concepts on

which such a development could be based. In this chapter, the intention is to put forth basic

challenges that have to be addressed in order to obtain relevant information in the context of

PBD and the DDSS proposed in this work.

3.2 Background in earthquake loss estimation and overview of the loss domain considered in this study

Building-specific loss estimation has been a focus area in the PBEE methodology proposed by

the Pacific Earthquake Engineering Research (PEER) Center since its inception in 1997 (e.g.,

Cornell and Krawinkler, 2000; Deierlein, 2004; Krawinkler and Miranda, 2004). In the PEER

PBEE methodology, loss is estimated through a three-step approach that associates seismic

hazard to building loss (i.e., seismic hazard and ground motion intensity to building response,

building response to building damage, and building damage to building loss). In PEER

terminology, building losses are categorized into monetary loss, downtime loss, and life loss.

These three categories of losses are estimated by utilizing a triple integral (based on the total

probability theorem) denoted as the “PEER framework equation”. This methodology has been

exercised in PEER Testbed studies on an old reinforced concrete building in Los Angeles

28

(Krawinkler, ed., 2005) and a university laboratory building (Comerio et. al., 2005), and has

been addressed in many professional papers and documents, such as Aslani and Miranda (2005)

and Baker and Cornell (2003). This dissertation is an effort within the PEER framework. It

addresses simplified PBEE, so the PEER terminology and corresponding approaches

incorporated by PEER for loss estimation are adopted.

Other recent studies on building specific loss estimation include the research conducted

by Kircher (2003), and Porter et. al., (2002). In Kircher (2003) a procedure for estimating loss in

welded steel moment-resisting frame buildings is presented. Porter et. al. (2002) shows the

sensitivity of building loss to uncertainty in basic variables involved in this estimation.

The building loss estimation methodology developed in PEER is similar to other loss

estimation methodologies in the sense that they all follow the 3-step approach from ground

motion hazard to building losses. However, the PEER methodology is different insofar that: (1)

damage states of building components are defined based on actual repair costs that will be

estimated by the contractor for replacement/repair, whereas in other approaches damage states

are defined in general terms with a poorly defined relation to repair/replacement costs (i.e.,

slight, moderate, extensive, and complete); (2) the PEER methodology is probability-based,

meaning that the estimated loss is presented in a probabilistic format that incorporates

propagation of uncertainty in different steps of the approach and from different sources of

uncertainty (aleatory and epistemic), and (3) the contribution of probability of collapse to

monetary loss is included in the PEER loss estimation methodology.

Building-specific loss estimation is concerned with monetary loss (direct loss),

downtime loss, and casualties. In a recent study conducted by Comerio (2005), downtime loss

for laboratories at the main campus of U.C. Berkeley is estimated by means of the PEER

approach. Researchers at PEER have also tried to address casualty rates for seismic events in old

reinforced concrete buildings ( Kano et al., 2006; Krawinkler, ed., 2005; Yeo and Cornell, 2003).

A similar effort has been conducted for the Kobe earthquake by Horie (2006). Although the

DDSS proposed in this research is general and can incorporate all three categories of loss, we

focus on monetary loss because of the very limited information available on downtime loss and

casualties. More specifically, we are concerned with the capital-related loss (i.e.,

repair/replacement cost of building components due to damage induced by ground motion), and

not the income-related loss (e.g., rental income loss, relocation expenses, etc.) (Kircher et. al.,

29

1997). Because of the importance of the life safety issue, as stated in Chapter 2, we use

probability of collapse as a surrogate to address the casualty issue.

In order to distinguish between monetary loss associated with non-collapse and monetary

loss induced by structural collapse, the loss domain is divided into two sub-domains, one

containing losses conditioned on collapse does not occur (i.e., NC sub-domain) and the other

containing losses conditioned on collapse does occur (i.e., C sub-domain). The total monetary

loss in the building conditioned on “NC” (i.e., total loss of NC sub-domain), is divided into

losses in individual “subsystems”. The definition of subsystems and the mathematical approach

for obtaining mean loss curves is introduced in Sections 3.3 and 3.4. In Section 3.5, two classes

of subsystems are introduced (i.e., story-level and building-level). The expected (mean) value of

total monetary loss of the building conditioned on ”C” (i.e., total loss of C sub-domain) is briefly

discussed in Section 3.6.

3.3 Definition of subsystems

The total monetary loss in a building after ground shaking is an aggregate of losses induced by

the ground motion in different building components. A “subsystem” is a collection of building

components whose losses are well represented by a single EDP. Building components are

grouped into subsystems according to factors such as functional use (e.g., structural,

nonstructural, content) and/or according to the sensitivity of the subsystem components to

engineering demand parameters, EDPs (e.g., interstory drift, floor acceleration). A similar

approach for dividing total loss of a building into losses of groups of building components has

been utilized in previous studies (e.g., Gunturi, 1993; Whitman et. al., 1997; Kircher et. al.,

1997) and is the basis of the loss estimation methodology implemented in the computer program

HAZUS. For each subsystem, the objective is to obtain a relationship between the expected

(mean) value of loss (i.e., total loss of subsystem) and a “most relevant” EDP, so that the latter

can be used in the DDSS. This EDP has to be well-correlated with losses in all components of

the subsystem, and has to be well-correlated with global structural response in order to permit

deduction of global design decisions.

Utilizing the subsystem approach, the total loss in a building is an aggregate of total loss

in individual subsystems. The DDSS discussed in Chapter 2 is in fact the process of such

aggregation (or disaggregation if we look at dividing the grand total loss in the building into

30

losses of individual subsystems). An important characteristic of this process is that the value of

loss in any subsystem is uncorrelated with the value of loss in other subsystems. As will be

shown in the next section, this characteristic is the consequence of the method we use to

aggregate losses in building components for obtaining the total loss in the building.

3.4 Methodology for development of mean loss curves for subsystems tems

In the previous section, we introduced the concept of “subsystems” and discussed different

characteristics of loss aggregation/disaggregation using the subsystem approach. In the context

of the DDSS, we need a general relationship between the expected (mean) value of loss in each

subsystem and the expected (mean) value of the EDP associated with that subsystem, denoted as

“subsystem mean loss curve“ (throughout our discussions about subsystems, all statements are

conditioned on collapse does not occur, [“NC”]). In Section 3.4.1, we show the mathematical

model and the process for developing such relationships. We show that three elementary

ingredients are needed for developing subsystem mean loss curves: (1) the expected (mean)

value of loss in the component given various damage states of that component; (2) a relationship

between the component damage states and the most relevant EDP that correlates well with the

damage in that component; and (3) a relationship between the component EDP and the general

(i.e., global) EDP considered for the subsystem.

3.4.1 Mathematical approach for development of mean loss curves for subsystems

The total loss in a building at a certain hazard level (“at a certain hazard level” in this work is

synonymous to “given the expected value of IM at that hazard level”) is an aggregate of losses in

building components at that hazard level. The simplest form of this aggregation is assuming that

the total loss in a building given IM is equal to the summation of losses in individual components

of the building given IM (i.e., Equation 3.1, and in expected [mean] form Equation 3.2). Such an

approach for aggregating losses in building components in order to obtain the total loss is

common for many loss estimation methodologies (Aslani and Miranda, 2005; Kircher et. al.,

1997; Porter, 2002).

( ) ( )1

| , | &N

jj

loss IM NC loss IM NC=

=∑ (3.1)

31

[ ]1 1

| & | & | &N N

j jj j

E loss IM NC E loss IM NC E loss IM NC= =

⎡ ⎤⎡ ⎤= =⎢ ⎥ ⎣ ⎦

⎣ ⎦∑ ∑ (3.2)

In Equation 3.1 and Equation 3.2, ( )| &loss IM NC is the value of total loss in building

given IM and conditioned on “NC”, ( )| &jloss IM NC is the loss in component j given IM

conditioned on “NC”, and N is the total number of components in the building. E[.] is the

expectation operator.

At this stage, we group the components of the building into several subsets (i.e., each

subset is a subsystem) as shown in Equation 3.3. The components in subsystem i, are a subset of

total components in the building. This means that we can assume that the total loss value in a

subsystem is equal to summation of loss values in individual components of that subsystem (i.e.,

Equation 3.4, and in expected [mean] form in Equation 3.5).

( ) ( )1

| & | &M

i

iloss IM NC loss IM NC

=

=∑ (3.3)

( ) ( )1

| & | &i

i

nij

jloss IM NC loss IM NC

=

=∑ (3.4)

1 1

| & | & | &i i

i

n ni i

j jj j

E loss IM NC E loss IM NC E loss IM NC= =

⎡ ⎤⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎣ ⎦⎣ ⎦

⎣ ⎦∑ ∑ (3.5)

In Equation 3.3, ( )| &iloss IM NC is the value of loss in subsystem i given IM and

conditioned on “NC”. It is assumed that a total of M subsystems exist. Equation 3.4 and Equation

3.5 describe the total value of loss in subsystem i (i.e., ( )| &iloss IM NC ), and loss in component j

of subsystem i (i.e, ( )| &ijloss IM NC ) for a given value of IM and conditioned on “NC”. It is

assumed that subsystem i has a number of ni components. If we assume that there is a one-to-one

relationship between the value of IM and the value of EDP associated with subsystem i (i.e.,

EDPi) we can rewrite Equation 3.4 and Equation 3.5 as Equation 3.6 and Equation 3.7,

respectively. The one-to-one relationship between IM and EDPi is provided in the form of mean

IM-EDP curves, which are discussed in Chapter 5.

( ) ( )1

| & | &i

i

ni i i

jj

loss EDP NC loss EDP NC=

=∑ (3.6)

1 1

| & | & | &i i

i

n ni i i i i

j jj j

E loss EDP NC E loss EDP NC E loss EDP NC= =

⎡ ⎤⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎣ ⎦⎣ ⎦

⎣ ⎦∑ ∑ (3.7)

32

In Equation 3.6 and Equation 3.7, ( )| &i iloss EDP NC is the value of loss in subsystem i

given the associated EDP (i.e., EDPi) and conditioned on “NC”, ( )| &i ijloss EDP NC is the loss of

component j in subsystem i, given EDPi conditioned on “NC”, and ni is the total number of

components in subsystem i.

Although EDPi is correlated with damage in component j in subsystem i, it is not necessarily the

EDP that is directly associated with loss in that component. For example, let’s assume that the

EDP we consider for the nonstructural-drift-sensitive-subsystem (NSDSS) of a building is the

average of maximum interstory drift ratios along the height of the building. The EDP that is

associated with damage/loss in a component that belongs to this subsystem (e.g., a partition in

the 3rd story) is the maximum interstory drift ratio of the third story. The information available (if

available) in the form of damage functions (i.e. damage fragility curves) and loss functions is

associated with the latter EDP and not the EDP associated with the subsystem (i.e., EDPi). For

this reason, | &i ijE loss EDP NC⎡ ⎤⎣ ⎦ is computed using the total probability theorem as follows:

( )0

| & | & | &i i i i i i ij j j j jE loss EDP NC E loss EDP NC P EDP EDP NC dEDP

∞

⎡ ⎤ ⎡ ⎤= ⋅ ⋅⎣ ⎦ ⎣ ⎦∫ (3.8)

where | &i ij jE loss EDP NC⎡ ⎤⎣ ⎦ is the expected value of loss in component j of subsystem i

conditioned on “NC” when it is subjected to a direct engineering demand parameter ijEDP .

( )| &i ijP EDP EDP NC is the probability of i

jEDP given the engineering demand parameter

associated with subsystem i (i.e., EDPi) conditioned on “NC”. The procedure to estimate such

probability is discussed in Chapter 5.

The expected value of loss in component j of subsystem i conditioned on “NC” when it is

subjected to ijEDP , | &i i

j jE loss EDP NC⎡ ⎤⎣ ⎦ , can be computed as a function of the expected (mean)

value of loss in the component when it is in different damage states and the probability

associated with being in that damage state given ijEDP . This is shown in Equation 3.9.

( )1

| & | & | &ijm

i i i i i ij j j jk jk j

kE loss EDP NC E loss DS NC P DS EDP NC

=

⎡ ⎤ ⎡ ⎤= ⋅⎣ ⎦ ⎣ ⎦∑ (3.9)

In Equation 3.9, ijkDS is damage state k in component j of subsystem i. For this

component, a total of ijm damage states exist. | &i i

j jkE loss DS NC⎡ ⎤⎣ ⎦ is the expected value of loss

33

in component j of subsystem i when it is in damage state ijkDS and conditioned on “NC”.

( )| &i ijk jP DS EDP NC is the probability of being in damage stated i

jkDS for component j of

subsystem i, given ijEDP , and conditioned on “NC”. Both terms on the right hand side of

Equation 3.9 (i.e., | &i ij jkE loss DS NC⎡ ⎤⎣ ⎦ , ( )| &i i

jk jP DS EDP NC ) are obtained from

experiment/analysis. Much work has been conducted in this respect (Kircher et. al., 1997;

Kircher, 2003; Aslani and Miranda, 2005) and much more research needs to be done to establish

such relationships. Merits of | &i ij jkE loss DS NC⎡ ⎤⎣ ⎦ , ( )| &i i

jk jP DS EDP NC are discussed in the

following sections. By substituting ingredients of Equation 3.7 from Equation 3.8 and Equation

3.9, we obtain Equation 3.10, which is the general equation for developing the mean loss

function for subsystem i.

( )( )

( )1 10

| & | & | &

| &

ii j

i

mni i i i i

j jk jk jj k

i i ij j

E loss EDP NC E loss DS NC P DS EDP NC

P EDP EDP NC dEDP

∞

= =

⎡ ⎤ ⎡ ⎤= ⋅⎣ ⎦⎣ ⎦

⋅ ⋅

∑ ∑∫ (3.10)

As shown in Section 3.3, building components can be grouped into subsystems according to

factors such as: functional use, sensitivity to a specific engineering demand parameter, spatial

location, etc. For instance, nonstructural components in a building whose damage is correlated

with drift (i.e., characterizing factor is sensitivity to drift) can be grouped into one subsystem

denoted as nonstructural-drift-sensitive-subsystem (NSDSS) for which the associated EDP could

be the average along the height of maximum interstory drift ratios for each ground motion in the

building. Combination of factors can also be considered for grouping building components.

Using the same example, we can group all the drift sensitive nonstructural components (i.e.,

characterizing factor is sensitivity to drift) in the third story (i.e., characterizing factor is spatial

location) denoted as NSDSS3. Several grouping criteria are suggested in Section 3.5 of this

chapter.

3.4.2 Component level mean loss functions

An essential part of a loss estimation methodology is obtaining the relation between the loss in a

component and its damage state (i.e., loss function). This relation could be deterministic

(Kircher, 2003; Porter, 2004) or probabilistic (Aslani and Miranda, 2005). In the deterministic

34

form a relation between the value of component loss and component damage state is provided. In

the probabilistic form, the loss function is the probability of incurring a certain value of loss

when a certain damage state is observed. As shown in Equation 3.10, we are interested in the

mean value of loss in component j of subsystem i when it is in damage state ijkDS .

Mathematically, both deterministic and probabilistic approaches provide us with enough

information for completing Equation 3.10. For generality, we explain both methods for obtaining

mean value of loss in a component given the component damage state.

Our focus in this research is on capital-related loss of buildings. In the context of

subsystem loss value, this means that we are interested in obtaining the replacement/repair cost

of components given their damage states. Replacement/repair cost of a component in a certain

damage state can be broken down into a base value and a number of adjustment factors that each

reflects a specific condition that triggers an increase or decrease in the price of

repair/replacement. Such a function can be written in the form of Equation 3.11 (the notation is

adopted from Mitrani-Reiser et. al., 2006)

( ) ( ),| |i i i i U i ij jk jk op i l h q j jk jkloss DS ds C C C C C loss DS ds= = ⋅ ⋅ ⋅ ⋅ ⋅ = (3.11)

In Equation 3.11, ( ), |i U i ij jk jkloss DS ds= is the unit-cost of replacement/repair of

component j in subsystem i when it is in damage state ijkds , and ( )|i i i

j jk jkloss DS ds= is the cost

(final cost with associated time stamp) of replacement/repair of component j in subsystem i when

it is in damage state ijkds . Each C-factor (e.g., iC ) is an adjustment factor reflecting a reason for

the contractor to increase/decrease the cost. More adjustment factors can be added to Equation

3.11 if needed. opC is the adjustment factor that represents the contractors’ profit and overhead,

iC is the inflation factor, LC is the building-location factor (e.g., for a building in a remote area

the replacement/repair cost increases), hC is the height factor (e.g., replacement/repair of

building components at higher elevation is more expensive than components at lower

elevations), and qC is the discount factor for quantity in replacement/repair cost.

The unit-cost of replacement/repair of component j of subsystem i, and C-factors are

essentially random variables. Their variability is due to our lack of knowledge, so it is considered

an epistemic uncertainty. For example, variability in estimation of unit-cost of

replacement/repair in a component given it’s damage state is due to limited amount of data

35

available from repair/replacement companies, and different procedures for repair/replacement of

the same damage have different costs (Aslani and Miranda, 2006). Equation 3.11 is rewritten as

Equation 3.12 in the logarithmic form. If we assume that the unit-cost of replacement/repair of

component j in subsystem i when it is in damage state ikds follows a cumulative lognormal

distribution in the form of Equation 3.13 (in which , ,i U i Uj jloss loss

σ β= ), and each adjustment C-factors

is lognormally distributed with logarithmic mean of ( )-factorLn Cμ and logarithmic dispersion of

( )-factorLn Cσ (i.e., ( ) -factor-factor CLn Cσ β= ) and are uncorrelated with each other, then the

replacement/repair cost of component j of subsystem i when it is in damage state ijkds follows a

cumulative lognormal distribution (i.e., Equation 3.14). The logarithmic mean (i.e., ( )ijLn loss

μ ) and

logarithmic dispersion (i.e., ( ) iijj lossLn loss

σ β= ) are obtained using Equation 3.15 and Equation 3.16,

respectively.

( ) ( ) ( ) ( ) ( ) ( ) ( ),i i Uj op i l h q jLn loss Ln C Ln C Ln C Ln C Ln C Ln loss= + + + + + (3.12)

( )( ) ( )

( )

,

,

,|, ,

|

|i U i

j jk

i U ij jk

i Uj Ln loss dsi U i U i

j j jkLn loss ds

Ln lP loss l DS ds

μ

σ

⎡ ⎤−⎢ ⎥≤ = =Φ⎢ ⎥⎢ ⎥⎣ ⎦

(3.13)

( )( ) ( )

( )

|

|

|i ij jk

i ij jk

ij Ln loss dsi i i

j j jkLn loss ds

Ln lP loss l DS ds

μ

σ

⎡ ⎤−⎢ ⎥≤ = =Φ⎢ ⎥⎢ ⎥⎣ ⎦

(3.14)

( ) ( ) ( ) ( ) ( ) ( ) ( ),i i Ui l hop qj j

Ln C Ln C Ln CLn C Ln CLn loss Ln lossμ μ μ μ μ μ μ= + + + + + (3.15)

( ) ,2 2 2 2 2 2 2

i Ui l h q jjCop Ci C C C lossLn loss

β β β β β β β= + + + + + (3.16)

The expected (mean) value of replacement/repair cost of component j of subsystem i

when it is in damage state ijkds , is obtained as shown in Equation 3.17.

( ) ( )2

| |

12

0

| | |i i i iLn loss dsj k loss dsj jki i i i i i

j jk j jk j jkE loss DS ds E loss ds dP loss ds e eμ β∞

⎡ ⎤ ⎡ ⎤= = ⋅ ≈ ⋅⎣ ⎦ ⎣ ⎦∫ (3.17)

As seen in Equation 3.17, in order to calculate the expected (mean) value of loss in

component j in subsystem i when it is in damage state ijkds , we need information about

( )|i ij jkLn loss DS ds

μ=

and |i i

j jkloss dsβ . At this time, we have very limited knowledge about ingredients of

36

Equation 3.17, therefore estimation of the expected (mean) value of component loss in the

probabilistic format is not yet feasible. For this reason, the deterministic form of the component

loss function is used. Using the same notation as in Equation 3.11, the expected

repair/replacement cost of component j in subsystem i when it is in damage state ijkds is found by

using Equation 3.18.

,| |i i i i U i ij jk jk op i l h q j jk jkE loss DS ds C C C C C E loss DS ds⎡ ⎤ ⎡ ⎤= = ⋅ ⋅ ⋅ ⋅ ⋅ =⎣ ⎦ ⎣ ⎦ (3.18)

In Equation 3.18 |i i ij jk jkE loss DS ds⎡ ⎤=⎣ ⎦ and , |i U i i

j jk jkE loss DS ds⎡ ⎤=⎣ ⎦ are the expected value of

replacement/repair cost of component j in subsystem i when it is in damage state ijkds , and the

expected value of unit-cost of replacement/repair of component j in subsystem i when it is in the

same damage state. C-factors used are all considered as average values.

3.4.3 Probability of being in a damage state for a component

Component damage fragility curves provide information on damage to the component as

function of an engineering demand parameter that correlates well with the component damage.

This information is in the form of probability of being in or exceeding a damage state given

EDP. It is assumed that such probability can be expressed in the form of a lognormal distribution

(Shinozuka et. al., 2000) as shown in Equation 3.19.

( )( ) ( )

( )|

ij

ij

ij Ln EDPi i i i

jk jk j jLn EDP

Ln edpP DS ds EDP edp

μ

σ

⎡ ⎤−⎢ ⎥≥ = =Φ⎢ ⎥⎢ ⎥⎣ ⎦

(3.19)

In Equation 3.19, ( )ijLn EDP

μ and ( )ijLn EDP

σ are the logarithmic mean and logarithmic

standard deviation ofi

jEDP . In order to find these two parameters, a maximum likelihood test

(Shinozuka et. al., 2000) or Kolmogorov-Smirnov goodness-of-fit test (Benjamin and Cornell,

1970) can be applied to available data obtained from previous events, experimental results, or

analytical results.

The probability of being in a damage state can be estimated as the arithmetic difference

between fragility functions corresponding to two consequent damage states as shown in Equation

3.20 (Aslani and Miranda, 2005).

37

( )

( )( )( ) ( )( )( )

1

1

|

1 | 0

| | 1

|

i i i ijk jk j j

i i i ijk j jj k

i i i i i i i ijk jk j j jk j jj k

i i i i ijk jk j j j

P DS ds EDP edp

P DS ds EDP edp k

P DS ds EDP edp P DS ds EDP edp k m

P DS ds EDP edp k m

+

+

= = =

⎧ − ≥ = =⎪⎪ ≥ = − ≥ = ≤ <⎨⎪⎪ ≤ = =⎩

(3.20)

In Equation 3.18, we have assumed that the component has a total of ijm damage states. k

= 0 is associated with the state of “no damage” in the component, and k = m represents the state

of “complete damage” in the component.

The source of variability in component damage fragility functions, in the form presented

in Equation 3.19, is randomness in behavior (aleatory). By far the largest source of epistemic

uncertainty is the judgment used by observers of tests in deciding when a component enters a

specified damage state. Aside from this source of epistemic uncertainty, Aslani and Miranda

(2006) consider two sources of epistemic uncertainty that affect the component fragility function.

One is the uncertainty caused by using a limited amount of data for obtaining the component

fragility function, and the second is the uncertainty introduced by using a pre-specified loading

protocol for obtaining the component fragility function. In order to incorporate the epistemic

uncertainty due to finite sample size in the component damage fragility curve, we inflate the

dispersion of the fragility function as shown in Equation 3.21.

2 2 2i i ld iTds Rds U dsβ β β= + (3.21)

In Equation 3.21, iTdsβ is the inflated dispersion of the fragility function, and

iRdsβ is the

dispersion of the fragility function when the effect of epistemic uncertainty is not considered

(i.e., iRdsβ = ( )i

jLn EDPσ ), and

ld iU dsβ is the dispersion due to finite-sample size and is estimated with

ld i iU ds Rds SNβ β= where NS is the sample size. Aslani & Miranda (2005) introduce a different

approach for incorporating the epistemic uncertainty resulting from the drift increments used in

loading protocols for obtaining component fragility curves. In this approach both dispersion and

median estimate of the median of EDP (i.e., ( )ˆ ijLn EDP

μ ) of the fragility curve change. Details of

this process can be obtained from Aslani & Miranda (2005).

3.5 Samples of subsystem mean loss curves

38

In the previous section, the general methodology for deriving mean loss curves for a subsystem

was introduced. For many cases, adequate information for obtaining such curves is not available

and much more research needs to be done to establish these relationships. In this section we

describe several strategies for grouping building components into subsystems and describe

qualitatively their form. Such mean loss curves are used in Chapter 7 of this dissertation in which

the implementation of the DDSS is illustrated.

3.5.1 Story-level subsystems and their mean loss curves

A story-level subsystem in the NC sub-domain includes a collection of building components that

are located in same story and whose loss can be described by a common EDP such as story drift

or floor acceleration. In order to compute the mean loss curve of such a subsystem we need, for

all component that make up the subsystem, sets of fragility curves that define the probability of

being in, or exceeding, damage states as a function of the subsystem EDP, conditioned on “NC”.

Also, we need sets of expected (mean) value of repair/replacement costs associated with each

damage state of all components in the subsystem. For instance, all the nonstructural drift

sensitive components of story i are grouped into one subsystem denoted as NSDSSi. The EDP

that we consider for this subsystem is the maximum interstory drift ratio in story i (i.e., iIDR ).

( )( )1 1

| , | , | ,ii j

iNSDSS i i i

mnNSDSS NSDSS NSDSS

i j jk jk ij k

E loss IDR NC E loss DS NC P DS IDR NC= =

⎡ ⎤ ⎡ ⎤= ⋅⎣ ⎦⎣ ⎦ ∑∑ (3.22)

In Equation 3.22, | ,iNSDSS ij jkE loss DS NC⎡ ⎤⎣ ⎦ is the expected (mean) value of loss in

component j of subsystem NSDSSi when it is in damage state iNSDSSjkDS , and

( )| ,iNSDSSjk iP DS IDR NC is the probability of being in damage state iNSDSS

jkDS given iIDR . The

difference between Equation 3.22 and Equation 3.10 is that in Equation 3.22 the term of the form

( )| ,i ijP EDP EDP NC that relates the EDP associated with the subsystem to the EDP of the

component is omitted. For the subsystem we are using in this example (i.e., NSDSSi), we assume

that the interstory drift ratio of story i is the same EDP that correlates well with damage in

components of this subsystem. This is the reason that the term ( )| ,i ijP EDP EDP NC and the

associated integration are omitted in Equation 3.22.

39

Completing Equation 3.22 for obtaining the mean loss curve for NSDSSi requires an

inventory of all nonstructural drift sensitive component of story i. For each component, expected

(mean) value of loss at different damage states, and probability of being in each of those damage

states given iIDR is needed. Only limited information of this form does exist at this moment,

which prevents us from obtaining mean loss curves in the fashion explained above. For

illustration purpose, development of a mean loss curve for a single partition is illustrated. Figure

3.1 shows a typical set of fragility curves for partitions using interstory drift ratio (IDR) as an

EDP (Taghavi and Miranda, 2003). Using Equation 3.20, the probability of being in each

damage state is plotted in Figure 3.2. Cost of repair/replacement per square foot for each damage

state is shown in Figure 3.3. The cost values are based on 2001 dollars and needs adjustment

based on the time that this information is being used. Using Equation 3.22 ( in = 1, ijm = 3) the

mean loss curve of a single partition is obtained and plotted in Figure 3.4.

Based on the insight we obtain from Figure 3.4, we have assumed a generic form for the

mean loss curve of story-level NSDSSi as shown in Figure 3.5. The vertical axis of this mean

loss curve is intentionally plotted without values because it depend on the inventory of

nonstructural drift sensitive components in the story. As seen in Figure 3.5, till a certain value of

drift, expectedly no loss will occur. At a certain level of drift (0.02 in Figure 3.5) we assume that

all the nonstructural drift sensitive components in the story have to be replaced. These two points

are connected with a straight line. Similar curves can be obtained for nonstructural acceleration

sensitive components and for structural components.

3.5.2 Building-level subsystems and their mean loss curves

Building-level subsystems are the most general class of subsystems. They incorporate building

components distributed over the full height of the building. Using this class of subsystems is

convenient because it relates a significant portion of total loss to a single EDP. Mean loss curves

of building-level subsystems relate the expected (mean) value of loss of the subsystem to an

EDP that expresses the global behavior of the building. This EDP is not the very EDP that

relates well to loss in individual components of the subsystem, however, it is closely related to it.

This means that for building level subsystems the term of the form ( )| ,i ijP EDP EDP NC , which

40

relates the EDP associated with the subsystem to the EDP of the component in the subsystem, is

not omitted.

For simplicity it is assumed that all structural components can be grouped into one

subsystem denoted as SS. The EDP that correlates with structural component loss depends on the

type of structural system. Figure 3.6 shows a generic form of a mean loss curve for a building-

level structural subsystem denoted as SS, for a moment-resisting frame (the vertical axis is

plotted without values because such values depend on the size of the building). For moment-

resisting frames we assume that the EDP associated with SS is the plastic drift ratio of the story

(total interstory drift ratio minus yield drift ratio). Other EDPs such as story ductility (total

interstory drift divided by story yield drift) and/or damage indices could also be considered. For

a shear wall, we assume that the EDP that correlates with structural damage is the plastic rotation

of the wall at the base of story i.

An important aspect of a SS mean loss curve is its discontinuity at a certain value of EDP

at which the mean loss curve jumps from a relatively small value (the value of all structural

components together is about 20% of cost of a building [Kircher et. al., 1997]) to the value of

total building loss (equal to the loss if collapse occurs, see Section 3.6). This jump occurs when

the owner considers the structure a total loss even though collapse has not occurred (e.g., large

residual deformation after the earthquake). Jumps also may be present in other mean curves if

large losses are associated with the attainment of specific EDP values.

3.6 Building loss at collapse

The value of building loss when collapse occurs can be estimated based on the replacement cost

of the whole building plus the cost of demolition/clean-up and design of a new building.

Adjustment factors (i.e., C-factors in Equation 3.11 and Equation 3.18) are different from the

“NC” case, meaning that the value of total loss when collapse occurs is not simply the

summation of losses of individual subsystems. The replacement cost of a building is different

from its market value (Aslani and Miranda, 2005) due to the fact that the market value of a

building includes a profit factor that is not applicable in loss estimation. By replacing the

collapsed building, same profit, or even more profit, is expected by the owner. Another factor

that increases the value of loss in the event of building collapse is the collateral damage that the

collapsed building imposes to other structures adjacent to it.

41

3.7 Summary

In this chapter we discussed the merits of the Loss Domain in the DDSS. We briefly explained

the state of knowledge in building specific loss estimation. The methodology presented in this

chapter is general, but we focus only on the monetary loss issues due to the insufficient

information available on downtime loss and casualties.

Upfront, the necessity of dividing building losses into two sub-domains of “collapse” and

“no-collapse” (i.e., denoted as “C” and “NC”, respectively) was put forward. We simplified the

relation between building losses in the NC sub-domain and building response parameters by

introducing the subsystem concept. Each “subsystem” is defined as a collection of building

components whose losses are well represented by a single building response parameter EDP.

Strategies for channeling building components into different subsystems and a methodology for

developing subsystem mean loss curves (i.e., relation between subsystem loss and the associated

EDP) were discussed in detail.

Two classes of subsystems were introduced: building-level subsystems and story-level

subsystems. Story-level subsystems in the NC sub-domain include a collection of building

components that are located in the same story and whose loss can be described by a common

EDP. A more global class subsystems is building-level subsystems in which the mean loss

curves relate the mean value of loss of the building-level subsystem to an EDP that expresses the

global behavior of the building. Although this class of subsystems is convenient to use because it

relates a significant portion of the total loss to a single EDP, there is an approximation involved

that makes employment of this type of subsystem limited to cases in which the response of the

building at different stories does not vary by much (i.e., a uniform behavior is expected).

At the end of the chapter, we briefly talked about the loss conditioned on collapse. We

argued that the value of total loss when collapse occurs is not simply the summation of losses of

individual subsystems, and other factors such as demolition and clean-up costs should be

considered.

-42-

0

0.25

0.5

0.75

1

0 0.005 0.01 0.015 0.02 0.025 0.03EDP[IDR]

P(D

M>d

m|E

DP)

Damage fragility curves for partitions

DS1: taping, pasting, painting

DS2: gypsum board replacement, taping, pasting, painting

DS3: gypsum board & frame replacement, taping, pasting, painting

Fig. 3.1 Damage fragility curves for partitions (information obtained from Taghavi &

Miranda 2003)

0

0.25

0.5

0.75

1

0 0.005 0.01 0.015 0.02 0.025 0.03EDP[IDR]

P(D

M=d

m|E

DP)

DS1: taping, pasting, painting

DS2: gypsum board replacement, taping, pasting, painting

DS3: gypsum board & frame replacement, taping, pasting, painting

Probability of being in each damage state for partitions

Fig. 3.2 Probability of being in different damage states for generic non-structural drift

sensitive components in Van Nuys Hotel Building (information obtained from Taghavi &

Miranda 2003)

43

Replacement/repair cost of a partition wall per ft2

(2001 dollars)

0

1

2

3

4

5

6

DS1 DS2 DS3

E[$l

oss|

DS=

ds] (

US

Dol

lars

/ft2 )

Fig. 3.3 Replacement/repair cost of partition wall for different damage states (2001

dollars) (information obtained from Taghavi & Miranda 2003)

0

1

2

3

4

5

6

0 0.005 0.01 0.015 0.02 0.025 0.03EDP[IDR]

E[lo

ss|E

DP]

(US

Dol

lars

/ft2 )

Mean loss curves for a partition wall per ft2

(2001 dollars)

Fig. 3.4 Mean loss curve for partition wall (2001 dollars)

44

0.0Ex

pect

ed S

ubsy

stem

$L

oss

0.005 0.015 0.0250.01 0.020.0 0.005 0.015 0.0250.01 0.02

IDR0.0

Expe

cted

Sub

syst

em

$Los

s0.005 0.015 0.0250.01 0.020.0 0.005 0.015 0.0250.01 0.020.0

Expe

cted

Sub

syst

em

$Los

s0.005 0.015 0.0250.01 0.020.0 0.005 0.015 0.0250.01 0.02

IDR

Fig. 3.5 Generic form of mean monetary loss curve for story-level non-structural drift

sensitive subsystem

0.0

Expe

cted

Sub

syst

em

$Los

s

0.005 0.015 0.0250.01 0.02

IDRp

0.0

Expe

cted

Sub

syst

em

$Los

s

0.005 0.015 0.0250.01 0.02

IDRp

Fig. 3.6 Generic form of mean monetary loss curve for building-level structural subsystem

- 45 -

4 STRUCTURAL SYSTEM DOMAIN: MODELING, PARAMETER SELECTION, AND ANALYSIS

4.1 Introduction

The decision making process in the simplified performance-based design/assessment

methodology described in Chapter 2 focuses on the Structural System Domain. Decisions on

suitable combinations of structural system, structural system material, and structural system

parameters need to be made in order to meet multiple performance objectives. This decision

making can be greatly facilitated by the availability of a database of engineering demand

parameters, EDPs, and collapse capacities obtained for various combinations of the

aforementioned structural decision variables. Such a database has been developed as part of this

study, and is discussed in Chapters 4, to 6. This chapter summarizes the range of structural

decision variables considered and the methodology utilized to develop the database. In Chapter 5

and Chapter 6 we discuss the sensitivity of EDPs and collapse capacities to variations in several

structural parameters. Some of the EDPs are utilized in Chapter 7 in the implementation of the

DDSS, and others are discussed in Chapters 5 and 6 as to their importance in design decision

making.

Parameters that affect the behavior of a building are numerous. This study opens a small

window to these parameters in order to investigate the sensitivity of the building response to a

finite set of important structural parameters. In this research, two basic structural systems are

studied: moment-resisting frames and reinforced concrete shear walls (denoted as shear walls).

Dual systems are not considered because of the large variety of combinations in structural

- 46 -

parameters needed to define their behavior. We will try to define the boundaries and scope of

structural systems considered in this study. Many simplifications are introduced along the

process of modeling these structural systems (e.g., disregard of soil structure interaction, use of

centerline models for modeling structural components, disregard of the gravity systems provided

in addition to moment frames or walls, etc.). Such simplifications had to be made in order to

make the effort manageable.

To model the structural systems used in this study, we use a component model that

incorporates stiffness and strength deterioration (monotonic and cyclic) (Ibarra et. al., 2005). A

new method for defining the backbone curve and cyclic deterioration parameters of this

component model is introduced. In this new method, elastic parameters of the component (i.e.,

initial stiffness, and yield rotation) are no longer used as the normalizing parameters for defining

the inelastic parameters of the backbone curve (i.e., ductility capacity, strain hardening, post-

capping stiffness ratio). A discussion on the component model used in this study is provided in

Section 4.3.

To identify the importance and quantify the effect of different structural parameter on the

response of structural systems considered in this study, we use simple mathematical models

denoted as “generic structures”. Two families of generic structures are devised: a family of

generic moment-resisting frames and a family of generic shear walls. Generic structures have

been used by many researchers (e.g., Medina and Krawinkler, 2003; Chintanapakdee and

Chopra, 2003; Seneviratna and Krawinkler, 1997; Esteva and Ruiz, 1989, etc.) to study the

behavior of structural systems. These generic structures are modeled with elastic elements and

rotational springs, which eliminates shear failures from considerations, but with this exception

permits incorporation of most of the important parameters that strongly affect structural

behavior. Ranges of variation of component parameters are defined based on experimental

studies performed by others and are discussed in detail in Section 4.5.

In order to systematically study the sensitivity of structural response to the variation of structural

parameters, a set of base case, and a set of variation to base case generic structural systems are

defined. For each base case the effect of variation of structural parameters on the response of the

structural system is studied in detail. These issues are addressed in Chapter 5 and Chapter 6.

- 47 -

4.2 Structural systems and behavior

As stated in Chapter 2, the objective of this research is to provide a simple tool that helps

engineers conduct a conceptual performance-based design for ordinary and regular buildings.

The outcome of conceptual design should be a small subset of building parameters that defines

desirable properties that strongly affect the building response in both the no-collapse and

collapse regimes. To accomplish this goal, we first need to identify those structural parameters

that significantly affect the building response/loss during an earthquake, and then investigate the

sensitivity of building response/loss to variations in important structural parameters.

4.2.1 Moment-resisting frames behavior under seismic loads

Compared to shear wall structures, moment-resisting frames are flexible structural systems in

which non-structural drift sensitive components are more prone to damage than nonstructural

acceleration sensitive components. The lateral deformation of moment-resisting frames is

influenced by the level of contribution of the shear-type behavior (Figure 4.1). The level of this

contribution depends on the variation of beam stiffnesses along the height of the moment-

resisting frame (Appendix A). For large contribution of shear-type deformations to the total

deformation of a moment-resisting frame, more of the drift (hence more of the damage) will

concentrate in the lower stories. Due to the importance of this issue on the distribution of damage

along the height of the moment-resisting frame, in this study we are considering the variation of

stiffness (and also strength) along the height of the moment-resisting frame.

Irregularity in plan and height of moment-resisting frames strongly affects behavior.

Sedarat and Bertero (1990) have addressed the issue of irregularity in plan and have shown that

disregarding the effect of eccentricity in plan underestimate the drift demand. Irregularity in

height was studied by Al-Ali and Krawinkler (1998) and Wood (1992). Both studies conclude

that irregularity in mass and stiffness along the height does not significantly affect the response

of the structure. However, Al-Ali and Krawinkler (1998) conclude that irregularity of strength

along the height in the form of a soft story substantially affects the response of the building.

There is much awareness of these problems in codes and design practice, and strict rules are

- 48 -

applied to obviate such problems. In this study the irregularity issues are not addressed, and the

focus is on structures that are regular in plan and elevation and, therefore, can be represented by

2-dimentional models.

Beams and column in moment-resisting frames undergo both shear and flexural

deformations. In this study we assume that the aspect (span to depth) ratio of structural elements

is sufficiently large so that shear deformations in individual structural elements are small

compared to flexural deformations and therefore can be neglected.

Shear failure of RC moment-resisting frame elements, especially columns with high axial

load, is definitely undesirable behavior. Recent studies have shown that the shear strength of a

RC member is a function of several parameters such as axial load ratio, level of deformation,

longitudinal steel ratio, and moment gradient of the member (Kowalsky and Priestley, 2000;

Lynn et. al., 1996). In this research we assume that adequate shear strength is provided in

moment-resisting frames so that shear failure does not have to be modeled. An approximate

method for incorporating the effect of shear failure is to reduce the flexural deformation capacity

of the RC member. This issue is discussed in Section 4.4.

Column to beam flexural strength ratio at joints of moment-resisting frames plays an

important role in the response of frame structures during a seismic event. Current design codes

require that for each joint the ratio of sum of column flexural strength to sum of beam flexural

strength be larger than 1.2 in RC moment-resisting frames (ACI 318-05) and about the same

value for steel moment-resisting frames (AISC 2005). Recent studies have shown that such

criteria do not provide adequate protection against column plastification and development of

story mechanisms. A value of 2.0 and more has been suggested for the column to beam flexural

strength ratio to prevent the formation of story mechanisms (Kara et. al., 2001). In this study, the

effect of this ratio on the structural response is studied. This issue is discussed in Section 4.4

We use a centerline model with concentrated plasticity to model moment-resisting

frames, i.e., Figure 4.2. This is done for simplicity and avoids the need to incorporate additional

variables (such as beam and column depth and joint shear behavior) in the analytical model of

generic frame structures. In Section 4.3 the general component model used for modeling beams

and columns is presented. As will be shown in Section 4.4, this component model can mimic

monotonic and cyclic behavior of moment-resisting frame elements with acceptable accuracy.

- 49 -

4.2.2 Shear walls behavior under seismic loads

Reinforced concrete shear walls are used commonly as the lateral load resisting system of

buildings. Many analytical and experimental studies have been performed in the past to address

the behavior of this structural system (e.g., Paulay, 2001; Paulay and Priestley, 1992; Vallenas

and Bertero, 1979; etc.). In general, shear wall buildings are stiffer than buildings whose lateral

load resisting system consists of moment-resisting frames. For this reason, shear wall buildings

provide good protection against damage to drift sensitive nonstructural components in the

building. However, for the same reason (i.e., large stiffness) shear walls attract higher floor

accelerations, which cause higher loss in acceleration sensitive nonstructural components.

In buildings with small to medium height the cross section dimensions of the wall is not

likely to change along the height of the building. Often, due to architectural constraints, openings

are needed which reduces the uniformity of stiffness and strength along the height of the wall.

Unless such openings are small and their effect on the deformation behavior and mode of failure

can be neglected, special considerations are needed in order to accommodate ductile behavior

(Ali and Wight, 1991; Paulay and Priestley, 1992). In this study we are concerned only with

cantilever shear walls in which the effect of openings (if any) on stiffness and strength is

negligible.

Cantilever shear walls undergo both shear-type and flexure-type deformations. The

contribution of each mode of deformation to the total deformation depends on the aspect ratio of

the shear wall, H/L, where H is the total height of the shear wall and L is the width of the shear

wall. This study is concerned only with shear walls with a relatively large aspect ratio (i.e.,

slender walls), hence the contribution of shear deformation to the total deformation of shear

walls is not considered

As a general rule in design of shear walls, shear failure should be prevented or postponed

in order to encourage ductile behavior. Figure 4.3 shows four typical modes of failure in shear

walls (Paulay and Priestley, 1992). Figures 4.3b and 4.3e show the type of desirable flexural

behavior where the flexural reinforcement has yielded in the plastic hinge zone. Undesirable

shear failure modes are illustrated in Figure 4.3c (diagonal shear failure) and Figure 4.3d (sliding

shear failure). For shear walls with a small aspect ratio the shear force needed to develop flexural

- 50 -

yielding at the base of the wall is large and, shear failure may become the driving mode of

failure. As stated previously, this study is concerned only with shear walls with a relatively large

aspect ratio, and it is assumed that adequate safety against shear failure is provided.

In slender shear walls during earthquakes, flexural yielding occurs usually at the base of

the wall (FEMA 306). Behavior of the wall after flexural yielding depends on the curvature

capacity and cyclic deterioration of stiffness and strength in the yielded region, which depend on

many factors such as cross section detailing, wall length, wall slenderness, moment gradient, and

axial and shear forces acting in the yielded region (Paulay and Priestley, 1992).

Several procedures have been proposed for analytical modeling of shear walls. Other than

finite element methods, three main modeling methods have been utilized by researchers in the

past: cantilever beam method, equivalent truss element method, and vertical line method.

Discussions about these models can be found in Vulcano and Bertero (1987). In this study we

use the cantilever beam method for modeling shear walls. This method assumes that the sectional

properties of the shear wall are concentrated in the vertical centerline of the wall. Each wall is

modeled as a series of beam elements as shown in Figure 4.4. Each element is defined as a

“component”, and it is assumed that each component represents the behavior of the associated

segment of the shear wall. This method assumes that plasticity is concentrated at the ends of

component. In Section 4.3 we introduce the general component model that we use for modeling

shear walls. This component model is developed based on the behavior of shear walls in

monotonic and cyclic tests and incorporates deterioration of stiffness and strength

(monotonically and cyclically).

4.3 Structural Component Model

An essential building block of an MDOF (or SDOF) system used in any seismic demand study

that addresses structural collapse is a structural component model that incorporates stiffness

degradation and strength deterioration (i.e., monotonic and cyclic deterioration). One of the

earliest studies on stiffness-degrading component models was performed by Clough and Johnson

(1966) and later modified by Mahin and Bertero (1975), in which the reloading stiffness is based

on the maximum displacement that has taken place in the previous loading history (i.e., Peak-

- 51 -

Oriented model). A more accurate model for RC components, which incorporates a trilinear

backbone curve and unloading stiffness degradation, is the well known Takeda model Takeda et.

al. (1970). In this study the bilinear peak-oriented model is utilized to define the basic hysteretic

response.

The component model used in this study is a modified version of the component

deterioration model proposed by Ibarra and Krawinkler (2005) (and Ibarra et al., 2005). This

component model has a trilinear backbone curve and incorporates strength and stiffness

deterioration based on the hysteretic energy dissipated when the component is subjected to cyclic

loading.

4.3.1 Backbone curve

Figure 4.5 illustrates characteristics of the trilinear backbone curve. Figure 4.5a shows the

original backbone curve used by Ibarra and Krawinkler (2005) and Figure 4.5b shows the

backbone curve with the new definitions used in this study. The backbone curve is depicted in

the form of moment-rotation but the relations are valid for any force-deformation response.

This backbone curve can be divided into three regions; (1) elastic, (2) post-yielding pre-

capping and (3) post-capping. The elastic region of the backbone curve is simply defined by the

component stiffness Ke and yielding moment My. (The yield moment may incorporate some

cyclic hardening if such is present during cyclic loading and is not incorporated by superimposed

hardening rules.) In the post-yielding pre-capping region the rate of increase in moment is much

smaller than in the elastic region. The extent of the post-yielding pre-capping region is defined

by the post yielding deformation capacity θp and capping strength Mc. For simplicity, throughout

this dissertation, θp is called plastic hinge rotation capacity*. After passing the Capping Point, we

enter the post-capping region where the component softens. The deformation at which the

strength drops to zero (this is a theoretical value that may never be attained because of residual

* In fact, post yielding rotation capacity consists of two deformations parts: (1)a plastic deformation part, and (2) an elastic deformation part cause by the increase in moment above My. As long as the elastic deformation part of the post yielding rotation capacity is small, we assume that the post yielding rotation is all plastic.

- 52 -

strength) is denoted as θu, and the increase in deformation from θc to θu is denoted as post-

capping deformation capacity θpc.

Compared to the original Ibarra/Krawinkler deterioration model (see notation in Figure

4.5a), the following changes are implemented in the definitions of important parameters that

define the properties of the deterioration model (see Figure 4.5b for new notation).

θp is used as reference value for deformation capacity rather than θc/θy: Conventionally the ductility ratio, defined as the ratio of “maximum” deformation to yield

deformation, has been used to assess component or structure performance. The term

“maximum” may be associated with a limit on acceptable damage or with a limit on acceptable

deterioration beyond which a component is considered to have “failed”. Acceptable performance

is then defined as the ductility capacity exceeding the ductility demand predicted by an analytical

model. The concept of ductility ratio is ingrained in many aspects of earthquake engineering, and

researchers and engineers alike use this term to evaluate component, story, and structure

behavior, mostly because it permits behavior description by means of a dimensionless quantity.

But like any dimensionless quantity, its absolute value depends on the value that is used for

normalization.

In the context of this research the emphasis is on simulating component and structure

behavior at all levels of performance, utilizing the component load-deformation deterioration

model discussed here. Thus, there is no specific ductility capacity that is associated with a

specific level of performance. But in the later discussed frame and wall parametric studies the

need exists to define parameters that “best” describe the load-deformation characteristics of the

components of which a structure exists. This is why a decision needs to be made whether or not

to retain the use of a “ductility capacity” as a basic measure of performance.

In past studies using the deterioration model discussed here (Ibarra and Krawinkler, 2005) the

ratio θc/θy had been used as a reference value for deformation capacity and has been referred to

as the backbone ductility capacity. Here it is argued to use the plastic deformation θp = θc - θy as

a basic measure of component deformation capacity instead of θc/θy. There are many reasons for

this change, some have to do with the component model itself and others have to do with sets of

parameters used later in the moment resisting frame and shear wall studies (see Sections 4.4 and

- 53 -

4.5). From the perspective of component modeling the following arguments can be made for this

change, particularly when considering that this research is concerned primarily with the effects

of deteriorating plastic hinge regions (in beams, columns, or wall segments) on structural

response.

• For a given component (whether steel or reinforced concrete) both θy and θc (or θp = θc -

θy) depend on the moment gradient (the effective M/V ratio), which varies a great deal

between structural configurations, and for a given configuration may vary significantly

within a loading history (in beams because of gravity load effects, and in beams and

particularly columns because of redistribution due to inelastic behavior). Thus, in

concept, a given component has neither a well defined θy nor a well defined θp (or θc). It

is then a matter of deciding which of the two parameters is more stable (less sensitive to

the effective M/V ratio and to other assumptions made in defining reference values).

Basic principles and an evaluation of experimental results indicate that θp is much more

stable than θy (and therefore more stable than μ = θp/θy). For flexural plastic hinge

regions in reinforced concrete components the following arguments are offered in support

of this claim.

• The quantity θy depends strongly on the moment gradient. For beam elements it is

conventional to base θy on the double curvature condition, i.e., to assume an elastic

rotational stiffness of 6EI/L, which implies M/V = L/2. Thus, for a given cross section, θy

is linear proportional to L, and it will be different again in a real situation if the

component is not in a double curvature condition.

• Even with the aforementioned definition of elastic stiffness, θy is still poorly defined for

reinforced concrete components because EI is poorly defined. Both the ACI code (ACI

318-05) and FEMA 356 recommend values for “effective” stiffness, but these values

(which differ between ACI 318 and FEMA 356) are often difficult to match with

experimental results.

- 54 -

• For shear walls the use of an elastic rotational stiffness of 6EI/L cannot be justified

because walls are not in a double curvature condition. Thus, a reference θy value will

have to be based on different stiffness assumptions.

• Based on a regression analysis of experimental data from tests of beams, columns, and

walls, Fardis and Biskinis (2003) have proposed a general equation relating the plastic

rotation capacity to many variables, including the ratio (M/V)/h, with h being the depth of

the section. Figure 4.6 shows the variation of θy and θp as a function of (M/V)/h. The

gray line with solid rectangles shows the ratio of θy that is obtained for a certain value of

(M/V)/h, to a reference value θy,ref that is obtained for (M/V)/h = 1 (i.e., values of θv/θy,ref

are read from the ordinate located on the right hand side of this figure). The black line

with solid diamonds show the ratio of θp/θp,ref and the values for this ratio are read on the

ordinate located on the right hand side of Figure 4.6. This figure shows clearly that θp is

much less sensitive to variations in M/V (for a given h) than θy, which is linearly

proportional to M/V regardless of assumptions on the elastic stiffness of the component.

For the reasons presented here, and for other reasons discussed in Sections 4.4 and 4.5, it

was decided to use θp rather than θc/θy as a basic measure of deformation capacity. In this

context it must be said that θp, defined as θc - θy, is only a reference value of the backbone curve

and it is not a plastic rotation in the classical sense because it ignores the presence of strain

hardening, see Figure 4.7

Mc/My is used to define the hardening region rather than αs:

The elastic portion of the backbone curve is followed by a branch with reduced stiffness, which

is conventionally denoted as the “strain hardening” branch. This branch extends until maximum

strength, Mc, is reached at θc. This portion of the backbone curve is conventionally defined by

the ratio of the reduced stiffness of the component after yielding Ks to the elastic stiffness of the

component Ke, and is denoted as the strain hardening ratio αs. In this study we use the capping

strength ratio Mc/My as the basic parameter to describe this branch of the backbone curve. In the

following, supporting arguments for this change in definition are provided.

- 55 -

• Basic principles indicate that stiffness of the strain hardening branch is strongly

dependant on the moment gradient and the hardening characteristics of the moment-

curvature relationship of the cross section (i.e., M-φ diagram).

• As mentioned previously, due to the poor definition of elastic stiffness of reinforced

concrete components, calibration of the strain hardening ratio αs becomes subjective.

• For shear walls with large elastic stiffness, calibration of αs and Mc/My shows that αs has

a larger variability than Mc/My (Sections 4.4 and 4.5).

Based on these arguments we decided to use Mc/My as the basic parameter rather than αs.

θpc/θp to define the softening region rather than αc:

Past studies have used the post capping stiffness ratio, αc, as a basic parameter for defining the

softening region of the backbone curve (Pekoz and Pincheira, 2006; Ibarra and Krawinkler,

2005). As discussed in the previous two cases (i.e., θp vs. θc/θy, and Mc/My vs. αs), the value of

the elastic stiffness of reinforced concrete components is ambiguous. For this reason, we rather

not use the elastic stiffness for normalizing basic parameters of the component backbone curve.

In the author’s opinion, the rotation capacity after the capping point is correlated with the plastic

hinge deformation capacity θp and not the yield rotation θy. For this reason we have assumed that

the basic parameter that reflects the characteristics of the post-capping region of the back bone

curve is the ratio of the post capping rotation capacity θpc to the plastic hinge rotation capacity

θp, denoted as θpc/θp. Reliable experimental data that could be used to calibrate θpc/θp is very

limited. Recently, Haselton et. al. (2006) have developed a database of trilinear backbone curves

by calibrating various cyclic load-deformation tests on reinforced concrete columns. The results

of this calibration are used for developing ranges of variation for θpc/θp as shown in Sections 4.4

and 4.5.

4.3.2 Hysteretic Model

The hysteretic model used in this study follows the rules of the Peak-Oriented model initially

proposed by Clough and Johnson (1966). This hysteretic model is shown in Figure 4.8. Other

- 56 -

hysteretic models (e.g., Pinching, and Bilinear) are not considered. Medina and Krawinkler

(2003), and Ibarra and Krawinkler (2005) showed that sensitivity of structural response

parameters (i.e., EDP and collapse capacity) to variation of hysteretic models is relatively small

except for Pinching hysteretic model with severe stiffness degradation. As long as the main focus

of this study is on design of new structures in which good detailing obviates severe pinching

behavior, we only consider the Peak-Oriented hysteretic model for structural component.

The component model used in this study incorporates four cyclic deterioration modes

once the yielding point is passed in cyclic loading. These four modes include: (1) Basic Strength

Deterioration, (2) Post-Capping Strength Deterioration, (3) Unloading Stiffness Deterioration,

(4) Accelerated Reloading Stiffness Deterioration. Each mode of cyclic deterioration is shown in

Figure 4.10. Details of for deterioration modes can be found in Ibarra & Krawinkler (2005) and

Ibarra et al. (2005).

4.3.3 Cyclic deterioration model

The cyclic deterioration rate of each mode of deterioration is controlled by the rule developed

initially by Rahnama and Krawinkler (1993) and slightly modified by the author. In the Rahnama

and Krawinkler (1993) deterioration model, deterioration in excursion i (i.e., βi) is a function of

the energy dissipated in that cycle (i.e., Ei), and a reference hysteretic energy dissipation capacity

of the component (i.e., Et). βi is obtained using Equation 4.1.

1

c

ii i

t jj

E

E Eβ

=

⎛ ⎞⎜ ⎟

= ⎜ ⎟− ∑⎜ ⎟

⎝ ⎠ (4.1)

In Equation 4.1 1

i

jj

E=

∑ is the sum of hysteretic energy dissipated in all previous

excursions, and the exponent c is an exponent that defines the rate of deterioration. Rahnama and

Krawinkler (1993) suggest that c is between 1.0 and 2.0. In this study, we use c = 1. Et is the

reference hysteretic energy dissipation capacity. Rahnama and Krawinkler (1993) suggested that

t y yE Mγ θ= (i.e., γ is a parameter that expresses the reference hysteretic energy dissipation

- 57 -

capacity as a function of the component’s yield moment and yield rotation). For reasons quoted

previously the yield rotation is not a good parameter for modeling of inelastic deformation

characteristics, and many arguments can be put forth for correlating the reference hysteretic

energy capacity of a component with the plastic deformation capacity rather than the yield

deformation. Thus, we postulate that Et is a multiple of Myθp, i.e., Et = λMyθp, where λ is a

parameter that needs to be calibrated using experimental results.

Four cyclic deterioration modes are considered in this study: Basic Strength

Deterioration, Post-Capping Strength Deterioration, Unloading Stiffness Deterioration, and

Accelerated Reloading Stiffness Deterioration. In the following each deterioration mode is

briefly explained with reference to Figure 4.10. Detailed information about these four modes of

deterioration can be found in Ibarra & Krawinkler (2005) and Ibarra et al. (2005).

Basic strength deterioration

In this deterioration mode (Figure 4.10a) once the yield moment is passed, the post-yielding

branch of the backbone curve at excursion i is translated towards the origin by an amount

equivalent to factor βs,i of the yield moment and strain-hardening slope of excursion i-1, i.e.,

Equations 4.2 and 4.3, respectively. βs,i is calculated according to Equation 4.1. It should be

noted that in this study we assume the yield moment and strain-hardening slope are equal in

positive and negative directions. We also assume that the deterioration parameter is equal for all

modes of deterioration.

( ), , , 11y i s i y iM Mβ −= − (4.2)

( ), , , 11s i s i s iK Kβ −= − (4.3)

Post-capping strength deterioration

In this mode of deterioration, once the yield moment is passed, the post-capping branch of the

backbone curve at excursion i is translated towards the origin by an amount equivalent to factor

βc,i of the reference moment Mref as shown in Figure 4.10b. This reference moment is the

intersection of the vertical axis with the projection of the post-capping branch.

- 58 -

( ), , , 11ref i c i ref iM Mβ −= − (4.4)

Unloading stiffness deterioration

In this deterioration mode, after passing the yield moment, the unloading stiffness at excursion i

is deteriorated by an amount equivalent to factor βk,i of the unloading stiffness of the previous

excursion i-1. Figure 4.10c shows this mode of deterioration. Unlike other modes of

deterioration, unloading stiffness deterioration occurs every half cycle and the hysteretic energy

dissipated in half cycle is used to update the unloading stiffness of the next cycles.

( ), , , 11u i k i u iK Kβ −= − (4.5)

Accelerated reloading stiffness

Once the yield moment is passed, the absolute value of the target rotation (which is defined as

the maximum positive or negative of rotation of past cycles) is increased in accordance with the

direction of loading (Figure 4.10d) as follows:

( ), , , 11t i a i t iθ β θ −= + (4.6)

4.4 Generic moment-resisting frames and range of structural parameters

Generic frames have been utilized by many researchers for assessing seismic response and

behavior of moment-resisting frames. In terms of geometry, generic frames used in the past can

be divided into two main categories: (1) single-bay generic frames, and (2) “fishbone” shape

generic frames. Many studies are reported in the literature, in which it is assumed that the

response of a multi-bay building can be simulated adequately by a single bay frame (e.g., Medina

and Krawinkler 2003; Chintanapakdee and Chopra, 2003; Esteva and Ruiz 1989.). Limitations of

this simplification are well known (e.g., Ruiz-Garcia, 2004). One important limitation is that at

each connection only one beam frames into two columns, which does not permit simulation of

realistic conditions at an interior joint.

“Fishbone” shape generic frames are a type of generic frames was used by Ogawa et. al.

(1999), Luco et. al., (2003), and Nakashima et. al (2002). In this simplification, a multi-bay

frame is modeled as a cantilever beam that at each floor level has two rotational springs

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connected to rollers on each side of the cantilever. A fundamental assumption in the

development of fishbone generic frames is that the rotations of joints at the same floor are

identical. Due to the limitations imposed by this assumption we did not use fishbone shaped

generic frames.

In this study we introduce and use a family of 3-bay generic frames. These generic

frames are expansions of the single-bay generic frames used in studies such as Ibarra &

Krawinkler (2005) and Medina and Krawinkler (2003). The main reason for using three bays

rather than one is that such generic frames have both interior and exterior columns, which

permits a realistic evaluation of the strong column - weak girder concept for both column types.

4.4.1 Geometry and number of stories of generic moment-resisting frames

Generic moment-resisting frames used in this study are two-dimensional centerline models that

cover the range of low-rise and mid-rise structures, i.e., we use frames with number of stories N

equal to 4, 8, 12, 16. The height of each story is assumed to be equal to 12’ and the bay span is

equal to 36’ (see Appendix A for arguments supporting these assumptions).

4.4.2 Fundamental Period of generic moment-resisting frame structures

For each number of stories, three fundamental periods are considered: T1 = 0.10N, 0.15N, and

0.2N. The parameter that relates the fundamental period to the number of stories is denoted as

“period coefficient” αt. Therefore, for moment-resisting frames αt = 0.10, 0.15, 0.20. This range

is selected based on data extracted from the database of frame structures periods provided by

Goel and Chopra (1997). To ensure that the contribution of non-structural elements on stiffness,

and hence structural period, is small, only structures with residential and office occupancy that

experienced ground motion acceleration more than 0.15g are selected from this database (as

reported by Bertero et. al., 1988, the period value increases by a factor of 1.7 to 2.0 from pre-

earthquake ambient conditions to pre-yielding conditions). Figure 4.11 shows the data point

extracted from the database of building periods developed by Goel and Chopra (1997) along with

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two lines of T = 0.1N and 0.2N. As seen, these two lines cover the range of variation of the

fundamental period of moment-resisting frames. This justifies the use of T1 = 0.1N and T1 =

0.2N. T1 = 0.15N is selected as an intermediate value to study the effect of variation of building

period on the response with a better resolution.

4.4.3 Viscous damping of generic moment-resisting frame structures

5% critical damping is assumed in the first mode and the third mode of vibration of generic

moment-resisting frames. This value for viscous damping is allowed in FEMA 356 and has been

used in previous studies such as Medina and Krawinkler (2003), and Ibarra and Krawinkler

(2005).

Although we have not considered viscous damping as a basic structural parameter, it is

acknowledged that a variation of this parameter independently affects all three domains of the

DDSS. For instance, smaller viscous damping results in a higher intensity measure (i.e., Sa(T1))

at a certain hazard level, which increases the deformation based response of the structural

systems and increases the associated loss in building components. Such effects are not

considered in this study.

4.4.4 Variation of stiffness and strength along the height of generic moment-resisting

frame structures

This study is concerned with regular moment-resisting frames. For this reason, the generic

frames used are regular in mass (i.e., each story has the same mass) and regular in stiffness (i.e.,

difference between the stiffness of adjacent stories is less than 60 percent of the story above or

less than 70 percent of the average stiffness of the three stories above [FEMA 450-1 2003]).

The same moment of inertia is assigned to columns and beams in one story. This

assumption, along with the assumption that the height of each story is 2.5 times the bay span,

ensures that the generic frames are representative of realistic field conditions (see Appendix A).

In Appendix A a detailed discussion of two different types of deformations in moment-resisting

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frames (i.e., shear-type deformations and flexural-type deformations) along with reasoning

behind the aforementioned assumptions are provided.

Three different cases for variation of beam stiffness and strength along the height of the

generic moment-resisting frames are considered. Details of the selection criteria are discussed in

Appendix A. These three alternatives are denoted as: Stiff. & Str. = “Shear”, “Int.”, and “Unif.”.

Figure 4.12 shows the ratio of the moment of inertia of beams at each floor to the

moment of inertia of the beams at the second floor (i.e., top of first story) for an 8-story generic

frame. Stiff. & Str. = “Shear” implies that the moment of inertia and bending strength of all

beams at a floor level is proportioned to the story shear force obtained from subjecting the

generic moment-resisting frame to the NEHRP lateral load pattern (this lateral load pattern is

period dependent, see FEMA 356). This alternative guarantees a straight line deflected shape

under the NEHRP load pattern. Stiff. & Str. = “Unif” implies that the beam moment of inertia

and bending strength is the same along the height of the structure, which represents a bounding

case for studying the effect of overstrength on structural response. In order to capture the

behavior of structural systems that fall in between these two bounding cases, the case of Stiff. &

Str. = “Int.” is introduced in which the stiffness bending strength of beams at each floor level is

the average of the stiffnesses of beams at the same floor level for Stiff. & Str. = “Shear” and

Stiff. & Str. = “Unif.”.

Figure 4.13 shows the effect of variation of stiffness along the height on the 1st mode

shape of the 8-story generic frame with αt = 0.15. As it is seen, the 1st mode shape of the case

with Stiff. & Str. = “Shear” is close to a straight line showing that the first mode load pattern and

the NEHRP load pattern do not differ by much.

4.4.5 Base shear strength of generic moment-resisting frame structures

The bending strength of beams at the second floor of the generic moment-resisting frame

structures is obtained by applying the NEHRP lateral load pattern (which is period dependent,

see FEMA 356); with yield base shear value Vy = γ W where γ is the yield base shear coefficient

and W is the seismic weight of the structure. γ is a function of the structure’s period i.e., γ =

Sa(T1)/Rμ. where Sa(T1) is the elastic spectral acceleration and Rμ is the ductility dependent

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reduction factor. For each value of fundamental period three values for γ are considered. These

three γ values are obtained by dividing a typical design response spectrum at the 10/50 hazard

level for the Los Angeles area on soil type D by three values of Rμ = 1.5, 3.0, 6.0 as shown in

Figure 4.14. The design spectrum is depicted with a solid black line. The inelastic spectra

associated with Rμ equal to 1.5, 3.0, and 6.0, are shown with gray lines. As seen in this figure, for

each fundamental period the range of variation of γ covers a reasonable range for studying the

effect of variation of this parameter on response of generic moment-resisting frames structures.

Bending strength of beams in upper floors is obtained by tuning the beam strength profile

of to beam stiffness profile discussed in Section 4.4.4. In this method, we assume that the

bending strength and initial stiffness are proportional.

4.4.6 Variation of column to beam strength ratio

Three different cases for variation of column to beam bending strength are considered in the

generic moment-resisting frames. These cases are denoted as: SCB = 1.2-1.2, 2.4-1.2, 2.4-2.4

and are schematically depicted in Figure 4.15. The first number represents the ratio of sum of

column strengths to sum of beam strength at an exterior joint of generic moment-resisting frame,

and the second number represents the same ratio for the interior joint.

The variations considered for the SCB factor covers a realistic range of column to beam

strength ratios. ACI 318-05 requires that this ratio be larger than 1.2. Kara et. al. (2001) suggest a

value equal to 2.0 or higher for this ratio.

4.4.7 Plastic hinge rotation capacity θp of generic moment-resisting frames components

In order to obtain proper realizations for the plastic hinge rotation capacity of beams and

columns used in generic moment-resisting frames, several sources were evaluated. FEMA 356

lists the acceptable plastic rotation for the “collapse prevention” performance objective as

between 0.5% and 2.5% for flexure-controlled reinforced concrete beams and as between 0.2%

and 2.0% for reinforced concrete columns. Figure 4.16 shows the plastic hinge rotation capacity

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of reinforced concrete columns obtained from the database of structural parameters developed by

Haselton (2006). Plastic hinge rotations of those tests with an axial load ratio less than 0.2 were

chosen for this statistical evaluation. (It is assumed that the generic frames used in this study are

not expected to carry much of the gravity load of the building and are primarily used for lateral

load resisting system). In Figure 4.16 the black solid line represents the lognormal distribution

fitted to the data points. Based on the data shown in Figure 4.16, the following three values for

variation of plastic hinge rotation are selected: θp = 1%, 3%, and 6%.

Rotational capacity of a steel member is highly correlated with the type of connection and

axial load on the member (Roeder et. al., 1989). For example, a study by Mander et al. (1994) on

top and seat angle connections showed that plastic rotation of steel members are sensitive to how

the bolts and nuts were oriented when tightened and is on the order of 3%. After the Northridge

Earthquake an extensive study on steel connection behavior was conducted (the SAC Steel

Project) in which the behavior of many steel connections was evaluated experimentally (FEMA

350 – FEMA 355). The range of variation of plastic hinge rotation proposed in the previous

paragraph seems to be adequate for steel beams and columns (FEMA 355F) with various

connection types. It is noted that in an ongoing study by Krawinkler and Lignos (2007) a

database of steel connection behavior is being created that incorporates results of many steel

connection test around the world. This database, upon availability, will be very helpful in

obtaining statistical measures for steel members plastic hinge rotation.

4.4.8 Post-capping rotation capacity ratio θpc/θp of generic moment-resisting frames

components

Due to the novelty of the concept of post-capping rotation capacity, test data for calibration of

this parameter is very limited. Many guidelines, such as FEMA 356, postulate a sharp drop in

strength after the capping point that stabilizes at a residual strength value (for both reinforced

concrete and steel members). FEMA-355F also suggests the same approach for addressing the

post capping stiffness. Some information on the post-capping rotation capacity of reinforced

concrete members is obtained from the database of calibrated components by Haselton et. al.,

(2006) (data of rectangular columns with axial load factor P/(f’cBH) less than 0.2 where P is the

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axial load, f’c is the concrete compressive strength, and B and H are the dimensions of the

section) were selected for this statistical evaluation). Figure 4.17 shows the variation of this

parameter along with the fitted lognormal distribution. Based on the data shown in Figure 4.17,

the following three values for variation of θpc/θp are selected: θpc/θp = 1, 5, 15.

4.4.9 Capping strength ratio Mc/My of generic moment-resisting frames components

As discussed in Section 4.3, Mc/My is the new parameter for defining the post-yielding region of

the backbone curve. Figure 4.18 shows the Mc/My values obtained from the database of

reinforced concrete column parameters calibrated by Haselton et. al (2006) (only data with axial

load ratio less than 0.2 is selected). Based on this data and the fact that the response of MDOF

systems is not very sensitive to the slope of the post-yielding branch of the backbone curve

(Rahnama & Krawinkler, 1993), we use Mc/My = 1.1 for all generic moment-resisting frames

members. This value is the average of data shown in Figure 4.18.

4.4.10 Cyclic deterioration parameter of generic moment-resisting frames components

In our previous discussion we argued that the yield rotation θy is not an ideal parameter for

normalization of structural component parameters (θy is moment gradient dependent and cannot

be defined rigorously). Based on this argument, we have modified the deterioration parameter

defined by Rahnama and Krawinkler (1993) (see Section 4.3). In order to find a proper range of

variation for this parameter, λ, we have used the database of structural parameters developed by

Haselton et. al., (2006). Figure 4.19 shows the variation of this parameter along with the fitted

lognormal distribution. Based on the data presented in Figure 4.19, , the following three values

for variation of λ are selected:λ = 10, 20, 50.

4.5 Generic shear walls and range of structural parameters

The shear walls considered in this study are flexural shear walls, i.e., their behavior is governed

by flexural-type deformations. Generic shear walls are modeled as cantilever members with

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equal masses lumped at each story level as shown in Figure 4.4. Coupled shear walls and squad

walls are not within the scope of this research. The generic shear walls used in this study are

modeled as a series of beam elements, one element per floor. Each element has a modulus of

elasticity E, cracked moment of inertia Ii, and length L (equal to the height of one story). Shear

deformations and shear failure are not considered. In order to separate the inelastic flexural

behavior and elastic flexural behavior in shear wall elements, each element is modeled with a

nonlinear zero-length spring element, and an elastic beam element as discussed in Appendix D.

The nonlinear zero-length spring is added to the bottom of each elastic beam element. Figure

4.20 shows the first two stories of a generic shear wall using the modeling rules described above.

4.5.1 Geometry and number of stories of generic shear walls

Generic shear walls are two-dimensional cantilever with number of stories, N = 4, 8, 12, 16.

These story numbers are selected to reasonably cover the range of low-rise and mid-rise

buildings. Centerline dimensions are used for structural elements. The height of each story is set

to 12’.

4.5.2 Fundamental Period of generic shear walls

We assume that the period of a generic shear wall is tied to the associated number of stories. The

parameter that relates the shear wall fundamental period to the number of stories of the shear

wall is denoted as the “period coefficient” αt. For the generic shear walls used in this study, three

realizations of structural period are used: T1 = 0.05N, 0.075N, and 0.1N (i.e. αt = 0.05, 0.075, and

0.1). These values are selected based on data from the database of shear wall building periods

developed by Goel and Chopra (1997). Only periods of residential and office buildings that were

obtained during a ground motion with PGA larger than 0.15g were selected from this database

(to ensure that the contribution of nonstructural components to fundamental period is minimal).

Figure 4.21 shows the data points along with two period bounds of T1 = 0.05N and T1 = 0.1N. It

- 66 -

is seen that most of the data points fall in between these bounds. Wallace and Moehle, (1992),

consider T1 = 0.05N and T1 = 0.1N as two good estimates for the period of shear wall building in

Chile and U.S, respectively. T1 = 0.075N is an intermediate period value than enables us to study

the effect of variation of building period of structures response with a better resolution.

4.5.3 Viscous damping of generic shear walls

For generic shear walls, 5% critical damping is assigned to the first mode and the third mode of

vibration. Consideration of 5% critical damping is allowed by FEMA 356 and has been the basis

of other research on nonlinear response of shear wall structures (e.g., Lee and Mosalam 2005;

Seneviratna and Krawinkler 1997). Other damping values, which may be justifiable for shear

wall configurations, are not considered in this study.

4.5.4 Bending strength of generic shear walls

The bending strength at the base of generic shear walls is obtained by applying the NEHRP

lateral load pattern with a base shear value of Vy. The resultant moment at the base of the shear

wall is considered as the bending strength My,base (i.e., My,base = Vy .H’ where H’ is the lever arm

of the NEHRP lateral load pattern applied to the shear wall) as shown in Figure 4.22. Vy is

calculated as Vy = γ.W where γ is the yield base shear coefficient and W is the seismic weight of

the shear wall. Three values of γ are considered for each period of the generic shear wall

structures. These γ values are obtained from the same spectra and the procedure discussed in

Section 4.4.5 for generic moment resisting frames, with the results shown in Figure 4.23.

4.5.5 Variation of stiffness and strength along the height of generic shear walls

In low-rise and mid-rise buildings, dimensions of cantilever shear walls usually do not change

with height. For this reason we have not considered a variation of stiffness along the height in the

generic shear wall models used in this study. This assumption is consistent with

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recommendations of FEMA 356 and the studies performed by other researchers (e.g., Paulay and

Uzumeri, 1975 ; Paulay and Priestley, 1992).

Bending moment demands along the height of flexural shear wall structures during earthquakes

are larger than bending strengths prescribed by codes (Krawinkler & Seneviratna, 1997; Blakely

et. al., 1975). The reason is that modes (shapes and periods) change radically once a plastic hinge

develops at the base of a shear wall. Two different cases for variation of bending strength along

the height of the generic shear walls are considered: Str. = “Unif”, “-0.05My,base / floor”.

In the first alternative, denoted as Str. = “Unif”, the strength of the shear wall is not

reduced along the height (i.e., uniform strength). The second alternative, denoted as Str. = “-

0.05My,base / floor”, represent a case in which the flexural strength of the shear wall is reduced

along the height at a rate equal to 5% of the base bending strength per floor. This alternative is

chosen to represent a reasonable reduction of flexural strength along the height in typical shear

walls. This pattern results in a bending strength of 80%, 60%, 40%, and 20% of My,base in the 4th,

8th, 12th, and 16th story, respectively, of the generic wall structures.

4.5.6 Plastic hinge rotation capacity θp of generic shear walls components

The range of variation for the plastic hinge rotation capacity of generic shear walls was obtained

using the database of structural parameters developed by Fardis and Biskinis (2003). This data is

plotted in Figure 4.24 along with the lognormal distribution curve fitted to the data points. As

seen, the median value of shear walls plastic hinge rotation is about 2% with a logarithmic

variation of 0.6. Also, as stated in FEMA 356, the acceptable plastic hinge rotation for the

“collapse prevention” performance level is between 0.2% and 1.5%. Based on these

observations, we consider three alternatives for plastic hinge rotation capacity: θp = 1%, 2%, and

3%.

4.5.7 Post-capping rotation capacity ratio θpc/θp of generic shear walls components

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Unfortunately, almost all previous shear wall tests do not push the shear wall far enough to

obtain a reliable post-capping rotation capacity. In order to investigate the range of θpc/θp in

which the response of a shear wall is sensitive, a 9-story shear wall structure with period of 0.9

seconds and plastic hinge rotation capacity of 2% is considered. Pushover curves for this

structure with various values of θpc/θp (i.e., between 0.4 and 9) are shown in Figure 4.25. Figure

4.26 shows the sensitivity of the median of collapse capacity to θpc/θp. As seen, the range in

which collapse capacity is sensitive to θpc/θp is between 0.4 and 3.0. For this reason, we consider

three realizations for θpc/θp in this range: θpc/θp = 0.5, 1, 3.

4.5.8 Capping strength ratio Mc/My of generic shear walls components

As mentioned in Section 4.3, normalizing the stiffness of the post-yielding branch of the

component backbone curve to the initial stiffness would not provide an attractable measure for

describing this part of the backbone curve. In the case of shear walls, as the initial stiffness of the

shear wall component is very large, the yield rotation is much smaller than the plastic hinge

rotation capacity. Such problems with normalized values to the elastic stiffness of the wall

component are common.

Test results obtained by Elnashai et. al (1990), Ali and Wight (1991), Oh et. al., (2002),

and Vallenas and Bertero (1979) show a large variation for strain hardening ratio (i.e., between

0% and 2.7%) and a rather smaller variation for capping strength ratio (i.e., 1.0 to 1.11. Based on

these results, we consider a single value for Mc/My = 1.10. This value obtained by only

considering those test results that have strain hardening larger than 0%.

4.5.9 Cyclic deterioration parameter λ of generic shear walls components

Experimental data on deterioration of shear wall, that fail in flexure are very limited. Since the

only shear wall structures addressed in this study are flexural shear walls, we have assumed the

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same variation for the deterioration parameter as we used for generic moment-resisting frames,

i.e., λ = 10, 20, and 50.

4.6 Development of database of structural response parameters

A comprehensive database of structural response parameters (EDPs and collapse capacities) for a

wide-range of combination in structural parameters are developed as part of this study. In the

previous sections, we introduced the structural parameters, their range of variation, and the

corresponding realizations used in this study. In this section our objective is to summarize the

combinations in structural parameters used to develop the aforementioned database and to

demonstrate the process for obtaining structural response parameters.

The number of combinations in structural response parameters is sizable in both generic

moment-resisting frames and generic shear walls. As seen in Sections 4.4 and 4.5, at least three

realizations for each structural parameter were introduced, which makes the number of

combination cases for k variables equal to approximately 3k. Performing nonlinear response

history analysis in the form of Incremental Dynamic Analysis at this scale would be extremely

time consuming. For this reason a number of base cases are introduced in which detailed

combinations in a subset of structural parameters are considered. Values of structural parameters

that are not considered in the subset of base case structural parameters are set to their center

value. Then, a number of variation cases are introduced for each base case. In each variation, one

of the structural parameters that was not considered in the subset of base case structural

parameters is changed and other structural parameters are kept unchanged. The following

subsections present information about base case and variation cases.

At the end of this section we introduce the methodology for obtaining structural response

parameters using nonlinear response history analysis. We use Incremental Dynamic Analysis

(Vamvatsikos and Cornell, 2002) to evaluate structural response parameters at each level of IM.

A brief discussion about this procedure is provided in Section 4.6.4

4.6.1 Base case generic structural systems

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Base cases are defined as the statistical combinations of three global structural parameters for

each type of structural system (generic moment-resisting frames and generic shear walls). These

parameters are: Number of stories N, period coefficient αt, and yield base shear coefficient γ.

Ranges of variation of these parameters were discussed in the preceding sections and are

summarized here for reference.

Structural parameters for base case generic moment-resisting frames

Number of stories: N = 4, 8, 12, 16

Period coefficient: αt = 0.1, 0.15, 0.2

Yield base shear coefficient: γ = 1

1 0.6min ,R T Rμ μ

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

& Rμ = 1.5, 3.0, 6.0

Variation of stiffness and strength along height: Stiff. & Str. = “Shear”.

Column to beam strength factor: SCB = 2.4-1.2

Plastic hinge rotation capacity of beams and column: θp = 0.03

Post-capping rotation capacity ratio of beams and column: θpc/θp = 5.0

Cyclic deterioration parameter: λ = 20

Structural parameters for base case generic shear walls

Number of stories: N = 4, 8, 12, 16

Period coefficient: αt = 0.05, 0.075, 0.1

Yield base shear coefficient: γ = 1

1 0.6min ,R T Rμ μ

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

& Rμ = 1.5, 3.0, 6.0

Variation of strength along height: Str. = “Unif”.

Plastic hinge rotation capacity of beams and column: θp = 0.02

Post-capping rotation capacity ratio of beams and column: θpc/θp = 1.0

Cyclic deterioration parameter: λ = 20

4.6.2 Variation in global modeling variables of base case generic structures

For base case generic moment-resisting frames

- 71 -

Variation of stiffness and strength along height: Stiff. & Str. = “Int.” , “Unif”

Column to beam strength factor: SCB = 1.2-1.2, 2.4-2.4

For base case generic shear walls

Variation of strength along height: Str. = “-0.05My,base / floor”

4.6.3 Variation in component variables of base case generic structures

For each base case structural system, variations in three component parameters are considered.

Figure 4.27 shows the layout of these variations with respect to the base case. As seen in this

figure, the base case is located in the center of these variations. Each variation is defined by a

change in only one of the following variables: plastic hinge rotation capacity θp, post-capping

rotation capacity θpc/θp, and deterioration parameter λ,. In Figure 4.27, a variation in each

component model parameter is marked with: “-1”, “0”, and “+1”. “0” represents the case in

which the value of the corresponding variable is set to the value for the base case. “-1” and “+1”

represent cases in which the value of the corresponding variable is set to the lower bound and

upper bound considered. As seen in Figure 4.27, for each base case six variations are considered.

Variation in component modeling variables in base case generic moment-resisting frames

Three generic moment-resisting frame component variables θp, θpc/θp, and λ are varied,

individually, as shown in Figure 4.27. The variations in above mentioned parameters in moment-

resisting frame components are:

Plastic hinge rotation capacity, θp = 1% & 6%

Post-capping rotation capacity ratio, θpc/θp = 1.0 & 15.0

Cyclic deterioration parameter, λ = 10 & 50

Variation in component modeling variables in base case generic shear walls

- 72 -

Three shear component variables θp, θpc/θp, and λ are varied, individually, as outlined in Figure

4.27. The variations in above mentioned parameters for shear walls are:

Plastic hinge rotation capacity, θp = 1% & 3%

Post-capping rotation capacity ratio, θpc/θp = 0.5 & 3.0

Cyclic deterioration parameter, λ = 10 & 50

4.6.4 Incremental Dynamic Analysis and determination of structural response

parameters

The base case structural systems and their variations are modeled in a modified version of the

DRAIN-2DX (1993) computer program in order to perform nonlinear response analysis. The

original DRAIN-2DX version has been modified by Medina and Krawinkler (2003), and Ibarra

and Krawinkler (2005), and includes the deteriorating component models described in Section

4.3 for various hysteresis behaviors (Bilinear, Peak-Oriented, and Pinching).

Incremental Dynamic Analysis, IDA, (Vamvatsikos and Cornell, 2002), is used to obtain

the structural response parameters. In this method, each generic structure is subjected to the

LMSR-N (Medina and Krawinkler, 2003) ground motion set, all scaled to a specific value of IM

(i.e., Sa(T1)), and nonlinear dynamic analysis is performed. The IM level is increased with small

increments until dynamic instability occurs due to P-Delta effects and component deterioration.

At each IM level, many structural response parameters, EDPs, are recorded and stored. These

EDPs are enlisted in Appendix E. Figure 4.28 shows typical IM-EDP curves obtained from IDAs

on an 8-story generic moment-resisting frame.

- 73 -

Fig. 4.1 Modes of deformation in structures (after Miranda, 1999)

(a) (b)

Fig. 4.2 Moment-resisting frames and corresponding structural model: (a) geometry, (b)

structural model

H

hColumn

component

Pure shear-type deformation

Pure flexural-type deformation

Combined shear-type & flexural-type deformations

Beam component

- 74 -

(a) (b) (c) (d) (e)

Fig. 4.3 Modes of failure in reinforced concrete shear walls: (a) shear wall loading during a

seismic event, (b) failure due to yielding of flexural reinforcement, (c) failure due to

diagonal tension, (d) failure due to sliding shear, and (e) failure due to shear/flexural

yielding (after Paulay and Priestley, 1992)

(a) (b) (c)

Fig. 4.4 Cantilever shear wall and corresponding structural model: (a) geometry, (b)

elastic and inelastic deformations, and (c) structural model

H

h Plastic hinge

Shear wall component θp = δp / H

δp

Possible plastic hinge

location

δy

- 75 -

yM

yθ cθ

M

θeK

s eKα

c eKα

Basic Parameters

eK

yM

sα

cα

c

y

θμ

θ=

Initial Stiffness

Yield Moment

Strain-Hardening Stiffness Ratio

Ductility Capacity

Post-Capping Stiffness Ratio

cc y y

y

θθ θ μθ

θ= =

( )1c y s e yM M Kα μ θ= + −

Capping Rotation

Capping Moment

cM

yy

e

MK

θ = Yield Rotation

c

y

θθ

uθ

Derived Parameters

cu c

c e

MK

θ θα

= +

Capping Point

Yielding Point

Post-CappingPost-Yielding Pre-CappingElastic

(a)

yM

yθ cθθ

eK

c

y

MM

Basic Parameters

eK

yM

c

y

MM

pc

p

θθ

pθ

Initial Stiffness

Yield Moment

Capping moment ratio

Plastic Hinge Rotation Capacity

Post-Capping Rotation Capacity Ratio

Derived Parameters

c y pθ θ θ= +

cc y

y

MM MM

=

Capping Rotation

Capping Moment

cM

yy

e

MK

θ = Yield Rotation

pθ pcθ

uθ

u c pcθ θ θ= +

Capping Point

Yielding Point

MPost-Capping

Post-Yielding Pre-CappingElastic

(b)

Fig. 4.5 Component back-bone curve and its parameters: (a) old definitions, (b) new

definitions

- 76 -

Sensitivity of yield rotation and plastic hinge rotation to (M/V)/h

0

1

2

3

4

5

6

0 1 2 3 4 5 6(M/V)/h

θ y/ θ

y,re

f

0

1

2

3

4

5

6

θ p/ θ

p,re

f

Yield rotation

Plastic hinge rotation

Fig. 4.6 Sensitivity of yield rotation and plastic hinge rotation to (M/V)/h

Definition of Plastic RotationP.H. region

θ

θ

δ

P

P

δp

δe

P.P.H.

Experiment:

Analytical Model:

L

Recorded quantities: P δ−Chord rotation Lθ δ= =End Moment M P L= =

P Mδ θ− ≈ −

3

&3 3

y ye e

P L M LEI EI

δ θ= =

p e

p e

δ δ δ

θ θ θ

= −

= −

3

3

3

pl

pl

PLEI

MLEI

δ δ

θ θ

= −

= −

Definition used in this study

M

θθpl

θplθpl

M

θplθpl

θpl

θp

θp

My

My

Fig. 4.7 Definition of θp and its approximation to plastic rotation.

- 77 -

-1

M

θ0

1

2

3

4

5

θ

N1

2

3

4

5

0

Fig. 4.8 Peak-Oriented hysteretic model used in this study

PEAK ORIENTED HYSTERETIC MODEL WITH

CYCLIC DETERIORATION

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

-4 -2 0 2 4 6Normalized Displacement

Nor

mal

ized

For

ce

Initial Backbone

Unloading Stiffnes Det.

Post- Capping Strength Det.

Basic Strength Det.

Accelerated Stiffness Det.

Fig. 4.9 Peak-Oriented hysteresis model with 4 modes of cyclic deterioration (After Ibarra

& Krawinkler 2005)

- 78 -

M

θ

Ku,1

0

Ke

My+

Ku,2

My-

12

3

45

6

7Interruption(disregard stiffness det)

Ke

M

θ

Ku,1

0

Ke

My+

Ku,2

My-

12

3

45

6

7Interruption(disregard stiffness det)

Ke

1

Ke

3

M1+

6

0

2

45

8

θco+

θc0-

θc1-

Ks,0

Ks,1+

Ks,1-

My+

My-

M1-

θc1+7 θ

M

Ks,0

1

Ke

3

M1+

6

0

2

45

8

θco+

θc0-

θc1-

Ks,0

Ks,1+

Ks,1-

My+

My-

M1-

θc1+7 θ

M

Ks,0

θ6

Δθt1-

Krel

My+

3

-

Ke

0

1 2

45

7

θc0+

OriginalEnvelope

θc0-

8 9M Δθt1

+

My

θ6

Δθt1-

Krel

MyMy+

3

-

Ke

0

1 2

45

7

θc0+

OriginalEnvelope

θc0-

8 9M Δθt1

+

MyMy

2

4

7

6

My+

3

Krel

0

1

θc0+θc1

+

OriginalEnvelope

θc1-

Mref,0+

θc0-

Mref,1+

Mref,1-

-

Ke

Mref,0-

θ

M

8

My

2

4

7

6

My+

3

Krel

0

1

θc0+θc1

+

OriginalEnvelope

θc1-

Mref,0+

θc0-

Mref,1+

Mref,1-

-

Ke

Mref,0-

θ

M

8

MyMy

(a) Basic strength deterioration

(c) Unloading stiffness deterioration

(b) Post-capping strength deterioration

(d) Accelerated reloading stiffness deterioration

Fig. 4.10 Peak-Oriented hysteresis model with 4 individual modes of cyclic deterioration

(After Ibarra & Krawinkler 2005)

- 79 -

Moment-resisting frame Period Estimation

0

0.5

1

1.5

2

0 4 8 12 16 20

Number of Stories

Perio

d

Data Points

T = 0.20N

T = 0.15N

T = 0.10N

Fig. 4.11 Relation between number of stories and period of generic moment-resisting

frames (Data obtained from Goel and Chopra, 1997)

Variation of stiffness along the height (MRF)N = 8, αt = 0.15

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1 1.2

Relative Beam Stiffness (Ibeam,i/Ibeam,1)

Floo

r

Stiff. & Str. = "Shear"

Stiff. & Str. = "Int."

Stiff. & Str. = "Unf."

Fig. 4.12 Variation of stiffness along the height of generic moment-resisting frame (N = 8,

αt = 0.15)

- 80 -

Effect of variation of stiffness along the height on 1st Mode Shape (MRF), N = 8, αt = 0.15

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1

1st Mode Shape Amplitude

Floo

r

Stiff. & Str. = "Shear"

Stiff. & Str. = "Int."

Stiff. & Str. = "Unf."

Fig. 4.13 Effect of variation of stiffness along the height of generic moment-resisting frame

(N = 8, αt = 0.15) on first mode shape

Design Response Spectra (MRF)ξ = 5%, Soil type D

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5Period

Sa(T

1)/g

Elastic Design SpectrumInelastic Design Spectrum, Rμ = 1.5Inelastic Design Spectrum, Rμ = 3.0Inelastic Design Spectrum, Rμ = 6.0

Values of γ considered in

generic moment-resisting frames

Fig. 4.14 Yield base shear coefficients γ = Vy /W for generic moment-resisting frames

- 81 -

My My My

My My My

1.2My

1.2My

1.2My 0.6My

0.6My

0.6My

My My My

My My My

1.2My

1.2My

1.2My 1.2My

1.2My

1.2My

My My My

My My My

2.4My

2.4My

2.4My 1.2My

1.2My

1.2My

SCB = 1.2-1.2

SCB = 2.4-1.2

SCB = 2.4-2.4

Exterior, story i Interior, story i

Exterior, Roof Interior, Roof





Fig. 4.15 Schematic representation of three variations of column strength in generic

moment-resisting frames (SCB = 1.2-1.2, 2.4-1.2, 2.4-2.4)

- 82 -

Range of Variation of θp for R/C ColumnsRectangular coluns, Axial load ratio < 0.2, cyclic loading

0

0.25

0.5

0.75

1

0 0.03 0.06 0.09 0.12θp

Prob

abili

ty o

f exc

eedi

ng (

p

Data Points

Fitted LognormalDistribution

θ p

Range of Variation of θp for R/C ColumnsRectangular coluns, Axial load ratio < 0.2, cyclic loading

0

0.25

0.5

0.75

1

0 0.03 0.06 0.09 0.12θp

Prob

abili

ty o

f exc

eedi

ng (

p

Data Points


θ p

Fig. 4.16 Variation of plastic hinge rotation capacity from column test results and the range

used in generic moment-resisting frames (data from Haselton et. al., 2006)

Range of Variation of θpc/θp for R/C ColumnsRectangular columns, Axial load ratio < 0.2, cyclic loading

0

0.25

0.5

0.75

1

0 5 10 15 20 25 30 35 40θpc/θp

Prob

abili

ty o

f exc

eedi

ng

pc/

p

Data Points


θ pc/θ

p

Fig. 4.17 Variation of post-capping rotation capacity ratio from column test results and the

range used in generic moment-resisting frames (data from Haselton et. al., 2006)

- 83 -

Range of Variation of Mc/My for R/C ColumnsRectangular coluns, Axial load ratio < 0.2, cyclic loading

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2Mc/My

Prob

abili

ty o

f exc

eedi

ng M

c/My Data Points

Average value usedin this study

Fig. 4.18 Variation of capping strength ratio from column test results and the average value

used in generic moment-resisting frame components (data from Haselton et. al., 2006)

Range of Variation of λ for R/C ColumnsRectangular columns, Axial load ratio < 0.2, cyclic loading

0

0.25

0.5

0.75

1

0 50 100 150 200

λ

Prob

abili

ty o

f exc

eedi

ng

Data Points


λ

Fig. 4.19 Variation of cyclic deterioration parameter from column test results (data from

Haselton et. al., 2006)

- 84 -

1pθ

1yθ

ipθ i

yθ

1θ

iθ

h

h

yM

1yθ 1

cθθ

eK

c

y

MM

cM

1pθ 1

pcθ

M

Generic Shear Wall Modeled

With Deteriorating Component

a

b

c d

yM

1,spyθ 1,

1, 1

spc

spy p

θθ θ≈ +

θ

speK

c

y

MM

cM

1pθ≈

M

1, 1 1,sp elpc pc cθ θ θ= +

yM

1,elyθ

θ

eleK

c

y

MM

cM

M

1,elcθ

Spring Backbone Curve Elastic Element Backbone Curve

1pθ

1yθ

ipθ i

yθ

1θ

iθ

1pθ

1yθ

ipθ i

yθ

1θ

iθ

h

h

yM

1yθ 1

cθθ

eK

c

y

MM

cM

1pθ 1

pcθ

M

yM

1yθ 1

cθθ

eK

c

y

MM

cM

1pθ 1

pcθ

M

Generic Shear Wall Modeled

With Deteriorating Component

a

b

c d

yM

1,spyθ 1,

1, 1

spc

spy p

θθ θ≈ +

θ

speK

c

y

MM

cM

1pθ≈

M


yM

1,elyθ

θ

eleK

c

y

MM

cM

M

1,elcθ


yM

1,spyθ 1,

1, 1

spc

spy p

θθ θ≈ +

θ

speK

c

y

MM

cM

1pθ≈

M


yM

1,elyθ

θ

eleK

c

y

MM

cM

M

1,elcθ


yM

1,spyθ 1,

1, 1

spc

spy p

θθ θ≈ +

θ

speK

c

y

MM

cM

1pθ≈

M


yM

1,elyθ

θ

eleK

c

y

MM

cM

M

1,elcθ


Fig. 4.20 Modeling of generic shear walls used in this study: (a) shear wall global model,

(b) shear wall component model, (c) inelastic spring in the shear wall component model,

and (d) elastic element in shear wall component model

- 85 -

Shear Wall Period Estimation

0

0.5

1

1.5

2

0 4 8 12 16 20

Number of Stories

Perio

dData Points

T = 0.100N

T = 0.075N

T = 0.050N

Fig. 4.21 Relation between number of stories and period of shear wall structures (Data

obtained from Goel and Chopra, 1997)

Fig. 4.22 Shear wall bending strength and yield base shear

yV

,y baseMyV

yV Wγ=

,y baseM WHγ ′=

H ′

H

- 86 -

Design Response Spectra (SW)ξ = 5%, Soil type D

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2Period

Sa(T

1)/g

Elastic Design SpectrumInelastic Design Spectrum, Rμ = 1.5Inelastic Design Spectrum, Rμ = 3.0Inelastic Design Spectrum, Rμ = 6.0

Values of γ considered in

generic shear walls

Fig. 4.23 Yield base shear coefficients γ = Vy /W for generic shear walls

Range of Variation of θp for Shear Walls

0

0.25

0.5

0.75

1

0 0.01 0.02 0.03 0.04 0.05 0.06θp

Prob

abili

ty o

f exc

eedi

ng

p

Data Points

Fitted lognormaldistribution

θ p

Range of Variation of θp for Shear Walls

0

0.25

0.5

0.75

1

0 0.01 0.02 0.03 0.04 0.05 0.06θp

Prob

abili

ty o

f exc

eedi

ng

p

Data Points

Fitted lognormaldistribution

θ p

Fig. 4.24 Variation of plastic hinge rotation capacity from shear wall test results and the

range used in generic shear walls (data from Fardis and Biskinis, 2003)

- 87 -

Post-Capping Rotation Capacity Effect on P.O.CN = 9, T1 = 0.9, θp = 2%, θpc/θp = var, Mc/My = 1.1

0.00

0.10

0.20

0.30

0.40

0.00 0.02 0.04 0.06 0.08

Roof Drift

V y/W

0.4 0.6

0.8 1

3 5

7 9

θpc/θp

Fig. 4.25 Sensitivity of pushover curve to post-capping rotation capacity ratio for a 9-story

generic shear wall structure, T = 0.9 sec.

Median of collapse capacity vs. θpc/θp (SW)N = 9, T = 0.9, γ = 0.33, θp=2%, θpc/θp=var., Mc/My=1.1

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10

θpc/θp

(Sa/

g)co

llaps

e

Fig. 4.26 Sensitivity of median of collapse capacity to post-capping rotation capacity ratio

for 9-story generic shear wall structure, T = 0.9 sec.

- 88 -

Fig. 4.27 Map of variation to the base case structural system due to three component

parameters θp, θpc/θp, and λ

Typical IM-EDP curve (MRF)N = 8, αt = 0.15, γ = 0.17, Stiff & Str = Shear, SCB = 2.4-2.4, ξ = 0.05

θp = 0.03, θpc/θp = 5, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3

0 0.01 0.02 0.03 0.04 0.05 0.06EDP[max.RDR]

IM[S

a(T 1

)]

Fig. 4.28 Typical IDA curves for an 8-story generic moment-resisting frame subjected to

the LMSR-N record set

Base +1

+1

+1

-1

-1

-1

θpc/θ

θp

λ

- 89 -

5 STRUCTURAL SYSTEM DOMAIN: RESPONSE OF STRUCTURAL SYSTEMS CONDITIONED ON NO-COLLAPSE

5.1 Introduction

The Structural System Domain contains engineering information of relevance in PBEE. In the

context of this dissertation it is the domain in which design alternatives in the Design Decision

Support System (DDSS) are evaluated and scrutinized to obtain suitable combinations of

structural system, structural material, and structural system parameters (combined denoted as

structural parameters) that simultaneously satisfy several performance objectives. The Structural

System Domain is divided into two sub-domains: the No-collapse sub-domain (NC sub-domain),

and the Collapse sub-domain (C sub-domain). This chapter discusses the merits of the NC sub-

domain, and Chapter 6 focuses on the C sub-domain.

The NC sub-domain of the Structural System Domain (denoted for brevity as the NC sub-

domain in the remainder of this chapter) contains information on relation between EDPs and IM

of different design alternatives. In the context of the DDSS, the EDPs of interest are those that

relate well with the various components of loss discussed in Chapter 3. For each subsystem, the

expected (mean) IM-EDP curves of different design alternatives are compared in order to make

informed decisions on basic structural parameters. As discussed in Chapter 2, we use mean

values of IM-EDP relationship in order to simplify the decision making process and leave the

full probabilistic evaluation of building losses to a subsequent rigorous performance-based

assessment of the fully designed building.

- 90 -

It has been shown by Aslani (2005) and Krawinkler (ed.) (2005) that the contribution of

losses in the NC sub-domain is dominant at higher probability hazard levels (i.e., more frequent

seismic events). This demonstrates the importance of the NC sub-domain and the need for

obtaining the mean IM-EDP curves for important subsystems that contribute most to the total

loss of a building (e.g., nonstructural drift sensitive subsystem and nonstructural acceleration

sensitive subsystem) for a wide range of structural parameters. A comprehensive database on IM-

EDP relationships has been developed in this study to support the DDSS summarized in Chapter

2. In this chapter we will illustrate the utility of this database on hand of examples that show the

sensitivity of EDPs to various structural parameters in the context of estimation of losses in drift

sensitive and acceleration sensitive nonstructural subsystems.

5.2 Statistical evaluation of EDP|IM

In Chapter 4 of this dissertation we discussed the process for obtaining the relation between EDP

and IM for the generic structural systems used in this study. In this process, IM-EDP curves are

obtained by subjecting the structural systems to a set of large-magnitude-small-distance (LMSR-

N) ground motions (Medina and Krawinkler, 2003) and perform Incremental Dynamic Analysis

(Vamvatsikos and Cornell, 2002). At each level of IM (i.e., Sa(T1)) the 40 ground motions of the

LMSR-N ground motion set are scaled to the same value of Sa(T1) and nonlinear response

analysis is performed to obtain structural response parameters. Figure 5.1 shows, with gray lines,

individual IDA (maximum roof drift ratio vs. Sa(T1)) curves for one of the generic shear wall

structure used in this study. Incremental Dynamic Analysis is carried out until dynamic

instability (collapse) occurs (see Chapter 6 for detailed information about collapse of structural

systems). The asterisk at the end of several of the IDA curves in Figure 5.1 shows the IM level at

which, for a specific ground motion, stability of the solution algorithm is obtained for the last

time. This point is denoted as the collapse capacity of the structure for the selected ground

motion.

In Figure 5.1 the results of nonlinear response analysis for Sa(T1) = 1.00g are shown with

solid black circles. These data points are statistically evaluated. In this research we have divided

the Structural System Domain into Collapse and No-Collapse sub-domains. In this chapter we

- 91 -

are only interested in the data points associated with No-Collapse (i.e. EDP|IM&NC). Assuming

that these data points are lognormally distributed (Equation 5.1), we obtain the median and

dispersion of this distribution as shown in Equation 5.2 and Equation 5.3, respectively.

( )( ) ( )

( )

| &

| &

| & Ln EDPIM NC

Ln EDPIM NC

Ln edpP EDP edp IM NC

μ

σ

⎡ ⎤−≥ =Φ⎢ ⎥

⎢ ⎥⎣ ⎦ (5.1)

( )( )| &ˆ | & exp Ln EDP IM NCEDP IM NC μ= (5.2)

( ), | & | &RC EDPIM NC Ln EDPIM NCβ σ= (5.3)

For reasons discussed in Chapter 2, mean values of structural response are used for

conceptual performance-based design and simplified performance-based assessment. The mean

value of EDP|IM&NC is obtained using Equation 5.4.

( ) 2, | &

1ˆ| & | & exp2 RC EDPIM NCEDP IM NC EDP IM NC β⎛ ⎞= ⋅ ⎜ ⎟

⎝ ⎠ (5.4)

The median and mean values of EDP|IM&NC are shown in Figure 5.1 using black lines

with solid black rectangles and triangles, respectively.

A comparison between the mean and median of EDP|IM&NC and EDP|IM (i.e., for both

collapse and not collapse cases) provides valuable information. To obtain the median and mean

values of EDP|IM we are using the “counted statistics” method (Medina & Krawinkler, 2003). In

this method the median of EDP|IM is obtained by averaging the 20th and 21st sorted values of

EDP|IM. The dispersion of EDP|IM is defined as the natural logarithm of the ratio of median of

EDP|IM to 16th percentile of EDP|IM. The 16th percentile of EDP|IM is the 6th sorted value of

EDP|IM. The mean value of EDP|IM is obtained by multiplying the median of EDP|IM in the

exponent of half the square of the dispersion of EDP|IM.

Figure 5.2 shows the comparison between median and mean of EDP|IM&NC and

EDP|IM for the same generic shear wall as used for Figure 5.1. The black lines with solid black

rectangles and triangles show the mean and median of EDP|IM&NC, respectively. The gray lines

with solid gray rectangles and triangles show the median of mean value of EDP|IM, respectively.

For small IM values, the difference between median (and mean) of EDP|IM&NC and median

(and mean) of EDP|IM is very small and is due to the difference between two methods for

finding the median and associated dispersion. However, as structural collapse occurs under a few

- 92 -

ground motions, the median and mean values of EDP|IM&NC are clearly smaller than median

and mean value of EDP|IM, which is due to using in the former case only data points associated

with ground motions under which collapse has not occurred.

The dispersion defined by Equation 5.3 is due to aleatory uncertainty caused by record-

to-record variability (for this reason we use the RC subscript in , | &RC EDP IM NCβ ). Epistemic

uncertainty due to lack of knowledge about the building’s real properties is not considered in the

mean value of EDP|IM&NC. Simultaneous consideration of the effects of epistemic and aleatory

uncertainties on the dispersion of EDP|IM&NC requires performing a Monte Carlo simulation in

which the properties of the mathematical model of the structural system are random variables

with associated distribution functions and a number of ground motions are employed that are

representative of the seismicity at the building’s location.

A simpler method for incorporating the effect of epistemic uncertainty in the total

dispersion of EDP|IM&NC is through the FOSM (First Order Second Moment) method. Using

this approach, the total dispersion of EDP|IM&NC (denoted as , | &equ EDP IM NCβ ) is obtained using

Equation 5.5. In this case, the mean of EDP|IM&NC is obtained using the total dispersion as

shown in Equation 5.6.

2 2, | & , | & , | &equ EDP IM NC RC EDP IM NC UC EDP IM NCβ β β= + (5.5)

( ) 2, | &

1ˆ| & | & exp2 equ EDP IM NCEDP IM NC EDP IM NC β⎛ ⎞= ⋅ ⎜ ⎟

⎝ ⎠ (5.6)

5.3 Sensitivity of EDPs related to nonstructural losses to variation of structural parameters in generic structures

The sensitivity of a selected number of EDPs that relate well with losses in nonstructural

subsystems to variations in structural parameters for a case study generic moment-resisting frame

and a case study generic shear wall are investigated in the following sections. The two case

studies are:

1. 8-story generic moment-resisting frame with fundamental period of T1 = 1.2 seconds (i.e.,

αt = 0.15), γ = 0.17, Stiff & Str. = Shear, SCB = 2.4-1.2, θp = 0.03, θpc/θp = 5.0, λ = 20,

and Mc/My = 1.1.

- 93 -

2. 8-story generic shear wall with fundamental period of T1 = 0.8 seconds (i.e., αt = 0.10), γ

= 0.17, Str. = Unif., θp = 0.02, θpc/θp = 1.0, λ = 20,and Mc/My = 1.1.

Nonstructural components can be divided into drift-sensitive and acceleration-sensitive

based on the EDP that correlates best with the damage induced during earthquakes. In Chapter 3

of this dissertation, two classes of subsystems were introduced: story-level subsystems, and

building-level subsystems. Mean loss curves for story-level subsystems relate the mean value of

loss in a selected number of components located in a story to a single EDP in the same story. On

the other hand, mean loss curves of building-level subsystems relate the mean value of loss in a

subset of building components to a global EDP of the structural system. Based on this

distinction, EDPs are divided into story-level EDPs and building level EDPs. For The sensitivity

study presented in this section focuses on both story-level EDPs and building-level EDPs.

Story-level EDPs considered in the sensitivity study are the maximum interstory drift

ratio in story i (denoted as ( )maxi

IDR ) and peak floor acceleration in story i (denoted as

( )iPFA ). Building-level EDPs considered in the sensitivity study are the average along the

height of maximum interstory drift ratios for each ground motion (denoted as ( )maxavg

IDR ) and

average along the height of peak floor accelerations for each ground motion (denoted as

( )avgPFA ). The sensitivity of a story-level EDP and the corresponding building-level EDP are

evaluated simultaneously in order to show the differences between the two and draw additional

conclusions.

5.3.1 Sensitivity of story-level and building-level drift-related EDPs to variation of

structural parameters for a case study generic moment-resisting frame structure

Sensitivity of ( )maxi

IDR and ( )maxavg

IDR of a case study generic moment-resisting frame to

six structural parameters is investigated in this study. These structural parameters include: 1)

yield base shear coefficient, γ, 2) column to beam strength ratio, SCB, 3) variation of stiffness

and strength along the height of the structural system, Stiff. & Str., 4) component plastic hinge

rotation capacity, θp, 5) component post-capping rotation capacity ratio, θpc/θp, and 6)

- 94 -

component cyclic deterioration parameter, λ. The sensitivity of the EDPs of the case study

generic moment-resisting frame to variation of each structural parameter is studied by keeping

all other parameters unchanged. Figure 5.3 shows the sensitivity of pushover curves of the case

study frame to variation in the above mentioned structural parameters.

Figures 5.4 to Figure 5.9 show the sensitivity of ( )maxi

IDR and ( )maxavg

IDR of the

case study moment-resisting frame to variation of the structural parameters. Each Figure consists

of six parts. Parts (a), (b), and (c) show the sensitivity of the mean value, conditioned on no-

collapse, of ( )maxi

IDR (denoted as ( )maxi

IDR ) for Sa(T1)/g = 0.25, 0.67, and 1.17,

respectively. These IM values are selected in order to demonstrate the sensitivity of the mean

value of this EDP at three relative intensities of [Sa(T1)/g]/γ = 1.5, 4.0, 7.0, respectively. The

results are presented for each story at each of the three selected relative intensities. Parts (d) and

(e) of each figure show the sensitivity of the mean value, conditioned on no-collapse, of

( )maxavg

IDR (denoted as ( )maxavg

IDR ), and the associated aleatory dispersion of

( )maxavg

IDR , respectively. Part (f) of each figure shows the median for all cases (i.e., collapse

and no-collapse cases) of ( )maxavg

IDR . This median value is obtained using counted statistics.

Effect of γ on story-level and building-level drift-related EDPs

Figures 5.4a to c show the values of ( )maxi

IDR for three values of γ = Vy/W = 0.08, 0.17, and

0.33, at Sa(T1)/g = 0.25, 0.67, and 1.17, respectively. By comparing these three figures we

conclude that the collapse mechanism is the development of a partial mechanism in the lower

stories of the structural system, which may involve a single story or multiple stories. At high

probability hazard levels (i.e., Figure 5.4a) the distribution of ( )maxi

IDR is almost uniform

over the height of the structure. As more inelastic behavior occurs (γ decreases or Sa(T1)/g

increases), larger P-Delta effects cause a concentration of drift in the lower stories (see the

pushover curve in Figure 5.3). This observation is valid regardless of the value of γ, and it is

concluded that in this case study additional yield base shear capacity does not change the

- 95 -

collapse mechanism but postpones the development of a partial mechanism to higher intensity

levels.

It should be noted that in Figure 5.4c the curve for γ = 0.08 is missing. The reason is that

profiles for ( )maxi

IDR are omitted once more that 50% of the structures have collapsed. This is

the case here and in several of the subsequent figures.

Figure 5.4d shows that the increase in ( )maxavg

IDR with Sa(T1)/g initially follows

similar patterns for all values of γ. But the curves start to deviate from one another at γ values at

which collapses start to occur, since ( )maxavg

IDR is based on no-collapse cases only. If no-

collapse cases as well as collapse cases would be considered, the curves for different γ values

would overlap. This is seen by inspecting Figure 5.4f, which shows the median of ( )maxavg

IDR

using all data. The curves overlap but terminate at very different values because the collapse

potential of the building is very much dependent on γ (see Chapter 6).

Figure 5.4e shows the dispersion in ( )maxavg

IDR for no-collapse cases. As seen, the

dispersion is small at small relative intensities (the differences are caused by higher mode

effects). The dispersion increases as the relative intensity is increased but saturates after a

number of ground motions have caused collapse.

Comparing Figures 5.4a, 5.4b, and 5.4c with Figure 5.4d, we conclude that using building

level subsystems for addressing drift-related losses at moderate and high relative intensities

(Figures 5.4b and 5.4c) involves an approximation because the deformation profile is not

uniform along the height.

Effect of SCB factor on story-level and building-level drift-related EDPs

Figure 5.5 shows the importance of the strong column – weak beam concept for three cases of

column to beam strength ratios of SCB = 2.4-2.4, 2,4-1.2, and 1.2-1.2. The importance of the

SCB factor, which will be emphasized in Chapter 6 dealing with collapse, is also evident from

Figures 5.5b and c. The smaller the SCB factor the larger are the story drifts in the lower stories.

- 96 -

In Figure 5.5c the curve for SCB 1.2-1.2 is missing because more than 50% of the structures

have collapsed at this intensity level.

Figure 5.5d shows the effect of the SCB factor on ( )maxavg

IDR . As seen in this figure,

for small values of ground motion intensity ( )maxavg

IDR is equal for all three moment-resisting

frames. As the ground motion intensity increases, the frame with SCB = 1.2-1.2 develops a

collapse mechanism sooner than other frames, and therefore the value of ( )maxavg

IDR increases

at a smaller rate because this mean value is obtained using only EDPs for ground motions that

have not yet caused collapse. A comparison of Figures 5.5d and 5.5f shows that the deviation

from the original trend of IM-EDP curves in Figure 5.6d is mostly due to obtaining the mean

value of response using only no-collapse data points.

Comparing the value of ( )maxavg

IDR with the value of ( )maxi

IDR at the same IM level

shows the approximation involved when using building-level EDPs rather than story-level EDPs

in the DDSS. Figures 5.5b and c show that as the intensity measure is increased, the value of

( )maxi

IDR becomes less uniform along the height of the structural system. Therefore, the value

of ( )maxavg

IDR is less representative of the overall behavior of the structural system. If the

value of loss does not linearly increase with EDP (see Chapters 3 and 7 for typical loss curves),

or the value of loss in different stories of the building varies by a considerable amount, the use of

story-level subsystems (and subsequently story-level EDPs ) in the DDSS provides much better

loss estimates.

Effect of Stiff & Str. on story-level and building-level drift-related EDPs

Figures 5.6a to c show the value of ( )maxi

IDR for three cases of variation of stiffness and

strength along the height of the structural system (i.e., Stiff. & Str. = Shear, Int., and Unif.), at

ground motion intensity level of Sa(T1) = 0.25, 0.67, and 1.17, respectively. It is evident that

using additional stiffness and strength in upper stories (i.e., Stiff. & Str. = Unif., and Int.) causes

significant changes in the deformation profile of the structural system. For Stiff. & Str. = Int. or

Unif. the drifts in upper stories are considerably smaller than those for Stiff. & Str. = Shear, at

- 97 -

the expense of a slightly larger drift in the bottom stories. Thus, from the perspective of

monetary loss it may be cost effective to provide additional stiffness and strength in the upper

stories (which is present in most frames because of gravity load design considerations). This is

evident also in the mean and median values shown in Figures 5.6d and f. But it needs to be

considered that the presence of higher strength and stiffness in the upper stories amplifies the

drifts in the lower stories, which in turn causes a slightly larger probability of collapse compared

with structures with Stiff. & Str. = Shear (see Section 6.5.1). This larger collapse probability

affects not only the potential for loss of life but also the monetary loss, which is the summation

of losses due to no-collapse and collapse events.

Effect of θp, θpc/θp, and λ on story-level and building-level drift-related EDPs

Figures 5.7, 5.8, and 5.9 show the sensitivity of ( )maxi

IDR and ( )maxavg

IDR to variation of

beams and columns backbone curve and cyclic deterioration parameters (i.e., θp, θpc/θp, and λ).

Inspection of these figure shows that drift demands are not sensitive to a variation of these

parameters - as long as no-collapses have occurred. Once collapses have occurred, the total loss

is computed by considering the no-collapse and the collapse sub-domains. If θp is small (e.g., θp

= 0.01), the no-collapse loss may be slightly smaller (see Figure 5.7d), but the loss due to a

sizeable probability of collapse will outweigh by far the small gains in the no-collapse domain.

5.3.2 Sensitivity of story-level and building-level acceleration -related EDPs to variation

of structural parameters for a case study generic moment-resisting frame

Sensitivities of ( )iPFA and ( )avg

PFA of the case study generic moment-resisting frame to the six

structural parameters described in Section 5.3.1 are investigated in this section and are illustrated

in Figures 5.10 to Figure 5.15. The layout of figures is identical to that used in the previous

section. In each figure, parts (a), (b), and (c) show the sensitivity of the mean value, conditioned

on no-collapse, of ( )iPFA (denoted as ( )i

PFA ) for three ground motion intensity levels of

Sa(T1)/g = 0.25, 0.67, and 1.17. Part (d), and (e) in each figure show the sensitivity of the mean

- 98 -

value, conditioned on no-collapse, of ( )avgPFA (denoted as ( )avg

PFA ) and the associated

aleatory dispersion of ( )avgPFA . Part (f) of each figure shows the median of

( )maxavg

IDR considering both collapse and no-collapse cases using counted statistics.

Effect of γ on story-level and building-level acceleration EDPs

Figure 5.10 shows that ( )iPFA and ( )avg

PFA are very sensitive to a variation of γ. The smaller

the yield base shear coefficient γ the smaller are the accelerations attracted in the building. Early

yielding in the lower stories due to small γ values at higher intensity levels can be considered as

a filter that reduces the acceleration demand in upper floors of the structure. Figure 5.10d shows

that by increasing the ground motion intensity, ( )avgPFA increases for all three γ values,

however, the rate of increase is reduced once the structural system develops a partial mechanism

in the lower stories. It is interesting to note that the dispersion in (PFA)i becomes smaller as the

ground motion intensity is increased.

Effect of SCB factor, Stiff & Str., θp, θpc/θp, and λ on story-level and building-level

acceleration EDPs

Figure 5.11 to 5.15 show the effect of variations in other structural parameters on the peak floor

accelerations. The effects of variations in all these parameters are small compared to those of γ or

the intensity measure Sa(T1). The general trend is stabilization of the statistical measures of peak

floor acceleration for systems with small γ (or large Sa(T1)), and a decrease in the dispersion.

5.3.3 Sensitivity of story-level and building-level drift-related EDPs to structural

parameters variation for a case study generic shear wall structure

In this section, the sensitivity of ( )maxi

IDR and ( )maxavg

IDR of an 8-story generic shear wall

structure to five structural parameters is investigated. These structural parameters include: 1)

yield base shear coefficient, γ, 2) variation of strength along the height of structural system, Str.,

- 99 -

3) component plastic hinge rotation capacity, θp, 4) component post-capping rotation capacity

ratio, θpc/θp, and 5) component deterioration parameter, λ. Figure 5.16 shows the sensitivity of

pushover curves of the case study shear wall to a variation in the aforementioned structural

parameters. Figures 5.17 to Figure 5.21 show the sensitivity of ( )maxi

IDR and ( )maxavg

IDR to

a variation of the structural parameters. Parts (a), (b), and (c) in each figure show the sensitivity

of ( )maxi

IDR for Sa(T1)/g = 0.38, 1.00, and 1.75 (i.e., [Sa(T1)/g]/γ = 1.5, 4.0, 7.0). Part (d), and

(e) in each figure show the sensitivity of ( )maxavg

IDR ) and the associated aleatory dispersion of

( )maxavg

IDR . Part (f) of each figure shows the median of ( )maxavg

IDR for all cases (i.e.,

collapse and no-collapse cases) obtained using counted statistics.

Effect of γ on story-level and building-level drift-related EDPs

Figure 5.17 shows the effect of the yield base shear coefficient γ on ( )maxi

IDR for the case

study shear wall. The general trend is that the drift demand in lower stories is smaller than the

drift demand in upper stories. This is as expected since only flexural-type deformations are

considered in shear walls and is seen in all the figures that show ( )maxi

IDR . The profiles of

maximum story drifts become more uniform the smaller the yield base shear (or the larger the

intensity of the ground motion), since the deflected shape changes from a pure flexural mode to a

more straight line shape. Higher modes do not have a large effect on the story drifts (at least not

for this 8-story shear wall structure).

Figure 5.17d shows the sensitivity of ( )maxavg

IDR to a variation of γ at different IM

levels. Comparing the values of ( )maxavg

IDR at Sa(T1)/g = 1.00 with ( )maxi

IDR in Figure

5.17b shows that ( )maxavg

IDR is a good representative of the overall behavior of the shear wall.

This is due to the uniform value of drift along the height of the shear wall.

Effect of Str. on story-level and building-level drift-related EDPs

- 100 -

Figure 5.18 shows that values of ( )maxi

IDR and ( )maxavg

IDR are insensitive to variation of

strength along the height of the shear wall. This is due to the small value of Mc/My and the small

decrease in bending strength along the height, which inhibit the spreading of plastic hinging

beyond the first story (at least for this 8-story structure).

Effect of θp, θpc/θp, and λ on story-level and building-level drift-related EDPs

Figures 5.19, 5.20, and 5.21 show the sensitivity of ( )maxi

IDR and ( )maxavg

IDR to variations

in θp, θpc/θp, and λ of the case study shear wall. Similar to the moment-resisting frame case

study, these figure show that drift demands are not sensitive to variations in component

backbone and deterioration parameters in the regime in which the number of collapse cases is

small. However, these figures show a significant dependence of the effects of parameter

variations on the level of IM at which dynamic instability (collapse) occurs.

5.3.4 Sensitivity of story-level and building-level acceleration-related EDPs to structural

parameters variation for a case study generic shear wall

Figures 5.22 to Figure 5.26 show the sensitivity of ( )iPFA and ( )avg

PFA to the five structural

parameters described in Section 5.3.3. Layout of the figures is the same as in the previous

section.

Effect of γ on story-level and building-level acceleration-related EDPs

Figures 5.22a to c show the effect of the yield base shear coefficient γ on ( )iPFA for three

intensities of Sa(T1)/g = 0.38, 1.00, 1.75. In all three figures it is seen that the contribution of

higher modes is significant in the acceleration response. The trend of variation of ( )iPFA along

the height shows a large value of acceleration demand at mid-height and at the roof level of the

shear wall.

- 101 -

The variation of ( )iPFA along the height is sensitive to the yield base shear coefficient.

For a given value of intensity measure, the acceleration along the height of the structure

increases with an increase in γ. This is due to the fact that the system with larger γ is subjected to

smaller nonlinear behavior, hence period elongation in not as dominant as it is for systems with

smaller γ.

Figure 5.22d shows the variation of ( )avgPFA with Sa(T1)/g for three values of γ. A

smaller value of γ will cause the shear wall to yield at a smaller value of IM, hence the

acceleration demand is reduced. Comparing the value of ( )avgPFA at Sa(T1)/g = 0.38, 1.00, and

1.75 with the corresponding ( )iPFA in Figures 5.22a, b, and c shows that ( )avg

PFA is not a

good representative of the behavior of the shear wall. This is due to the non-uniform acceleration

demand along the height of the case study shear wall.

Effect of Str., θp, θpc/θp, and λ on story-level and building-level acceleration-related EDPs

As Figures 5.23, to 5.26 show, the sensitivity of ( )iPFA and ( )avg

PFA to variations in all these

parameters is small. Similar to the case study moment-resisting frame, it is concluded that

acceleration demands of the case study shear wall are not sensitive to variations in strength over

the height and in component backbone and deterioration parameters.

5.4 Summary

The write-up and figures presented in this chapter provide a glimpse into the wealth of

information available in the IM-EDP database generated as part of this research. Information of

the type illustrated here is available for a wide range of frame and wall structures as discussed in

Section 4.6. The database contains individual and statistical data from IDAs, for a wide range of

EDPs that goes far beyond that utilized directly in the DDSS discussed in Chapter 7. It serves as

a resource for understanding seismic behavior of frame and wall structures and for quantifying

design parameters that may greatly affect the performance of a building at all levels of interest,

- 102 -

ranging from minor loss to collapse of the structural system. Some of these parameters, such as

interstory drift and floor acceleration, are used directly in the DDSS. Other parameters quantified

in the database provide assistance for designing structural systems in a manner that avoids failure

modes that may adversely affect the seismic response at large inelastic deformations. Examples

of phenomena that can be assessed quantitatively from the information contained in the IM-EDP

database include:

• Strong column – weak beam concept for moment resisting frames

• Moment and shear demands at column splice locations of moment resisting frames

• Plastic rotation demands at column bases

• Overturning moment demands for column design in moment resisting frames

• Assessment of structure P-Delta effects in the inelastic range of behavior for flexible

moment resisting frame structures

• Shear force demands in wall structures designed for flexural hinging at the wall base

• Bending strength demands along the height of shear walls

• Energy demands for energy based design of frames and shear wall structures

• Assessment of Rμ factors for design for collapse safety

These and many more design challenges can be addressed through utilization of the IM-

EDP database generated in this research. These issues are not addressed in this dissertation as

they are important for design but are not central to the theme of simplified performance-based

earthquake engineering.

- 103 -

EDP-IM Curves (SW)N = 8, T = 0.8, γ = 0.25, Stiff = Unif. Str. = -0.05My,base, ξ = 0.05

θp = 0.02, θpc/θp = 1, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

0 0.01 0.02 0.03 0.04 0.05EDP[max.RDR]

IM[S

a(T 1

)]

Individual IDA curveslast point used before collapse

Mean no-collapse

Median no-collapse

Fig. 5.1 Incremental Dynamic Analysis and corresponding mean and median value of the

EDP

EDP-IM Curves (SW)N = 8, T = 0.8, γ = 0.25, Stiff = Unif. Str. = -0.05My,base, ξ = 0.05

θp = 0.02, θpc/θp = 1, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

0 0.01 0.02 0.03 0.04 0.05EDP[max.RDR]

IM[S

a(T 1

)]

Mean no-collapse

Individual IDA curveslast point used before collapse

Median no-collapse

Meanall

Medianall

Fig. 5.2 Difference between mean and median value of EDP conditioned on no-collapse and

the mean and median value of EDP for all data

- 104 -

N = 8, T1 = 1.2, γ = var., Stiff. & Str. = Shear, SCB = var., ξ = 0.05 θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0.00

0.05

0.10

0.15

0.20

0 0.02 0.04 0.06 0.08 0.1Roof Drift δroof/H

Vy/W

2.4-2.4SCB

1.2-1.2 2.4-1.2

Effect of SCB on Pushover Curve (MRF)

N = 8, T1 = 1.2, γ = 0.17, Stiff. & Str. = var., SCB = 2.4-1.2, ξ = 0.05 θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0.00

0.05

0.10

0.15

0.20


Vy/W

Stiff. & Str.SheaInt.Unif.

Effect of Stiff. & Str.on Pushover Curve (MRF)N = 8, T1 = 1.20, γ = var., Stiff. & Str. = Shear, SCB = 2.4-1.2, ξ = 0.05

θp = 0.03, θpc/θp = 5.0, λ = var., Mc/My = 1.1

0.00

0.05

0.10

0.15

0.20


Vy/W

λ10 20 50

Effect of λ on Pushover Curve (MRF)

N = 8, T1 = 1.20, γ = var., Stiff. & Str. = Shear, SCB = 2.4-1.2, ξ = 0.05 θp = 0.03, θpc/θp = var., λ = 20, Mc/My = 1.1

0.00

0.05

0.10

0.15

0.20


Vy/W

θpc/θp1.0 5.0 15.0

Effect of θpc/θp on Pushover Curve (MRF)

N = 8, T1 = 1.20, γ = var., Stiff. & Str. = Shear, SCB = 2.4-1.2, ξ = 0.05 θp = 0.03, θpc/θp = 5.0, λ = var., Mc/My = 1.1

0.00

0.10

0.20

0.30

0.40


Vy/W

γ0.08 0.17 0.33

Effect of γ on Pushover Curve (MRF)N = 8, T1 = 1.20, γ = var., Stiff. & Str. = Shear, SCB = 2.4-1.2, ξ = 0.05

θp = var., θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0.00

0.05

0.10

0.15

0.20


Vy/W

θp0.060.030.01

Effect of θp on Pushover Curve (MRF)

(a) (d)

(b) (e)

(c) (f)

Fig. 5.3 Pushover curves for case study moment resisting frame with variation in structural

parameters

- 105 -

(a) (d)

(b) (e)

(c) (f)

Effect of γ on Med. of maxIDRavg|IM (MRF)N = 8, T1 = 1.2, γ = var. ,Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05

θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

0 0.01 0.02 0.03 0.04Median of (maxIDR)avg

Sa(T

1)/g

γ = 0.17

γ = 0.08

γ = 0.33

Effect of γ on maxIDRavg|IM & NC (MRF)N = 8, T1 = 1.2, γ = var. ,Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05

θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

0 0.01 0.02 0.03 0.04(maxIDR)avg

Sa(T

1)/g

γ = 0.17

γ = 0.08

γ = 0.33

Effect of γ on Disp. of maxIDRavg|IM & NC (MRF)N = 8, T1 = 1.2, γ = var. ,Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05

θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4 0.5Dispersion of (maxIDR)avg

Sa(T

1)/g

γ = 0.17

γ = 0.08

γ = 0.33

Effect of γ on maxIDRi|IM & NC (MRF) [Sa(T1) = 0.25]N = 8, T1 = 1.2, γ = var. ,Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05

θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.01 0.02 0.03 0.04(maxIDR)i

Stor

y N

umbe

r

γ = 0.17

γ = 0.08

γ = 0.33


θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.01 0.02 0.03 0.04(maxIDR)i

Stor

y N

umbe

r

γ = 0.33

γ = 0.17

γ = 0.08


θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.01 0.02 0.03 0.04(maxIDR)i

Stor

y N

umbe

r

γ = 0.33

γ = 0.17

γ = 0.08

Fig. 5.4 Effects of γ on drift demands of case study moment-resisting frame

- 106 -

(a) (d)

(b) (e)

(c) (f)

N = 8, T1 = 1.2, γ = 0.17 ,Stiff.&Str. = Shear, SCB = var. , ξ = 0.05 θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5


Sa(T

1)/g

Effect of SCB on Med. of maxIDRavg|IM (MRF)

SCB = 2.4-1.2

SCB = 1.2-1.2

SCB = 2.4-2.4

N = 8, T1 = 1.2, γ = 0.17 ,Stiff.&Str. = Shear, SCB = var. , ξ = 0.05 θp = 0.03, θpc/θp = 5 , λ = 20 , Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.01 0.02 0.03 0.04(maxIDR)i

Stor

y N

umbe

r

Effect of SCB on (maxIDR)i|IM & NC (MRF) [Sa(T1) = 0.25]

SCB = 2.4-1.2

SCB = 1.2-1.2

SCB = 2.4-2.4


1

2

3

4

5

6

7

8

0 0.01 0.02 0.03 0.04(maxIDR)i

Stor

y N

umbe

r

Effect of SCB on maxIDRi|IM & NC (MRF) [Sa(T1) = 0.67]

SCB = 2.4-1.2

SCB = 1.2-1.2

SCB = 2.4-2.4


1

2

3

4

5

6

7

8

0 0.01 0.02 0.03 0.04(maxIDR)i

Stor

y N

umbe

r

Effect of SCB on maxIDRi|IM & NC (MRF) [Sa(T1) = 1.17]

SCB = 2.4-1.2

SCB = 1.2-1.2

SCB = 2.4-2.4

N = 8, T1 = 1.2, γ = 0.17 ,Stiff.&Str. = Shear , SCB = var., ξ = 0.05 θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

0 0.01 0.02 0.03 0.04(maxIDR)avg

Sa(T

1)/g

Effect of SCB on maxIDRavg|IM & NC (MRF)

SCB = 2.4-1.2

SCB = 1.2-1.2

SCB = 2.4-2.4

N = 8, T1 = 1.2, γ = 0.17 , Stiff.&Str. = Shear, SCB = var., ξ = 0.05 θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5


Sa(T

1)/g

Effect of SCB on Disp. of maxIDRavg|IM & NC (MRF)

SCB = 2.4-1.2SCB = 1.2-1.2

SCB = 2.4-2.4

Fig. 5.5 Effects of SCB factor on drift demands of case study moment-resisting frame

- 107 -

(a) (d)

(b) (e)

(c) (f)

N = 8, T1 = 1.2, γ = 0.17 ,Stiff.&Str. = var., SCB = 2.4-1.2, ξ = 0.05 θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5


Sa(T

1)/g

Effect of Stiff.&Str. on Med. of maxIDRavg|IM (MRF)

Stiff.&Str. = Int.

Stiff.&Str. = Unif.

Stiff.&Str. = Shear

N = 8, T1 = 1.2, γ = 0.17 , Stiff.&Str. = var., SCB = 2.4-1.2, ξ = 0.05 θp = 0.03, θpc/θp = 5 , λ = 20 , Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.01 0.02 0.03 0.04(maxIDR)i

Stor

y N

umbe

r

Effect of Stiff.&Str. on maxIDRi|IM & NC (MRF) [Sa(T1) = 0.25]

Stiff.&Str. = Int.

Stiff.&Str. = Unif.

Stiff.&Str. = Shear


1

2

3

4

5

6

7

8

0 0.01 0.02 0.03 0.04(maxIDR)i

Stor

y N

umbe

r


Stiff.&Str. = Int.

Stiff.&Str. = Unif.

Stiff.&Str. = Shear


1

2

3

4

5

6

7

8

0 0.01 0.02 0.03 0.04(maxIDR)i

Stor

y N

umbe

r


Stiff.&Str. = Int.

Stiff.&Str. = Unif.

Stiff.&Str. = Shear


0

0.5

1

1.5

2

2.5

0 0.01 0.02 0.03 0.04(maxIDR)avg

Sa(T

1)/g

Effect of Stiff.&Str. on maxIDRavg|IM & NC (MRF)

Stiff.&Str. = Int.

Stiff.&Str. = Unif.

Stiff.&Str. = Shear

N = 8, T1 = 1.2, γ = 0.17 , Stiff.&Str. = var., SCB = 2.4-1.2, ξ = 0.05 θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5


Sa(T

1)/g

Effect of Stiff.&Str. on Disp. of maxIDRavg|IM & NC (MRF)

Stiff.&Str. = Int.

Stiff.&Str. = Unif.

Stiff.&Str. = Shear

Fig. 5.6 Effects of Stff.& Str. parameters on drift demands of case study moment-resisting

frame

- 108 -

(a) (d)

(b) (e)

(c) (f)

N = 8, T1 = 1.2, γ = 0.17 ,Stiff.&Str. = Shear, SCB = 2.4-1.2 , ξ = 0.05 θp = var., θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5


Sa(T

1)/g

Effect of θp on Med. of maxIDRavg|IM (MRF)

θp = 0.03

θp = 0.01

θp = 0.06

N = 8, T1 = 1.2, γ = 0.17 ,Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05 θp = var., θpc/θp = 5.0, λ = 20, Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.01 0.02 0.03 0.04(maxIDR)i

Stor

y N

umbe

r

θp = 0.03

θp = 0.01

θp = 0.06

Effect of θp on maxIDRi|IM & NC (MRF) [Sa(T1) = 0.25]


1

2

3

4

5

6

7

8

0 0.01 0.02 0.03 0.04(maxIDR)i

Stor

y N

umbe

r

θp = 0.03

θp = 0.01

θp = 0.06



1

2

3

4

5

6

7

8

0 0.01 0.02 0.03 0.04(maxIDR)i

Stor

y N

umbe

r

θp = 0.03

θp = 0.01

θp = 0.06


N = 8, T1 = 1.2, γ = 0.17 ,Stiff.&Str. = Shear , SCB = 2.4-1.2, ξ = 0.05 θp = var., θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

0 0.01 0.02 0.03 0.04(maxIDR)avg

Sa(T

1)/g

Effect of θp on maxIDRavg|IM & NC (MRF)

θp = 0.03

θp = 0.01

θp = 0.06

N = 8, T1 = 1.2, γ = 0.17 , Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05 θp = var., θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5


Sa(T

1)/g

Effect of θp on Disp. of maxIDRavg|IM & NC (MRF)

θp = 0.03θp = 0.01

θp = 0.06

Fig. 5.7 Effects of component θp on drift demands of case study moment-resisting frame

- 109 -

(a) (d)

(b) (e)

(c) (f)

N = 8, T1 = 1.2, γ = 0.17 ,Stiff.&Str. = Shear, SCB = 2.4-1.2 , ξ = 0.05 θp = 0.03, θpc/θp = var., λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5


Sa(T

1)/g

Effect of θpc/θp on Med. of maxIDRavg|IM (MRF)

θp/θp = 5

θp/θp = 1

θpc/θp = 15

N = 8, T1 = 1.2, γ = 0.17, Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05 θp = 0.03, θpc/θp = var., λ = 20, Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.01 0.02 0.03 0.04(maxIDR)i

Stor

y N

umbe

rEffect of θpc/θp on maxIDRi|IM & NC (MRF) [Sa(T1) = 0.25]

θpc/θp = 5

θpc/θp = 1

θpc/θp = 15


1

2

3

4

5

6

7

8

0 0.01 0.02 0.03 0.04(maxIDR)i

Stor

y N

umbe

r

Effect of θpc/θp on maxIDRi|IM & NC (MRF) [Sa(T1) = 0.67]

θpc/θp = 5

θpc/θp = 1

θpc/θp = 15


1

2

3

4

5

6

7

8

0 0.01 0.02 0.03 0.04(maxIDR)i

Stor

y N

umbe

r

θpc/θp = 5

θpc/θp = 1

θpc/θp = 15

Effect of θpc/θp on maxIDRi|IM & NC (MRF) [Sa(T1) = 1.17]

N = 8, T1 = 1.2, γ = 0.17 ,Stiff.&Str. = Shear , SCB = 2.4-1.2, ξ = 0.05 θp = 0.03, θpc/θp = var., λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

0 0.01 0.02 0.03 0.04(maxIDR)avg

Sa(T

1)/g

Effect of θpc/θp on maxIDRavg|IM & NC (MRF)

θp/θp = 5

θp/θp = 1

θpc/θp = 15

N = 8, T1 = 1.2, γ = 0.17 , Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05 θp = 0.03, θpc/θp = var., λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5


Sa(T

1)/g

Effect of θpc/θp on Disp. of maxIDRavg|IM & NC (MRF)

θp/θp = 5

θp/θp = 1

θpc/θp = 15

Fig. 5.8 Effects of component θpc/θp on drift demands of case study moment-resisting frame

- 110 -

(a) (d)

(b) (e)

(c) (f)

N = 8, T1 = 1.2, γ = 0.17 ,Stiff.&Str. = Shear, SCB = 2.4-1.2 , ξ = 0.05 θp = 0.03, θpc/θp = 5.0, λ = var., Mc/My = 1.1

0

0.5

1

1.5

2

2.5


Sa(T

1)/g

Effect of λ on Med. of maxIDRavg|IM (MRF)

λ = 20

λ = 10

λ = 50

N = 8, T1 = 1.2, γ = 0.17 ,Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05 θp = 0.03, θpc/θp = 5 , λ = var , Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.01 0.02 0.03 0.04maxIDR

Stor

y N

umbe

r

Effect of λ on maxIDR|IM & NC (MRF) [Sa(T1) = 0.25]

λ = 20

λ = 10

λ = 50


1

2

3

4

5

6

7

8

0 0.01 0.02 0.03 0.04maxIDR

Stor

y N

umbe

r

Effect of λ on maxIDR|IM & NC (MRF) [Sa(T1) = 0.67]

λ = 20

λ = 10

λ = 50


1

2

3

4

5

6

7

8

0 0.01 0.02 0.03 0.04(maxIDR)i

Stor

y N

umbe

r

λ = 20

λ = 10

λ = 50

Effect of λ on E(maxIDRi|IM & NC) (MRF) [Sa(T1) = 1.17]

N = 8, T1 = 1.2, γ = 0.17 ,Stiff.&Str. = Shear , SCB = 2.4-1.2, ξ = 0.05 θp = 0.03, θpc/θp = 5.0, λ = var., Mc/My = 1.1

0

0.5

1

1.5

2

2.5

0 0.01 0.02 0.03 0.04(maxIDR)avg

Sa(T

1)/g

Effect of λ on maxIDRavg|IM & NC (MRF)

λ = 20

λ = 10

λ = 50

N = 8, T1 = 1.2, γ = 0.17 , Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05 θp = 0.03, θpc/θp = 5.0, λ = var., Mc/My = 1.1

0

0.5

1

1.5

2

2.5


Sa(T

1)/g

Effect of λ on Disp. of maxIDRavg|IM & NC (MRF)

λ = 20

λ = 10

λ = 50


study moment-resisting frame

- 111 -

(a) (d)

(b) (e)

(c) (f)

Effect of γ on Med. of PFAavg|IM (MRF)N = 8, T1 = 1.2, γ = var. ,Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05

θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2Median of (PFA)avg

Sa(T

1)/g

γ = 0.17

γ = 0.08

γ = 0.33

Effect of γ on PFAavg|IM & NC (MRF)N = 8, T1 = 1.2, γ = var. ,Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05

θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2(PFA)avg

Sa(T

1)/g

γ = 0.17

γ = 0.08

γ = 0.33

Effect of γ on Disp. of PFAavg|IM & NC (MRF)N = 8, T1 = 1.2, γ = var. ,Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05

θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4 0.5Dispersion of (PFA)avg

Sa(T

1)/g

γ = 0.17

γ = 0.08

γ = 0.33

Effect of γ on PFAi|IM & NC (MRF) [Sa(T1) = 0.25]N = 8, T1 = 1.2, γ = var., Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05

θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2(PFA)i

Floo

r Num

ber

γ = 0.17

γ = 0.08

γ = 0.33


θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2(PFA)i

Floo

r Num

ber

γ = 0.33

γ = 0.17

γ = 0.08


θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2(PFA)i

Floo

r Num

ber

γ = 0.33

γ = 0.17

γ = 0.08

Fig. 5.10 Effects of γ on acceleration demands of case study moment-resisting frame

- 112 -

(a) (d)

(b) (e)

(c) (f)

N = 8, T1 = 1.2, γ = 0.17 ,Stiff.&Str. = Shear, SCB = var. , ξ = 0.05 θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2Median of maxIDRavg

Sa(T

1)/g

Effect of SCB on Med. of PFAavg|IM (MRF)

SCB = 2.4-1.2

SCB = 1.2-1.2

SCB = 2.4-2.4

N = 8, T1 = 1.2, γ = 0.17, Stiff.&Str. = Shear, SCB = var. , ξ = 0.05 θp = 0.03, θpc/θp = 5 , λ = 20 , Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2(PFA)i

Floo

r Num

ber

Effect of SCB on PFAi|IM & NC (MRF) [Sa(T1) = 0.25]

SCB = 2.4-1.2

SCB = 1.2-1.2

SCB = 2.4-2.4


1

2

3

4

5

6

7

8

0 0.5 1 1.5 2(PFA)i

Floo

r Num

ber


SCB = 2.4-1.2

SCB = 1.2-1.2

SCB = 2.4-2.4


1

2

3

4

5

6

7

8

0 0.5 1 1.5 2(PFA)i

Floo

r Num

ber


SCB = 2.4-1.2

SCB = 1.2-1.2

SCB = 2.4-2.4

N = 8, T1 = 1.2, γ = 0.17 ,Stiff.&Str. = Shear , SCB = var., ξ = 0.05 θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2PFAavg

Sa(T

1)/g

Effect of SCB on PFAavg|IM & NC (MRF)

SCB = 2.4-1.2

SCB = 1.2-1.2

SCB = 2.4-2.4

N = 8, T1 = 1.2, γ = 0.17 , Stiff.&Str. = Shear, SCB = var., ξ = 0.05 θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4 0.5Dispersion of maxIDRavg

Sa(T

1)/g

Effect of SCB on Disp. of PFAavg|IM & NC (MRF)

SCB = 2.4-1.2SCB = 1.2-1.2

SCB = 2.4-2.4

Fig. 5.11 Effects of SCB factor on acceleration demands of case study moment-resisting

frame

- 113 -

(a) (d)

(b) (e)

(c) (f)


0

0.5

1

1.5

2

2.5


Sa(T

1)/g

Effect of Stiff.&Str. on Med. of PFAavg|IM (MRF)

Stiff.&Str. = Int.

Stiff.&Str. = Unif.

Stiff.&Str. = Shear

N = 8, T1 = 1.2, γ = 0.17 ,Stiff.&Str. = var., SCB = 2.4-1.2, ξ = 0.05 θp = 0.03, θpc/θp = 5 , λ = 20 , Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2(PFA)i

Floo

r Num

ber

Effect of Stiff.&Str. on PFAi|IM & NC (MRF) [Sa(T1) = 0.25]

Stiff.&Str. = Int.

Stiff.&Str. = Unif.

Stiff.&Str. = Shear


1

2

3

4

5

6

7

8

0 0.5 1 1.5 2(PFA)i

Floo

r Num

ber


Stiff.&Str. = Int.

Stiff.&Str. = Unif.

Stiff.&Str. = Shear


1

2

3

4

5

6

7

8

0 0.5 1 1.5 2(PFA)i

Floo

r Num

ber


Stiff.&Str. = Int.

Stiff.&Str. = Unif.

Stiff.&Str. = Shear


0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2(PFA)avg

Sa(T

1)/g

Effect of Stiff.&Str. on PFAavg|IM & NC (MRF)

Stiff.&Str. = Int.

Stiff.&Str. = Unif.

Stiff.&Str. = Shear

N = 8, T1 = 1.2, γ = 0.17 , Stiff.&Str. = var., SCB = 2.4-1.2, ξ = 0.05 θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5


Sa(T

1)/g

Effect of Stiff.&Str. on Disp. of PFAavg|IM & NC (MRF)

Stiff.&Str. = Int.

Stiff.&Str. = Unif.

Stiff.&Str. = Shear

Fig. 5.12 Effects of Stiff. & Str. Parameters on acceleration demands of case study moment-

resisting frame

- 114 -

(a) (d)

(b) (e)

(c) (f)

N = 8, T1 = 1.2, γ = 0.17 ,Stiff.&Str. = Shear, SCB = 2.4-1.2 , ξ = 0.05 θp = var., θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5


Sa(T

1)/g

Effect of θp on Med. of PFAavg|IM (MRF)

θp = 0.03

θp = 0.01

θp = 0.06

N = 8, T1 = 1.2, γ = 0.17, Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05 θp = var., θpc/θp = 5.0, λ = 20, Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2(PFA)i

Floo

r Num

ber

θp = 0.03

θp = 0.01

θp = 0.06

Effect of θp on PFAi|IM & NC (MRF) [Sa(T1) = 0.25]


1

2

3

4

5

6

7

8

0 0.5 1 1.5 2(PFA)i

Floo

r Num

ber

θp = 0.03

θp = 0.01

θp = 0.06



1

2

3

4

5

6

7

8

0 0.5 1 1.5 2(PFA)i

Floo

r Num

ber

θp = 0.03

θp = 0.01

θp = 0.06


N = 8, T1 = 1.2, γ = 0.17 ,Stiff.&Str. = Shear , SCB = 2.4-1.2, ξ = 0.05 θp = var., θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2(PFA)avg

Sa(T

1)/g

Effect of θp on PFAavg|IM & NC (MRF)

θp = 0.03

θp = 0.01

θp = 0.06

N = 8, T1 = 1.2, γ = 0.17 , Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05 θp = var., θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5


Sa(T

1)/g

Effect of θp on Disp. of PFAavg|IM & NC (MRF)

θp = 0.03θp = 0.01

θp = 0.06

Fig. 5.13 Effects of component θp on acceleration demands of case study moment-resisting

frame

- 115 -

(a) (d)

(b) (e)

(c) (f)

N = 8, T1 = 1.2, γ = 0.17 ,Stiff.&Str. = Shear, SCB = 2.4-1.2 , ξ = 0.05 θp = 0.03, θpc/θp = var., λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5


Sa(T

1)/g

Effect of θpc/θp on Med. of PFAavg|IM (MRF)

θp/θp = 5

θp/θp = 1

θpc/θp = 15


1

2

3

4

5

6

7

8

0 0.5 1 1.5 2(PFA)i

Floo

r Num

ber

Effect of θpc/θp on PFAi|IM & NC (MRF) [Sa(T1) = 0.25]

θpc/θp = 5

θpc/θp = 1

θpc/θp = 15


1

2

3

4

5

6

7

8

0 0.5 1 1.5 2(PFA)i

Floo

r Num

ber


θpc/θp = 5

θpc/θp = 1

θpc/θp = 15


1

2

3

4

5

6

7

8

0 0.5 1 1.5 2(PFA)i

Floo

r Num

ber


θpc/θp = 5

θpc/θp = 1

θpc/θp = 15

N = 8, T1 = 1.2, γ = 0.17 ,Stiff.&Str. = Shear , SCB = 2.4-1.2, ξ = 0.05 θp = 0.03, θpc/θp = var., λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2(PFA)avg

Sa(T

1)/g

Effect of θpc/θp on PFAavg|IM & NC (MRF)

θp/θp = 5

θp/θp = 1

θpc/θp = 15

N = 8, T1 = 1.2, γ = 0.17 , Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05 θp = 0.03, θpc/θp = var., λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5


Sa(T

1)/g

Effect of θpc/θp on Disp. of PFAavg|IM & NC (MRF)

θp/θp = 5

θp/θp = 1

θpc/θp = 15

Fig. 5.14 Effects of component θpc/θp on acceleration demands of case study moment-

resisting frame

- 116 -

(a) (d)

(b) (e)

(c) (f)

N = 8, T1 = 1.2, γ = 0.17 ,Stiff.&Str. = Shear, SCB = 2.4-1.2 , ξ = 0.05 θp = 0.03, θpc/θp = 5.0, λ = var., Mc/My = 1.1

0

0.5

1

1.5

2

2.5


Sa(T

1)/g

Effect of λ on Med. of PFAavg|IM (MRF)

λ = 20

λ = 10

λ = 50

N = 8, T1 = 1.2, γ = 0.17, Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05 θp = 0.03, θpc/θp = 5 , λ = var. , Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2(PFA)i

Floo

r Num

ber

Effect of λ on PFAi|IM & NC (MRF) [Sa(T1) = 0.25]

λ = 20

λ = 10

λ = 50


1

2

3

4

5

6

7

8

0 0.5 1 1.5 2(PFA)i

Floo

r Num

ber


λ = 20

λ = 10

λ = 50


1

2

3

4

5

6

7

8

0 0.5 1 1.5 2(PFA)i

Floo

r Num

ber

λ = 20

λ = 10

λ = 50


N = 8, T1 = 1.2, γ = 0.17 ,Stiff.&Str. = Shear , SCB = 2.4-1.2, ξ = 0.05 θp = 0.03, θpc/θp = 5.0, λ = var., Mc/My = 1.1

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2(PFA)avg

Sa(T

1)/g

Effect of λ on PFAavg|IM & NC (MRF)

λ = 20

λ = 10

λ = 50

N = 8, T1 = 1.2, γ = 0.17 , Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05 θp = 0.03, θpc/θp = 5.0, λ = var., Mc/My = 1.1

0

0.5

1

1.5

2

2.5


Sa(T

1)/g

Effect of λ on Disp. of PFAavg|IM & NC (MRF)

λ = 20

λ = 10

λ = 50

Fig. 5.15 Effects of component cyclic deterioration parameter λ on acceleration demands of

case study moment-resisting frame

- 117 -

(a) (d)

(b) (e)

(c)

N = 8, T1 = 0.80, γ = var. , Str. = Unif., ξ = 0.05 θp = 0.02, θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0.00

0.10

0.20

0.30

0.40

0.50

0.60


Vy/W

γ0.13 0.25 0.50

Effect of γ on Pushover Curve (SW)

N = 8, T1 = 0.80, γ = 0.25 , Str. = Unif., ξ = 0.05 θp = var., θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0.00

0.05

0.10

0.15

0.20

0.25

0.30


Vy/W

θp0.01 0.02 0.03

Effect of θp on Pushover Curve (SW)

N = 8, T1 = 0.80, γ = 0.25, Str. = Unf., ξ = 0.05 θp = 0.02, θpc/θp = var., λ = 20, Mc/My = 1.1

0.00

0.05

0.10

0.15

0.20

0.25

0.30


Vy/W

θpc/θp0.5 1.0 3.0

Effect of θpc/θp on Pushover Curve (SW)

N = 8, T1 = 0.80, γ = 0.25 , Str. = Unf., ξ = 0.05 θp = 0.02, θpc/θp = 1.0, λ = var., Mc/My = 1.1

0.00

0.05

0.10

0.15

0.20

0.25

0.30


Vy/W

λ10 20 50

Effect λ on Pushover Curve (SW)

N = 8, T1 = 0.80, γ = 0.25 , Str. = var. , ξ = 0.05 θp = 0.02, θpc/θp = 1.0, λ = 20., Mc/My = 1.1

0.00

0.05

0.10

0.15

0.20

0.25

0.30


Vy/W

Str.Unf. -0.05My,base/story

Effect of Str. on Pushover Curve (SW)

Fig. 5.16 Pushover curves for case study shear wall with variation in structural parameters

- 118 -

(a) (d)

(b) (e)

(c) (f)

Effect of γ on Med. of maxIDRavg|IM (SW)N = 8, T1 = 0.8, γ = var. ,Str. = Unif., ξ = 0.05

θp = 0.02, θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3

0 0.01 0.02 0.03Median of (maxIDR)avg

Sa(T

1)/g

γ = 0.25

γ = 0.13

γ = 0.50

Effect of γ on maxIDRavg|IM & NC (SW)N = 8, T1 = 0.8, γ = var. ,Str. = Unif., ξ = 0.05

θp = 0.02, θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3

0 0.01 0.02 0.03(maxIDR)avg

Sa(T

1)/g

γ = 0.25

γ = 0.13

γ = 0.50

Effect of γ on Disp. of maxIDRavg|IM & NC (SW)N = 8, T1 = 0.8, γ = var., Str. = Unif., ξ = 0.05

θp = 0.02, θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4 0.5 0.6Dispersion of (maxIDR)avg

Sa(T

1)/g

γ = 0.25

γ = 0.13

γ = 0.50

Effect of γ on maxIDRi|IM & NC (SW) [Sa(T1) = 0.38]N = 8, T1 = 0.8, γ = var.,Str. = Unif, ξ = 0.05

θp = 0.02, θpc/θp = 1, λ = 20, Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.01 0.02 0.03(maxIDR)i

Stor

y N

umbe

r

γ = 0.25

γ = 0.13

γ = 0.50


θp = 0.02, θpc/θp = 1, λ = 20, Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.01 0.02 0.03(maxIDR)i

Stor

y N

umbe

r

γ = 0.50

γ = 0.25

γ = 0.13


θp = 0.02, θpc/θp = 1, λ = 20, Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.01 0.02 0.03(maxIDR)i

Stor

y N

umbe

r

γ = 0.50

γ = 0.25

γ = 0.13

Fig. 5.17 Effects of γ on drift demands of case study shear wall

- 119 -

(a) (d)

(b) (e)

(c) (f)

N = 8, T1 = 0.8, γ = 0.25 ,Str. = var., ξ = 0.05 θp = 0.02, θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3


Sa(T

1)/g

Effect of Stiff.&Str. on Med. of maxIDRavg|IM (SW)

Str. = -0.05My,base/floor

Str. = Unif.


0

0.5

1

1.5

2

2.5

3

0 0.01 0.02 0.03(maxIDR)avg

Sa(T

1)/g

Effect of Str. on maxIDRavg|IM & NC (SW)


Str. = Unif.

N = 8, T1 = 0.8, γ = 0.25 , Str. = var., ξ = 0.05 θp = 0.02, θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3


Sa(T

1)/g

Effect of Str. on Disp. of maxIDRavg|IM & NC (SW)


Str. = Unif.

N = 8, T1 = 0.8, γ = 0.25, Str. = var., ξ = 0.05 θp = 0.02, θpc/θp = 1 ,λ = 20 , Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.01 0.02 0.03(maxIDR)i

Stor

y N

umbe

rEffect of Str. on maxIDRi|IM & NC (SW) [Sa(T1) = 0.38]


Str. = Unif.


1

2

3

4

5

6

7

8

0 0.01 0.02 0.03(maxIDR)i

Stor

y N

umbe

r

Effect of Str. on maxIDRi|IM & NC (SW) [Sa(T1) = 1.00]


Str. = Unif.


1

2

3

4

5

6

7

8

0 0.01 0.02 0.03(maxIDR)i

Stor

y N

umbe

r

Effect of Str. on maxIDRi|IM & NC (SW) [Sa(T1) = 1.75]


Str. = Unif.

Fig. 5.18 Effects of strength distribution on drift demands of case study shear wall

- 120 -

(a) (d)

(b) (e)

(c) (f)

N = 8, T1 = 0.8, γ = 0.25 ,Str. = ,Unif , ξ = 0.05 θp = var., θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3


Sa(T

1)/g

Effect of θp on Med. of maxIDRavg|IM (SW)

θp = 0.02

θp = 0.01

θp = 0.03

N = 8, T1 = 0.8, γ = 0.25 ,Str. = ,Unif , ξ = 0.05 θp = var., θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3

0 0.01 0.02 0.03(maxIDR)avg

Sa(T

1)/g

Effect of θp on maxIDRavg|IM & NC (SW)

θp = 0.02

θp = 0.01

θp = 0.03

N = 8, T1 = 0.8, γ = 0.25 , Str. = Shear, ξ = 0.05 θp = var., θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3


Sa(T

1)/g

Effect of θp on Disp. of maxIDRavg|IM & NC (SW)

θp = 0.02θp = 0.01

θp = 0.03

N = 8, T1 = 0.8, γ = 0.25, Str. = Unif., ξ = 0.05 θp = var. , θpc/θp = 1 , λ = 20 , Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.01 0.02 0.03(maxIDR)i

Stor

y N

umbe

r

θp = 0.02

θp = 0.01

θp = 0.03

Effect of θp on maxIDRi|IM & NC (SW) [Sa(T1) = 0.38]


1

2

3

4

5

6

7

8

0 0.01 0.02 0.03(maxIDR)i

Stor

y N

umbe

r

θp = 0.02

θp = 0.01

θp = 0.03



1

2

3

4

5

6

7

8

0 0.01 0.02 0.03(maxIDR)i

Stor

y N

umbe

r

θp = 0.02

θp = 0.01

θp = 0.03


Fig. 5.19 Effects of component θp on drift demands of case study shear wall

- 121 -

(a) (d)

(b) (e)

(c) (f)

N = 8, T1 = 0.8, γ = 0.25 ,Str. = Unif., ξ = 0.05 θp = 0.02, θpc/θp = var., λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3


Sa(T

1)/g

Effect of θpc/θp on Med. of maxIDRavg|IM (SW)

θp/θp = 1

θp/θp = 0.5

θpc/θp = 3

N = 8, T1 = 0.8, γ = 0.25 , Str. = Unif. , ξ = 0.05 θp = 0.02, θpc/θp = var., λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3

0 0.01 0.02 0.03(maxIDR)avg

Sa(T

1)/g

Effect of θpc/θp on maxIDRavg|IM & NC (SW)

θp/θp = 1

θp/θp = 0.5

θpc/θp = 3

N = 8, T1 = 0.8, γ = 0.25 , Str. = Unif., ξ = 0.05 θp = 0.02, θpc/θp = var., λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3


Sa(T

1)/g

Effect of θpc/θp on Disp. of maxIDRavg|IM & NC (SW)

θp/θp = 1

θp/θp = 0.5

θpc/θp = 3

N = 8, T1 = 0.8, γ = 0.25, Str. = Unif., ξ = 0.05 θp = 0.02, θpc/θp = var. , λ = 20 , Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.01 0.02 0.03maxIDR

Stor

y N

umbe

rEffect of θpc/θp on maxIDR|IM & NC (SW) [Sa(T1) = 0.38]

θpc/θp = 1

θpc/θp = 0.5

θpc/θp = 3


1

2

3

4

5

6

7

8

0 0.01 0.02 0.03(maxIDR)i

Stor

y N

umbe

r

Effect of θpc/θp on maxIDRi|IM & NC (SW) [Sa(T1) = 1.00]

θpc/θp = 1

θpc/θp = 0.5

θpc/θp = 3


1

2

3

4

5

6

7

8

0 0.01 0.02 0.03(maxIDR)i

Stor

y N

umbe

r

θpc/θp = 1

θpc/θp = 0.5

θpc/θp = 3

Effect of θpc/θp on maxIDRi|IM & NC (SW) [Sa(T1) = 1.75]

Fig. 5.20 Effects of component θpc/θp on drift demands of case study shear wall

- 122 -

(a) (d)

(b) (e)

(c) (f)

N = 8, T1 = 0.8, γ = 0.25 ,Str. = Unif., ξ = 0.05 θp = 0.02, θpc/θp = 1.0, λ = var., Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3


Sa(T

1)/g

Effect of λ on Med. of maxIDRavg|IM (SW)

λ = 20

λ = 10

λ = 50

N = 8, T1 = 0.8, γ = 0.25 ,Str. = Unif. , ξ = 0.05 θp = 0.02, θpc/θp = 1.0, λ = var., Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3

0 0.01 0.02 0.03(maxIDR)avg

Sa(T

1)/g

Effect of λ on maxIDRavg|IM & NC (SW)

λ = 20

λ = 10

λ = 50

N = 8, T1 = 0.8, γ = 0.25 , Str. = Unif., ξ = 0.05 θp = 0.03, θpc/θp = 5.0, λ = var., Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3


Sa(T

1)/g

Effect of λ on Disp. of maxIDRavg|IM & NC (SW)

λ = 20

λ = 10

λ = 50

N = 8, T1 = 0.8, γ = 0.25, Str. = Unif., ξ = 0.05 θp = 0.02, θpc/θp = 1 , λ = var , Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.01 0.02 0.03(maxIDR)i

Stor

y N

umbe

r

Effect of λ on maxIDRi|IM & NC (SW) [Sa(T1) = 0.38]

λ = 20

λ = 10

λ = 50


1

2

3

4

5

6

7

8

0 0.01 0.02 0.03(maxIDR)i

Stor

y N

umbe

r


λ = 20

λ = 10

λ = 50


1

2

3

4

5

6

7

8

0 0.01 0.02 0.03(maxIDR)i

Stor

y N

umbe

r

λ = 20

λ = 10

λ = 50



study shear wall

- 123 -

(a) (d)

(b) (e)

(c) (f)

Effect of γ on Med. of PFAavg|IM (SW)N = 8, T1 = 0.8, γ = var. ,Str. = Unif., ξ = 0.05

θp = 0.02, θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3


Sa(T

1)/g

γ = 0.25

γ = 0.13

γ = 0.50

Effect of γ on PFAavg|IM & NC (SW)N = 8, T1 = 0.8, γ = var., Str. = Unif, ξ = 0.05

θp = 0.02, θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2(PFA)avg

Sa(T

1)/g

γ = 0.25

γ = 0.13

γ = 0.50

Effect of γ on Disp. of PFAavg|IM & NC (SW)N = 8, T1 = 0.8, γ = var. ,Str. = Unif., ξ = 0.05

θp = 0.02, θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4 0.5 0.6Dispersion of (PFA)avg

Sa(T

1)/g

γ = 0.25

γ = 0.13

γ = 0.50

Effect of γ on PFAi|IM & NC (SW) [Sa(T1) = 0.38]N = 8, T1 = 0.8, γ = var.,Str. = Unif, ξ = 0.05

θp = 0.02, θpc/θp = 1, λ = 20, Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3(PFA)i

Floo

r Num

ber γ = 0.25

γ = 0.13

γ = 0.50


θp = 0.02, θpc/θp = 1, λ = 20, Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3(PFA)i

Floo

r Num

ber

γ = 0.50

γ = 0.25

γ = 0.13


θp = 0.02, θpc/θp = 1, λ = 20, Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3(PFA)i

Floo

r Num

ber

γ = 0.50

γ = 0.25

γ = 0.13

Fig. 5.22 Effects of γ on acceleration demands of case study shear wall

- 124 -

(a) (d)

(b) (e)

(c) (f)


0

0.5

1

1.5

2

2.5

3


Sa(T

1)/g

Effect of Str. on Med. of PFAavg|IM (SW)


Str. = Unif.


0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2(PFA)avg

Sa(T

1)/g

Effect of Str. on PFAavg|IM & NC (SW)


Str. = Unif.

N = 8, T1 = 0.8, γ = 0.25 , Str. = var., ξ = 0.05 θp = 0.02, θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3


Sa(T

1)/g

Effect of Str. on Disp. of PFAavg|IM & NC (SW)


Str. = Unif.

N = 8, T1 = 0.8, γ = 0.25, Str. = var., ξ = 0.05 θp = 0.02, θpc/θp = 1, λ = 20, Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3(PFA)i

Floo

r Num

ber

Effect of Str. on PFAi|IM & NC (SW) [Sa(T1) = 0.38]


Str. = Unif.


1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3(PFA)i

Floo

r Num

ber


Str. = -0.05My,base/floorStr. = Unif.


1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3(PFA)i

Floo

r Num

ber



Str. = Unif.

Fig. 5.23 Effects of strength distribution on acceleration demands of case study shear wall

- 125 -

(a) (d)

(b) (e)

(c) (f)

N = 8, T1 = 0.8, γ = 0.25, Str. = Unif, ξ = 0.05 θp = var., θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3


Sa(T

1)/g

Effect of θp on Med. of PFAavg|IM (SW)

θp = 0.03

θp = 0.01

θp = 0.06

N = 8, T1 = 0.8, γ = 0.25 ,Str. = Unif., ξ = 0.05 θp = var., θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2(PFA)avg

Sa(T

1)/g

Effect of θp on PFAavg|IM & NC (SW)

θp = 0.02

θp = 0.01

θp = 0.03

N = 8, T1 = 0.8, γ = 0.25 , Str. = Unif., ξ = 0.05 θp = var., θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3


Sa(T

1)/g

Effect of θp on Disp. of PFAavg|IM & NC (SW)

θp = 0.02θp = 0.01

θp = 0.03


1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3(PFA)i

Floo

r Num

ber

θp = 0.02

θp = 0.01

θp = 0.03

Effect of θp on PFAi|IM & NC (SW) [Sa(T1) = 0.38]


1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3(PFA)i

Floo

r Num

ber

θp = 0.02

θp = 0.01

θp = 0.03



1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3(PFA)i

Floo

r Num

ber

θp = 0.02

θp = 0.01

θp = 0.03


Fig. 5.24 Effects of component θp on acceleration demands of case study shear wall

- 126 -

(a) (d)

(b) (e)

(c) (f)


0

0.5

1

1.5

2

2.5

3


Sa(T

1)/g

Effect of θpc/θp on Med. of PFAavg|IM (SW)

θp/θp = 1

θp/θp = 0.5

θpc/θp = 3

N = 8, T1 = 0.8, γ = 0.25 ,Str. = Unif. , ξ = 0.05 θp = 0.02, θpc/θp = var., λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2(PFA)avg

Sa(T

1)/g

Effect of θpc/θp on PFAavg|IM & NC (SW)

θp/θp = 1

θp/θp = 0.5

θpc/θp = 3

N = 8, T1 = 0.8, γ = 0.25 , Str. = Unif., ξ = 0.05 θp = 0.02, θpc/θp = var., λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3


Sa(T

1)/g

Effect of θpc/θp on Disp. of PFAavg|IM & NC (SW)

θp/θp = 1

θp/θp = 0.5

θpc/θp = 3


1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3(PFA)i

Floo

r Num

ber

Effect of θpc/θp on PFAi|IM & NC (SW) [Sa(T1) = 0.38]

θpc/θp = 1

θpc/θp = 0.5

θpc/θp = 3


1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3(PFA)i

Floo

r Num

ber


θpc/θp = 1

θpc/θp = 0.5

θpc/θp = 3


1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3(PFA)i

Floo

r Num

ber

θpc/θp = 1

θpc/θp = 0.5

θpc/θp = 3


Fig. 5.25 Effects of component θpc/θp on acceleration demands of case study shear wall

- 127 -

(a) (d)

(b) (e)

(c) (f)


0

0.5

1

1.5

2

2.5

3


Sa(T

1)/g

Effect of λ on Med. of PFAavg|IM (SW)

λ = 20

λ = 10

λ = 50

N = 8, T1 = 0.8, γ = 0.25 ,Str. = Unif. , ξ = 0.05 θp = 0.02, θpc/θp = 1.0, λ = var., Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2(PFA)avg

Sa(T

1)/g

Effect of λ on PFAavg|IM & NC (SW)

λ = 20

λ = 10

λ = 50


0

0.5

1

1.5

2

2.5


Sa(T

1)/g

Effect of λ on Disp. of PFAavg|IM & NC (SW)

λ = 20

λ = 10

λ = 50

N = 8, T1 = 0.8, γ = 0.25 ,Str. = Unif., ξ = 0.05 θp = 0.02, θpc/θp = 1 , λ = var. , Mc/My = 1.1

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3(PFA)i

Floo

r Num

ber

Effect of λ on PFAI|IM & NC (SW) [Sa(T1) = 0.38]

λ = 20

λ = 10

λ = 50


1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3(PFA)i

Floo

r Num

ber

Effect of λ on PFAi|IM & NC (SW) [Sa(T1) = 1.00]

λ = 20

λ = 10

λ = 50


1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3(PFA)i

Floo

r Num

ber

λ = 20

λ = 10

λ = 50

Effect of λ on PFAi|IM & NC (SW) [Sa(T1) = 1.75]

Fig. 5.26 Effects of component cyclic deterioration parameter λ on acceleration demands of

case study shear wall

- 128 -

- 129 -

6 STRUCTURAL SYSTEM DOMAIN: ASSESSMENT OF PROBABILITY OF COLLAPSE AND DESGIN FOR COLLAPSE SAFETY


The primary objective of earthquake resistant design is to protect life safety, and it is generally

acknowledged that complete or partial collapse is the primary (but not the only) source of loss of

lives in severe earthquakes. Nevertheless, as shown in Chapter 2, collapse is also a contributor to

direct and indirect monetary losses. As stated in recent studies (Krawinkler, ed., 2005), collapse

can be a major contributor to monetary losses for older buildings with a non-conforming

structural system (e.g., old non-ductile reinforced concrete buildings). In the design decision

process, collapse prevention is one of the major constraints. Traditionally, collapse is assumed to

be associated with an acceptable story drift or component plastic hinge rotation. Such an

approach ignores redistribution of damage and does not account for the ability of the structural

system to sustain deformations without collapse that are significantly larger than those associated

with loss in resistance of individual elements.

In this chapter, several aspects of assessment of probability of collapse and design for

collapse safety are discussed. In this regard, two representations of the collapse potential of

buildings are considered: the probability of collapse at discrete hazard levels, and the MAF of

collapse (Mean Annual Frequency of collapse). The basic ingredient for obtaining the collapse

potential of buildings is a collapse fragility curve, which expresses the probability of collapse as

a function of the selected intensity measure, IM, (the spectral acceleration at the fundamental

- 130 -

period of the building, Sa(T1) is used in this study as the IM). In this chapter, we first show the

process for obtaining the collapse fragility curve of a structural system. It should be noted

upfront that in this study, collapse is considered as the loss of dynamic sidesway stability of the

structural system. Other sources of collapse such as loss of vertical carrying capacity or

cascading (progressive) collapse are not considered.

A detailed discussion on the effect of different sources of uncertainty (i.e., aleatory and

epistemic) on collapse fragility curve is presented. As will be emphasized in Section 6.3, in this

research the aleatory uncertainty and randomness are used interchangeably. The only source of

aleatory uncertainty considered in this study is due to record-to-record variability. Using

concepts and methods presented in this chapter, assessment of the collapse potential of buildings

and design for collapse safety, including both sources of uncertainty (i.e., aleatory and

epistemic), are made possible.

Section 6.5 is devoted to a detailed evaluation of the sensitivity of collapse fragility

curves to variations of structural parameters for an 8-story frame and an 8-story shear wall

structure. The objective is to illustrate the effect of different design decisions on the collapse

fragility curve. Although the results of this sensitivity study are case specific, they provide

general guidance on sensitivity to structural parameters by extrapolation.

In order to facilitate the selection of appropriate structural parameters for decision

making for collapse protection, the database of collapse fragility curves for the generic frame and

shear wall structures discussed in Chapter 4 are exploited to derive closed form solutions for

median and dispersion of collapse fragility curves. This is achieved through a multi-variate

regression analysis summarized in Section 6.6. Results of these regression analyses show good

fits to the data, which indicates that the regressed values of ηc and βRC (i.e., median and

dispersion due to record-to-record variability of collapse fragility curve) can be utilized for

choosing appropriate structural systems and system parameters.

6.2 Collapse fragility curves

In most codes and guidelines, a building’s collapse capacity is correlated with a structural

response parameter, such as roof or story drift. As shown by Ibarra (2003), evaluation of

- 131 -

structural response parameters near collapse becomes very sensitive to many factors such as

assumptions made in the structural model, type of elements used in the structural model, and the

computer program employed for analysis. For this reason, in this research “collapse capacity” is

not defined with a response parameter but directly with the ground motion intensity at which the

building will become dynamically instable due to structural elements deterioration and P-Δ

effects (denoted as Sac since Sa(T1) is used here as the IM). The collapse capacity for a distinct

ground motion record is obtained by incrementing the intensity of that ground motion and

performing nonlinear response history analysis until a Sac level is reached at which dynamic

instability occurs.

This process is shown in Figure 6.1a for an 8-story generic moment-resisting frame

subjected to the LMSR-N record set. The solid black circle at the end of each IDA curve shows

the last point at which the solution converged. The projection of each solid black circle on the

vertical axis, illustrated with a solid gray circle, shows the collapse capacity of this building for

an individual record. The cumulative distribution function, assuming a lognormal distribution, of

these spectral acceleration values that correspond to structural collapse is defined as the

“collapse fragility curve” and is shown with a heavy black line in Figure 6.1a. A more

conventional form of the collapse fragility curve for this generic moment-resisting frame is

shown in Figure 6.1b with a solid black line along with the associated data points plotted with

gray circles.

6.3 Aleatory and epistemic uncertainties in probability of collapse

Sources of uncertainty in the collapse capacity are differentiated into aleatory and epistemic.

Aleatory uncertainty and randomness are used interchangeably in this study. The only source of

aleatory uncertainty considered throughout this work is ground motion record-to-record

variability. The epistemic uncertainty is mainly due to lack of knowledge about the building’s

real properties (e.g., uncertainty in material, stiffness, strength, and deterioration properties of

elements in the mathematical model of the building, and uncertainty due to inability to

incorporate all elements that may contribute to lateral strength and stiffness). Simultaneous

consideration of the effects of epistemic and aleatory uncertainties on the collapse capacity

- 132 -

necessitates performing a Monte Carlo simulation using a mathematical model of the structure in

which the properties of elements are simulated as random variables with certain distribution

functions, and using ground motions that are representative of the seismicity at the building’s

location. Such an approach would be very elaborate.

As stated by Cornell et. al. (2002), and Shinozuka et. al. (2000), a simple way of de-

convolving the effects of epistemic and aleatory uncertainties on the collapse capacity is to

assume that the median of the collapse capacity is a random variable and the dispersion due to

record-to-record variability is independent from the dispersion due to epistemic uncertainty in

the structural model. In order to implement this simplification and find the probability of

collapse given Sac, one needs to perform IDAs using representative ground motions and the

structure’s mathematical model with the properties of elements set to their median value. Using

such an approach provides the collapse fragility curve with median value of the collapse

capacity, Cη , and the dispersion due to record to record variability, βRC (“Randomness in

collapse Capacity). In order to incorporate the effect of epistemic uncertainty, it is assumed that

the median of the collapse capacity, Cη , is a random variable. For simplicity it is assumed that

the random variable Cη is lognormally distributed with median (median estimate of median of

collapse capacity) of ˆCη (see footnote♣)2and dispersion of UCβ (“Uncertainty in collapse

Capacity”). Accurate estimation of UCβ involves Monte Carlo simulation using accurate

distribution function for structural members or simply using the FOSM (First Order Second

Moment) method. Use of FOSM for estimation of UCβ in collapse capacity of structural system

has been utilized by Ibarra and Krawinkler (2005) and was shown that UCβ is in the order of 0.4.

The confidence Y in any estimate of Cη , denoted as YCη , is defined as the probability that

the actual median value of collapse capacity is greater than YCη . The values of Y

Cη for Y

confidence and the probability of collapse given Sac can be obtained by knowing ˆCη , UCβ , RCβ

and using normal distribution tables i.e., Equation. 6.1 and Equation 6.2.

♣This is the same median value of collapse capacity that was obtained by performing IDA using the mathematical model of the structure with properties of its members set to their median values and the set of representative ground

- 133 -

( ) ( ) ( )ˆActual

YC CY

C CUC

Ln LnP Y

η ηη η

β

⎛ ⎞−⎜ ⎟= = Φ⎜ ⎟⎝ ⎠

(6.1)

( )( ) ( )

| C

C

YC

RC

aa

Ln S LnP C S

η

β

⎛ ⎞−⎜ ⎟= Φ⎜ ⎟⎝ ⎠

(6.2)

The graphical presentation of the effect of the epistemic uncertainty, as described in the

previous paragraph, is shown in Figure 6.2. In this figure the fragility curve drawn with black

solid line is the collapse fragility curve as obtained from fitting the lognormal distribution to the

data obtained from IDAs using representative ground motions and the model of the structure in

which the properties of beams and columns are set to their median values. The effect of

epistemic uncertainty is that the median of this fragility curve is shifted to the left according to

the dispersion due to epistemic uncertainty, UCβ , and the confidence level sought (more than

50%). For instance, in this figure we have assumed that UCβ = 0.4. The fragility curve drawn

with a dark gray line is associated with 84% confidence and the fragility curve drawn with a light

gray line is associated with 90% confidence. As the confidence level increases, for a given value

of UCβ , the estimate of median of collapse capacity decreases. Also, as the UCβ increases, for a

given confidence level, the estimate of median of collapse capacity decreases. For the 84%

confidence, as shown in Figure 6.2, there is only 100%-84% = 16% probability that the actual

value of median of collapse capacity is less than 84%Cη . For 90% confidence, there is 10%

probability that the true value of median of collapse capacity be less than 90%Cη , which shows the

reason for 90%Cη being less than 84%

Cη .

Another method for incorporating the effect of epistemic uncertainty in the estimation of

probability of collapse is by not decreasing the estimate of median of collapse capacity but

inflating the dispersion due to record to record variability, βRC, to square-root-of-the-sum-of-the-

squares of βRC and βUC as shown in Equation 6.3. We denote this method as the “mean method”

motions. We did not use this notation (i.e. ˆCη ) previously because we had not described that the median of collapse capacity itself is a random variable.

- 134 -

in contrast with the “confidence method” described above because it provides a mean estimate of

the probability of collapse. Such a fragility curve is represented by Equation 6.4 and shown in

Figure 6.2 with black dashed line. The confidence level Y associated with the probability of

collapse Pc obtained from this fragility curve at a given value of Sac can be obtained using

Equation 6.5.

2 2

EQU RC UCβ β β= + (6.3)

( )( ) ( )50%

| C

C

C

EQU

aa

Ln S LnP C S

ηβ

⎛ ⎞−⎜ ⎟= Φ⎜ ⎟⎝ ⎠

(6.4)

( ) ( )50%

1 C C

UC EQU

EQU RC

aLn S LnY

ηβ ββ β

⎛ ⎞⎜ ⎟−⎜ ⎟= − Φ⎜ ⎟⋅⎜ ⎟−⎝ ⎠

(6.5)

6.4 Design for tolerable probability of collapse at discrete hazard levels and MAF of collapse

Obtaining the probability of collapse at discrete hazard levels, and the Mean Annual Frequency,

(MAF) of collapse, adheres to two dependent and equally important relationships; that between

the seismic hazard and ground motion intensity measure (i.e. seismic hazard curve), and that

between the ground motion intensity measure and the probability of collapse (i.e. collapse

fragility curve). The seismic hazard curve contains the return period dependent description of the

ground motion intensity. The intensity measure could be a scalar or a vector quantity (Baker and

Cornell, 2004). In the process of development of such curves, aleatory and epistemic

uncertainties are involved, which makes the seismic hazard at any intensity measure at any

seismic hazard level to be a random variable. It has been proposed by Cornell (2002) and Jalayer

(2003) that the aleatory and epistemic uncertainties in the hazard are assumed to be dealt with by

using the mean hazard curve, denoted as 1( ( ))Sa Sa Tλ .

As discussed in Chapter 2 and in previous section, for a design alternative and

representative ground motion records for the location of the building, one can develop collapse

- 135 -

fragility curves by performing Incremental Dynamic Analyses. Such an exercise has been

conducted for the generic structural systems used in this study. We assume that the spectral

acceleration associated with collapse, Sac, is a lognormally distributed random variable with a

random median Cη and dispersion RCβ . We also assume that the random median of Cη is

lognormally distributed with median of ˆCη and dispersion UCβ . Using this information, one can

obtain, for a given design alternative, the probability of collapse at a hazard level (denoted as PR)

with Y confidence by following these steps:

1. Using the median estimate of the median of collapse capacity, ˆCη , and the associated

dispersion due to epistemic uncertainty, UCβ , calculate an estimate of the median of

collapse capacity with Y confidence, denoted as YCη , using Equation 6.6:

ˆ UC YKYC Ce βη η −= (6.6)

where YK is the standardized Gaussian variate associated with the probability Y of not

being exceeded. The value of YK for different confidence levels is shown in Figure 6.3.

2. Using the mean hazard curve obtained for the building, find the spectral acceleration

value that corresponds to the hazard level PR for which the probability of collapse is

being calculated, RPSa . It should be emphasized that this is an simplified approach in

which mean estimate of spectral acceleration at the target hazard level is used to obtain

the probability of collapse with certain confidence.

3. Calculate the probability of collapse for hazard level PR with Y confidence using

Equation 6.7:

( )( ) ( )

|PR

PR

YC

RC

aa

Ln S LnP C S

η

β

⎛ ⎞−⎜ ⎟= Φ⎜ ⎟⎝ ⎠

(6.7)

In order to estimate the MAF of collapse with Y confidence, one needs to integrate the

collapse fragility curve that is obtained for the median of collapse capacity with Y confidence, YCη , over the mean hazard curve as shown in Equation 6.8:

( | ) ( )YSaC YSa

a aP C s d sλ λ= ∫ (6.8)

- 136 -

In Equation 6.8, ( )|YaP C s is the probability distribution function of Sac for Y

confidence. Jalayer (2003) has introduced Equation 6.9 as a closed form solution for Equation 6.7:

( ) ( ) ( ) ( ) ( )2 2ˆ| exp 1 2 expYC Y Sa Sa C RC Y UCSa a aP C s d s k K kλ λ λ η β β⎡ ⎤⎡ ⎤⎡ ⎤ ⎡ ⎤= =∫ ⎣ ⎦⎣ ⎦⎣ ⎦⎣ ⎦ (6.9)

The simplified expression on the right-hand side of Equation 6.9 contains the MAF of the

spectral acceleration associated with the 50% probability of collapse, ˆ( )Sa Cλ η , and two terms that

account for aleatory and epistemic uncertainties in the computation of the collapse capacity. As

seen in Equation 6.9, the epistemic uncertainty in the hazard curve is assumed to be considered

by using the mean hazard curve. The second term on the right hand side of Equation 6.9 accounts

for the aleatory uncertainty and contains the slope of the hazard curve in the log-log domain, k,

at the referenced spectral acceleration value ˆCη , and the dispersion, βRC, due to record-to-record

variability in the collapse fragility curve. The third term on the right hand side of Equation 6.9

accounts for the epistemic uncertainty. If one wants to disregard the effect of epistemic

uncertainty in the assessment of probabilities of collapse (i.e. not include a confidence statement

for such estimations), one could simply assign KY = 0.0 in Equation 6.7 and Equation 6.9. In

such case, the confidence level will be 50%.

Using the “mean method” for obtaining the probability of collapse given IM and

integrating this collapse fragility curve over the mean hazard curve, we obtained the mean

estimate of the mean annual frequency of collapse as shown in Equation 6.10:

( ) ( ) ( ) ( )2 2ˆ| exp 1 2YC Sa Sa C EQUSa a aP C s d s kλ λ λ η β⎡ ⎤⎡ ⎤= =∫ ⎣ ⎦ ⎣ ⎦ (6.10)

The formulation discussed in the previous paragraphs along with the database of collapse

capacity of generic structures used in this study (Chapter 4) can be used in the DDSS in order to

conduct a conceptual design based on target performance objectives. In this manner design for

collapse safety at discrete hazard levels and for the life time of the building in the form of MAF

of collapse is greatly facilitated.

6.5 Sensitivity of collapse fragility curves to structural parameters variation in generic structures

- 137 -

In the next two sections the sensitivity of collapse fragility curves to variations in structural

parameters of a case study generic moment-resisting frame and a case study generic shear wall

are investigated. The goal is to identify important structural parameters that greatly affect the

median and dispersion of collapse capacity (i.e., ηc, and βRC). The two case studies are:

1. 8-story generic moment-resisting frame with fundamental period of T1 = 1.2 seconds (i.e.,

αt = 0.15), γ = var., Stiff & Str. = Shear, SCB = 2.4-1.2, θp = 0.03, θpc/θp = 5.0, λ = 20,

and Mc/My = 1.1.

2. 8-story generic shear wall with fundamental period of T1 = 0.8 seconds (i.e., αt = 0.10), γ

= var., Str. = Unif., θp = 0.02, θpc/θp = 1.0, λ = 20, and Mc/My = 1.1.

The sensitivity of ηc and βRC to variation of each structural parameter is studied by

keeping all other parameters unchanged and varying the associated parameter and the yield base

shear coefficient γ.

6.5.1 Sensitivity of collapse fragility curves to structural parameters variation in case

study generic moment-resisting frame

Sensitivity of the case study generic moment-resisting frame to several structural parameters is

shown in Figures 6.4 to 6.9. Each figure consists of 4 parts. Part (a) and part (b) of each figure

show the sensitivity of median of collapse capacity ηc to variation of a single structural

parameter and the yield base shear coefficient γ. Part (c) of each figure shows the sensitivity of

βRC to variation of the structural parameters. Pushover curves of structures considered in each

sensitivity study are illustrated in part (d) of each figure.

Effect of plastic hinge rotation capacity θp

Figure 6.4 shows the sensitivity of median of collapse capacity ηc and the associated aleatory

dispersion βRC to variation of plastic hinge rotation capacity θp for the case study 8-story generic

moment-resisting frame with a yield base shear coefficients γ = 0.33, 0.17, and 0.08,

respectively. Figure 6.4d shows the associated pushover curves for the abovementioned generic

moment-resisting frames. As seen in Figure 6.4a, for a given value of plastic hinge rotation

- 138 -

capacity, θp, the median of collapse capacity increases almost linearly with the yield base shear

coefficient γ.

We define global post-yielding drift capacity (see Figure 6.4d) as the amount of drift

between the yielding point and the point where the pushover curve reaches zero strength. For

small values of θp, the difference between the global post-yielding drift capacities of 8-story

generic moment-resisting frames with different yield base shear coefficient is small. This is due

to the small deformation capacity in beams and columns of the first story of the frame structure

with small θp where the story mechanism is initiated that leads to structural collapse. Therefore,

the increase in the median of collapse capacity obtained by increasing γ is relatively small (see

the gray line in Figure 6.4a). The opposite is true for large values of θp (i.e., black line in Figure

6.4a).

Figure 6.4b shows the sensitivity of median of collapse capacity to variation of θp for

given values of γ. This figure shows that the median of collapse capacity increases with θp. The

rate of this increase, given γ, is not uniform and is reduced for larger values of θp. This is due to

the fact that for small values of θp = 0.01 (and for the constant value of θpc/θp = 5.0), the post-

capping rotation capacity of the beams and column is small which does not give the opportunity

to the structure to redistribute the nonlinear behavior. However, increasing θp to 0.02 improves

the behavior by providing more deformation capacity in beams and columns of the first floor

where the collapse is initiated. This effect is also seen in the pushover curves associated with θp

= 0.01 and 0.02 in Figure 6.4d.

By increasing θp from 0.02 to 0.03 we will not get the same increase in ηc. This is due to

the significance of the P-Delta effects for large values of θp (and for the constant value of Mc/My

= 1.1). For large values of θp, P-Delta effects reduce the post-yielding stiffness of structural

members to negative values (see pushover curves in Figure 6.4d), and affect the collapse

potential of the structure significantly.

The domination of P-Delta effects is even larger for structures with small γ values. As

seen in Figure 6.4b, the rate of increase in ηc is small for the structure whose γ = 0.08. From the

above discussion, we conclude that in order to effectively reduce the probability of collapse

(increase ηc), we need to increase both the plastic hinge rotation capacity (i.e. good detailing)

- 139 -

and the yield base shear strength coefficient γ, especially when large P-Delta effects cause a

negative post-yielding stiffness in the pushover curve.

Figure 6.4c shows the record-to-record dispersion βRC. As seen in this figure, the effect of

θp and γ is negligible on the dispersion and βRC is close to 0.4 for all cases.

Effect of post-capping rotation capacity ratio θpc/θp

Figure 6.5b shows the sensitivity of ηc to variation of θpc/θp for given values of γ. It is seen that

the median of collapse capacity increases with θpc/θp. The rate of this increase depends of the

value of γ and range of variation of θpc/θp. For small values of γ, ηc is not sensitive to θpc/θp, due

to domination of P-Delta effects. Figure 6.5d shows the pushover curves of frames with γ = 0.08

and θpc/θp = 1.0, 5.0, 15.0. As seen in this figure, the effect of P-Delta in reducing the post

yielding stiffness is large and increasing θpc/θp does not affect the global behavior of the

structure by much.

For larger values of γ, ηc is increased with θpc/θp. The increment in ηc is larger between

θpc/θp = 1.0 and 5.0, than θpc/θp = 5.0 and 15.0. For small values of θpc/θp, the post-capping

rotation capacity of beams and column is small and causes a sudden failure after passing the

capping point. This effect could be seen in pushover curves associate with θpc/θp = 1.0 in Figure

6.5d. For larger values of θpc/θp, ηc is increased due to the increase in post-capping deformation

capacity of beams and columns. The increase in the post-capping deformation capacity of beams

and columns, θpc, is not proportional to the increase in θpc/θp due to P-Delta effects. The

reduction in post-capping deformation capacity due to P-Delta effect is larger for large values of

θpc/θp. For this reason, the rate of increase in ηc is reduced with θpc/θp.

Effect of cyclic deterioration parameter λ

Figure 6.6 shows the effect of the cyclic deterioration parameter λ on median of collapse

capacity ηc and associated dispersion βRC for the case study 8-story generic moment-resisting

frame. The effect of variation in λ on the median of collapse capacity is less important than that

- 140 -

of the previous two parameters (i.e., plastic hinge rotation capacity θp and post capping rotation

capacity ratio θpc/θp). As seen in Figure 6.6, the effect of cyclic deterioration in increasing the

median of collapse capacity is larger when the cyclic deterioration parameter is increased from λ

= 10 to λ = 20 than from λ = 20 to λ = 50. This is due to the change in the cause of failure from

dominance of P-Delta (for larger values of λ) to dominance of deterioration in structural

members (for smaller values of λ). The dispersion βRC is also insensitive to this parameter.

Effect of variation of stiffness and strength along the height

The effect of a gradual variation of stiffness and strength along the height on the median of

collapse capacity, as shown in Figure 6.7b, is small, particularly for small values of γ. In general,

providing additional stiffness and strength along the height of moment-resisting frames slightly

decreases the median collapse capacity. It concentrates the inelastic demand in the lower stories

and limits the redistribution of inelastic deformations along the height of the structure. In such a

case the structure may develop a story (or multiple stories) mechanism and lateral instability

(i.e., collapse) occurs earlier. As seen in the pushover curves of Figure 6.7d, the reduction in

global post-capping drift capacity due to an increase in stiffness and strength along the height is

largest for the case where the yield base shear coefficient is large (i.e., γ = 0.33). Similar to

previous cases, the dispersion insensitive to a gradual variation in strength and stiffness and is

equal to 0.4.

Effect of column to beam strength ratio

Figure 6.8b shows that the effect of SCB ratio on ηc is significant. This increase is due to a

change of failure mechanism. For larger values of the SCB ratio, column hinging is postponed

and therefore the structure is more capable of redistributing the nonlinear deformations between

beams. For small SCB ratios, local (single or multiple stories) mechanisms form rather early, a

phenomenon that cannot be observed from the presented results but is seen from pushover

deflection profiles not shown here. From all the parameters investigated, the strong column –

weak beam factor has the largest effect on the collapse capacity.

- 141 -

An increase in SCB ratio deemphasizes the effect of P-Delta in reduction of ηc, especially

in structures with small γ. As seen in Figure 6.8b, increasing SCB ratio from SCB = 1.2-1.2 (as

specified in the present ACI code) to SCB = 2.4-2.4 for γ = 0.08, increases ηc by 90%. This

increment for γ = 0.17 and 0.33 is equal to 65%. P-Delta reduces the stiffness of structural

members, especially the stiffness of columns in the lower stories of the structure. This effect

leads to development of a story mechanism in lower stories of the structure. By increasing the

column strength, this phenomenon is postponed and nonlinear behavior is focused more into the

beams.

Effect of P-Delta

Although the effect of P-Delta on reduction of ηc especially for structure with small base shear

strength was discussed previously, the importance of P-delta effects on the collapse capacity is

isolated from other parameters and illustrated in Figure 6.9. This figure shows the variation of

ηc, βRC, and associated pushover curves with and without P-Delta effects. As seen in Figure 6.9d,

the difference between the post-yielding global drift capacities of cases with and without P-Delta

increases as γ is decreased. For this reason the effect of P-Delta is larger for cases in which γ is

small.

Figure 6.9b shows the effect of P-Delta on reduction of ηc for three values of γ. The

reduction in ηc due to P-Delta effects is in the range of 65%, 30%, and 10% for γ = 0.08, 0.17,

and 0.33, respectively. As mentioned previously, P-Delta reduces the stiffness of structural

members in lower stories and concentrates nonlinear behavior in the lower stories of the

structural system. This effect, refrains the structure to use its potential deformation capacity in

different stories to absorb the energy of the ground motion.

6.5.2 Sensitivity of collapse fragility curves to structural parameters variation in case

study generic shear wall

- 142 -

For the case study generic shear wall, sensitivity of median collapse capacity ηc and βRC to

several structural parameters are investigated and shown in Figures 6.10 to 6.14. Layout of the

figures is similar to the layout presented in the previous section.

Effect of plastic hinge rotation capacity θp

Figure 6.10 shows the sensitivity of median of collapse capacity ηc and the associated aleatory

dispersion βRC to variation of plastic hinge rotation capacity θp for the case study 8-story generic

shear wall with a yield base shear coefficients γ = 0.5, 0.25, and 0.13. As seen in Figure 6.10a,

for each value of plastic hinge rotation capacity θp, the rate of increase in median collapse

capacity with γ is almost constant and equal to 30%.

Figure 6.10b shows the sensitivity of ηc to θp for given values of base yield base shear

coefficient γ. The general trend is that the median of collapse capacity increases linearly with θp.

This observation in contrast with Figure 6.4b where illustrates the sensitivity of ηc to the θp,

shows that plastic hinge rotation capacity has a larger effect on ηc in shear walls.

Dominance of P-Delta effects in reducing the median of collapse capacity is smaller for shear

walls. Comparison between Figure 6.10b and Figure 6.4b (or associated pushover curves in

Figure 6.10d and Figure 6.4d) shows that the reduction in the rate of increase in ηc with θp is

smaller for shear walls. This is due to dominance of flexural-deformation in shear walls and will

be discussed subsequently.

As seen in Figure 6.10c, the dispersion βRC is almost constant and equal to 0.4 for all

cases but the shear wall with large strength of γ = 0.5 and small plastic hinge rotation capacity of

θp = 0.01. This structure has a brittle behavior (i.e., small θp and rapid fall of strength after the

capping point) which gives no chance for stabilization after the capping point is passed, therefore

the dispersion is small.

- 143 -

Effect of post-capping rotation capacity ratio θpc/θp

Figure 6.11 shows the effect of post-capping rotation capacity ratio θpc/θp on median of collapse

capacity ηc and the associated aleatory dispersion βRC for the case study shear wall. General

trends show that ηc varies linearly with θpc/θp (see Figure 6.11b). This observation compared to

the observation we made in the case of frames (Figure 6.5b) shows that the effect of P-Delta in

reducing ηc is smaller for shear walls.

Figure 6.11d shows the pushover curves for shear walls used for investigating the

sensitivity of ηc to θpc/θp. Inspection of this figure shows that the amount roof drift after passing

the capping point (denoted as global post-capping drift capacity) in shear walls with same θpc/θp

and θp, is larger when γ is small. This is due to the unloading of the shear wall after passing the

capping point where all the elastic deformation (i.e., in this case elastic drift ratio) of the shear

wall is subtracted from the post-capping deformation of the plastic hinge region. Shear wall with

larger yield base shear coefficient will have a larger yield drift; therefore, the reduction to

component post-capping rotation capacity θpc is larger. Consequently, the global post-capping

drift capacity of shear walls with larger yield base shear coefficient γ is smaller than the same

quantity in shear walls with smaller base shear coefficient γ.

Dispersion βRC is relatively insensitive to the parameter θpc/θp unless for systems

designed with large base shear strength γ and small θpc/θp. For such systems, the reduction in

post-capping global drift capacity is large in proportion to the component post-capping rotation

capacity. Therefore, as the system passes the capping point it fails rapidly and leaves no extra

capacity which results in small dispersion in collapse capacity.

Effect of cyclic deterioration parameter λ

Figure 6.12 shows the effect of cyclic deterioration parameter λ on median of collapse capacity

ηc for the 8-story generic shear wall. The effect this parameter on median collapse capacity ηc is

less important than plastic hinge rotation capacity θp and post capping rotation capacity ratio

θpc/θp. Unless for shear walls with small values of λ in which collapse is consequence of

component cyclic deterioration, value of λ does not affect ηc significantly. For shear walls with

- 144 -

large λ, collapse is a consequence of P-Delta or component deformation capacity. These

observations are in agreement with our previous conclusion in moment-resisting frames.

Effect of cyclic deterioration on dispersion βRC is benign unless for shear walls with

small λ (i.e., Figure 6.12c). For such shear walls, rapid deterioration of the deformation capacity

of the plastic hinge region cases a sudden collapse, which is translated into a small dispersion in

collapse capacity. .

Effect of reduction of bending strength along the height

As seen in Figure 6.13, the effect of reduction in bending strength along the height on the median

of collapse capacity ηc and its associated aleatory dispersion βRC for the case study shear wall

was very small. The pushover curves show no difference between the load-deformation

relationships of two systems. This is due to the ratio of capping moment to yield moment that we

used for this study (i.e., Mc/My = 1.1). Using this assumption, moment at the base where usually

the plastic hinge occurs does not increase enough to cause yielding in upper levels in most of the

cases. Also, higher modes deformations did not develop bending moments larger than the

bending strength of shear wall segments.

Effect of P-Delta

Comparison between Figure 6.14b and 6.9b shows that the effect of P-Delta in reducing ηc is

significantly larger for moment-resisting frames than for shear walls. As seen in Figure 6.14b,

for shear walls with small yield base shear coefficient, the difference between the ηc obtained by

including and not including P-Delta effects is larger than other cases. This is due to the higher

effect of P-Delta on systems with small γ (see Figure 6.14d).

On reason for smaller P-Delta effects in shear walls than moment-resisting frames is the

deformation profile of these structural systems. Shear walls deformation is mostly flexural-type

(Figure 4.1) in which most of the deformation is concentrated at the top of the structure where

the amount of axial load is small. On the other hand, the deformation of moment-resisting frames

is mostly shear-type in which most of the deformation is concentrated in lower stories where

- 145 -

columns are subjected to maximum of axial forces. For this reason, the reduction in global

stiffness of moment-resisting frames is larger than shear walls which results in development of

story mechanism in lower stories of these structural systems.

6.6 Sensitivity of collapse fragility curves to ground motion ε

In Chapter 2 of this dissertation a brief discussion about the Hazard Domain and choice of IM

was presented. A new vector based IM presented by Baker and Cornell (2005) (i.e., Sa(T1) and ε)

has shown to be better representative for ground motion intensity. As stated by Baker and

Cornell (2005), the practice of scaling up ground motions without consideration of ε (zero-

epsilon on average) is likely to result in over-estimation of the demand on the structure.

Additionally, disaggregation of the ground motion hazard shows that ground motions with small

mean annual frequency of exceedance (rare events) are all positive-epsilon motions and vice

versa.

Due to the importance of design for collapse safety and the fact the IM used in this study

has been the scalar Sa(T1), obtaining the sensitivity of median collapse capacity to Sa(T1) and ε is

essential. For this reason, we extend the sensitivity study presented in Section 6.5 for a moment-

resisting frame and shear wall to study the effect of ε on median of collapse capacity. For this

reason, the values of ε for the LMSR-N ground motions were found using two spectral

acceleration prediction equations (attenuation law) presented by Boore et., al. (1997) (denoted as

BJF) and Abrahamson and Silva (1997) (denoted as AS97).

Figure 6.15a shows the effect of ε obtained from BJF on the median collapse capacity of

the case study moment-resisting frame. Solid black circles in this figure are plotted as a pair of ε

and Sa(T1) associated with collapse for a certain ground motion. ε is obtained using the M and R

of the associated ground motion using BJF. The solid gray line depicts the median of collapse

capacity obtained without consideration of ε as discussed in Section 6.2. As seen, this value is

constant and independent from ε. The solid black line in Figure 6.16a show the fitted curve with

the form of ηc = b0eb1 using the least square method to the data points. Figure 6.15b shows the

same information but using a ε obtained from AS97.

- 146 -

Figure 6.15 show that consideration of ε significantly affects the prediction of ηc for the

case study moment-resisting frame. At ε = 0, the estimate of ηc using BJF or AS97 provides

almost the same value as ηc obtained by not considering ε. This shows that median collapse

capacities obtained in this study without consideration ε are not biased (zero-epsilon on average).

On the other hand, it is seen that for low probability hazard level (rare events) the median

collapse capacity obtained by ignoring the effect of ε leads to an underestimation of ηc. For

instance, ignoring effect of ε on median collapse capacity at an hazard level associated with ε = 2

leads to an underestimation of ηc by 50% using BJF and 45% using AS97 for estimating of

median of collapse capacity with ε.

Figure 6.16 shows the same information for the case study shear wall. Similar

observations and conclusions made for the case study moment-resisting frame can be seen in this

figure. For example, at a hazard level with ε = 1.5, underestimation of ηc is about 40% using

either BJF or AS97 for estimation of collapse capacity. It is important to note that using the

information provided in Figure 6.15 and Figure 6.16 should be restricted to the range of ε used

for regression. Accuracy of estimates of median collapse capacity using extrapolation of these

relations is matter of faith.

6.7 Development of closed-form equations for estimation of median and dispersion of collapse fragility curves of generic structures

Using the database of collapse capacities obtained for generic moment-resisting frames and shear

walls, closed-form equations are derived to estimate the median of the collapse capacity ηc and

the associated aleatory dispersion βRC as a function of structural parameters. Several regression

models with diverse combinations of predictor variables were considered to obtain the simplest

yet accurate closed-form equation for estimation of the median collapse capacity ηc. Details of

the multivariate regression analysis and the criteria for reducing regression models with several

predictor variables to simpler models with fewer predictor variables are presented in Appendix

C.

An attempt was made also to obtain closed-form equations for estimates of the aleatory

dispersion βRC in the collapse fragility. An evaluation of data shows that the average of βRC for

- 147 -

all combinations of structural parameters in generic moment-resisting frames and generic shear

walls is equal to 0.38 and 0.48, respectively. In a previous study, in which single-bay generic

moment-resisting frames were used, Ibarra and Krawinkler (2005) state that βRC is close to 0.4

for several combinations of structural parameters. The dispersion of estimates of βRC (i.e.,

dispersion of dispersion) is equal to 0.05 for generic moment-resisting frames and 0.07 for

generic shear walls. Due to the small dispersion in estimates of βRC, we conclude that βRC = 0.4

for generic moment-resisting frames and βRC = 0.5 for generic shear walls are good estimates for

aleatory uncertainty in collapse capacity of these structural systems.

6.7.1 Development of closed-form equation for estimation of median of collapse capacity

of generic moment-resisting frame structure

Based on the evaluation of the extensive database on median of collapse capacities of generic

moment-resisting frames we conclude that this parameter has a strong nonlinear dependence on

the number of stories N and the period coefficient αt. For this reason, data was discriminated by

number of stories N and period coefficient αt in 12 bins (i.e., 4 number of stories N, each with 3

number of period coefficients αt) as shown in Figure 6.17. Each bin contains 21 data points that

are medians of collapse capacity for combinations of yield base shear coefficient γ, component

plastic hinge rotation capacity θp, component post-capping rotation capacity ratio θpc/θp, and

component cyclic deterioration parameter λ. Variation of stiffness and strength along the height

(Stiff. & Str.) and strong column – weak beam ratio (SCB) are not used in the regression model

because they were considered as qualitative variables.

A first-order regression model based on four predictor variables was fitted to median

collapse capacities in each bin to serve as a starting point. This regression model is called the

“full model” and is shown in Equation 6.11. Linear multivariate regression analysis was used to

obtain the regression coefficients in each bin. This process is addressed in Appendix C.

- 148 -

( ) ( )

( )

( )

0, ,

, ,

, ,

( , ) ( , )

( , ) ( , )

( , ) ( , )

p

pc p

c MRF T MRF T p

pcMRF T MRF T

p

MRF T T FM MRF

Ln a N a N Ln

a N Ln a N Ln

a N Ln N

θ

θ θ λ

γ

η α α θ

θα α λ

θ

α γ ε α

= + ⋅ +

⎛ ⎞⋅ + ⋅ +⎜ ⎟⎜ ⎟

⎝ ⎠⋅ +

(6.11)

In Equation 6.11 the a factors are the regression coefficients and are obtained by

performing multivariate regression analysis in each bin of data (note the dependence on N and αt

for each a factor). The quantity ,( , )T FM MRFNε α is the regression error of this model and is

obtained for each bin. As long as we use the least square method for obtaining the regression

coefficient, ,( , )T FM MRFNε α is a zero mean random variable with variance of 2, ( , )FM MRF TNσ α .

The regression model presented in Equation 6.11 has five predictors in each bin of data.

In order to obtain a simpler equation with a smaller number of predictors, we introduce a

“reduced model” with one predictor for each bin as shown in Equation 6.12.

( ) ( ) ( )

( )

0,

,

( , ) 0.32 0.08 0.08

0.73 ( , )

pcc MRF T p

p

T RM MRF

Ln b N Ln Ln Ln

Ln N

θη α θ λ

θ

γ ε α

⎛ ⎞= + + + +⎜ ⎟⎜ ⎟

⎝ ⎠+

(6.12)

or in a simpler form and in the arithmetic domain that could be used as a design aid

formulation:

( )0.08

0.32 0.08 0.730,ˆ exp ( , ) pc

c MRF T pp

b Nθ

η α θ λ γθ

⎛ ⎞= ⋅ ⋅ ⋅ ⋅⎜ ⎟⎜ ⎟

⎝ ⎠ (6.13)

In Equation 6.12 and 6.13 0, ( , )MRF Tb N α is the only predictor and is obtained in each bin

of data by performing multivariate regression analysis. Values of 0, ( , )MRF Tb N α are shown in

Figure 6.18. Figure 6.18a shows 0, ( , )MRF Tb N α as a function of number of stories N and period

coefficient αt, and Figure 6.18b shows the same parameter as a function of first mode period T1

(i.e., T1 = N*αt) and period coefficient αt.

In Equation 6.13, ˆcη is the median estimate of the median of collapse capacity. As shown

in Equation 6.12, this estimate has a regression error of ( , )RM TNε α . This error term is zero

- 149 -

mean with variance of 2 2, ,( , ) ( , )RM MRF T est MRF TN Nσ α β α= . , ( , )est MRF TNβ α is shown in Figure

6.19. As seen in this figure, the estimation dispersion is relatively small, between 0.1 and 0.13,

and is much smaller than βRC for generic moment-resisting frames (i.e., βRC = 0.4).

Figure 6.20 and Figure 6.21 compare the predictions obtained from Equation 6.13 with

the data from which it is derived. Figure 6.22 shows the ratio of computed median of collapse

capacity (i.e. data) to estimated median collapse capacity using Equation 6.12 as function of

various structural parameters. Figure 6.23, Figure 6.24, and Figure 6.25, show the comparison

between the estimated value of median collapse capacity and the computed one as function of θp,

θpc/θp, and λ, respectively. Based on these figures, we conclude that Equation 6.12 is a good

estimator for median of collapse capacity in generic moment-resisting frames in the range of

parameters for which the regression was performed. Range of variations of parameters

considered for this regression was discussed in Chapter 4.

6.7.2 Development of closed-form equation for estimation of median of collapse capacity

of generic shear wall structures

Using the approach presented in Section 6.6.1, we obtain Equation 6.14 for estimation of the

median of collapse capacity of generic shear wall structures. Equation 6.15 is the equivalent of

Equation 6.14 in the arithmetic domain. The map of discrimination of data based on number of

stories N and period coefficient αt is shown in Figure 6.26.

( ) ( ) ( )

( )

0,

,

( , ) 0.66 0.17 0.13

0.33 ( , )

pcc SW T p

p

T RM SW

Ln b N Ln Ln Ln

Ln N

θη α θ λ

θ

γ ε α

⎛ ⎞= + + + +⎜ ⎟⎜ ⎟

⎝ ⎠+

(6.14)

( )0.17

0.66 0.13 0.330,ˆ exp ( , ) pc

c SW T pp

b Nθ

η α θ λ γθ

⎛ ⎞= ⋅ ⋅ ⋅ ⋅⎜ ⎟⎜ ⎟

⎝ ⎠ (6.15)

Values of 0, ( , )SW Tb N α are shown in Figure 6.27. The estimation error term

,( , )T RM SWNε α is zero mean with dispersion of , ( , )est SW TNβ α as shown in Figure 6.28. As seen

in this figure, the estimation dispersion is between 0.03 and 0.07, which is much smaller than βRC

- 150 -

for generic shear walls (βRC = 0.5). Scatter plots in which the predictions of Equation 6.15 and

the data points are compared are shown in Figure 6.29 and Figure 6.30. Figure 6.31 shows the

ratio of computed median of collapse capacity to estimated median collapse capacity using

Equation 6.15 as function of various structural parameters. Finally, figure 6.32, Figure 6.33, and

Figure 6.34, show the difference between the estimated value of median collapse capacity and

the computed one as function of θp, θpc/θp, and λ, respectively. Based on these figures, we

conclude that the Equation 6.14 is a good estimator for median of collapse capacity in the range

of parameters for which the regression was performed (Chapter 4).

The efficiency of Equation 6.15 for obtaining the median of collapse capacity of slender

shear walls can be evaluated through a simple example. For instance, the median collapse

capacity of a 7-story slender shear wall with the period coefficient αt = 0.1, plastic hinge rotation

capacity θp = 0.025, post capping rotation capacity ratio θpc/θp = 3.0, cyclic deterioration

parameter λ = 40, and base shear coefficient γ = 0.3, is obtained by substituting the

corresponding values in Equation 6.15. The value of 0, (7,0.1)SWb is obtained from Figure 6.27a

and is equal to 3.75.

( )( )( )( )( )0.66 0.17 0.13 0.33ˆ exp 3.75 0.025 3.0 40 0.3 4.9c gη = = (6.16)

6.8 Summary

In this chapter, we tried to provide concepts and tools for design for collapse safety and

assessment of probability of collapse. Two representations for measuring the collapse potential

of buildings were discussed and methods for obtaining such metrics were presented. These two

measures are probability of collapse given IM and mean annual frequency of collapse. We

showed the process for obtaining the collapse fragility curve which is the basic ingredient for

calculation of collapse potential of a building.

Effect of two sources of variability (i.e., aleatory and epistemic) on collapse fragility

curve and consequently on MAF of collapse was discussed in detail. Two methods were

presented for this purpose: “confidence method” and “mean method”. We discussed each method

- 151 -

and provided step-by-step procedure for incorporating both source of uncertainty in calculation

of probability of collapse given IM and MAF of collapse.

Sensitivity of collapse fragility curve parameters (i.e., median of collapse capacity and

associated aleatory uncertainty) to variation of structural parameters were discussed for two case

study structures in detail. Sources for structural collapse were discussed and general trends were

investigated. It is concluded that in moment-resisting frames, P-Delta is the number one cause of

collapse specially in structures with small yield base shear coefficient. Another equally important

parameter that affects the collapse potential of moment-resisting frames is the ratio of column to

beam strength. It was shown that increasing this parameter from 1.2 (ACI suggestion) to 2.4

increases the median of collapse capacity by 90%. Other causes of collapse are heavy cyclic

deterioration or small plastic hinge rotation capacity of structural components, which become

more important for shear walls.

Sensitivity of median collapse capacity to ε was investigated in Section 6.6. In this

research, we have used a scalar IM (i.e.,Sa(T1)) and by performing this sensitivity study we tried

to investigate the effect of not considering ε as an auxiliary IM. It was seen that that median of

collapse capacity could be underestimate by 50% for hazard levels with small probability of

occurrence.

Using the database of building response parameters, we tried to develop closed-form

equations for that not only facilitate the design and assessment processes, but also helps in

understanding the major trends and importance of certain parameters in changing the collapse

potential of the structural system. The goal was to use the simplest from of equation with least

estimation error. Comparison between estimates and real data show that equations presented are

in good accordance with the data.

- 152 -

Obtaining the collapse fragility curve (MRF)N = 8, T1 = 1.2, γ = 0.17, Stiff & Str = Shear, SCB = 2.4-2.4, ξ = 0.05

θp = 0.03, θpc/θp = 5, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07EDP[max.RDR]

IM[S

a(T 1

)]

Individual IDA curveslast point before collapse

Projection of last point before collapse

Collapse fragility curve

(a)

Obtaining the collapse fragility curve (MRF)N = 8, T1 = 1.2, γ = 0.17, Stiff & Str = Shear, SCB = 2.4-2.4, ξ = 0.05

θp = 0.03, θpc/θp = 5, λ = 20, Mc/My = 1.1

0

0.25

0.5

0.75

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5IM[Sa(T1)]

Prob

abili

ty o

f Col

laps

e

Data points

Collapse fragility curve

(b)

Fig. 6.1 Obtaining collapse fragility curve with Incremental Dynamic Analysis: a) obtaining

data point, b) collapse fragility curve

- 153 -

Uncertainty and collapse fragility curve (MRF)N = 8, αt = 0.15, γ = 0.17, Stiff & Str = Shear, SCB = 2.4-2.4, ξ = 0.05

θp = 0.03, θpc/θp = 5, λ = 20, Mc/My = 1.1

0

0.25

0.5

0.75

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5IM[Sa(T1)]

Prob

abili

ty o

f Col

laps

e

Collapse Fragility Curves

50% Confidence84% Confidence90% Confidence

Mean method

( )ˆPDF of ( ),c c UCN Lnη η β→

50%ˆc cη η=84%cη

90%cη

, T1 = 1.2

Fig. 6.2 Uncertainty and collapse fragility curve

Standardized Gaussian Variate, KY

0

0.5

1

1.5

2

2.5

3

3.5

4

0.5 0.6 0.7 0.8 0.9 1Confidence level Y

KY

Fig. 6.3 Standard Gaussian variate KY for different confidence levels

- 154 -

Plastic Hinge Rotation Capacity Effect on ηc (MRF)N = 8, T1 = 1.2, γ = var.,Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05


0

0.5

1

1.5

2

2.5

3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35Base shear coefficient γ

c (S

a/g)

θp = 0.06

θp = 0.03

θp = 0.01

(a) (c)

(b) (d)

N = 8, T1 = 1.2, γ = var., Stiff. & Str. = Shear, SCB = 2.4-1.2, ξ = 0.05 θp = var., θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0.00

0.10

0.20

0.30

0.40

0 0.03 0.06 0.09 0.12Roof Drift δroof/H

Vy/W

θp

γ 0.060.030.01

0.33

0.17

0.08

Plastic Hinge Rotation Capacity Effect on Pushover Curve (MRF)Plastic Hinge Rotation Capacity Effect on ηc (MRF)N = 8, T1 = 1.2, γ = var.,Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05


0

0.5

1

1.5

2

2.5

3

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07Plastic hinge rotation capacity θp

c (Sa

/g)

γ = 0.33

γ = 0.17

γ = 0.08

Plastic Hinge Rotation Capacity Effect on βRC (MRF)N = 8, T1 = 1.2, γ = var.,Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05


0

0.25

0.5

0.75

1


R

C

θp = 0.06

θp = 0.03

θp = 0.01

ηηηη

ββ

Fig. 6.4 Sensitivity of collapse fragility curve parameters to plastic hinge rotation capacity

θp and base shear coefficient γ in case study generic moment-resisting frame: a&b)

sensitivity of ηc, c) sensitivity of βRC, d) Pushover curve

- 155 -

(a) (c)

(b) (d)

N = 8, T1 = 1.2, γ = var., Stiff. & Str. = Shear, SCB = 2.4-1.2, ξ = 0.05 θp = 0.03, θpc/θp = var., λ = 20, Mc/My = 1.1

0.00

0.10

0.20

0.30

0.40


Vy/W

θpc/θp

γ 1 5 15

Post-Cap. Rotation Capacity Effect on Pushover Curve (MRF)

0.33

0.17

0.08

Post-Cap. Rotation Capacity Effect on βRC (MRF)N = 8, T1 = 1.2, γ = var.,Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05

θp = 0.03, θpc/θp = var., λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3


c (S

a/g)

θpc/θp= 15

θp/θp= 5

θp/θp= 1

Post-Cap. Rotation Capacity Effect on ηc (MRF)N = 8, T1 = 1.2, γ = var.,Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05


0

0.5

1

1.5

2

2.5

3

0 5 10 15Post-capping rotation capacity ratio θpc/θp

c (Sa

/g)

γ = 0.33

γ = 0.17

γ = 0.08

Post-Cap. Rotation Capacity Effect on ηc (MRF)N = 8, T1 = 1.2, γ = var.,Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05


0

0.25

0.5

0.75

1


R

C

θpc/θp= 15

θp/θp= 5

θp/θp= 1

ηηηη

ββ

Fig. 6.5 Sensitivity of collapse fragility curve parameters to post-capping plastic hinge

rotation capacity θpc/θp and base shear coefficient γ in case study generic moment-resisting

frame: a&b) sensitivity of ηc, c) sensitivity of βRC, d) Pushover curve

- 156 -

(a) (c)

(b) (d)


0.00

0.10

0.20

0.30

0.40


Vy/W

λγ 10 20 50

0.33

0.17

0.08

Cyclic Deterioration Effect on Pushover Curve (MRF)

Cyclic Deterioration Effect on ηc (MRF)N = 8, T1 = 1.2, γ = var.,Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05


0

0.5

1

1.5

2

2.5

3


c (S

a/g)

λ = 50

λ = 20

λ = 10

Cyclic Deterioration Effect on ηc (MRF)N = 8, T1 = 1.2, γ = var.,Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05


0

0.5

1

1.5

2

2.5

3

0 10 20 30 40 50 60

Cyclic deterioration parameter λ

c (Sa

/g)

λ = 50

λ = 20

λ = 10

Cyclic Deterioration Effect on βRC (MRF)N = 8, T1 = 1.2, γ = var.,Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05


0

0.25

0.5

0.75

1


R

C

λ = 50

λ = 20

λ = 10

ηηηη

ββ

Fig. 6.6 Sensitivity of collapse fragility curve parameters to cyclic deterioration parameter

λ and base shear coefficient γ in case study generic moment-resisting frame: a&b)


- 157 -

(a) (c)

(b) (d)

N = 8, T1 = 1.2, γ = var., Stiff. & Str. = var., SCB = 2.4-1.2, ξ = 0.05 θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0.00

0.10

0.20

0.30

0.40


Vy/W

Stiff. & Str.γ ShearInt.Unf.

Variation of Stiff. and Str. Effect on Pushover Curve (MRF)

0.33

0.17

0.08

Variation of Stiff. and Str. Effect on ηc (MRF)N = 8, T1 = 1.2, γ = var.,Stiff.&Str. = var., SCB = 2.4-1.2, ξ = 0.05

θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3


÷ c (S

a/g)

Stiff.&Str. = Shear

Stiff.&Str. = Int.

Stiff.&Str. = Unif.

Variation of Str. and Stif. Effect on βRC (MRF)N = 8, T1 = 1.2, γ = var.,Stiff.&Str. = var., SCB = 2.4-1.2, ξ = 0.05

θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.25

0.5

0.75

1


÷R

C

Stiff.&Str. = Shear

Stiff.&Str. = Int.

Stiff.&Str. = Unif.

ηηηη

ββ

Variation of Str. and Stif. Effect on ηc (MRF)N = 8, T1 = 1.2, γ = var.,Stiff.&Str. = var., SCB = 2.4-1.2, ξ = 0.05

θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4

÷ c(S

a/g)

γ = 0.33

γ = 0.17

γ = 0.08

Stiff.&Str. = Shear

Stiff.&Str. = Int.

Stiff.&Str. = Unif.

Fig. 6.7 Sensitivity of collapse fragility curve parameters to variaton of stiffness and

strength along the height and base shear coefficient γ in case study generic moment-

resisting frame: a&b) sensitivity of ηc, c) sensitivity of βRC, d) Pushover curve

- 158 -

(a) (c)

(b) (d)

N = 8, T1 = 1.2, γ = var., Stiff. & Str. = Shear, SCB = var., ξ = 0.05 θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0.00

0.10

0.20

0.30

0.40


Vy/W

SCBγ 1.2-1.2 2.4-1.2 2.4-2.4

0.33

0.17

0.08

Strong Column Beam Ratio Effect on Pushover Curve (MRF)

Strong Column Beam Ratio Effect on ηc (MRF)N = 8, T1 = 1.2, γ = var.,Stiff.&Str. = Shear, SCB = var., ξ = 0.05

θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3


c (S

a/g)

SCB = 2.4-2.4

SCB = 2.4-1.2

SCB = 1.2-1.2

Strong Column Beam Ratio Effect on ηc (MRF)N = 8, T1 = 1.2, γ = var.,Stiff.&Str. = Shear, SCB = var., ξ = 0.05

θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4

∪c (

Sa/g

)

γ = 0.33

γ = 0.17

γ = 0.08

SCB = 1.2-1.2 SCB = 2.4-1.2 SCB = 2.4-2.4

Strong Column Beam Ratio Effect on βRC (MRF)N = 8, T1 = 1.2, γ = var.,Stiff.&Str. = Shear, SCB = var., ξ = 0.05

θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.25

0.5

0.75

1


∪R

C

SCB = 2.4-2.4

SCB = 2.4-1.2

SCB = 1.2-1.2

ηηηη

ββ

Fig. 6.8 Sensitivity of collapse fragility curve parameters to column to beam strength ratio

and base shear coefficient γ in case study generic moment-resisting frame: a&b) sensitivity

of ηc, c) sensitivity of βRC, d) Pushover curve

- 159 -

(a) (c)

(b) (d)


0.00

0.10

0.20

0.30

0.40


Vy/W

WithP-Deltaγ

0.33

0.17

0.08

P-Delta Effect on Pushover Curve (MRF)

WithoutP-Delta

P-Delta Effect on ηc (MRF)N = 8, T1 = 1.2, γ = var., Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05

θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3


�c

(Sa/

g)

Without P-Delta

With P-Delta

P-Delta Effect on ηc (MRF)N = 8, T1 = 1.2, γ = var.,Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05

θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.5

1

1.5

2

2.5

3

0 1 2 3

�c (

Sa/g

)

γ = 0.33

γ = 0.17

γ = 0.08

With P-DeltaWithout P-Delta

P-Delta Effect on βRC (MRF)N = 8, T1 = 1.2, γ = var., Stiff.&Str. = Shear, SCB = 2.4-1.2, ξ = 0.05

θp = 0.03, θpc/θp = 5.0, λ = 20, Mc/My = 1.1

0

0.25

0.5

0.75

1

0 0.1 0.2 0.3 0.4

Base shear coefficient γ

�R

C

Without P-Delta

With P-Delta

ηηηη

ββ

Fig. 6.9 Sensitivity of collapse fragility curve parameters to P-Delta effects and base shear

coefficient γ in case study generic moment-resisting frame: a&b) sensitivity of ηc, c)

sensitivity of βRC, d) Pushover curve

- 160 -

Plastic Hinge Rotation Capacity Effect on ηc (SW)N = 8, T1 = 0.10, γ = var., Str. = Unf., ξ = 0.05


0

1

2

3

4

5

6

0 0.1 0.2 0.3 0.4 0.5 0.6

Base shear coefficient γ

⎠ c (S

a/g)

θp = 0.03

θp = 0.02

θp = 0.01

(a) (c)

(b) (d)

N = 8, T1 = 0.10, γ = var., Str. = Unf., ξ = 0.05 θp = var., θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0.00

0.20

0.40

0.60


Vy/W

θp

γ 0.01 0.02 0.03

0.50

0.25

0.13

Plastic Hinge Rotation Capacity Effect on P.O.C (SW)Plastic Hinge Rotation Capacity Effect on ηc (SW)N = 8, T1 = 0.10, γ = var.,Str. = Unf., ξ = 0.05


0

1

2

3

4

5

6

0.00 0.01 0.02 0.03 0.04Plastic hinge rotation capacity θp

⎠ c (S

a/g)

γ = 0.50

γ = 0.25

γ = 0.13

Plastic Hinge Rotation Capacity Effect on βRC (SW)N = 8, T1 = 0.10, γ = var., Str. = Unf., ξ = 0.05


0

0.25

0.5

0.75

1

0 0.1 0.2 0.3 0.4 0.5 0.6Base shear coefficient γ

⎠R

C

θp = 0.03

θp = 0.02

θp = 0.01

ηηηη

ββ

0.80.8 0.80.8

0.80.80.80.8

Fig. 6.10 Sensitivity of collapse fragility curve parameters to plastic hinge rotation capacity

and base shear coefficient γ in case study generic shear wall: a&b) sensitivity of ηc, c)


- 161 -

Post-Cap. Rotation Capacity Effect on ηc (SW)N = 8, T1 = 0.10, γ = var., Str. = Unf., ξ = 0.05


0

1

2

3

4

5

6


¬ c (S

a/g)

θpc/θp= 3.0

θpc/θp= 1.0

θpc/θp= 0.5

(a) (c)

(b) (d)

N = 8, T1 = 0.10, γ = var., Str. = Unf., ξ = 0.05 θp = 0.02, θpc/θp = var., λ = 20, Mc/My = 1.1

0.00

0.20

0.40

0.60


Vy/W

θpc/θp

γ 0.5 1.0 3.0

0.50

0.25

0.13

Post-Cap. Rotation Capacity Effect on P.O.C (SW)

Post-Cap. Rotation Capacity Effect on βRC (SW)N = 8, T1 = 0.10, γ = var., Str. = Unf., ξ = 0.05


0

0.25

0.5

0.75

1


R

C

θpc/θp= 3.0

θpc/θp= 1.0

θpc/θp= 0.5

Post-Cap. Rotation Capacity Effect on ηc (SW)N = 8, T1 = 0.10, γ = var.,Str. = Unf., ξ = 0.05 θp = 0.02, θpc/θp = var., λ = 20, Mc/My = 1.1

0

1

2

3

4

5

6

0 1 2 3 4Post-capping rotation capacity ratio θpc/θp

⎠ c (S

a/g)

γ = 0.50

γ = 0.25

γ = 0.13

ηηηη

ββ

0.80.8 0.80.8

0.80.80.80.8

Fig. 6.11 Sensitivity of collapse fragility curve parameters to post-capping rotation capacity

ratio and base shear coefficient γ in case study generic shear wall: a&b) sensitivity of ηc, c)


- 162 -

(a) (c)

(b) (d)

N = 8, T1 = 0.10, γ = var., Str. = Unf., ξ = 0.05 θp = 0.02, θpc/θp = 1.0, λ = var., Mc/My = 1.1

0.00

0.20

0.40

0.60


Vy/W

λγ 10 20 50

0.50

0.25

0.13

Cyclic Deterioration Effect on P.O.C (SW)

Cyclic Deterioration Effect on ηc (SW)N = 8, T1 = 0.10, γ = var., Str. = Unf., ξ = 0.05 θp = 0.02, θpc/θp = 1.0, λ = var., Mc/My = 1.1

0

1

2

3

4

5

6


( c (S

a/g)

λ = 50

λ = 20

λ = 10

Cyclic Deterioration Effect on ηc (SW)N = 8, T1 = 0.10, γ = var.,Str. = Unf., ξ = 0.05 θp = 0.02, θpc/θp = 1.0, λ = var., Mc/My = 1.1

0

1

2

3

4

5

6

0 10 20 30 40 50 60Cyclic deterioration parameter λ

⎝ c (S

a/g)

γ = 0.50

γ = 0.25

γ = 0.13

Cyclic Deterioration Effect on βRC (SW)N = 8, T1 = 0.10, γ = var., Str. = Unf., ξ = 0.05 θp = 0.02, θpc/θp = 1.0, λ = var., Mc/My = 1.1

0

0.25

0.5

0.75

1


⎝R

C

λ = 50

λ = 20

λ = 10

ηηηη

ββ

0.80.8 0.80.8

0.80.80.80.8

Fig. 6.12 Sensitivity of collapse fragility curve parameters to cyclic deterioration parameter

and base shear coefficient γ in case study generic shear wall: a&b) sensitivity of ηc, c)


- 163 -

Red. of Bend. Mom. Along Height Effect on βRC (SW)N = 8, T1 = 0.10, γ = var., Str. = var., ξ = 0.05

θp = 0.02, θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0

0.25

0.5

0.75

1


R

C

Str. = Unif.

Str. = -0.05Mybase/floor

(a) (c)

(b) (d)

Red. of Bend. Mom. Along Height Effect on ηc (SW)N = 8, T1 = 0.10, γ = var., Str. = var., ξ = 0.05

θp = 0.02, θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0

1

2

3

4

5

6


Ξ c (S

a/g)

Str. = Unif.

Str. = -0.05Mybase/floor

Red. of Bend. Mom. Along Height Effect on ηc (SW)N = 8, T1 = 0.10, γ = var.,Str. = var., ξ = 0.05

θp = 0.02, θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0

1

2

3

4

5

6

0 1 2 3

Ξc (

Sa/g

)

γ = 0.50

γ = 0.25

γ = 0.13

Str. = Unif.Str. = -0.05Mybase/floor

ηη

β

0.8 0.8

0.80.8 N = 8, T1 = 0.10, γ = var., Str. = var. , ξ = 0.05 θp = 0.02, θpc/θp = 1.0, λ = 20., Mc/My = 1.1

0.00

0.20

0.40

0.60


Vy/W

Str.γ Unf. -0.05M y,base/floor

0.50

0.25

0.13

Red. of Bend. Mom. Along Height Effect on P.O.C (SW)

Fig. 6.13 Sensitivity of collapse fragility curve parameters to reduction of bending strength

along the height and base shear coefficient γ in case study generic shear wall: a&b)


- 164 -

(a) (c)

(b) (d)

N = 8, T1 = 0.10, γ = var., Str. = var. , ξ = 0.05 θp = 0.02, θpc/θp = 1.0, λ = 20., Mc/My = 1.1

0.00

0.20

0.40

0.60


Vy/W

γ

0.50

0.25

0.13

P-Delta Effect on P.O.C (SW)

WithP-Delta

WithoutP-Delta

P-Delta Effect on ηc (SW)N = 8, T1 = 0.10, γ = var., Str. = var., ξ = 0.05

θp = 0.02, θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0

1

2

3

4

5

6


÷ c (S

a/g)

Without P-Delta

With P-Delta

P-Delta Effect on ηc (SW)N = 8, T1 = 0.10, γ = var.,Str. = var., ξ = 0.05

θp = 0.02, θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0

1

2

3

4

5

6

0 1 2 3

÷ c (S

a/g)

γ = 0.50

γ = 0.25

γ = 0.13

With P-DeltaWithout P-Delta

P-Delta Effect on βRC (SW)N = 8, T1 = 0.10, γ = var., Str. = var., ξ = 0.05

θp = 0.02, θpc/θp = 1.0, λ = 20, Mc/My = 1.1

0

0.25

0.5

0.75

1


÷R

C

Without P-Delta

With P-Delta

ηηηη

ββ

0.80.8 0.80.8

0.80.80.80.8

Fig. 6.14 Sensitivity of collapse fragility curve parameters to P-Delta effects and base shear

coefficient γ in case study generic shear wall: a&b) sensitivity of ηc, c) sensitivity of βRC, d)

Pushover curve

- 165 -

Effect of ε (BJF) on ηc (MRF)N = 8, T 1 = 1.2, γ = 0.17, Str.&Stiff = Shear, SCB = 2.4-1.2, ξ = 0.05

θp = 0.03, θpc/θp = 5.0, λ = 20, M c/My = 1.1

0.0

1.0

2.0

3.0

4.0

5.0

-3 -2 -1 0 1 2 3ε

c (S

a/g)

Individual collapse capacities

No epsilon

With epsilon

Effect of ε (BJF) on ηc (MRF)N = 8, T 1 = 1.2, γ = 0.17, Str.&Stiff = Shear, SCB = 2.4-1.2, ξ = 0.05

θp = 0.03, θpc/θp = 5.0, λ = 20, M c/My = 1.1

0.0

1.0

2.0

3.0

4.0

5.0

-3 -2 -1 0 1 2 3ε

c (S

a/g)


No epsilon

With epsilon

Effect of ε (AS97) on ηc (MRF)N = 8, T 1 =1.2, γ = 0.17, Siff. & Str. = Shear, SCB = 2.4-1.2, ξ = 0.05

θp = 0.03, θpc/θp = 5.0, λ = 20, M c/My = 1.1

0.0

1.0

2.0

3.0

4.0

5.0

-3 -2 -1 0 1 2 3ε

c (S

a/g)


No epsilon

With epsilon

(a)

(b)

0.32ˆ 1.43c e εη =

0.26ˆ 1.48c e εη =

ˆ 1.38cη =

ˆ 1.38cη =

ηηηη

Fig. 6.15 Sensitivity of median of collapse capacity to ε in case study moment-resisting

frame: a) ε from BJF, and b) ε from AS97

- 166 -

Effect of ε (AS97) on ηc (SW)N = 8, T 1 = 0.80, γ = 0.25, Str. = Unf., ξ = 0.05

θp = 0.02, θpc/θp = 1.0, λ = 20, M c/My = 1.1

0.0

2.0

4.0

6.0

8.0

10.0

-2 -1 0 1 2ε

c (S

a/g)


No epsilon

With epsilon

Effect of ε (AS97) on ηc (SW)N = 8, T 1 = 0.80, γ = 0.25, Str. = Unf., ξ = 0.05

θp = 0.02, θpc/θp = 1.0, λ = 20, M c/My = 1.1

0.0

2.0

4.0

6.0

8.0

10.0

-2 -1 0 1 2ε

c (S

a/g)


No epsilon

With epsilon

(a)

(b)

0.36ˆ 3.07c e εη =

0.31ˆ 3.03c e εη =

ˆ 3.02cη =

ˆ 3.02cη =

ηηηη

Effect of ε (BJF) on ηc (SW)N = 8, T 1 = 0.80, γ = 0.25, Str. = Unf., ξ = 0.05

θp = 0.02, θpc/θp = 1.0, λ = 20, M c/My = 1.1

0.0

2.0

4.0

6.0

8.0

10.0

-2 -1 0 1 2ε

c (S

a/g)


No epsilon

With epsilon

Effect of ε (BJF) on ηc (SW)N = 8, T 1 = 0.80, γ = 0.25, Str. = Unf., ξ = 0.05

θp = 0.02, θpc/θp = 1.0, λ = 20, M c/My = 1.1

0.0

2.0

4.0

6.0

8.0

10.0

-2 -1 0 1 2ε

c (S

a/g)


No epsilon

With epsilon

ηη

Fig. 6.16 Sensitivity of median of collapse capacity to ε in case study shear wall: a) ε from

BJF, and b) ε from AS97

- 167 -

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.13

γ = 0.25γ = 0.50

γ = 0.08

γ = 0.17γ = 0.33

γ = 0.13

γ = 0.25γ = 0.50

γ = 0.06

γ = 0.13γ = 0.25

γ = 0.06

γ = 0.11γ = 0.22

γ = 0.08

γ = 0.17γ = 0.33

γ = 0.04

γ = 0.08γ = 0.17

γ = 0.04

γ = 0.08γ = 0.17

γ = 0.06

γ = 0.13γ = 0.25

γ = 0.03

γ = 0.06γ = 0.13

αt = 0.1 αt = 0.15 αt = 0.2

N =

4N

= 8

N =

12

N =

16

Base Case0.03,5,20

0.06

50

15

0.01

1

10

θpc/θc

θp

λ

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.13

γ = 0.25γ = 0.50

γ = 0.13

γ = 0.25γ = 0.50

γ = 0.08

γ = 0.17γ = 0.33

γ = 0.08

γ = 0.17γ = 0.33

γ = 0.13

γ = 0.25γ = 0.50

γ = 0.13

γ = 0.25γ = 0.50

γ = 0.06

γ = 0.13γ = 0.25

γ = 0.06

γ = 0.13γ = 0.25

γ = 0.06

γ = 0.11γ = 0.22

γ = 0.06

γ = 0.11γ = 0.22

γ = 0.08

γ = 0.17γ = 0.33

γ = 0.08

γ = 0.17γ = 0.33

γ = 0.04

γ = 0.08γ = 0.17

γ = 0.04

γ = 0.08γ = 0.17

γ = 0.04

γ = 0.08γ = 0.17

γ = 0.04

γ = 0.08γ = 0.17

γ = 0.06

γ = 0.13γ = 0.25

γ = 0.06

γ = 0.13γ = 0.25

γ = 0.03

γ = 0.06γ = 0.13

γ = 0.03

γ = 0.06γ = 0.13

αt = 0.1 αt = 0.15 αt = 0.2

N =

4N

= 8

N =

12

N =

16

Base Case0.03,5,20

0.06

50

15

0.01

1

10

θpc/θc

θp

λ

Base Case0.03,5,20

0.06

50

15

0.01

1

10

θpc/θc

θp

λ

Base Case0.03,5,20

0.06

50

15

0.01

1

10

θpc/θc

θp

λ

Base Case0.03,5,20

Base Case0.03,5,20

Base Case0.03,5,20

0.06

50

15

0.01

1

10

θpc/θc

θp

λ

0.06

50

15

0.01

1

10

θpc/θc

θp

λ


collapse capacity in generic moment-resisting frames

- 168 -

Base factor b0,MRF for Moment-Resisting Frames

0.0

1.0

2.0

3.0

4.0

5.0

0 4 8 12 16Number of stories N

b 0,M

RF

αt = 0.10

αt = 0.15

αt = 0.20

(a)

Base factor b0,MRF for Moment-Resisting Frames

0.0

1.0

2.0

3.0

4.0

5.0

0 0.5 1 1.5 2 2.5 3 3.5First Mode Period T1

b 0,M

RF

αt = 0.10

αt = 0.15

αt = 0.20

(b)

Fig. 6.18 Base factor b0,MRF for estimation of median of collapse capacity in moment-

resisting frames

- 169 -

Estimation Error in Median of Collapse Capacity Moment-Resisting Frames

0.00

0.05

0.10

0.15

0.20

0 4 8 12 16Number of Stories N

β est

,MR

F

αt = 0.10

αt = 0.15

αt = 0.20

(a)

Estimation Error in Median of Collapse Capacity Moment-Resisting Frames

0.00

0.05

0.10

0.15

0.20

0 0.5 1 1.5 2 2.5 3 3.5First Mode Period T1

β est

,MR

F

αt = 0.10

αt = 0.15

αt = 0.20

(b)


moment-resisting frames

- 170 -

Estimated vs. Computed Median of Collapse CapacityMoment-Resisting Frame, N = 8, T1 = 1.6, 1 predictor

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7Estimated median collapse capacity

Com

pute

d m

edia

n co

llaps

e ca

paci

ty


0

1

2

3

4

5

6

7


Com

pute

d m

edia

n co

llaps

e ca

paci

ty


0

1

2

3

4

5

6

7


Com

pute

d m

edia

n co

llaps

e ca

paci

ty


0

1

2

3

4

5

6

7


Com

pute

d m

edia

n co

llaps

e ca

paci

ty


0

1

2

3

4

5

6

7


Com

pute

d m

edia

n co

llaps

e ca

paci

tyEstimated vs. Computed Median of Collapse CapacityMoment-Resisting Frame, N = 4, T1 = 0.4, 1 predictor

0

1

2

3

4

5

6

7


Com

pute

d m

edia

n co

llaps

e ca

paci

ty

(a)

(b)

(c)

(d)

(e)

(f)

0

0.5

1

1.5

2

0 0.5 1 1.5 2

Fig. 6.20 Scatter plots for median of collapse capacity in moment-resisting frames: (a) N = 4

& T1 = 0.4sec., (b) N = 4 & T1 = 0.6sec., (c) N = 4 & T1 = 0.8sec., (d) N = 8 & T1 = 0.8sec., (e)

N = 8 & T1 = 1.2sec., (f) N = 8 & T1 = 1.6sec.

- 171 -


0

1

2

3

4

5

6

7


Com

pute

d m

edia

n co

llaps

e ca

paci

ty


0

1

2

3

4

5

6

7


Com

pute

d m

edia

n co

llaps

e ca

paci

ty


0

1

2

3

4

5

6

7


Com

pute

d m

edia

n co

llaps

e ca

paci

ty


0

1

2

3

4

5

6

7


Com

pute

d m

edia

n co

llaps

e ca

paci

ty


0

1

2

3

4

5

6

7


Com

pute

d m

edia

n co

llaps

e ca

paci

tyEstimated vs. Computed Median of Collapse CapacityMoment-Resisting Frame, N = 12, T1 = 1.2, 1 predictor

0

1

2

3

4

5

6

7


Com

pute

d m

edia

n co

llaps

e ca

paci

ty

(a)

(b)

(c)

(d)

(e)

(f)

0

0.5

1

1.5

2

0 0.5 1 1.5 2

0

0.5

1

1.5

2

0 0.5 1 1.5 2

0

0.25

0.5

0.75

1

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

0

0.25

0.5

0.75

1

0 0.25 0.5 0.75 1


1.2sec., (b) N = 12 & T1 = 1.8sec., (c) N = 12 & T1 = 2.4sec., (d) N = 16 & T1 = 1.6sec., (e) N =

16 & T1 = 2.4sec., (f) N = 16 & T1 = 3.2sec.

- 172 -

0.7

0.8

0.9

1

1.1

1.2

1.3

0 20 40 600.7

0.8

0.9

1

1.1

1.2

1.3

0 10 20

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.5 1

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.05 0.1

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.9 1.8 2.7 3.60.7

0.8

0.9

1

1.1

1.2

1.3

0 4 8 12 16

N

θpc/θp λθp

T1 γ

Fig. 6.22 Ratio of computed median collapse capacity to estimated median collapse capacity

for variation in structural parameters of moment-resisting frames: (a) N, (b) T1, (c) γ, (d)

θp, (e) θpc/θp, (f) λ

- 173 -

(a) (d)

(b) (e)

(c) (f)

Estimated vs. Computed of ηc (MRF)N = 4, T1 = 0.4, γ = var.,Stif. & Str. = Unif., SCB = 2.4-1.2, ξ = 0.05


0

2

4

6

8

10

0 0.02 0.04 0.06Plastic hinge rotation capacity θp

c (Sa

/g)

Estimate Data

γ = 0.17γ = 0.33γ = 0.67



0

2

4

6

8

10


c (Sa

/g)

Estimate Data

γ = 0.17γ = 0.33γ = 0.67



0

2

4

6

8

10


c (Sa

/g)

Estimate Data

γ = 0.13γ = 0.25γ = 0.50



0

0.5

1

1.5

2


c (Sa

/g)

Estimate Data

γ = 0.06γ = 0.13γ = 0.25



0

0.5

1

1.5

2


c (Sa

/g)

Estimate Data

γ = 0.04γ = 0.08γ = 0.17



0

0.5

1

1.5

2


c (Sa

/g)

Estimate Data

γ = 0.03γ = 0.06γ = 0.13

ηc

ηcηc

ηc

ηc

ηc ηc

ηc

ηc

ηc η cη c


capacity for moment-resisting frames as a function of θp: (a) N = 4 T1 = 0.4, (b) N = 4 T1 =

0.6, (c) N = 4 T1 = 0.8, (d) N = 16 T1 = 1.6, (e) N = 16 T1 = 2.4, (f) N = 16 T1 = 3.2

- 174 -

(a) (d)

(b) (e)

(c) (f)


θp = 0.03 , θpc/θp = var., λ = 20, Mc/My = 1.1

0

2

4

6

8

10

0 5 10 15 20Plastic hinge rotation capacity θpc/θp

÷ c (S

a/g)

Estimate Data

γ = 0.17γ = 0.33γ = 0.67



0

2

4

6

8

10


÷ c (S

a/g)

Estimate Data

γ = 0.17γ = 0.33γ = 0.67



0

2

4

6

8

10


÷ c (S

a/g)

Estimate Data

γ = 0.13γ = 0.25γ = 0.50



0

0.5

1

1.5

2


÷c (

Sa/g

)

Estimate Data

γ = 0.06γ = 0.13γ = 0.25



0

0.5

1

1.5

2


÷c (

Sa/g

)Estimate Data

γ = 0.04γ = 0.08γ = 0.17



0

0.5

1

1.5

2


÷ c (S

a/g)

Estimate Data

γ = 0.03γ = 0.06γ = 0.13

ηc

ηcηc

ηc

ηc

ηc

ηc

ηc

ηc

ηc

ηc

ηc


capacity for moment-resisting frames as a function of θpc/θp: (a) N = 4 T1 = 0.4, (b) N = 4 T1

= 0.6, (c) N = 4 T1 = 0.8, (d) N = 16 T1 = 1.6, (e) N = 16 T1 = 2.4, (f) N = 16 T1 = 3.2

- 175 -

(a) (d)

(b) (e)

(c) (f)



0

2

4

6

8

10

0 20 40 60Cyclic Deterioration Parameter λ

c (Sa

/g)

Estimate Data

γ = 0.17γ = 0.33γ = 0.67



0

2

4

6

8

10


c (Sa

/g)

Estimate Data

γ = 0.17γ = 0.33γ = 0.67



0

2

4

6

8

10


c (Sa

/g)

Estimate Data

γ = 0.13γ = 0.25γ = 0.50



0

0.5

1

1.5

2


c (Sa

/g)

Estimate Data

γ = 0.06γ = 0.13γ = 0.25



0

0.5

1

1.5

2


c (Sa

/g)

Estimate Data

γ = 0.04γ = 0.08γ = 0.17



0

0.5

1

1.5

2


c (Sa

/g)

Estimate Data

γ = 0.03γ = 0.06γ = 0.13

ηc

ηcηc

ηc

ηc

ηc

ηc

ηc

ηc

ηc

ηc

ηc


capacity for moment-resisting frames as a function of λ: (a) N = 4 T1 = 0.4, (b) N = 4 T1 =

0.6, (c) N = 4 T1 = 0.8, (d) N = 16 T1 = 1.6, (e) N = 16 T1 = 2.4, (f) N = 16 T1 = 3.2

- 176 -

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.13

γ = 0.25γ = 0.50

γ = 0.11

γ = 0.22γ = 0.44

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.08

γ = 0.17γ = 0.33

γ = 0.08

γ = 0.17γ = 0.33

γ = 0.13

γ = 0.25γ = 0. 50

γ = 0.06

γ = 0.13γ = 0.25

αt = 0.05 αt = 0.075 αt = 0.1

N =

4N

= 8

N =

12

N =

16

Base Case0.02,1,20

0.03

50

3

0.01

0.5

10

θpc/θc

θp

λ

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.13

γ = 0.25γ = 0.50

γ = 0.13

γ = 0.25γ = 0.50

γ = 0.11

γ = 0.22γ = 0.44

γ = 0.11

γ = 0.22γ = 0.44

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.17

γ = 0.33γ = 0.66

γ = 0.08

γ = 0.17γ = 0.33

γ = 0.08

γ = 0.17γ = 0.33

γ = 0.08

γ = 0.17γ = 0.33

γ = 0.08

γ = 0.17γ = 0.33

γ = 0.13

γ = 0.25γ = 0. 50

γ = 0.13

γ = 0.25γ = 0. 50

γ = 0.06

γ = 0.13γ = 0.25

γ = 0.06

γ = 0.13γ = 0.25

αt = 0.05 αt = 0.075 αt = 0.1

N =

4N

= 8

N =

12

N =

16

Base Case0.02,1,20

0.03

50

3

0.01

0.5

10

θpc/θc

θp

λ

Base Case0.02,1,20

0.03

50

3

0.01

0.5

10

θpc/θc

θp

λ

Base Case0.02,1,20

0.03

50

3

0.01

0.5

10

θpc/θc

θp

λ

Base Case0.02,1,20

Base Case0.02,1,20

Base Case0.02,1,20

0.03

50

3

0.01

0.5

10

θpc/θc

θp

λ

0.03

50

3

0.01

0.5

10

θpc/θc

θp

λ


collapse capacity in generic shear wall

- 177 -

Base factor b0,SW for Shear Walls

0.0

1.0

2.0

3.0

4.0

5.0

0 4 8 12 16Number of stories N

b 0,S

W

αt = 0.05

αt = 0.075

αt = 0.10

(a)

Base factor b0,SW for Shear Walls

0.0

1.0

2.0

3.0

4.0

5.0

0 0.5 1 1.5 2First Mode Period T1

b 0,S

W

αt = 0.05

αt = 0.075

αt = 0.10

(b)

Fig. 6.27 Base factor b0,SW for estimation of median of collapse capacity in shear walls

- 178 -

Estimation Error in Median of Collapse Capacity Shear Walls

0.00

0.05

0.10

0.15

0.20

0 4 8 12 16Number of Stories N

β est

,SW

αt = 0.05

αt = 0.075

αt = 0.10

(a)

Estimation Error in Median of Collapse Capacity Shear Walls

0.00

0.05

0.10

0.15

0.20

0 0.5 1 1.5 2First Mode Period T1

β est

,SW

αt = 0.05

αt = 0.075

αt = 0.10

(b)

Fig. 6.28 Estimation error (epistemic) in estimation of median of collapse capacity in shear

walls

- 179 -

Estimated vs. Computed Median of Collapse CapacityShear Wall, N = 8, T1 = 0.8, 1 predictor

0

2

4

6

8

10

0 2 4 6 8 10Estimated median collapse capacity

Com

pute

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n co

llaps

e ca

paci

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0

2

4

6

8

10


Com

pute

d m

edia

n co

llaps

e ca

paci

ty


0

2

4

6

8

10


Com

pute

d m

edia

n co

llaps

e ca

paci

ty


0

2

4

6

8

10


Com

pute

d m

edia

n co

llaps

e ca

paci

ty


0

2

4

6

8

10


Com

pute

d m

edia

n co

llaps

e ca

paci

tyEstimated vs. Computed Median of Collapse Capacity

Shear Wall, N = 4, T1 = 0.2, 1 predictor

0

2

4

6

8

10


Com

pute

d m

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n co

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paci

ty

(a)

(b)

(c)

(d)

(e)

(f)


0.2sec., (b) N = 4 & T1 = 0.3sec., (c) N = 4 & T1 = 0.4sec., (d) N = 8 & T1 = 0.4sec., (e) N = 8

& T1 = 0.6sec., (f) N = 8 & T1 = 0.8sec.

- 180 -


0

2

4

6

8

10


Com

pute

d m

edia

n co

llaps

e ca

paci

ty


0

2

4

6

8

10


Com

pute

d m

edia

n co

llaps

e ca

paci

ty


0

2

4

6

8

10


Com

pute

d m

edia

n co

llaps

e ca

paci

ty


0

2

4

6

8

10


Com

pute

d m

edia

n co

llaps

e ca

paci

ty


0

2

4

6

8

10


Com

pute

d m

edia

n co

llaps

e ca

paci

tyEstimated vs. Computed Median of Collapse Capacity

Shear Wall, N = 12, T1 = 0.6, 1 predictor

0

2

4

6

8

10


Com

pute

d m

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n co

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paci

ty

(a)

(b)

(c)

(d)

(e)

(f) Fig. 6.30 Scatter plots for median of collapse capacity in shear walls: (a) N = 12 & T1 =

1.2sec., (b) N = 12 & T1 = 1.8sec., (c) N = 12 & T1 = 2.4sec., (d) N = 16 & T1 = 1.6sec., (e) N =

16 & T1 = 2.4sec., (f) N = 16 & T1 = 3.2sec.

- 181 -

0.8

0.9

1

1.1

1.2

1.3

0 4 8 12 16

N

0.8

0.9

1

1.1

1.2

1.3

0 4 8 12 16

N

0.8

0.9

1

1.1

1.2

1.3

0 0.5 1 1.5 2T1

0.8

0.9

1

1.1

1.2

1.3

0 0.5 1 1.5 2T1

0.8

0.9

1

1.1

1.2

1.3

0 0.02 0.04

θp

0.8

0.9

1

1.1

1.2

1.3

0 0.02 0.04

θp

0.8

0.9

1

1.1

1.2

1.3

0 2 4

θpc/θp

0.8

0.9

1

1.1

1.2

1.3

0 2 4

θpc/θp

0.8

0.9

1

1.1

1.2

1.3

0 20 40 60

λ

0.8

0.9

1

1.1

1.2

1.3

0 20 40 60

λ

0.8

0.9

1

1.1

1.2

1.3

0 0.5 1γ

0.8

0.9

1

1.1

1.2

1.3

0 0.5 1γ

Fig. 6.31 Ratio of computed median collapse capacity to estimated median collapse capacity

for variation in structural parameters of shear walls: (a) N, (b) T1, (c) γ, (d) θp, (e) θpc/θp, (f)

λ

- 182 -

(a) (d)

(b) (e)

(c) (f)

Estimated vs. Computed of ηc (SW)N = 4, T1 = 0.2, γ = var.,Str. = Unif., ξ = 0.05


0

2

4

6

8

10

0 0.01 0.02 0.03 0.04Plastic hinge rotation capacity θp

c (S

a/g)

Estimate Data

γ = 0.17γ = 0.33γ = 0.67



0

2

4

6

8

10


c (S

a/g)

Estimate Data

γ = 0.17γ = 0.33γ = 0.67



0

2

4

6

8

10


c (Sa

/g)

Estimate Data

γ = 0.17γ = 0.33γ = 0.67



0

2

4

6

8

10


c (Sa

/g)

Estimate Data

γ = 0.13γ = 0.25γ = 0.50



0

2

4

6

8

10


c (Sa

/g)

Estimate Data

γ = 0.08γ = 0.17γ = 0.33



0

2

4

6

8

10


c (Sa

/g)

Estimate Data

γ = 0.06γ = 0.13γ = 0.25

η cη cηc

ηc

ηc

ηc

ηc

ηc

η cη c η cη c


capacity for shear walls as a function of θp: (a) N = 4 T1 = 0.2, (b) N = 4 T1 = 0.3, (c) N = 4

T1 = 0.4, (d) N = 16 T1 = 0.8, (e) N = 16 T1 = 1.2, (f) N = 16 T1 = 1.6

- 183 -

(a) (d)

(b) (e)

(c) (f)

Estimated vs. Computed of ηc (SW)N = 4, T1 = 0.2, γ = var.,Str. = Unif., ξ = 0.05 θp = 0.02, θpc/θp = var., λ = 20, Mc/My = 1.1

0

2

4

6

8

10

0 1 2 3Post-Capping Rotation Capacity Ratio θpc/θp

c (Sa

/g)

Estimate Data

γ = 0.17γ = 0.33γ = 0.67


0

2

4

6

8

10


c (Sa

/g)

Estimate Data

γ = 0.17γ = 0.33γ = 0.67


0

2

4

6

8

10


c (Sa

/g)

Estimate Data

γ = 0.17γ = 0.33γ = 0.67



0

2

4

6

8

10


c (Sa

/g)

Estimate Data

γ = 0.13γ = 0.25γ = 0.50



0

2

4

6

8

10


c (Sa

/g)

Estimate Data

γ = 0.08γ = 0.17γ = 0.33



0

2

4

6

8

10


c (Sa

/g)

Estimate Data

γ = 0.06γ = 0.13γ = 0.25

η cη cηc

ηc

ηc

ηc

ηc

ηc

η cη c η cη c


capacity for shear walls as a function of θpc/θp: (a) N = 4 T1 = 0.2, (b) N = 4 T1 = 0.3, (c) N =

4 T1 = 0.4, (d) N = 16 T1 = 0.8, (e) N = 16 T1 = 1.2, (f) N = 16 T1 = 1.6

- 184 -

(a) (d)

(b) (e)

(c) (f)

Estimated vs. Computed of ηc (SW)N = 4, T1 = 0.2, γ = var.,Str. = Unif., ξ = 0.05 θp = 0.02, θpc/θp = 1.0, λ = var., Mc/My = 1.1

0

2

4

6

8

10


c (Sa

/g)

Estimate Data

γ = 0.17γ = 0.33γ = 0.67


0

2

4

6

8

10


c (Sa

/g)

Estimate Data

γ = 0.17γ = 0.33γ = 0.67


0

2

4

6

8

10


c (Sa

/g)

Estimate Data

γ = 0.17γ = 0.33γ = 0.67


0

2

4

6

8

10


c (Sa

/g)

Estimate Data

γ = 0.13γ = 0.25γ = 0.50


0

2

4

6

8

10


c (Sa

/g)

Estimate Data

γ = 0.08γ = 0.17γ = 0.33


0

2

4

6

8

10


c (Sa

/g)

Estimate Data

γ = 0.06γ = 0.13γ = 0.25

η cη cηc

ηc

ηc

ηc

ηc

ηc

η cη c η cη c


capacity for shear walls as a function of λ: (a) N = 4 T1 = 0.2, (b) N = 4 T1 = 0.3, (c) N = 4 T1

= 0.4, (d) N = 16 T1 = 0.8, (e) N = 16 T1 = 1.2, (f) N = 16 T1 = 1.6

- 185 -

7 IMPLEMENTATION OF PROPOSED DESIGN DECISION SUPPORT SYSTEM

7.1 Introduction

In previous chapters we discussed various aspects of a proposed Simplified Performance-Based

Design (denoted as Simplified PBD) methodology in order to conduct conceptual design based

on target performance objectives. In Chapter 2 the general methodology for Simplified PBD was

discussed. We introduced a simple graphical tool (i.e., Design Decision Support System, DDSS)

in which conceptual performance-based design is accomplished by focusing on the following

three domains: Hazard Domain, Structural System Domain, and Loss Domain. Merits of the

Hazard and Loss Domains were discussed in detail in Chapters 2 and 3, respectively. Detailed

discussions about information provided in the Structural System Domain were presented in

Chapters 4, 5, and 6.

In this Chapter the DDSS is implemented in the conceptual design of an 8-story office

building located in Los Angeles. Multiple design objectives at various hazard levels are

considered: (1) acceptable expected $loss of $500,000 at the 50/50 hazard level, (2) 10%

tolerable probability of collapse at the 2/50 hazard level, and (3) tolerable mean annual

frequency, MAF, of collapse of 0.0002. Design for acceptable downtime loss and tolerable

number of casualties is outside the scope of this study and is not implemented in this example.

As stated in Chapter 2, the DDSS is based on mean values of ground motion intensity, building

response parameters, and losses. In this example, mean values are obtained by only considering

the dispersion due to record-to-record variability (i.e. aleatory uncertainty). At the end of this

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chapter we will perform conceptual design for collapse safety by incorporating both sources of

variability, aleatory (record-to-record) as well as epistemic.

In the following sections we demonstrate how the DDSS is implemented in order to

obtain the “best” structural system and combination of structural parameters that satisfy the

design criteria (in the context of this research the term “best” is limited to choices among the

structural systems investigated and to loss considerations only, i.e., differences in construction

costs are not considered). First, the DDSS is implemented using building-level subsystems and

evaluating a group of four moment-resisting frame design alternatives and a group of three shear

wall design alternatives. Each design alternative is evaluated and compared with other design

alternatives in its group. Then, the “best” moment-resisting frame alternative and the “best”

shear wall alternative are compared in order to decide on the “best” structural system and

structural parameters combination. Second, we will repeat the conceptual performance-based

design by utilizing the DDSS with story-level subsystems. We will show the advantage of using

story-level subsystems in the DDSS. In the last section of this chapter we will show the effect of

incorporating the epistemic uncertainty in the design for collapse safety.

7.2 Implementation of DDSS for conceptual design using building-level subsystems

7.2.1 Information to be provided in the Hazard Domain.

As stated in Chapter 2, Sa(T1) is one of the intensity measures widely used by researchers and

engineers. For the location of this building, spectral accelerations at the 50/50, 10/50, and 2/50

hazard levels for various periods are obtained from site specific hazard analysis. If so needed, for

example for MAF estimations, this data is supplemented by hazard curves regressed through

these three IM values. These hazard curves are obtained by fitting a “power function” to

available data points (see Chapter 2 for details). Figure 7.1 shows example hazard curves along

with spectral accelerations at the 50/50, 10/50, and 2/50 hazard levels. The hazard curves are

drawn in a 90o counterclockwise rotated coordinate system, with the IM axis vertical and the

frequency axis horizontal and pointing to the left, in order to conform to the format displayed

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later in the DDSS. As seen in Figure 7.1, using the regressed hazard curves may results in very

large and unrealistic spectral acceleration values at long return periods, which is due to the type

of function used for this regression. The information at very long return periods will not affect

significantly the mean annual monetary loss (which is controlled by more frequent events), but

may have a large effect on the MAF of collapse. For performance evaluation at discrete hazard

levels the actual values of spectral acceleration are used rather than the values from the hazard

curve.

7.2.2 Information to be provided in the Loss Domain.

In the first implementation we are using building-level subsystems. We assume that for monetary

loss estimation purpose the building can be divided into three building-level subsystems; a

nonstructural drift sensitive subsystem (NSDSS), a nonstructural acceleration sensitive

subsystem (NSASS), and a structural subsystem (SS). It is assumed that nonstructural

components in a building are known prior to the structural design process and do not depend on

the type of structural system (i.e., architectural drawings become available before structural

design decisions have to be made). Based on these assumptions, NSDSS and NSASS are known

and can be quantified before the structural design process. The value of loss in SS depends on the

structural system yet to be designed, however, it is usually a relatively small contributor to total

value of the building.

Figure 7.2 shows the assumed mean $loss-EDP curves for the NSDSS and NSASS

subsystems. The EDP associated with $loss in NSDSS is the average of the maximum interstory

drift ratios over the height, ( )maxavg

IDR and the EDP associated with $loss in NSASS is the

average of peak floor accelerations over the height, ( )avgPFA . As mentioned before, The DDSS

is concerned with mean values of EDPs, hence the mean values of EDPs associated with losses

in the NSDSS and NSASS, ( )maxavg

IDR and ( )avgPFA , are shown on the abscissa of Figure

7.2a and 7.2b, respectively. As stated in Chapter 3, there is little hard data behind the assumed

mean $loss-EDP curves; they are mostly based on judgment.

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Figure 7.3 shows samples of mean $loss-EDP curves for the SS subsystem. The $loss in

SS depends on the type of structural system and associated structural parameters. If moment-

resisting frames are used as the structural system, the average of maximum plastic interstory drift

ratios over the height ( )maxavg

pIDR is considered an appropriate EDP for the SS subsystem. For

shear wall design alternatives, the maximum plastic hinge rotation over the height ( )maxmax pθ is

considered an appropriate EDP. ( )maxavg

pIDR and ( )maxmax pθ are used as the mean values of

aforementioned EDPs, respectively, to relate the mean value of the SS subsystem loss to mean

value of associated EDP. As discussed in Chapter 3, the jump in the mean $loss-EDP curve for

SS from a relatively small value to the value of total loss (usually the replacement cost in present

dollars) incorporates the owner’s or engineer’s decision to demolish the building even though the

structure has not collapsed. Jumps also may be present in other loss-EDP curves if large losses

are associated with the attainment of specific EDP values. Note that the SS mean $loss curves

start rising as soon as the EDP becomes larger than zero because inelastic deformation quantities

are used as EDPs.

As mentioned in Section 7.1, in this study we are not concerned with downtime losses

and casualties because insufficient information is available to develop the associated loss curves.

When such information becomes available, the DDSS can be generalized to incorporate

downtime and casualty performance objectives.

7.2.3 Information to be provided in the Structural System Domain.

As discussed in Chapter 2, the Structural System Domain contains mean IM-EDP curves and

collapse fragility curves of design alternatives. Such relationships are obtained using the

database of IM-EDP relationships and collapse fragility curves for generic moment-resisting

frames and generic shear walls developed as part of this study (a list of EDPs recorded for

generic moment-resisting frames and generic shear walls is presented in Appendix F). Details on

the characteristics of these generic structures along with the range of variation of structural

parameters are presented in Chapter 4. In Chapters 5 and Chapter 6 the behavior of structural

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systems due to variation of structural parameters is discussed. These discussions are intended to

assist in the understanding of the effects of various design decisions on building behavior.

For this design example, IM-EDP relationships for each subsystem and design alternative

are obtained from the aforementioned database.

7.2.4 Implementation of DDSS using building-level subsystems.

The implementation of the DDSS, using four moment-resisting frame alternatives and three

shear wall alternatives for an 8-story office building, is illustrated in Figure 7.4 and Figure 7.5,

respectively. For each type of structural system we compare the design alternatives in terms of

expected value of $loss, and choose a “best” solution. In Figure 7.6, the “best” moment-resisting

frame solution and the “best” shear wall solution are compared in order to make an informed

decision on the structural system and corresponding structural parameters. Basic system

parameters for moment-resisting frame alternatives are: fundamental period T1, base shear

strength coefficient γ = Vy/W, variation of stiffness and strength along the height of the structure

(denoted as Stiff. & Str. as discussed in Chapter 4), and column to beam bending strength ratio

(denoted as SCB factor). Basic system parameters for shear wall alternatives are T1, γ, and

variation of strength along the height (i.e., denoted as Str. as discussed in Chapter 4). For all

moment-resisting frames design alternatives the plastic hinge rotation capacity θp, post-capping

rotation capacity ratio θpc/θp, and the deterioration parameter λ, are set to 3%, 5, and 20,

respectively. For shear wall design alternatives the parameters θp, θpc/θp, and λ are set to 2%, 1,

and 20, respectively.

It is assumed that expected building losses can be disaggregating into losses in three

building subsystems, i.e., the previously discussed SSDSS, NSASS, and SS subsystems, and that

the expected losses in these subsystems are represented by the mean $loss-EDP curves shown in

the $Loss Domain of Figure 7.4.

Figure 7.4 shows the design decision making process for limiting $loss at the 50/50

hazard level to a target value of $500,000 using moment-resisting frame alternatives. As seen in

Loss Domain, the NSDSS has the largest contribution to the value of the building and is likely to

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control damage at the 50/50 hazard level. Therefore, a trial “design point” is identified by

assuming that the total acceptable mean value of $loss equal to $500,000 comes from the

NSDSS. Design point(s) are found in the Structural System Domain by the intersection of the IM

for the 50/50 hazard and the EDP associated with $500,000 loss in NSDSS. As stated in Section

7.2.1 target periods for various structural systems have to be pre-selected since the Sa(T1) hazard

curve depend on the period T1, which is one of the structural parameters in the design decision

making process. Four design alternatives are represented by the expected (mean) IM-EDP curves

and the collapse fragility curves shown in the Structural System Domain of Figure 7.4. These

curves are obtained from the database of generic moment-resisting frame structures developed as

part of this research. The logic behind choosing each design alternative and merits of selecting

one of them as the “best” moment-resisting frame design alternative is described in the following

paragraphs.

Our starting point is the NSDSS mean $loss curve, and we want to limit the expected

value of $loss in this subsystem to $500,000. To achieve this target, we need to consider a design

alternative that limits the EDP associated with the mean value $loss in the NSDSS. As the lower

left portion of Figure 7.4 shows, the EDP associated with a mean loss of $500,000 is rather small

(0.003), which up front points towards the need for a stiff structure. From the IM-EDP database

it is found that only frames with T1 ≤ 0.8 sec. fulfill this target performance objective at the 50/50

hazard level. The site mean hazard curve for T1 = 0.8 sec. is shown with a thin black line in the

Hazard Domain of Figure 7.4. The intersection of the Sa(0.8) value for the 50/50 hazard and the

EDP associated with $500,000 loss in NSDSS in the Structural System Domain is our first

design point.

As the first design alternative, denoted as GF1, we consider an 8-story moment-resisting

frame with T1 = 0.8 seconds, a yield base shear coefficient of γ = 0.5 (this coefficient is not the

design base shear coefficient since the reference strength is the global yield strength of the

structure), stiffness and strength variation along the height tuned to the story shear force pattern

for the structure when subjected to the NEHRP load pattern (i.e., Stiff. & Str. = Shear), and a

column to beam strength ratio of 2.4 for exterior columns and 1.2 for interior columns (i.e., SCB

= 2.4-1.2). Mean IM-EDP curves and the collapse fragility curve for this design alternatives are

shown with thin black lines in the Structural System Domain of Figure 7.4. The SS mean loss

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curve for this design alternative is shown with a thin black line in the SS sub-domain of the Loss

Domain.

Going back to NSDSS in the Structural System Domain, we see that the mean IM-EDP

curve for this design alternative barely falls to the left of the design point, meaning that the mean

value of $loss in NSDSS at the 50/50 hazard level is very close to the acceptable mean value of

$loss at this hazard level. Extending the line associated with the 50/50 hazard level to the

NSASS and SS mean IM-EDP curves of the Structural System Domain and reading the

associated mean values of $loss in the Loss Domain, we see that at this hazard level the mean

$loss in NSASS and SS is zero Extending the line associated with 50/50 hazard level further to

the Collapse Sub-Domain of the Structural System Domain also shows that the probability of

collapse for the GF1 design alternative at the 50/50 hazard level is practically zero. Based on

these observations, we reach the conclusion that design alternative GF1 satisfies the first

performance objective, which is to limit the mean value of $loss at the 50/50 hazard level to

$500,000. However, this alternative is not very attractive in the sense that it is relatively

expensive (base shear coefficient of γ = 0.5 along with T1 = 0.8 translates into large beam and

column section sizes). Also, GF1 causes substantial NSASS $loss at the 10/50 and 2/50 hazard

levels (these values are obtained by drawing the lines associated with Sa(0.8) at the 10/50 and

2/50 hazard levels and reading the mean values of $loss associated with NSASS in the Loss

Domain).

In order to explore more cost effective options, we consider two alternatives, GF2 and

GF3, with a yield base shear coefficient of γ = 0.25. For GF2 we increase the column to beam

strength ratio of the interior columns to 2.4 (i.e., SCB = 2.4-2.4, which makes the structure much

less susceptible to plastic hinging in the columns) whereas for GF3 we increase the stiffness and

strength in the stories above the first one (i.e., Stiff. & Str. = Int. as discussed in Chapter 4).

Other structural parameters are kept the same as in GF1 for both the GF2 and GF3 design

alternatives. Corresponding mean IM-EDP curves and collapse fragility curves are shown with

thick black lines for GF2 and with thick gray lines for GF3 (thick red lines in PDF file). Since

the base shear coefficient of design alternatives GF2 and GF3 is half that of design alternative

GF1, and therefore the construction cost of GF2 and GF3 is less than the cost of GF1, the SS

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mean loss curves for these design alternatives (shown with thick black line for GF2 and with

thick gray line for GF3) are below that that for alternative G1.

The effectiveness of design alternatives GF2 and GF3, compared to GF1, is illustrated in

Figure 7.4. As seen in the NSDSS subdomain of the Structural System Domain, the value of

( )maxavg

IDR for design alternatives GF1 and GF2 at the 50/50 hazard level is the same.

Continuing the line associated with the 50/50 hazard level in the Structural System Domain and

reading associated mean $loss values of NSASS and SS in the Loss Domain shows that both

design alternatives result in almost zero mean $loss in NSASS and SS. The probability of

collapse for both systems is also practically zero. Overall, at the 50/50 hazard level, both GF2

and GF3 alternatives are satisfactory.

The difference between design alternatives GF2 and GF3 is observed by inspecting losses

at the 10/50 and 2/50 hazard levels. Due to additional stiffness in the upper stories the GF3

design alternative results in significantly smaller ( )maxavg

IDR than design alternatives GF1 and

GF2, which in turn indicates much smaller NSDSS loss and seems to favor GF3 as the best

alternative. However, GF3 has a larger probability of collapse at the 10/50 and 2/50 hazard levels

(larger than 10% at 2/50 hazard level). This is due to the additional stiffness and strength in

upper stories that does not allow redistribution of inelastic deformations among all structural

members. This phenomenon causes a concentration of inelastic behavior in the lower stories of

the structure, which results in early exhaustion of the deformation capacity of structural members

in the lower stories, and hence a higher probability of collapse. Since the probability of collapse

for design alternative GF3 at the 2/50 hazard level is clearly larger than 10%, this design

alternative is not acceptable.

At the 10/50 and 2/50 hazard levels design alternative GF2 exhibits a mean value of $loss

in NSASS smaller than GF1. This is an improvement from GF1 to GF2. The mean value of $loss

in NSDSS and SS at the 2/50 hazard level is larger for GF2 than GF1 but this is considered

acceptable because at the 2/50 hazard level we are concerned more with the life safety/collapse

issue than the mean value of $loss. Finally, both design alternatives GF1 and GF2 provide an

acceptable margin against the 10% probability of collapse at the 2/50 hazard level.

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Putting all these considerations together, we conclude that design alternative GF2 is the

“best” moment-resisting frame design of the alternatives considered here for the 8-story office

building. In the context of this dissertation, the term “best” refers primarily to the achievement of

pre-specified loss targets (monetary loss and collapse potential) without consideration of

potential downtime losses, which is outside the scope of this dissertation. Little attention is paid

here to up-front construction cost (even though a frame structure with γ = 0.25 is certainly much

more cost effective than one with γ = 0.5, which again favors alternative GF2 over GF1). It is

likely that this “best” moment-resisting frame alternative is economically difficult to justify. A

frame structure with T1 = 0.1N (T1 = 0.8 sec. for an 8-story building) is very stiff and requires

large members, whether the material is steel or reinforced concrete. But the cost issue is outside

the scope of this dissertation. It is up to the engineer to evaluate cost considerations, and it is not

a limitation of the proposed procedure. If the construction cost of a frame structure with T1 = 0.8

sec. is deemed too high, other moment frame systems with a longer period can be considered,

with IM-EDP data taken from the database discussed in Chapter 4. But the information provided

in Figure 7.4 shows that a frame with T1 > 0.8 sec. will result in NSDSS losses that will exceed

the target value of $500,000 at the 50/50 level.

For illustration, we consider a fourth design alternative, that being an 8-story moment-

resisting frame with T1 = 1.6 seconds. This alternative, denoted as GF4, has the following

structural parameters: T1 = 1.6, γ = 0.25, Stiff. & Str. = Shear, and SCB = 2.4-1.2. The yield base

shear coefficient of γ = 0.25 is relatively large for T1 = 1.6, which makes GF4 comparable with

GF1 in the sense that both designs result in relatively strong structures. The mean hazard curve

for T1 = 1.6 is shown in the Hazard Domain of Figure 7.4 (thin gray line, or thin red line in PDF

file), and the appropriate mean IM-EDP curves, collapse fragility curve, and mean $loss curve of

SS are shown in the Structural System Domain and Loss Domain of this figure. As expected, the

NSDSS loss at the 50/50 level is very large (greater than 2 Mill. Dollars), which makes this

design alternative economically unfeasible from a loss perspective. In addition, this alternative

would result in an unacceptable probability of collapse at the 10/50 and 2/50 hazard levels.

In the previous paragraphs we discussed four design alternatives using moment-resisting

frames as the structural system for the design example. We found out that design alternative GF2

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is the “best" moment-resisting frame alternative from a loss perspective. In the following, we

consider three shear wall design alternatives.

Figure 7.5 shows the DDSS using shear walls as the structural system for the building

under consideration. Buildings whose lateral load resisting system consists of shear walls usually

have a shorter period than buildings whose lateral system consists of moment-resisting frames.

For purposes of comparison, and considering that the GF2 design alternative with T1 = 0.8

achieves the target value of mean annual loss limitation of $500,000, we first explore shear wall

design alternatives with the same fundamental period. The appropriate mean hazard curve is

illustrated in the Hazard Domain of Figure 7.5. The “design point” in the Structural System

Domain is obtained from the intersection of the horizontal line associated with Sa(0.8) for the

50/50 hazard level and the vertical line associated with the max avgIDR for the $500,000 mean

value of $loss in NSDSS.

For the first design alternative, denoted as GW1, we consider an 8-story shear wall

system with T1 = 0.8 seconds, yield base shear coefficient γ = 0.5, uniform stiffness along the

height, and reduction of strength along the height of 0.05My,base per story (My,base is the yield

moment at the base of the wall obtained from applying the NEHRP load pattern with a base

shear value of γW). Corresponding mean IM-EDP curves, collapse fragility curves, and mean SS

$loss curves are shown with thin black lines in the Structural System Domain and $Loss domains

of Figure 7.5. Focusing on NSDSS, we observe that the mean IM-EDP curve for this design

alternative passes almost through the design point. Following the 50/50 hazard IM-EDP-loss

paths we observe that at this hazard level the mean values of $loss in NSASS and SS are close to

zero. Following the 50/50 hazard IM to collapse fragility curve path we observe that the

probability of collapse for this design alternative is practically zero. Thus, alternative GW1

fulfills all design objectives, and it also shows relatively small losses at the 10/50 and 2/50

hazard levels.

However, design GW1 has to be considered an expensive design because it utilizes γ =

Vy/W = 0.5. For this reason we consider two other design alternatives, denoted as GW2 and

GW3. For design alternative GW2, we keep T1 = 0.8 sec. but reduce the yield base shear

coefficient to γ = 0.25. All other structural parameters are the same as for GW1. Design

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alternative GW3 is a shear wall with T1 = 0.4 (i.e., stiffer shear wall), γ = 0.17, uniform stiffness

along the height, and reduction of strength along the height of 0.05My,base per story.

Corresponding mean IM-EDP curves, collapse fragility curves, and mean SS $loss curves are

shown with thick black lines and thin gray (red in the PDF file) lines for GW2 and GW3,

respectively. Inspection of the three design alternatives GW1, GW2, and GW3 shows that the

design alternative GW2 is the “best” shear wall design. GW3 is considered an over-designed

alternative because the corresponding mean $loss values in NSDSS, NSASS, and SS, and the

probability of collapse at the 2/50 hazard level are much smaller than required. GW2 fulfills all

performance targets but has half the base shear strength of GW1, i.e., it is considered a much

more cost-effective design. It also shows comparable performance to GW1 at the 10/50 and 2/50

hazard levels.

In Figure 7.6 the “best” frame and wall design alternatives for the 8-story building are

compared. These two design alternatives are GF2 and GW2. Both design alternatives limit mean

value of $loss in NSDSS to $500,000, with no additional $loss in the NSASS and SS subsystems

at the 50/50 hazard level. The probability of collapse for both alternatives at the 50/50 hazard

level is practically zero meaning that no additional $loss is incurred because of building collapse.

Continuing our comparison to the 10/50 and 2/50 hazard levels, we see that the moment-resisting

frame alternative GF2 has a better performance at higher hazard levels (i.e., at both the 10/50 and

2/50 hazard levels the mean value of $loss in NSDSS and NSASS, and the probability of

collapse, are smaller for GF2).

This indicates that GF2 is the most desirable design -- based on expected losses. But the

caveat is the construction cost of the structural system, which likely is much higher for a frame

system than for a wall system with the same period (0.8 sec. in this example). The construction

cost issue is not addressed here. If it is considered, the wall alternative GW2 may come out as the

winner. The advantage of the proposed DDSS is that other alternatives, i.e., frames with longer

T1 and smaller γ, and walls with shorter T1 and different γ, can be evaluated for $loss and

collapse probability in the illustrated manner by taking advantage of the mean IM-EDP data

discussed in Chapter 5

The merits of design alternatives GF2 and GW2 can also be assessed in terms of the

expected annual loss. As mentioned in Chapter 2, the expected annual $loss for each design

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alternative is obtained by numerically integrating the mean $loss-IM curve over the associated

hazard curve. For each design alternative, the mean $loss-IM curve is constructed by finding the

associated mean value of $loss at different IM levels. Figure 7.7 shows the mean $loss-IM curves

for GF2 and GW2, marked with solid black diamonds and solid black rectangles, respectively.

The expected value of $loss conditioned on collapse and on no-collapse are also shown in this

figure to illustrate the contribution of each condition (collapse and no-collapse) to the total value

of mean $loss. In Figure 7.8, $loss-IM curves for GF2 and GW2 are shown side-by-side with the

associated hazard curve (which is the same for both cases because the fundamental period is the

same). Numerical integration is performed and it is calculated that the expected annual $loss for

the two design alternatives GF2 and GW2 is equal to $17,000 and $22,000, respectively.

An alternative or additional performance objective may be the criterion that the MAF of

collapse should be limited to a target value, such as 0.0002. The MAF can be evaluated from

Equation 6.9. For design alternatives GF2 and GW2 the slope of the hazard curve associated

with T1 = 0.8 in the log-log domain is about 2.16 (k = 2.16). The median of the collapse capacity,

ηc, for GF2 and GW2 is equal to 3.56g and 3.10g, respectively (median and dispersion due to

aleatory uncertainty of collapse capacity is obtained from the database of structural response

parameters developed for this study). Using the T = 0.8 sec. hazard curve the value of the mean

annual frequency of exceedance associated with the median collapse capacity, λ(ηc), is equal to

9.53E-5 and 1.30E-4, respectively. The dispersion due to randomness (aleatory uncertainty), βRC,

for GF2 and GW2 is equal to 0.4 and 0.5, respectively. Thus, using Equation 6.9, the MAF of

collapse for design alternatives GF2 and GW2 is found from Equation 7.1 and 7.2, respectively.

As seen, the MAF of collapse for GW2 is larger than the tolerable value (i.e., 0.0002), which

makes this design alternative undesirable. [This does not mean that a shear wall design is

undesirable; but it means that either the base shear strength of the walls or the plastic rotation

capacity of the walls has to be increased. Data discussed in Chapter 6 can be utilized to explore

other alternatives.]

( )2 29.53 5 exp 1 2 2.16 0.40 0.000138 /C E yrλ ⎡ ⎤= − × × =⎡ ⎤⎣ ⎦ ⎣ ⎦ (7.1)

( )2 21.3 4 exp 1 2 2.16 0.50 0.000233 /C E yrλ ⎡ ⎤= − × × =⎡ ⎤⎣ ⎦ ⎣ ⎦ (7.2)

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The decision on a final “best” choice of a structural system is a matter of further

exploration, utilizing the data discussed in Chapter 5 and Chapter 6, the DDSS system illustrated

in this chapter, and construction cost considerations that are outside the scope of this dissertation.

Considering only the alternatives illustrated in this chapter, and disregarding construction cost

considerations, the conclusion would be that a moment-resisting frame with T1 = 0.8, γ = 0.25,

Stiff. & Str. = Shear, and SCB = 2.4,2.4, θp = 3%, θpc/θp = 5.0, and λ = 20, is the “best” solution

that meets the stipulated performance objectives. This conclusion is based on evaluating a few

discrete alternatives and using building-level subsystems.

7.3 Implementation of DDSS for conceptual design using story-level subsystems

In this section, we implement the DDSS to identify the best structural system and associated

basic structural parameters using story-level subsystems. The process starts with gathering

information on hazard curves and mean $loss curves for story-level subsystems. Detailed

discussion on the required ingredients for the DDSS was provided in Sections 7.2.1, 7.2.2, and

7.2.3. Therefore, we only review the important differences between ingredients of the building-

level subsystem approach and the story-level subsystem approach in this section.

It is assumed that for monetary loss estimation purposes each story is divided into a Non-

Structural Drift Sensitive Subsystem (i.e., NSDSSi), Non-Structural Acceleration Sensitive

Subsystem (i.e., NSASSi), and Structural Subsystem (i.e., SSi). Parameter i represent a

story/floor level. EDPs associated with $loss in the NSDSSi, and the NSASSi are: the maximum

inter-story drift ratio in story i (i.e., max iIDR ), and the peak floor acceleration at floor i (i.e.,

iPFA ), respectively. For moment-resisting frame and shear wall design alternatives, maximum

plastic drift ratio of story i (i.e., ( )maxi

pIDR ) and maximum plastic hinge rotation in story i (i.e.,

( )maxi

pθ are considered as relevant EDPs for SSi monetary loss.

The DDSS using story-level subsystems has three important and practical advantages.

First, by using story-level subsystems we can incorporate different mean $loss curves for

different stories. This means that if for any reason the value of a single story is clearly different

from others, its effect could be directly incorporated in the design process. Second, we will

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obtain a more accurate measure of total loss because a story-level EDP is a more accurate

predictor of the demand than a structure-level EDP, which averages the EDP over the height.

Third, by working with story-level subsystems we gain the flexibility of distributing losses in

different stories evenly or unevenly. This means that we can set the design strategy to either

obtain identical loss values at every story, or allow concentration of loss in one story and reduce

loss in others.

For simplicity, we assume here that nonstructural components in all stories have the same

value. Based on this assumption, mean $loss curves for the NSDSSi and the NSASSi have a

similar shape to mean $loss curves for the building-level NSDSS and NSASS subsystems. The

difference between the two sets is in the mean value of loss given EDP, which for the story-level

subsystems is one-eight of that of the 8-story building.

Implementation of the DDSS based on story-level subsystems is illustrated in Figures 7.9,

7.10, and 7.11. Figures 7.9 and 7.10 show the implementation for the same moment-resisting

frame alternatives (GF1 to GF4) and shear wall alternatives (GW1 to GW3) evaluated in the

building-level DDSS. In Figure 7.11, we compare the two “best” designs and draw conclusions

accordingly. As mentioned previously, we do not consider downtime losses and causalities due

to lack of information. The DDSS is capable of incorporating such losses upon availability of

sufficient data.

Figure 7.9 shows the implementation of the DDSS considering the four moment-resisting

frame alternatives GF1, GF2, GF3, and GF4. Because of space limitations and in order to

maintain focus on salient aspects of the design alternatives, we will concentrate on the following

three story/floor levels: 1st story/2nd floor (representative of a lower level story/floor), 4th

story/5th floor (representative of a mid-level story/floor), and 8th story/Roof (representative of

an upper level story/floor). For each story/floor level, three subsystems of NSDSSi, NSASSi, and

SSi (i represents the story/floor level) are introduced. Therefore, the DDSS scheme consists of

nine subsystems. Due to the large number of subsystems, Figure 7.9 is sub-divided into Figures

7.9a, 7.9b, and 7.9c, each corresponding to a story/floor level. The Collapse sub-domain is the

same for each of the figures.

We start our evaluation process by comparing mean values of $loss in the NSDSSi for

the four moment-resisting frame design alternatives in Figure 7.9a, b, and c. In terms of

- 199 -

behavior, GF4 has the largest drift at the 50/50 hazard level in all stories. The corresponding

mean $loss value of this alternative is much larger than $500,000/8 at all levels, which eliminates

GF4 from further consideration.

Alternatives GF1, GF2, and GF3 show almost identical loss values at the 50/50 hazard

level in NSDSS1 and NSASS1, and small mean value of $loss in SS1. By performing the same

exercise for intermediate stories in Figure 7.9b and for upper stories in Figure 7.9c, we conclude

that GF1, GF2, and GF3 are acceptable design alternatives for limiting the value of $loss in each

story to $500,000/8 at the 50/50 hazard level. A distinction between the alternatives becomes

evident at the 10/50 and 2/50 hazard levels. For instance, at the 10/50 hazard level alternative

GF3 causes more structural damage and nonstructural drift sensitive damage in lower stories

than design alternative GF2 and GF1. This is a direct effect of the additional stiffness and

strength in upper stories that creates concentration of deformation and nonlinear behavior in

lower stories.

Design alternative GF1 initiates noticeable nonstructural acceleration sensitive damage at

mid-stories and upper-stories of the structure compared to design alternative GF2. This is due to

the high strength of this structure that postpones inelastic behavior and allows higher modes to

increase the acceleration response. Such a design would be unacceptable if the contribution of

nonstructural acceleration sensitive subsystem loss to total monetary loss is large compared to

the contribution of nonstructural drift sensitive loss (for instance, in a museum). The design

alternative GF2 is preferred to GF1 because of this observation and its lower construction cost.

The Collapse Sub-Domain evaluation is identical to that for the building subsystem case

because collapse of a story is considered collapse of the structure. For this reason, GF3 is

eliminated as a feasible alternative.

Figure 7.10 shows the implementation of DDSS for the three shear wall alternatives

GW1, GW2, and GW3 at the same three story/floor levels as used for the moment-resisting

frame alternatives. The evaluation process starts by comparing mean values of $loss in the

NSDSS of the 1st, 4th, and 8th story (Figures 7.10a, b, and c). For all three alternatives the mean

value of $loss in NSDSS1 at the 50/50 hazard level is relatively small (clearly less than

$500,000/8) because the maximum interstory drift ratio in the first story (i.e., ( ) .1max

stIDR ) of

shear wall buildings is smaller than the maximum interstory drift ratio in other stories (see

- 200 -

Chapter 5). For intermediate stories, mean values of NSDSS $loss for all alternatives are

approximately equal to $500,000/8, and in upper stories, mean values of $loss for all alternatives

are again about equal but larger than $500,000/8, which compensates for the reduction in mean

value of NSDSS $loss observed in the lower stories.

Performing the same exercise for NSASS story-level subsystems shows that the mean

value of $loss in the upper stories (NSASS8) at the 50/50 hazard level is considerable for GW1

and GW2. Since the acceptable $500,000 loss has been exhausted in the nonstructural drift

sensitive subsystems, there is no allowance for additional $loss in the NSASS. For this reason,

design alternatives GW1 and GW2 are not satisfactory. This observation was not made in the

DDSS based on building-level subsystems. When we used building-level subsystems, the EDP

we considered for the NSASS losses was the average of peak floor acceleration along the height

of the building, ( ) .avgPFA . By averaging the peak floor accelerations along the height, we

reduced the seismic demand to a value that corresponds to zero $loss in NSASS ( ( ) .avgPFA is

smaller than 0.5g, the level at which NSASS begins to experience losses).

The conclusion is that the use of building-level subsystems in the DDSS, which is

convenient insofar that it simplifies the number of subsystems that have to be considered, may

lead to erroneous interpretations if the EDP associated with a subsystem loss is not distributed

uniformly among stories of the building.

Based on the aforementioned observations we choose GW3 as the “best” shear wall

design alternative among those considered in this evaluation. It is noteworthy that the SSi

Structural System sub-domain for all shear wall alternatives is blank for stories 4 and 8. The

reason is that inelastic deformation occurs only in the first story in which a flexural hinge forms

in the shear walls.

In Figure 7.11 the “best” frame and wall design alternatives (i.e., GF2 and GW3) are

compared. Both design alternatives satisfy the performance objectives of expected $loss less than

$500,000 at the 50/50 hazard level and the collapse probability less than 0.002 at the 2/50 level.

But by comparing the performance of the frame alternative with that of the wall alternative at the

50/50, 10/50, and 2/50 hazard levels, and considering the higher construction cost value of the

frame alternative, we conclude that the wall alternative is clearly the preferred.

- 201 -

As an additional performance objective it is stated that the MAF of collapse should be

smaller than 0.0002. The MAF of collapse for GF2 was evaluated previously (Equation 7.1) as

0.000138. The MAF of collapse of the new “best” shear wall alternative, GW3, is obtained in

Equation 7.3 using Equation 6.9 (i.e., k = 2.54, ηc = 4.88g, λ(ηc) = 5.01E-5, and βRC = 0.5). The

MAF of collapse using design alternative GW3 is smaller than the tolerable value (i.e., 0.0002)

which makes design alternative GW3 acceptable.

( )2 25.01 5 exp 1 2 2.54 0.50 0.000112 /C E yrλ ⎡ ⎤= − × × =⎡ ⎤⎣ ⎦ ⎣ ⎦ (7.3)

We conclude that using story-level subsystems, design alternative GW3 (i.e., shear wall,

T1 = 0.4, γ = 0.17, Stiff. = Unif., Str. = -0.05My,base/floor, θp = 2%, θpc/θp = 1, and λ = 20) is the”

best” solution among the solutions pursued and meets all three performance objectives. As

mentioned before, this decision is made based on limited number of design alternatives without

considering construction costs, potential downtime costs, and actual life losses with which can

radically change the decision.

7.4 Incorporating the effect of epistemic uncertainty in design for collapse safety

In the previous section we obtained two design alternatives (i.e., GF2 and GW3) that satisfy the

performance objectives as stated in Section 7.1. We used mean IM-EDP curves and collapse

fragility curves that incorporate only the aleatory uncertainty as the source of variability in the

structure’s response and collapse capacity evaluation. In this section we take the collapse

evaluation one step further by considering also epistemic uncertainty in the performance

objective for tolerable probability of collapse.

As shown in Chapter 6, there are two methods for incorporating the epistemic uncertainty

in the probability of collapse. In the first method, denoted as “mean method”, it is assumed that

the median of the collapse fragility, ηc, is not affected by the additional source of variability,

however, the dispersion of the collapse fragility is inflated from βRC to 2 2RC UCβ β+ (i.e., Equation

6.3), where βRC is the dispersion due to aleatory uncertainty and βUC is the dispersion due to

epistemic uncertainty. In the second method, denoted as “confidence method”, it is assumed that

the dispersion of the collapse fragility, βRC, is not affected by the epistemic uncertainty, however,

- 202 -

the median, ηc, is a random variable with a lognormal distribution. The median of this lognormal

distribution is the median estimate of the median of collapse capacity, 50%Cη ,your super and

subscripts are not legible and the dispersion of this lognormal distribution is βUC. Given the

lognormal distribution associated with median of collapse capacity and the value of confidence

level sought, Y%, the median of collapse capacity associated with confidence level Y% is

obtained, %YCη . The collapse fragility curve constructed using %Y

Cη and βRC is the collapse fragility

curve associated with Y% confidence.

Details of the brief explanation provided here can be found in Chapter 6. Figures 7.12

and 7.13 show fragility curves with different confidence levels for design alternatives GF2 and

GW3, respectively. The median estimate of the median of collapse capacity, 50%Cη and dispersion

due to aleatory uncertainty, βRC, are equal to the values we used previously in the DDSS and are

obtained from the database of collapse capacities and dispersions due to aleatory uncertainty of

generic structures developed for this study. The dispersion due to epistemic uncertainty, βUC, is

set to 0.4 (Krawinkler et. al., 2005). The collapse fragility curve denoted with “(Mean)” is the

fragility curve obtained using the “mean method” for which the associated confidence level is

calculated and noted (Equation 6.5). The collapse fragility curve denoted with “50%

Confidence” is the same collapse fragility curve that we used in the DDSS procedure, which did

not include the effect of epistemic uncertainty. The collapse fragility curve associated with 90%

confidence was obtained using the “confidence method” by setting the confidence level to 90%

(Equation 6.7).

On the left side of each figure the associated hazard curve for the design alternative is

provided. Following the line associated with the 2/50 hazard level, we see a significant increase

in probability of collapse because of incorporation of epistemic uncertainty in obtaining the

probability of collapse. This effect is better depicted in Figure 7.14 where we show the variation

of probability of collapse at the 2/50 hazard level for design alternatives GF2 and GW3 as a

function of confidence level Y%. As seen in this figure, incorporation of epistemic uncertainty in

the form of confidence levels increases the probability of collapse at the 2/50 hazard level

exponentially. The large increase in probability of collapse with incorporation of confidence

levels is due to the large value of βUC.

- 203 -

If the performance target is a tolerable probability of collapse of 10% at the 2/50 hazard

level in the mean sense, then by using Figure 7.12 and 7.11 for design alternatives GF2 and

GW3, respectively, we can show that both design alternatives are acceptable. In both cases the

probability of collapse is slightly less than 10%. But if a 90% confidence level is associated with

this performance target, then both design alternatives would fail the test.

By integrating the collapse fragility curve associated with Y% confidence over the mean

hazard curve, we can obtain the MAF of collapse associated with Y% confidence. By integrating

the collapse fragility curve obtained using the “mean method”, we obtain the mean of MAF of

collapse (i.e., or loosely speaking MAF of collapse). A close form solution for this integration is

provided in Chapter 6 (i.e., Equation 6.9). In Equations 7.1 and 7.3 we computed the MAF of

collapse without incorporating the confidence level (Ky = 0.0 in Equation 6.8). Figure 7.15 shows

the variation of MAF of collapse for design alternatives GF2 and GW3 (for βUC = 0.4). The rate

of increase in MAF of collapse is large and is due to the large value of βUC . Using Figure 7.15

we can see that, for example, only design alternative GW3 is acceptable if the performance target

is a tolerable mean of MAF of collapse of 0.0002. As seen in Figure 7.15, the value of MAF of

collapse for GW3 with 69% confidence level is less than 0.0002, whereas the MAF of collapse

for GF2 with 71% confidence level is larger than 0.0002.

7.5 Concluding remarks

In this chapter we tried to illustrate the potential of the proposed Design Decision Support

System (DDSS). The ingredients needed for implementation are:

• Mean hazard curves that define the site hazard in terms of a specific IM (Sa(T1) is used

in this implementation) and for the period (or period range) that defines the IM for the

design alternative to be considered,

• Mean loss curves for building or story subsystems, which relate the mean (expected)

loss in a subsystem to an EDP that (1) correlates well with the loss in the subsystem,

and (2) can be evaluated in a systematic manner for the design alternatives to be

considered,

- 204 -

• Mean IM-EDP relationships for all EDPs in the loss domain for the design alternatives

to be considered, and

• Collapse fragility curves for the design alternatives to be considered.

The work discussed in this dissertation focuses on the evaluation of mean IM-EDP

relationships and collapse fragility curves. A database has been developed that contains specific

information on these relationships for generic moment resisting frame and wall structures. The

process described in Chapters 4 to 6 can be followed if the structural systems under

consideration fall outside the range of parameters evaluated in this study.

In the proposed DDSS the aforementioned ingredients are represented graphically in the

three-domain charts illustrated in Figures 7.4 to 7.11. There are great advantages to the graphical

three-domain representation. It permits, for a potential design alternative, an

instantaneous/simultaneous inspection of consequences ($loss and collapse probability) at

various hazard levels and an evaluation of the loss incurred in the individual subsystems. It

permits, for a set of potential design alternatives, an instantaneous/simultaneous evaluation of the

relative advantages and disadvantages of the alternatives, and in this manner greatly facilitates

design decision making. The frequently asked and much debated questions about stiffness versus

flexibility, and strength versus ductility, can be addressed in the context of expected losses at

various hazard levels and in the context of building occupancy, which will dictate the relative

value of the nonstructural drift sensitive subsystem (NSDSS), the nonstructural acceleration

sensitive subsystem (NSASS), and the structural subsystem (SS). The DDSS will facilitate the

decision making for appropriate structural systems for buildings based on subsystem values and

functionality, which will be very different, for instance, for hospitals, office building, residential

buildings, and museums.

It must be emphasized that the DDSS provides only mean estimates of losses and does

not account for dispersions, except in the assessment of the probability of collapse. In the latter

context, aleatory as well as epistemic uncertainties can be considered, as discussed in Section 7.4

The process illustrated in Figures 7.4 to 7.11 can be used also for a quick performance

assessment, avoiding the usually large analytical effort involved in accounting for uncertainties

and their propagation from hazard and ground motion modeling all the way up to loss estimation

and decision making.

- 205 -

It should be emphasized, that the DDSS presented in this chapter along with the database

of structural response parameters should not to be used for black-and-white design decisions.

This is due to several simplifications devised in the concepts and data behind the DDSS, which

were discussed in associated chapters (i.e., Chapters 2, 3, 4). Another important issue that

certainly needs to be considered is the upfront cost of construction, which in some cases is more

important than the cost of repair/replacement of the structural system. Ultimately, the author

wants to emphasize that the DDSS should be used as decision support tool and not as a decision

criterion.

-206-

0.5

1.0

1.5

2.0

3E-36E-39E-31.2E-31.5E-3

50/50 10/50 2/50

Mean Annual Frequency of Exceedance, λ(Sa(T1)/g)

Sa(T

1)/g

0.5

1.0

1.5

2.0

3E-36E-39E-31.2E-31.5E-3

50/50 10/50 2/50

Mean Annual Frequency of Exceedance, λ(Sa(T1)/g)

Sa(T

1)/g

Fig. 7.1 Mean hazard curves for T1 = 0.4, 0.8, 1.6 for the location of the design example.

(a) (b)

Fig. 7.2 Mean $loss curves for building-level subsystems: (a) NSDSS, (b) NSASS

0.0

Expe

cted

Sub

syst

em

$Los

s ( i

n m

illio

ns)

6.0

4.0

2.0

0.00.005 0.015 0.0250.01 0.02

8.0

10.0

12.0

max avgIDR

0.0

Expe

cted

Sub

syst

em

$Los

s ( i

n m

illio

ns)

6.0

4.0

2.0

0.00.005 0.015 0.0250.01 0.02

8.0

10.0

12.0

max avgIDR

0.5 1.0 1.5 2.00.0

6.0

4.0

2.0

8.0

10.0

12.0

Expe

cted

Sub

syst

em

$Los

s ( i

n m

illio

ns)

avgPFA0.5 1.0 1.5 2.00.0

6.0

4.0

2.0

8.0

10.0

12.0

Expe

cted

Sub

syst

em

$Los

s ( i

n m

illio

ns)

avgPFA

- 207 -

(a) (b)

Fig. 7.3 Sample of mean $loss curves for building-level SS subsystems: (a) moment-resisting

frame, (b) shear wall

6.0

4.0

2.0

8.0

10.0

12.0

0.02 0.030.01

Expe

cted

Sub

syst

em

$Los

s ( i

n m

illio

ns)

( ) .max

avgpIDR

6.0

4.0

2.0

8.0

10.0

12.0

0.02 0.030.01

Expe

cted

Sub

syst

em

$Los

s ( i

n m

illio

ns)

( ) .max

avgpIDR

6.0

4.0

2.0

8.0

10.0

12.0

0.015 0.020.01

Expe

cted

Sub

syst

em

$Los

s ( i

n m

illio

ns)

( )max.max pθ

6.0

4.0

2.0

8.0

10.0

12.0

0.015 0.020.01

Expe

cted

Sub

syst

em

$Los

s ( i

n m

illio

ns)

( )max.max pθ

- 208 -

0

1

0

11.0

Mean IM-EDP Curvesfor Design Alternatives

0.0

Expe

cted

Sub

syst

em

$Los

s ( i

n m

illio

ns)

EDP=Avg. of max. story drift ratios,

(maxIDR)avg.

EDP=Avg. of max. floor accelerations,

(PFA)avg.(g)

6.0

4.0

2.0

0.00.005 0.015 0.0250.01 0.02

8.0

10.0

0.5 1.0 1.5 2.00.0

2/50

T1=0.8sec.

T1=1.6sec.

50/50

Sa(T1)/g Sa(T1)/g

10/50

2.0

1.5

0.5

2.0

1.5

1.0

0.5

0.0 0.01 0.02 0.5 1.0 1.5 2.00.0 20%10%0%

1.5

2.0

1.0

0.5

Sa(T1)/g

( )| &E EDP IM NC

$Loss Curves( )$ | &E Loss EDP NC

12.0

1.0

2.0

1.5

0.5

EDP=Avg. of max. story plastic IDR,

(maxIDRp)avg.

Sa(T1)/g

( | )P C IM

Collapse Fragility Curves for Design

Alternatives

0.02 0.03

Expected Total $Loss (in millions)

(PFA)avg. (g)(maxIDR)avg. (maxIDRp)avg.

( )$ |E Loss C$Loss Value

λ(Sa(T1)/g) P(C|Sa(T1)/g)

NSDSS Bldg. level

NSDSS Bldg. level

(Collapse)(No Collapse)

λ(IM)

Hazard Curves for

Design Alternatives

6.0

4.0

2.0

8.0

10.0

12.0

6.0

4.0

2.0

8.0

10.0

12.0

Hazard Domain

Structural System Domain

$Loss Domain

Expe

cted

Tot

al $

Loss

at

Col

laps

e ( i

n m

illio

ns)

6.0

4.0

8.0

10.0

12.0

2.0

0.0

0.01

NSASS Bldg. level

NSASS Bldg. level

SS Bldg. level

SS Bldg. level

0.02 0.030.010.5

8-story Frame, T1 = 0.8 sec., γ = 0.25, Stiff. & Str. = Int., SCB = 2.4,1.2

8-story Frame, T1 = 0.8 sec., γ = 0.50, Stiff. & Str. = Shear, SCB = 2.4,1.2


8-story Frame, T1 = 1.6 sec., γ = 0.25, Stiff. & Str. = Int., SCB = 2.4,1.2 All SYSTEMS θp = 3%, θpc/θp = 5, λ = 20

GF1

GF2

GF3

GF4





GF1

GF2

GF3

GF4

Fig. 7.4 Example of DDSS implementation in conceptual design with performance targets

at discrete hazard levels, using building-level subsystems and moment-resisting frame

structural systems GF1, GF2, GF3, and GF4

- 209 -

1.0


0.0

Expe

cted

Sub

syst

em

$Los

s ( i

n m

illio

ns)


(maxIDR)avg.


(PFA)avg.(g)

6.0

4.0

2.0

0.00.005 0.015 0.0250.01 0.02

8.0

10.0

0.5 1.0 1.5 2.00.0

Sa(T1)/g Sa(T1)/g2.0

1.5

0.5

2.0

1.5

1.0

0.5

0.0 0.01 0.02 0.5 1.0 1.5 2.00.0 20%10%0%

1.5

2.0

1.0

0.5

Sa(T1)/g

( )| &E EDP IM NC


12.0

1.0

2.0

1.5

0.5

EDP=Max. of max. plastic rotation,

(maxθp)max.

Sa(T1)/g

( | )P C IM


Alternatives

0.02


(PFA)avg.(maxIDR)avg. (maxθp)max.


P(C|Sa(T1)/g)

NSDSS Bldg. level

NSDSS Bldg. level


λ(IM)

Hazard Curves for

Design Alternatives

6.0

4.0

2.0

8.0

10.0

12.0

6.0

4.0

2.0

8.0

10.0

12.0

Hazard Domain


$Loss Domain

Expe

cted

Tot

al $

Loss

at

Col

laps

e ( i

n m

illio

ns)

6.0

4.0

8.0

10.0

12.0

2.0

0.0

0.01

NSASS Bldg. level

NSASS Bldg. level

SS Bldg. level

SS Bldg. level

0.020.01

8-story Wall, T1 = 0.8 sec., γ = 0.50, Stiff. = Unif., Str. = -0.05My,base / floor

All SYSTEMS θp = 2%, θpc/θp = 1, λ = 20



GW1

GW2

GW3





GW1

GW2

GW3

50/50 10/50

λ(Sa(T1)/g) 2/50

T1=0.4sec.

T1=0.8sec.


at discrete hazard levels, using building-level subsystems and shear wall structural systems

GW1, GW2, and GW3

- 210 -

1.0


0.0

Expe

cted

Sub

syst

em

$Los

s ( i

n m

illio

ns)


(maxIDR)avg.


(PFA)avg.(g)

6.0

4.0

2.0

0.00.005 0.015 0.0250.01 0.02

8.0

10.0

0.5 1.0 1.5 2.00.0

2/50

T1=0.8sec.

50/50

Sa(T1)/g

10/50

2.0

1.5

0.5

2.0

1.5

1.0

0.5

0.0 0.01 0.02 0.5 1.0 1.5 2.00.0 20%10%0%

1.5

2.0

1.0

0.5

Sa(T1)/g

( )| &E EDP IM NC


12.0

1.0

2.0

1.5

0.5

Sa(T1)/g

( | )P C IM


Alternatives


(maxIDR)avg.



NSDSS Bldg. level

NSDSS Bldg. level


λ(IM)

Hazard Curves for

Design Alternatives

6.0

4.0

2.0

8.0

10.0

12.0

6.0

4.0

2.0

8.0

10.0

12.0

Hazard Domain


$Loss Domain

Expe

cted

Tot

al $

Loss

at

Col

laps

e ( i

n m

illio

ns)

6.0

4.0

8.0

10.0

12.0

2.0

0.0

NSASS Bldg. level

NSASS Bldg. level

SS Bldg. level

SS Bldg. level

(PFA)avg. (g)(PFA)avg. (g) EDP for SS

EDP for SS

8-story Wall, T1 = 0.8 sec., γ = 0.25, Stiff. = Unif., Str. = -0.05My,base / floor θp = 2%, θpc/θp = 1, λ = 20

8-story Frame, T1 = 0.8 sec., γ = 0.25, Stiff. & Str. = Shear, SCB = 2.4,2.4 θp = 3%, θpc/θp = 5, λ = 20

GF2

GW2 8-story Wall, T1 = 0.8 sec., γ = 0.25, Stiff. = Unif., Str. = -0.05My,base / floor

θp = 2%, θpc/θp = 1, λ = 20


GF2

GW2

Sa(T1)/g


at discrete hazard levels, using building-level subsystems and comparing “best” shear wall

(GW2) and “best” moment-resisting frame(GF2) alternatives

- 211 -

Expected $loss given IM, E($loss|IM)design alternatives GF2 & GW2

0

0.5

1

1.5

2

0 2 4 6 8 10 12Expected $loss (in millions)

Sa(T

1)/g

Series2 Series1 Series3

Series5 Series4 Series6

E($loss|IM & C)

E($loss|IM & C)

E($loss|IM & NC)

E($loss|IM & NC)

E($loss|IM)

E($loss|IM)

GF2

GW2

Fig. 7.7 Expected value of $loss given IM for “best” moment-resisting frame (GF2) and

“best” shear wall(GW2) alternatives

Sa(T1)/g

2.0

1.5

1.0

0.5

λ(Sa(T1)/g) E($Loss|Sa(T1)/g) (in millions)

1262

Design for E($Loss) design alternative GF2 & GW2

Mean $Loss-IM curves

Design Alternative GF2

Design Alternative GW2

4 8 1010/5050/50 2/50

( ) ( ) ( )$ $ | IMIM

E loss E loss im d imλ= ∫

( ) $17000/ for GF2$

$22000/ for GW2year

E lossyear

≈⎧= ⎨≈⎩

Sa(T1)/g

2.0

1.5

1.0

0.5


1262





4 8 1010/5050/50 2/50

( ) ( ) ( )$ $ | IMIM


( ) $17000/ for GF2$

$22000/ for GW2year

E lossyear

≈⎧= ⎨≈⎩

Sa(T1)/g

2.0

1.5

1.0

0.5


1262





4 8 1010/5050/50 2/50

( ) ( ) ( )$ $ | IMIM


( ) $17000/ for GF2$

$22000/ for GW2year

E lossyear

≈⎧= ⎨≈⎩

Fig. 7.8 MAF of $loss for “best” moent-resisting frame (GF2) and “best” shear wall (GW2)

alternatives

- 212 -

0

1

0

1

00

1.0


0.0

Expe

cted

Sub

syst

em

$Los

s ( i

n m

illio

ns)

EDP=Avg. of max. 1st story drift ratios,

(maxIDR)st.1

EDP=Avg. of max. 2nd floor accelerations,

(PFA)fl.2(g)

0.75

0.50

0.25

0.00.005 0.015 0.0250.01 0.02

1.00

1.25

0.5 1.0 1.5 2.00.0


1.5

0.5

2.0

1.5

1.0

0.5

0.0 0.01 0.02 0.5 1.0 1.5 2.00.0 20%10%0%

1.5

2.0

1.0

0.5

Sa(T1)/g

( )| &E EDP IM NC


1.50

1.0

2.0

1.5

0.5

EDP=Avg. of max. 1st story plastic IDR,

(maxIDRp)st.1

Sa(T1)/g

( | )P C IM


Alternatives

0.02 0.03


(PFA)fl.2(g)(maxIDR)st.1 (maxIDRp)st.1


P(C|Sa(T1)/g)

NSDSS Story levelNSDSS Story level


λ(IM)

Hazard Curves for

Design Alternatives

Hazard Domain


$Loss Domain

Expe

cted

Tot

al $

Loss

at

Col

laps

e ( i

n m

illio

ns)

6.0

4.0

8.0

10.0

12.0

2.0

0.0

0.01

NSASS Story levelNSASS Story level

SS Story level

SS Story level

0.02 0.030.01

0.75

0.50

0.25

1.00

1.25

1.50

0.75

0.50

0.25

1.00

1.25

1.50





GF1

GF2

GF3

GF4





GF1

GF2

GF3

GF4

2/50

T1=0.8sec.

T1=1.6sec.

50/50 10/50

λ(Sa(T1)/g)

Fig. 7.9a Example of DDSS implementation in conceptual design with performance targets

at discrete hazard levels, using story-level subsystems and moment-resisting frame

structural systems GF1, GF2, GF3, and GF4, (STORY 1)

- 213 -

0

1

0

11.0


0.0

Expe

cted

Sub

syst

em

$Los

s ( i

n m

illio

ns)

EDP=Avg. of max. 1th story drift ratios,

(maxIDR)st.4

EDP=Avg. of max. 5th floor accelerations,

(PFA)fl.5(g)

0.75

0.50

0.25

0.00.005 0.015 0.0250.01 0.02

1.00

1.25

0.5 1.0 1.5 2.00.0


1.5

0.5

2.0

1.5

1.0

0.5

0.0 0.01 0.02 0.5 1.0 1.5 2.00.0 20%10%0%

1.5

2.0

1.0

0.5

Sa(T1)/g

( )| &E EDP IM NC


1.50

1.0

2.0

1.5

0.5


(maxIDRp)st.4

Sa(T1)/g

( | )P C IM


Alternatives

0.02 0.03


(PFA)fl.5(g)(maxIDR)st.4 (maxIDRp)st.4


P(C|Sa(T1)/g)



λ(IM)

Hazard Curves for

Design Alternatives

Hazard Domain


$Loss Domain

Expe

cted

Tot

al $

Loss

at

Col

laps

e ( i

n m

illio

ns)

6.0

4.0

8.0

10.0

12.0

2.0

0.0

0.01


SS Story level

SS Story level

0.02 0.030.01

0.75

0.50

0.25

1.00

1.25

1.50

0.75

0.50

0.25

1.00

1.25

1.50





GF1

GF2

GF3

GF4





GF1

GF2

GF3

GF4

2/50

T1=0.8sec.

T1=1.6sec.

50/50 10/50

λ(Sa(T1)/g)

Fig. 7.9b Example of DDSS implementation in conceptual design with performance targets



- 214 -

0

1

0

11.0


0.0

Expe

cted

Sub

syst

em

$Los

s ( i

n m

illio

ns)


(maxIDR)st.8

EDP=Avg. of max. Roof accelerations,

(PFA)Roof(g)

0.75

0.50

0.25

0.00.005 0.015 0.0250.01 0.02

1.00

1.25

0.5 1.0 1.5 2.00.0


1.5

0.5

2.0

1.5

1.0

0.5

0.0 0.01 0.02 0.5 1.0 1.5 2.00.0 20%10%0%

1.5

2.0

1.0

0.5

Sa(T1)/g

( )| &E EDP IM NC


1.50

1.0

2.0

1.5

0.5


(maxIDRp)st.8

Sa(T1)/g

( | )P C IM


Alternatives

0.02 0.03


(PFA)Roof(g)(maxIDR)st.8 (maxIDRp)st.8


P(C|Sa(T1)/g)



λ(IM)

Hazard Curves for Design

Alternatives

Hazard Domain


$Loss Domain

Expe

cted

Tot

al $

Loss

at

Col

laps

e ( i

n m

illio

ns)

6.0

4.0

8.0

10.0

12.0

2.0

0.0

0.01


SS Story level

SS Story level

0.02 0.030.01

0.75

0.50

0.25

1.00

1.25

1.50

0.75

0.50

0.25

1.00

1.25

1.50





GF1

GF2

GF3

GF4





GF1

GF2

GF3

GF4

2/50

T1=0.8sec.

T1=1.6sec.

50/50 10/50

λ(Sa(T1)/g)

Fig. 7.9c Example of DDSS implementation in conceptual design with performance targets



- 215 -

1.0


0.0

Expe

cted

Sub

syst

em

$Los

s ( i

n m

illio

ns)


(maxIDR)st.1

EDP=Avg. of max. 2nd

floor accelerations,

(PFA)fl.2 (g)

0.75

0.50

0.25

0.00.005 0.015 0.0250.01 0.02

1.00

1.25

0.5 1.0 1.5 2.00.0


1.5

0.5

2.0

1.5

1.0

0.5

0.0 0.01 0.02 0.5 1.0 1.5 2.00.0 20%10%0%

1.5

2.0

1.0

0.5

Sa(T1)/g

( )| &E EDP IM NC


1.50

1.0

2.0

1.5

0.5

EDP=Max. of 1st story plastic rotation,

(maxθp)st. 1.

Sa(T1)/g

( | )P C IM


Alternatives

0.02


(PFA)fl.2(g)(maxIDR)st. 1 (maxθp)st. 1


P(C|Sa(T1)/g)



λ(IM)

Hazard Curves for

Design Alternatives

Hazard Domain


$Loss Domain

Expe

cted

Tot

al $

Loss

at

Col

laps

e ( i

n m

illio

ns)

6.0

4.0

8.0

10.0

12.0

2.0

0.0

0.01


SS Story level

SS Story level

0.020.01

50/50 10/50

λ(Sa(T1)/g) 2/50

T1=0.4sec.

T1=0.8sec.





GW1

GW2

GW3





GW1

GW2

GW3

0.75

0.50

0.25

1.00

1.25

1.50

0.75

0.50

0.25

1.00

1.25

1.50


targets at discrete hazard levels, using story-level subsystems and shear wall structural

systems GW1, GW2, and GW4, (STORY 1)

- 216 -

1.0


0.0

Expe

cted

Sub

syst

em

$Los

s ( i

n m

illio

ns)


(maxIDR)st.4



(PFA)fl.5 (g)

0.00.005 0.015 0.0250.01 0.02 0.5 1.0 1.5 2.00.0


1.5

0.5

2.0

1.5

1.0

0.5

0.0 0.01 0.02 0.5 1.0 1.5 2.00.0 20%10%0%

1.5

2.0

1.0

0.5

Sa(T1)/g

( )| &E EDP IM NC


1.0

2.0

1.5

0.5


(maxθp)st. 4.

Sa(T1)/g

( | )P C IM


Alternatives

0.02


(PFA)fl.5(g)(maxIDR)st. 4 (maxθp)st. 4


P(C|Sa(T1)/g)



λ(IM)

Hazard Curves for

Design Alternatives

Hazard Domain


$Loss Domain

Expe

cted

Tot

al $

Loss

at

Col

laps

e ( i

n m

illio

ns)

6.0

4.0

8.0

10.0

12.0

2.0

0.0

0.01


SS Story level

SS Story level

0.020.01

50/50 10/50

λ(Sa(T1)/g)

T1=0.4sec.

T1=0.8sec.

2/50





GW1

GW2

GW3





GW1

GW2

GW3

0.75

0.50

0.25

1.00

1.25

1.50

0.75

0.50

0.25

1.00

1.25

1.50

0.75

0.50

0.25

1.00

1.25

1.50




- 217 -

1.0


0.0

Expe

cted

Sub

syst

em

$Los

s ( i

n m

illio

ns)


(maxIDR)st.8



(PFA)Roof (g)

0.00.005 0.015 0.0250.01 0.02 0.5 1.0 1.5 2.00.0


1.5

0.5

2.0

1.5

1.0

0.5

0.0 0.01 0.02 0.5 1.0 1.5 2.00.0 20%10%0%

1.5

2.0

1.0

0.5

Sa(T1)/g

( )| &E EDP IM NC


1.0

2.0

1.5

0.5


(maxθp)st. 8.

Sa(T1)/g

( | )P C IM


Alternatives

0.02


(PFA)Roof(g)(maxIDR)st. 8 (maxθp)st. 8


P(C|Sa(T1)/g)



λ(IM)

Hazard Curves for

Design Alternatives

Hazard Domain


$Loss Domain

Expe

cted

Tot

al $

Loss

at

Col

laps

e ( i

n m

illio

ns)

6.0

4.0

8.0

10.0

12.0

2.0

0.0

0.01


SS Story level

SS Story level

0.020.01

50/50 10/50

λ(Sa(T1)/g)

T1=0.4sec.

T1=0.8sec.

2/50





GW1

GW2

GW3





GW1

GW2

GW3

0.75

0.50

0.25

1.00

1.25

1.50

0.75

0.50

0.25

1.00

1.25

1.50

0.75

0.50

0.25

1.00

1.25

1.50




- 218 -

0

1

0

1.0


0.0

Expe

cted

Sub

syst

em

$Los

s ( i

n m

illio

ns)


(maxIDR)st.1

EDP=Avg. of max. 2nd floor accelerations,

(PFA)fl.2(g)

0.75

0.50

0.25

0.00.005 0.015 0.0250.01 0.02

1.00

1.25

0.5 1.0 1.5 2.00.0


1.5

0.5

2.0

1.5

1.0

0.5

0.0 0.01 0.02 0.5 1.0 1.5 2.00.0 20%10%0%

1.5

2.0

1.0

0.5

Sa(T1)/g

( )| &E EDP IM NC


1.50

1.0

2.0

1.5

0.5

Sa(T1)/g

( | )P C IM


Alternatives


(PFA)fl.2(g)(maxIDR)st.1


P(C|Sa(T1)/g)



λ(IM)

Hazard Curves for

Design Alternatives

Hazard Domain


$Loss Domain

Expe

cted

Tot

al $

Loss

at

Col

laps

e ( i

n m

illio

ns)

6.0

4.0

8.0

10.0

12.0

2.0

0.0


SS Story level

SS Story level

0.75

0.50

0.25

1.00

1.25

1.50

0.75

0.50

0.25

1.00

1.25

1.50

2/50λ(Sa(T1)/g)



GF2


θp = 2%, θpc/θp = 1, λ = 20


GF2

GW3

EDP for SS

EDP for SS

T1=0.8sec.

50/50 10/50

T1=0.4sec.


targets at discrete hazard levels, using story-level subsystems and comparing “best” shear

wall (GW3) and “best” moment resisting frame (GF2) alternatives (STORY 1)

- 219 -

1.0


0.0

Expe

cted

Sub

syst

em

$Los

s ( i

n m

illio

ns)


(maxIDR)st.4

EDP=Avg. of max. 5th floor accelerations,

(PFA)fl.5(g)

0.75

0.50

0.25

0.00.005 0.015 0.0250.01 0.02

1.00

1.25

0.5 1.0 1.5 2.00.0


1.5

0.5

2.0

1.5

1.0

0.5

0.0 0.01 0.02 0.5 1.0 1.5 2.00.0 20%10%0%

1.5

2.0

1.0

0.5

Sa(T1)/g

( )| &E EDP IM NC


1.50

1.0

2.0

1.5

0.5

Sa(T1)/g

( | )P C IM


Alternatives


(PFA)fl.5(g)(maxIDR)st.4


P(C|Sa(T1)/g)



λ(IM)

Hazard Curves for

Design Alternatives

Hazard Domain


$Loss Domain

Expe

cted

Tot

al $

Loss

at

Col

laps

e ( i

n m

illio

ns)

6.0

4.0

8.0

10.0

12.0

2.0

0.0


SS Story level

SS Story level

0.75

0.50

0.25

1.00

1.25

1.50

0.75

0.50

0.25

1.00

1.25

1.50

λ(Sa(T1)/g) EDP for SS

EDP for SS



GF2


θp = 2%, θpc/θp = 1, λ = 20


GF2

GW3

T1=0.8sec.

50/50 10/50

T1=0.4sec.




- 220 -

1.0


0.0

Expe

cted

Sub

syst

em

$Los

s ( i

n m

illio

ns)


(maxIDR)st.8

EDP=Avg. of max. Roof accelerations,

(PFA)Roof(g)

0.75

0.50

0.25

0.00.005 0.015 0.0250.01 0.02

1.00

1.25

0.5 1.0 1.5 2.00.0


1.5

0.5

2.0

1.5

1.0

0.5

0.0 0.01 0.02 0.5 1.0 1.5 2.00.0 20%10%0%

1.5

2.0

1.0

0.5

Sa(T1)/g

( )| &E EDP IM NC


1.50

1.0

2.0

1.5

0.5

Sa(T1)/g

( | )P C IM


Alternatives


(PFA)Roof(g)(maxIDR)st.8


P(C|Sa(T1)/g)



λ(IM)

Hazard Curves for Design

Alternatives

Hazard Domain


$Loss Domain

Expe

cted

Tot

al $

Loss

at

Col

laps

e ( i

n m

illio

ns)

6.0

4.0

8.0

10.0

12.0

2.0

0.0


SS Story level

SS Story level

0.75

0.50

0.25

1.00

1.25

1.50

0.75

0.50

0.25

1.00

1.25

1.50

2/50λ(Sa(T1)/g) EDP for SS

EDP for SS



GF2


θp = 2%, θpc/θp = 1, λ = 20


GF2

GW3

T1=0.8sec.

50/50 10/50

T1=0.4sec.




- 221 -

Sa(T1)/g

2.0

1.5

1.0

0.5

λ(Sa(T1)/g)

2/5010/50

Collapse fragility curves

Design Alternative GF250% Confidence

90% Confidence

Design for tolerable probability of collapse at 2/50 design alternative GF2

(Mean) 71% Confidence

P(C|Sa(T1)/g)

30%20%10%

Sa(T1)/g

2.0

1.5

1.0

0.5

λ(Sa(T1)/g)

2/5010/50


Design Alternative GF250% Confidence

90% Confidence

Design for tolerable probability of collapse at 2/50 design alternative GF2


P(C|Sa(T1)/g)

30%20%10%


aleatory uncertainties for design alternative GF2

Sa(T1)/g

2.0

1.5

1.0

0.5


30%20%10%2/5010/50

Design for tolerable probability of collapse at 2/50 design alternative GW3


Design Alternative GW350% Confidence

90% Confidence


Sa(T1)/g

2.0

1.5

1.0

0.5


30%20%10%2/5010/50

Design for tolerable probability of collapse at 2/50 design alternative GW3


Design Alternative GW350% Confidence

90% Confidence



aleatory uncertainties for design alternative GW3

- 222 -

Effect of confidence level on P(C) at 2/50 GF2 & GW3 desgin alternatives

0.00

0.10

0.20

0.30

0.50 0.60 0.70 0.80 0.90 1.00

Confidence level (Y)

Prob

abili

ty o

f col

laps

e at

2/5

0 ha

zard

leve

l for

con

fiden

ce le

vel Y

Desgin Alternative GF2

Desgin Alternative GW3

Fig. 7.14 Effect of confidence level on probability of collapse at 2/50 hazard level for design

alternatives GF2 and GW3

Effect of confidence level on MAF of collapse GF2 & GW3 desgin alternatives

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

2.5E-04

3.0E-04

3.5E-04

4.0E-04

4.5E-04

0.50 0.60 0.70 0.80 0.90 1.00

Confidence level (Y)

MA

F of

col

laps

e fo

r con

fiden

ce Y

Desgin Alternative GF2

Desgin Alternative GW3

Fig. 7.15 Effect of confidence level on MAF of collapse for design alternatives GF2 and

GW3

-223-

8 SUMMARY AND CONCLUSIONS

This dissertation summarizes a simplified approach to Performance-Based Earthquake

Engineering (PBEE), intended to facilitate implementation of PBEE in engineering practice.

PBEE is concerned with design, assessment, and construction of buildings (or other engineering

facility) whose performance should comply with objectives expressed by stakeholders (owner,

user, society). This research concentrates on performance-based design, PBD, and performance-

based assessment, PBA of buildings. The goal is to improve the understanding of PBEE and

provide engineers and researchers with tools that make PBEE less labor intensive and more user-

friendly. In the subsequent sections the major impacts of this research are summarized.

8.1 Development of a framework for Simplified PBEE

In the proposed simplified PBEE approach, losses are separated into direct ($) loss, downtime

loss, and life loss. The approach is accomplished by explicitly considering the following three

domains: Hazard Domain, Structural System Domain, and Loss Domain. In each domain, mean

values of relationships are established. In the Hazard Domain, mean IM values at specific return

periods (or complete hazard curves) are established for the specific site and for targeted

fundamental periods of feasible structural systems. In this research the spectral acceleration at

the 1st mode period of the structural system is selected as the IM. The proposed methodology is

capable of incorporating other scalar or vector IM as is discussed in Chapter 2 of this

dissertation.

The Structural System Domain is divided into two sub-domains. One is conditioned on

collapse does not occur (NC sub-domain) and contains mean relationships between the IM and

appropriate EDPs, for the given building (in case of PBA) or for structural systems that present

feasible design alternatives (in case of PBD). These relationships can be obtained by means of

- 224 -

Incremental Dynamic Analyses (IDAs), using a set of ground motions representative for the site

hazard. Issues associated with this sub-domain are discussed in Chapter 5 of this dissertation.

The other sub-domain in the Structural System Domain is conditioned on collapse does

occur (C sub-domain) in which collapse fragility curves for the structural systems are presented.

Collapse may contribute significantly to direct loss, in the amount of total loss of building times

probability of collapse at a given hazard level. Discussions about this sub-domain are presented

in Chapters 6 of this dissertation.

Similar to the Structural System Domain, the Loss Domain is divided into two sub-

domains according to the occurrence of collapse. The NC sub-domain of the Loss Domain

contains mean loss curves that are pertinent for the specific building. These loss curves should be

simple enough to facilitate practical use, but detailed enough to provide insight into the sources

of losses and quantitative enough to provide realistic estimates of losses as a function of a

relevant EDP. Quantification of such mean loss curves is the great challenge that needs to be

addressed through research. At this time only rough estimates can be provided.

To facilitate loss aggregation the loss curves must not be too local (i.e., not at the

component level), but to facilitate flexibility, use of judgment, and evaluation of impact, they

also must not be too global (e.g., not lumping all losses into a single mean loss curve). It is

recommended to break up the building inventory into subsystems that can be quantified

separately and that exhibit sensitivity to specific EDPs. One option is to divide the inventory into

the following three subsystems: a nonstructural drift sensitive system (NSDSS), a nonstructural

acceleration sensitive system (NSASS), and a structural system (SS). The mean loss – EDP

curves for each of these subsystems could be estimated at the story level or at the global building

level. Selection of subsystems is subjective and depends on many factors, one of them being the

availability of data for relating the expected value of loss to the expected value of the specific

EDP considered for that subsystem. The C-sub-domain of the loss domain contains the expected

value of the loss of the building conditioned on collapse occurs. Discussion about the Loss

Domain is presented in Chapter 3 of this dissertation.

The difference between Simplified PBD and Simplified PBA is in the flow of

information. In Simplified PBA one can start from the Hazard Domain and for a given hazard

level find the expected value of IM on the mean hazard curve. Building response can be obtained

from IDAs that leads to the development of IM|EDP&NC curves. Given the relation between the

expected value of loss and the most relevant EDP for each subsystem, and using the expected

- 225 -

value of EDP obtained from the mean IM|EDP&NC at the IM level obtained in the previous step,

the expected value of loss, conditioned on collapse does not occur, for this subsystem is

calculated. This process can be completed for all subsystems considered in the building. The

addition of losses in the individual subsystems is the value of total loss of the building

conditioned on collapse does not occur at the selected IM level..

Simultaneously, the probability of collapse at the selected IM level is obtained using the

collapse fragility curve. Given this collapse probability and previously obtained values of total

loss of the building conditioned on collapse does occur, and total loss of the building conditioned

on collapse does not occur, the total loss of the building is obtained.

The flow of information for Simplified PBD is different from that for Simplified PBA

because an effective structural system has yet to be found based on specific performance

objectives. The procedure is to enter the hazard curve using the hazard level at which a

performance objective is to be fulfilled and to enter the mean loss curve with the acceptable loss

(the targeted performance objective) to find the corresponding expected EDP. The point in the

NC sub-domain of the Structural System Domain defined by the so obtained expected IM and

EDP values can be considered as a “design target point”. Using mean IM-EDP relationships

developed for generic structural systems as a part of this study will tell which structural systems

and combination of structural parameters provide a solution that fulfills the stated performance

objective.

Alternatively, by finding the expected losses and the probability of collapse at various

hazard levels, and integrating each over the hazard curve, the expected annual $loss and the

Mean Annual Frequency (MAF) of collapse can be obtained. These metrics can be used as an

averaged performance objective.

Due to the importance of assessment of the collapse potential of buildings and design for

collapse safety, special attention is dedicated to this topic. The effects of aleatory and epistemic

uncertainties on the probability of collapse and MAF of collapse are discussed in detail in

Chapter 6 of this dissertation.

Implementation of the aforementioned methodology for conceptual PBD and simplified

PBA of an 8-story building is discussed in detail in Chapter 7. In this implementation both

building-level and story-level subsystems are considered and comparisons are made between the

outcomes of the DDSS.

- 226 -

8.2 Introduction of Subsystem concept

One of the contributions of this study is the introduction of the subsystem concept for

aggregating the losses in individual building components, or disaggregating the total loss of a

building. A “subsystem” is defined as a collection of building components whose losses are well

represented by a single EDP. Strategies for grouping building components into different

subsystems and a methodology for developing subsystem mean loss curves are discussed in

detail in Chapter 3.

Two classes of subsystems are introduced in this dissertation for the purpose of

simplifying design and assessment of buildings. These two classes of subsystems are building-

level subsystems and story-level subsystems. Building-level subsystems are used in order to

relate losses in a group of building components to a single global EDP of the building. Although

this class of subsystems is convenient to use as it relates a significant portion of the total loss to a

single EDP, the approximation involved in loss estimation limits the employment of this class of

subsystems to cases in which the response of the building in different stories does not vary by

much.

Story-level subsystems are more localized subsystems that include a collection of

building components located in the same story and whose losses can be described by a common

EDP. The approximation involved in loss estimation by using building-level subsystems does

not exist when using story-level subsystems. However, the number of subsystems that need to be

considered increases substantially. Procedures and equations for obtaining mean loss curves for

building-level and story-level subsystems are provided in Chapter 3 of this dissertation.

8.3 Introduction of a new method for describing structural components behavior (monotonic and cyclic)

The method proposed by Ibarra et al. (2005) for defining the backbone curve and cyclic

deterioration parameters of structural component model is based on the elastic properties of

components (initial stiffness and yield rotation). Due to the large ambiguity and uncertainty in

defining these properties it has been difficult to obtain stable values for parameters that describe

the inelastic behavior of components.

An alternative method for defining the backbone curve and cyclic deterioration

parameters of structural component model is introduced in Chapter 4 of this dissertation. In this

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method, elastic properties of the component are no longer used as the normalizing parameters for

defining the inelastic properties of the backbone curve. Parameters such as: ductility capacity,

strain hardening ratio, post-capping stiffness ratio, and cyclic deterioration parameter based on

yield rotation, are replaced with plastic hinge rotation capacity, capping strength ratio, post-

capping rotation capacity ratio, and cyclic deterioration parameter based on plastic hinge rotation

capacity. Calibrations of test results show stable values for the new parameters as shown in

Chapter 4.

8.4 Development of a comprehensive database of structural response parameters for combinations of structural systems parameters

In order to provide supporting information for the DDSS, a comprehensive database of structural

response parameters (EDPs and collapse capacities) for a wide-range of combination of

structural parameters for moment-resisting frames and shear walls is developed as part of this

study. Structural parameters that are considered as variables in this database are chosen among

those global and local structural parameters that are believed to have significant effect on the

response of the structural systems. The range of variation of structural parameters considered for

developing the database are obtained based on calibration of many test results conducted by

several researchers.

8.5 Assessment of sensitivity of structural response parameters to variation of structural parameters

In Chapter 5 of this dissertation the sensitivity of several EDPs to variations in structural

parameters for a base case generic moment-resisting frame and a base case generic shear walls

are presented. EDPs selected for this sensitivity study include: maximum interstory drift at

individual stories ( )maxi

IDR , average along the height of maximum interstory drift ratio for

different ground motions ( )maxavg

IDR , peak floor acceleration at each floor ( )iPFA , and

average along the height of peak floor acceleration for different ground motions ( )avgPFA . The

sensitivity study focused on mean values of EDPs conditioned on collapse does not occur in

order to provide useful information for the DDSS. For the range of parameters investigated, the

following observations were made from this sensitivity study.

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• In both generic moment-resisting frame and generic shear wall the effect of yield base

shear coefficient on variation of ( )maxi

IDR , ( )maxavg

IDR , ( )iPFA , and ( )avg

PFA is

considerable for IM levels at which substantial inelastic behavior of the structural system

is observed.

• In the moment-resisting frame, ( )maxi

IDR and ( )maxavg

IDR become sensitive to the

column to beam strength ratio (SCB ratio) when the response is highly inelastic. The

SCB ratio affects development of a local mechanism that may cause early collapse,

especially in cases with SCB = 1.2-1.2 (present code recommendation)

• In the moment-resisting frame, ( )maxi

IDR and ( )maxavg

IDR are sensitive to the

variation of stiffness and strength along the height of the structural system (Stiff. & Str.).

This parameter affects the shape of the first mode of the structural system, which directly

affects the deformed shape of the structural system.

• The effect of variations in structural component parameters on the aforementioned EDPs

is benign. These parameters have a significant effect on the response of the structure

only close to collapse.

In Chapter 6 of this dissertation a sensitivity study of the median collapse capacity ηc to

variations of structural parameters is presented. Closed-form equation for estimations of ηc for

moment-resisting frames and shear walls are developed, and the efficiency of these equations is

demonstrated in several figures. The major conclusions obtained from a limited sensitivity study

using a moment-resisting frame and a shear wall are as follows:

• P-Delta is the main reason for collapse of structural systems. The effect of P-Delta is

smaller for shear walls than moment-resisting frames.

• The effect of P-Delta in developing local (single or multiple story) mechanism that

accelerate collapse is larger for structures with smaller yield base shear coefficient.

• The effect of the column to beam strength ratio on the median collapse capacity of

moment-resisting frames is large. It is observed that increasing this parameter (i.e., SCB)

from 1.2 to 2.4 increases the median of collapse capacity by 90%.

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8.6 Concluding remarks and suggestions for future work

In this study. many assumptions are made in the development of the simplified PBEE

methodology and of supporting information for the DDSS. For one, the DDSS provides only

mean estimates of losses and does not account for the dispersion of different random variables

involved in the process. Secondly, the mean loss - EDP curves utilized in the implementation are

based on limited data and on much judgment. Much more research is needed to quantify such

loss curves more accurately. Thirdly, the DDSS addresses only losses and pays only lip service

to the up-front construction cost of various design alternatives. Finally, the author wants to

emphasize that the DDSS is intended to serve as a decision support tool and not as a decision

criterion.

Areas for future research on topics related to this dissertation can be divided into 4 major

categories. These categories and specific topics for future research are as follows:

1. Enhancement in the general approach for simplified performance-based earthquake

engineering:

• Enhancement of the current PBD process by using statistical measures other than

mean values in order to accomplish a more effective PBD procedure.

• Packaging the DDSS presented in this dissertation for educational/professional

use.

• Enhancement of the PBD process by incorporating optimization techniques to

obtain an optimum design.

2. Enhancements in the Hazard Domain:

• Use of other scalar IMs or vector IMs for addressing the ground motion intensity

given the hazard level.

• Use of representative ground motions for special sites such as buildings located

on soft-soils and/or near faults.

3. Enhancement of information provided in the Structural System Domain

• Enhancement of the current database of EDP|IM for broader realizations and more

combinations of structural parameters

• Enhancement of the current database of EDP|IM by incorporating structural

models that are representative of buildings with control devices.

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• Enhancement of the current database of EDP|IM by using component models that

are representative of structural components made of new materials.

4. Development of information needed in the Loss Domain

• Development of loading protocols for experimentation leading to damage state

fragility curves.

• Development of mean loss curves for various subsystems that can be used in the

DDSS.

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REFERENCES

ACI Committee 318, 2005. Building Code Requirements for Structural Concrete (ACI 318-05) and

Commentary (ACI 318R-05). American Concrete Institute, Farmington Hills, MI. 443 pages.

AISC 2002, Seismic Provisions for Structural Steel Buildings, American Institute of Steel Construction,

Chicago, IL.

ATC-43, 1999. Evaluation of earthquake damaged concrete and masonry wall buildings. Report FEMA

306 ,Federal Emergency Management Agency, Washington, DC.

ATC-63, 2004. Quantification of Building System Performance and Response Parameters. Applied

Technology Council, Redwood City, California.

American Society of Civil Engineers (ASCE), 2000. Prestandard and commentary for the seismic

rehabilitation of buildings. Report FEMA 356 ,Federal Emergency Management Agency, Washington,

DC.

Abrahamson, N. A., Silva, W. J., 1997. Empirical response spectral attenuation relations for shallow

crustal earthquakes, Seismological Research Letters, 68, 94-126.

Alavi, B., Krawinkler, H., 2001. Effects of near-fault ground motions on frame structures. Report No.

138, The John A. Blume Earthquake Engineering Center, Department of Civil & Environmental

Engineering, Stanford University, Stanford, California.

Alavi, B., Krawinkler, H., 2004. Behavior of moment-resisting frame structures subjected to near-fault

ground motions. Earthquake Engineering and Structural Dynamics, 33(6), 687-706.

Al-Ali, A. A. K., Krawinkler, H., 1998. Effect of vertical irregularities on seismic behavior of building

structures. Report No. 130, The John A. Blume Earthquake Engineering Center, Department of Civil &

Environmental Engineering, Stanford University, Stanford, California.

Ali, A., Wight, J. K., 1991. RC Structural Walls with Staggered Door Openings. ASCE Journal of

Structural Engineering, 117(5), 1514-1531.

Applied Technology Council, 1985. Earthquake damage evaluation data for California (ATC-13),

Redwood City, California.

Aslani, H., 2005. Probabilistic earthquake loss estimation and loss disaggregation in buildings. Ph.D

Dissertation (under supervision of Prof. Eduardo Miranda), Stanford University, Stanford California.

Aslani, H., Miranda, E., 2006. Assessment and propagation of epistemic uncertainty in loss estimation.

Proceedings of the 8th U.S. National Conference on Earthquake Engineering, San Francisco, California,

San Francisco, California, Paper No. 1123.

Bazzurro, P., Cornell, C. A., 2001. Vector-valued probabilistic seismic hazard analysis. Seismological

Research Letters, 72(2), pp 273.

- 232 -

Baker, J. W., Cornell, C. A., 2003. Uncertainty specification for loss estimation using FOSM method.

Report No. PEER 2003/07, Pacific Earthquake Engineering Research Center, University of California at

Berkeley, Berkeley, California.

Baker J. W., Cornell C. A., 2005. Vector-valued ground motion intensity measure consisting of spectral

acceleration and epsilon. Earthquake Engineering & Structural Dynamics, 34(10), 1193-1217

Baker J. W., Cornell C. A., 2006. Spectral shape, epsilon and record selection. Earthquake Engineering &

Structural Dynamics (in press)

Benjamin, J, R., Cornell, C. A., 1970. Probability, statistics, and decision for civil engineers. 4th edition,

Boston: McGraw-Hill, Inc.

Bernal, J., 1994. Viscous damping in inelastic structural response. ASCE Journal of Structural

Engineering, 120(4), 1240-1254.

Bertero, V. V., Bendimerad, M. F., Shah, H. C., 1988. Fundamental period of reinforced concrete

moment-resisting frame structures. Report No. 87, The John A. Blume Earthquake Engineering Center,

Department of Civil & Environmental Engineering, Stanford University, Stanford, California

Blakeley, R. W. G., Cooney, R. C., Megget, L. M., 1975. Seismic shear loading at flexural capacity in

cantilever wall structures. Bulletin of New Zealand National Society for Earthquake Engineering, 8(4),

278-290.

Blume, J. A., 1968. Dynamic characteristics of multi-story buildings. ASCE Journal of Structural

Division, 94(ST2), 377–402.

Bommer, J. J., Acevedo, A. B., 2004. The use of real earthquake accelerograms as input to dynamic

analysis. Journal of Earthquake Engineering, 8 (Special Issue1), 43-91.

Boore, D. M., Joyner, W. B., Fumal, T. E. 1997. Equations for estimating horizontal response spectra and

peak acceleration from Western North American earthquakes: a summary of recent work, Seismological

Research Letters, 68(1), 128-153.

Building Seismic Safety Council (BSSC), 2001. Seismic design criteria for steel moment-frame

structures. Report FEMA 350 –FEMA 355 , CD-ROM, Federal Emergency Management Agency,

Washington, DC.

Building Seismic Safety Council (BSSC), 2003. NEHRP recommended provisions for seismic regulations

for new buildings and other structures. Report FEMA 450-1, Federal Emergency Management Agency,

Washington, DC.

Carr, A. J., 2003. RUAUMOKO Inelastic Dynamic Analysis Program, Dept. of Civil Engineering,

University of Canterbury, Christchurch, New Zealand.

- 233 -

Chintanapakdee, C., Chopra, A. K., 2003. Evaluation of modal pushover analysis procedure using

vertically ‘regular’ and irregular generic frames. Report No. EERC 2003/03, Earthquake Engineering

Research Center, University of California at Berkeley, Berkeley, California.

Clough, R. W., Johnson, S. B., 1966. Effect of stiffness degradation on earthquake ductility requirements.

Proceedings of Japan Earthquake Engineering Symposium.

Comerio, M. C. ed. 2005. PEER Testbed Study on a Laboratory Building: Exercising Seismic

Performance Assessment. Report No. PEER 2005/12, Pacific Earthquake Engineering Research

Center,University of California at Berkeley, Berkeley, California.

Comerio, M. C., 2006. Performance engineering and disaster recovery: limiting downtime. Proceedings

of the 8th U.S. National Conference on Earthquake Engineering, San Francisco, California, Paper No.

531.

Cordova, P. P., Deierlein, G. G., Mehanny, S. S., Cornell, C. A., 2000. Development of a two-parameter

seismic intensity measure and probabilistic assessment procedure. Proceedings of the 2nd U.S.-Japan

Workshop on Performance-Based Earthquake Engineering for Reinforced Concrete Building Structures,

Sapporo, Japan.

Cornell CA, 1968. Engineering seismic risk analysis. Bulletin of the Seismological Society of America,

58, 1583-1606.

Cornell C. A., Krawinkler H., 2000. Progress and challenges in seismic performance assessment. PEER

News, April 2000.

Cornell, C. A., Luco, N., 2001. Ground motion intensity measures for structural performance assessment

at near-fault sites. Proceedings U.S.-Japan Joint Workshop and Third Grantees Meeting, U.S.-Japan

Cooperative Research on Urban Earthquake Disaster Mitigation, University of Washington, Seattle,

Washington.

Cornell, C. A., Jalayer, F., Hamburger, RO., Foutch, DA., 2002. Probabilistic basis for 2000 SAC Federal

Emergency Management Agency steel moment frame guidelines. ASCE Journal of Structural

Engineering, 128(4), 526-533.

Deierlein, G., 2004. Overview of a comprehensive framework for earthquake performance assessment.

Proc. International Workshop on Performance-Based Seismic Design – Concepts and Implementation,

Bled, Slovenia, 15-26.

Elnashai, A., Pilakoutas, K., 1990. Experimental behaviour of reinforced concrete walls under earthquake

loading, Earthquake Engineering and Structural Dynamics, 19(3), 389-407.

Esteva, L., Ruiz, S. E., 1989. Seismic failure rates of multistory frames. ASCE Journal of Structural

Engineering,.115(2), 268-284.

- 234 -

Fardis, M. N., Biskinis, D. E., 2003. Deformation capacity of RC members, as controlled by flexure or

shear, Proceedings of the International Symposium Honoring Shunsuke Otani in: Performance-Based

Engineering for Earthquake Resistant Reinforced Concrete Structures: A Volume Honoring Shunsuke

Otani, 511-530.

Goel, R. K., Chopra, A. K., 1997. Vibration properties of buildings determined from recorded earthquake

motions, Report UCB/EERC-97/14, Earthquake Engineering Research Center, University of California at


Gunturi, S., 1993. Building-specific earthquake damage estimation. Ph.D Dissertation (under supervision

of Prof. Haresh Shah), Dept. of Civil and Environmental Engineering, Stanford University, Stanford

California

Hall, J., 2006. Problems encountered from the use (or misuse) of Rayleigh damping. Earthquake

Engineering and Structural Dynamics, 35(5), 525-545.

Hart, G., Vasudevan, R., 1975. Earthquake design of buildings: damping. ASCE Journal of Structural

Division, 101(ST1), 11-30.

Haselton, C. B., Liel, A. B., Taylor Lange, S., Deierlein, G.G., 2006. Beam-column element model

calibrated for predicting flexural response leading to global collapse of RC frame buildings, PEER Report

2006, Pacific Earthquake Engineering Research Center, University of California at Berkeley, Berkeley,

California.

Horie, K. Hayashi, H., Maki, N., Okimura., T., Tanaka, S., Torii, N., 2006. Development of seismic

vulnerability functions for building collapse without survival space for accurate human casualty

estimation. Procedings of the 8th U.S. National Conference on Earthquake Engineering, San Francisco,

California, Paper No. 837.

Hutchinson, T. C., Chai, Y. H, Boulanger, R. W., Idriss, I. M., 2004. Inelastic seismic response of

extended pile-shaft-supported bridge structures. Earthquake Spectra, 20(4), 1057-1080.

Ibarra, L. F., Krawinkler, H., 2005. Global collapse of frame structures under seismic excitations. Report

No. PEER 2005/06, Pacific Earthquake Engineering Research Center, University of California at


Ibarra L. F., Medina R. A., Krawinkler H., 2005. Hysteretic models that incorporate strength and

stiffness deterioration. Earthquake Engineering and Structural Dynamics, 34(12), 1489-1511.

International Code Council, 2000. International Building Code 2000, ICC, Birmingham, AL.

Jalayer, F., 2003. Direct probabilistic seismic analysis: implementing non-linear dynamic assessments.

Ph.D. dissertation (under supervision of Prof. Allin Cornell), Department of Civil & Environmental


- 235 -

Kano, M. et. al., 2006. Development of casualty models for non-ductile concrete frame structures for use

in PEER’s performance-based earthquake engineering framework. Proceedings of the 8th U.S. National

Conference on Earthquake Engineering, San Francisco, California, Paper No. 917.

Kara, K. L., Bracci, J. M., 2001. Seismic evaluation of column-to-beam strength ratios in reinforced

concrete frames. ACI Structural Journal, 98(6), 843-851.

Kim, T. W., Foutch, D., LaFave, J., 2005. A practical model for seismic analysis of reinforced concrete

shear wall buildings. Journal of Earthquake Engineering, 9(3), 393-417.

Kircher, C. A., 1997. Estimation of earthquake losses to buildings. Earthquake Spectra, 19(2), 365-384.

Kircher, C. A., 2003. Earthquake loss estimation methods for welded steel moment-frame buildings.

Earthquake Spectra, 19(2), 365-384.

Kowalsky, M. J, Priestley, M. J. N, 2000, Improved analytical model for shear strength of circular

reinforced concrete columns in seismic regions. ACI Structural Journal, 97(3), 388-396.

Kramer, S. L., 1996. Geotechnical Earthquake Engineering, Prentice Hall.

Krawinkler, H., 2002. A general approach to seismic performance assessment. Proc. International

Conference on Advances and New Challenges in Earthquake Engineering Research, ICANCEER, Hong

Kong, Vol 3,173-180.

Krawinkler, H. (ed), 2005. Van Nuys hotel building testbed report: Exercising seismic Performance

Assessment. Report No. PEER 2005/11, Pacific Earthquake Engineering Research Center,University of

California at Berkeley, Berkeley, California.

Krawinkler, H., 2006. Importance of good nonlinear analysis. Proc., 2006 Annual Meeting of the Los

Angeles Tall Buildings Structural Design Council, Los Angeles.

Krawinkler, H., Lignos, D., 2007. A database for modeling deterioration in beams and columns subjected

to cyclic bending moments. to be presented at the 4th NEES annual meeting.

Krawinkler, H., Miranda, E., 2004. Performance-based earthquake engineering. Earthquake Engineering:

from engineering seismology to performance-based engineering. Chapter 9, Bozorgnia Y, Bertero VV

(eds); CRC Press: Boca Raton, 9-1 to 9-59.

Lee, T. H., Mosalam, K. M., 2005. Seismic demand sensitivity of reinforced concrete shear-wall building

using FOSM method. Earthquake Engineering and Structural Dynamics, 34(14), 1719-1736.

Luco, N., 2002. Probabilistic seismic demand analysis, SMRF connection fractures, and near source

effects. Ph.D. dissertation (under supervision of Prof. Allin Cornell), Department of Civil &


- 236 -

Luco, N., Mori, Y., Funahashi, Y., Cornell, C. A., Nakashima, M., 2003. Evaluation of predictors of non-

linear seismic demands using 'fishbone' models of SMRF buildings. Earthquake Engineering &

Structural Dynamics, 32(14), 2267-2288.

Lynn, A. C., Moehle, J. P., Mahin, S. A., Holmes, W. T., 1996. Seismic evaluation of existing reinforced

concrete building columns. Earthquake Spectra, 12(4), 715-739.

Mahin, S. A., Bertero, V. V., 1975. An evaluation of some methods for predicting seismic behavior of

reinforced concrete buildings. Report No. EERC 75/05, Earthquake Engineering Research Center,

University of California, Berkeley, California.

Mander, J. B., Chen, S. S., Pekcan, G., 1994. Low-cycle fatigue behavior of semi-rigid top-and-seat angle

connections. AISC Engineering Journal, 31(3), 111-122.

Medina, R., Krawinkler, H., 2003. Global collapse of frame structures under seismic excitations. Report

No. PEER 2003/15, Pacific Earthquake Engineering Research Center, University of California at


Miranda, E., 1999. Approximate seismic lateral deformation demands in multistory buildings. ASCE

Journal of Structural Engineering, 125(4), 417-425.

Miranda, E. Taghavi, S., Aslani, H., 2002. Some fragility curves for the Van Nuys testbed. presentation

for the PEER testbed meeting, Stanford California.

Mitrani-Reiser, M., Haselton, C., Goulet, C., Porter, K., Beck, J., Deierlein, G., 2006. Evaluation of the

seismic performance of a code-conforming reinforced-concrete frame building – part II: loss estimation.

Proceedings of the 8th U.S. National Conference on Earthquake Engineering, San Francisco, California,

Paper No. 969.

Moehle, J. P., and Deierlein, G. G. 2004. A framework for performance-based earthquake engineering.

Proceedings of 13th World Conference on Earthquake Engineering, Paper No. 679, Vancouver, CA.

Nakashima, M., Ogawa, K., Inoue, K., 2002. Generic frame model for simulation of earthquake responses

of steel moment frames. Earthquake Engineering and Structural Dynamics, 31(3), 671-692.

Nakashima, M., Ogawa, K., Inoue, K., 2002. Evaluation of predictors of non-linear seismic demands

using fishbone models of SMRF buildings. Earthquake Engineering and Structural Dynamics, 32(14),

2267-2288.

National Institute of Building Sciences (NIBS), 1999. HAZUS: Earthquake Loss Estimation

Methodology, Technical Manuals I, II, & III, prepared for the Federal Emergency Management Agency,

Washington, D.C.

Neter, J., Kutner, M., Nachtsheim, C. J., Wasserman, W., 1996. Applied linear statistical models. 4th

edition, Boston: McGraw-Hill, Inc.

- 237 -

Ogawa, K., Kamura, H., Inoue, K., 1999. Modeling of moment resisting frame to fishbone-shape for

response analysis. Journal of Structural and Construction Engineering (AIJ). 521, 119-126 (in Japanese).

Oh, Y. H., Han, S. W., Lee, L. H., 2002. Effect of boundary element details on the seismic deformation

capacity of structural walls. Earthquake Engineering and Structural Dynamics, 31(8), 1583-1602.

OpenSees, 2002. Open System for Earthquake Engineering Simulation, http://opensees.berkeley.edu/,

Pacific Earthquake Engineering Research Center, University of California at Berkeley, Berkeley,

California.

Pekoz, H. A., Pincheira, J. A., 2006. Effect of hysteretic shape on the seismic response of stiffness and

strength degrading SDOF systems. Proceedings of the 8th U.S. National Conference on Earthquake

Engineering, San Francisco, California, Paper No. 456.

Paulay, T., 2001. Seismic response of structural walls: recent developments. Canadian Journal of Civil

Engineering, 28(6), 922-937.

Paulay, T., Priestley, M. J. N., 1992. Seismic design of reinforced concrete and masonry buildings. John

Wiley & Sons, 1st edition, New York, N.Y.

Paulay, T., Uzumeri, S. M., 1975. A critical review of the seismic design provisions for ductile shear

walls of the Canadian code and commentary. Canadian Journal of Civil Engineering, 2(4), 592-601.

Pilakoutas, K., Elnashai, A., 1995. Cyclic behavior of reinforced concrete cantilever walls, part I:

experimental results, ACI Structural Journal, 92(3), 271-281.

Porter, K. A. et. al., 2002. Sensitivity of building loss estimates to major uncertain variables, Earthquake

Spectra, 18(4), 719-743.

Porter, K. A., Beck, J. L., Shaikhutdinov, R., 2004. Simplified estimation of economic seismic risk for

buildings. Earthquake Spectra, 20(4), 1239-1263.

Prakash, V., Powell, G. H., Campbell, S., 1993. DRAIN-2DX: basic program description and user guide.

Report No. UCB/SEMM-93/17. University of California at Berkeley, Berkeley, California.

Rahnama, M., Krawinkler, H., 1993. Effects of soft soils and hysteresis model on seismic demands.

Report No. 108, The John A. Blume Earthquake Engineering Center, Dept. of Civil and Environmental


Roeder, C. W., Carpenter, J. E., Taniguchi, H., 1989. Predicted ductility demands for steel moment

resisting frames. Earthquake Spectra, 5(2), 409-427.

Rosenblueth, E., 1979. Predicted ductility demands for steel moment resisting frames. ASCE Journal of

the Engineering Mechanics Division, 105(EM1), 177-187.

Ruiz-Garcia, J., 2004. Performance-Based assessment of existing structures accounting for residual

displacements. Ph.D. dissertation (under supervision of Prof. Eduardo Miranda), Department of Civil &


- 238 -

Samprit, C., Hadi, H., Price, B., 2000. Regression analysis by example. 3rd edition, New York: John

Wiley & Sons, Inc.

Sedarat, H., Bertero, V. V., 1990. Effects of torsion on the linear and nonlinear seismic response of

structures. Report No. EERC 1990/12, Earthquake Engineering Research Center, University of California

at Berkeley, Berkeley, California.

Seneviratna, G. D. P. K., Krawinkler, H., 1997. Evaluation of inelastic MDOF effects for seismic design.

Report No. 120, The John A. Blume Earthquake Engineering Center, Dept. of Civil and Environmental


Shome, N., Cornell, C. A., 1999. Probabilistic seismic demand analysis of nonlinear structures. Report

No. RMS-35, Reliability of Marine Structures Program, Dept. of Civil Engineering, Stanford University.

Shinozuka, M., Feng, M., Jongheon, L., Naganuma, T., 2000. Statistical analysis of fragility curves.

ASCE Journal of Structural Engineering, 126(12), 1224-1232.

Song, J., Pincheira, J., 2000. Spectral displacement demands of stiffness and strength degrading systems.

Earthquake Spectra 16(4), 817-851.

Stewart, J. P., Chiou, S. J., Bray, J. D., Graves, R. W., Somerville, P. G., Abrahamson, N. A., 2001.

Ground motion evaluation procedures for performance-based design. Report No. PEER 2001/09, Pacific

Earthquake Engineering Research Center, University of California at Berkeley, Berkeley, California.

Takeda, T., Sozen, M., Nielsen, N., 1970. Reinforced concrete response to simulated earthquakes. ASCE

Journal of Structural Division, 96(12), 2557-2573.

Taghavi, S., Miranda, E., 2003. Response assessment of nonstructural building elements. Report PEER

2003/05, Pacific Earthquake Engineering Center, University of California at Berkeley, Berkeley,

California.

Tothong, P., 2006. Improved probabilistic seismic demand analysis using advanced scalar ground motion

intensity measures, attenuation relationships, and near-fault effects Ph.D. dissertation (under supervision

of Prof. Allin Cornell), Department of Civil & Environmental Engineering, Stanford University, Stanford,

California.

Vallenas, J. M., Bertero, V. V., Popov, E. P., 1979. Hysteretic behavior of reinforced concrete structural

walls. Report No. EERC 1979/28, Earthquake Engineering Research Center, University of California at


Vamvatsikos, D., Cornell, C. A., 2002. Incremental Dynamic Analysis, Earthquake Engineering and

Structural Dynamics, 31(3), 491-514.

Vulcano, A., Bertero, V. V., 1987. Analytical models for predicting the lateral response of RC shear

walls: Evaluation of their reliability. Report No. EERC 1987/19, Earthquake Engineering Research

Center, University of California at Berkeley, Berkeley, California.

- 239 -

Wallace, J. W., Moehle, J. P., 1992. Ductility and detailing requirements of bearing wall buildings. ASCE

Journal of Structural Engineering, 118(6), 1625-1644.

Wen, Y. K., 1976. Method for random vibration of hysteretic systems. ASCE Journal of Engineering

Mechanics Division, 102(EM2), 249-263.

Whitman, R. V., Anagnos, T., Kircher, CA., Lagorio, H. J, Lawson, R. S., 1997. Development of a

national earthquake loss estimation methodology. Earthquake Spectra, 13(4), 643-661.

Wood, S. L., 1992. Seismic response of R/C frames with irregular profiles. ASCE Journal of Structural

Engineering, 118(2), 545-566.

Yeo, G. L., Cornell, C. A., 2003. Building-specific seismic fatality estimation methodology. Proceedings

of the 9th International Conference on Applications of Statistics and Probability in Civil Engineering,

Vol. 1, 765-771.

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APPENDIX A VARIATION OF STIFFNESS AND STRENGTH ALONG THE HEIGHT OF GENERIC MOMENT-RESISTING FRAMES

As discussed in Chapter 4 of this dissertation, we have considered several assumptions for

proportioning the stiffness and strength of beams and columns along the height of generic

moment-resisting frames. This appendix summarizes the supporting facts and observations for

such assumptions.

A.1 Relation between stiffness of beams and columns in moment-resisting frame

structures

The lateral deformation of buildings is composed of shear-type deformations and flexural-type

deformations as shown in Figure A.1 (repeated in Figure 4.2). To show the relative contribution

of these two types of deformation in the total lateral deformation of a structure, Blume (1968)

introduced a dimensionless parameter ρ, which is defined as the ratio of the sum of the stiffness

ratios of all the beams at the floor closest to the midheight of the moment-resisting frame to the

sum of the stiffness ratio of all the columns at the same floor:

b

story i beams bi

c

story i colmns c

EILEIL

ρ =∑

∑ (A.1)

ρ equal to zero represents a fully flexural-type deformation (e.g. lateral deformation of a

shear wall structure or moment-resisting frames with very stiff columns and relatively flexible

beams where lateral deformation is due to column deformation) and ρ equal to infinity represents

a fully shear-type deformation (e.g. lateral deformation of a moment-resisting frame with very

- 242 -

stiff beams and relatively flexible columns where lateral deformation is due to beam rotation).

An intermediate value for ρ represents a moment-resisting frame which the lateral deformation is

due to deformation in both columns and beams. Miranda (1999) introduced the parameter α to

describe the participation of shear-type and flexural-type deformation in a structure. A value of α

equal to zero, and infinity represent fully flexural-type, and fully shear-type deformation

behavior respectively. The effect of α on deformation demand of structures was studied by

Miranda (1999), and Miranda and Reyes (2002). The influence of α on acceleration demand of

structures was studied by Miranda and Taghavi (2005) and Taghavi and Miranda (2005).

In order to assess the value of ρ for frame structures, the variation along the height of this

parameter in SAC steel structures was studied as shown in Figure A.2 (Gupta and Krawinkler,

1999). Table A.1 shows the value of ρ averaged along the height in the SAC structures. As seen

in Figure A.2, there is a trend that ρ is reduced along the height of these structures. The average

of ρ for all of the SAC structures is about 0.3. For simplicity, the moment of inertia and length of

beams and columns of moment-resisting frames used in this study are tuned such that the value

of ρ along the height is equal to 0.3. We will show that variation along the height of ρ in

moment-resisting frames has a negligible effect on structure’s deformation demand as long as the

average along the height of ρ is large enough that the contribution of shear-type deformation is

larger than the flexural-deformation.

A.2 Variation of stiffness along the height of moment-resisting frame structures

Structural stiffness depends on the absolute value of stiffness of structural elements and their

configuration in the structure. In order to find typical relations between the stiffness of structural

elements, the stiffness configuration of a number of structures are studied. Figure A.3 shows the

variation along the height of beams stiffness ratio in floor i normalized to beam stiffness ratio in

the first floor of SAC structures. Figure A.4 shows the variation along the height of columns

stiffness ratio in story i normalized to columns stiffness ratio in the first floor of SAC structures.

Both graphs show the relative height on the abscissa.

It is observed in Figure A.3 that for steel moment resisting frames, beam stiffness is

reduced in the upper stories of the structure. The same observation extends to column stiffness

although the trend is not as clear as for beams. The sudden decrease in the column stiffness ratio

- 243 -

in the first story of the 9-story and 20-story SAC buildings is due to an increase in the height of

this floor (from 13’ to 18’) while the section is the same as what is used in the second story.

Based on these observations and the fact that most of the lateral deformation of moment-resisting

frames is due to beams deformation, we decided to use three alternatives for variation along the

height of beam moment of inertia in the generic moment-resisting frames used in this study.

These three alternatives are denoted as: “Shear”, “Unif.”, and “Int.”, as shown in Figure A.5:

Alt. 1) In this alternative, moment of inertia of all beams at a floor level are proportioned to

the story shear force obtained from subjecting the generic moment-resisting frame to

the NEHRP lateral load pattern (this lateral load pattern is period dependent, see

FEMA 356). This alternative represents a seismic design for drift limitation and

guarantees that the structure will have a straight line deformed shape when subjected

to the NEHRP lateral load pattern.

Alt. 2) Beams moment of inertia is uniform along the height of the generic moment-resisting

frame (“Unif.”). This alternative represents those structural designs in which the

designer uses a similar cross section for beams in several stories for simplicity in

design and construction. Such decisions are arbitrary and depend on many factors

such as availability of the structural material, cost of using joints with different

detailing, etc. This alternative represents a boundary case for using similar cross

sections in several stories of the moment-resisting frame.

Alt. 3) In this alternative, variation along the height of beams moment of inertia in generic

moment-resisting frames is the average of the first and second alternatives. As the

difference between the previously mentioned two alternatives is large, this alternative

is introduced as an intermediate case in order to study the effect of variation along the

height of beam stiffness in generic moment-resisting frames on structural response

more clearly.

As seen in Figure A.4, moment of inertia of columns is reduced along the height of the

structure in 9-story and 20-story frames and is kept constant for 3-story frames. As long as the

deformation in every story of a moment-resisting frame is mainly due to the rotation of beams,

- 244 -

we assume that variation along the height of column moment of inertia has a secondary effect on

the structural response. This characteristic of moment-resisting frames supports our previous

assumption of using a constant ρ along the height of the base case generic moment-resisting

frames. For simplicity, we will assume that the moment of inertia of columns in any story is

equal to the moment of inertia of beams of the top-floor of this story. If we assume that the beam

length is 2.5 times the length of columns in a 3-bay moment-resisting frame, then the value of ρ

at each story, ρi, is equal to 0.3:

32.5 0.34

b

story i beams bi

c

story i colmns c

I IL LI I

LL

ρ = = =∑

∑ (A.2)

This is almost equal to the average of ρ for the SAC structures, as stated in the previous

section. In order to show that the deformation response of generic moment-resisting frames with

different average ρ values are not significantly different and show that the variation along the

height of beam moment of inertia is the important parameter that affects the deformation

response of generic moment-resisting frames, we have studied the first mode shape of a 9-story

single bay moment-resisting frame.

Figure A.6 shows the comparison between the first mode shape of the 9-story structure

with two variations along the height of beam moment of inertia, “Shear” and “Unif.”, and two

values for the average ρ (the value of ρ is constant along the height of structure, which means

that the variation along the height of column moment of inertia is similar to the beams). It is seen

that the first mode shape is more sensitive to variation along the height of beams moment of

inertia than to the value of average ρ. From this observation it is concluded that variation along

the height of beam moment of inertia has a larger effect on the deformation response of generic

moment-resisting frames and justifies our choice of introducing two boundary cases of “Shear”

and “Unif.”, and the intermediate case, “Int.”, for variation along the height of beams moment of

inertia.

In order to show that variation along the hight of ρ (which translates to unequal variation

along the height of column moment of inertia and beam moment of inertia) has smaller effect on

the deformation response of a moment-resisting frame than variation along the height of beam

moment of inertia, the first mode shape of the 9-story generic moment-resisting frame was

computed and plotted in Figure A.7 for two variations along the height of beam moment of

- 245 -

inertia, “Shear” and “Unif.”, and two variations along the height of column moment of inertia, Ic

= Ib and Ic = constant. Ic = Ib denotes the case in which beam and column moment of inertia are

equal at each story (i.e. variation along the height of beam and column moment of inertia is the

same), and Ic = constant represents the case in which column moment of inertia is kept constant

along the height of the moment-resisting frame. In all cases, the average of ρ is equal to 0.3.

The results show that variation along the height of beam moment of inertia has more

influence on the response of a moment-resisting frame structure than the variation along the

height of column moment of inertia (or ρ). This conclusion is supported by the fact that most of

the deformation in moment-resisting frames is due to beam rotation and less due to column

deformations. It should be noted that this conclusion is valid while the value of ρ is in the

domain that the shear-type deformations have larger contribution to the total lateral deformation

of the structure.

A.3 Variation of strength along the height of moment-resisting frame structures

In the previous section, we described different alternatives for variation of stiffness along the

height of base case generic moment-resisting frames. We showed that for frame structures

(where most of the deformation is due to beams rotation) the effect of variation along the height

of beam moment of inertia on the first mode shape of the 9-story structure is larger than variation

along the height of ρ. For this reason, we proposed three alternatives for variation along the

height of beam moment of inertia, which consists of two boundary cases of “Shear” and “Unif.”,

and an intermediate case of “Int.”. In this section we discuss alternatives for variation along the

height of beam and column strength in the base case generic moment-resisting frames. For

simplicity, we will first focus on variation along the height of beam strength and will introduce

alternatives for variation along the height of this parameter. Next, we will discuss columns

strength and its variation along the height of the structure.

In a steel moment-resisting frame, variation along the height of beam strength, Mpb, could

be translated into variation along the height of beam’s Plastic Section Modulus, Zb. Zb and the

Elastic Section Modulus, Sb, of a steel cross section are related with a parameter called “Shape

Factor”, k, which is a constant for a specific shape of beam’s cross section. For an I shape steel

cross sections (which most of beams in a steel moment-resisting frame have this cross section

- 246 -

shape), k is between 1.12 to 1.17. For practical reasons, in order to reduce the moment of inertia

of beams, Ib, typically, the depth of the beam is kept constant and the area of flanges are reduced.

This means that variation along the hight of Ib and Sb, and subsequently Zb, are the same.

We can conclude that for most cases, variation along the height of beam strength and

stiffness for steel moment-resisting frames are equal. This means that two boundary cases for

variation of beam stiffness along the height of “Shear” and “Unif.”, and an intermediate case of

“Int.”, may also represent variation along the height of beam strength for steel moment-resisting

frames. Variation along the height of beam strength and beam stiffness might not be the same in

real steel moment-resisting frames. In this research, we have tried to bracket variations along the

height of beam stiffness and assumed that the stiffness and strength are proportional for

simplicity.

In reinforced concrete moment-resisting frames, except for the first and last floor, the

depth of the beams are typically kept the same and the reinforcement is change if needed.

Therefore, we can assume that the variation along the height of beam strength and stiffness for a

reinforced concrete moment-resisting frame are equal. In reality, variation along the height of

beam strength in reinforced concrete moment-resisting frames does not necessarily follow the

variation along the height of beam stiffness because there are lots ambiguity in definition of

stiffness and strength for reinforced concrete cross sections. For one, there is no specific value

for stiffness of a reinforced concrete beam. The stiffness of such beam is very large before the

first crack is initiated. After initiation of the first crack, the stiffness is reduced slowly till the

tension reinforcement yields (in case of under-reinforced sections) or concrete crushes in the

compression zone (in case of over-reinforced sections). Also, there is no specific value for the

yield moment because of the highly nonlinear behavior of the section in the region where tension

reinforced yields or concrete crushes in the compression zone. As stated for the case of steel

beams, we have tried to bracket variations along the height of beam stiffness (or better say the

equivalent stiffness of reinforced concrete beams) and assumed that the stiffness and strength are

proportional for simplicity. Based on this discussion we define the indicator for variation of

stiffness and strength along the height of generic moment resisting frames as “Stiff. & Str.”, and

the three alternatives for this variable are Stiff. & Str. = Shear, Int., Unif.

As stated in section A.2, we have assumed that the variation along the height of column

stiffness is the same as variation along the height of beam stiffness. So the three cases of

variation along the height of beam stiffness (i.e. “Shear”, “Int”, and “Unif.”)) do define the

- 247 -

variation along the hight of column stiffness. We define the strength of columns framing to a

joint based on the strength of beams framing to the same joint. Based on Section 21.4.2.2 of ACI

318-02 (2002), the sum of columns moment capacity that frame into a joint shall be more that

1.2 times the sum of beams framing into the same joint. Using this approach for defining the

strength of columns at each joint, we consider three alternatives for the value of column strength

framing to a joint based on the strength of beams framing to that joint. Figure A.8 shows these

three alternatives (i.e. SCB = 1.2-1.2, 2.4-1.2, 2.4-2.4). In SCB = 1.2-1.2, the sum of columns

moment capacity that frame into an exterior or interior joint is 1.2 times the sum of beams

framing into the same joint. This alternative is considered to show the effect of uniform column

to beam strength ratio for the limit introduced by ACI. As seen in Figure A.8, using this

criterion, the strength of columns framing into exterior and interior joints in the same floor is

different. Such difference in the strength of columns framing into joints of same floor motivated

us to introduce another alternative where columns framing into exterior and interior joints in the

same floor have the same strength. This alternative is SCB = 2.4-1.2 and the relative value of

beam and column yield moments are shown in Figure A.8. As seen, the interior columns have

the same strength in SCB = 1.2-1.2 and SCB = 2.4-1.2. The strength of exterior columns in SCB

= 2.4-1.2 is two times the strength of exterior columns in SCB = 1.2-1.2. Finally, in order to

study a case with stronger columns, we introduce SCB = 2.4-2.4 where the strong column weak

beam coefficient is equal to 2.4 for both interior and exterior columns.

- 248 -

Table A.1: Average of � for SAC structures

Avg. ρ N = 3 N = 3 N = 3 Los Angeles 0.35 0.47 0.28

Seattle 0.27 0.17 0.23

Fig. A.1 Modes of deformation in structures (after Miranda 1999).

Pure shear-type deformation

Pure flexural-type deformation

Combined shear-type & flexural-type deformations

- 249 -

Variation along the height of ρ in SAC structures (pre-northridge desgin)

0

0.25

0.5

0.75

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ρ

z/H

LA 3-story LA 9-story LA 20-story

SE 3-story SE 9-story SE 20-story

Fig. A.2 Variation along the height of ρ in SAC structures (pre-Northridge design)

Variation along the height of beam stiffness in SAC structures (pre-northridge desgin)

0

0.25

0.5

0.75

1

0 0.5 1 1.5 2

z/H

LA 3-story LA 9-story LA 20-storySE 3-story SE 9-story SE 20-story

Fig. A.3 Variation along the height of beam stiffness ratio in SAC structures (pre-

Northridge design)

- 250 -

Variation along the height of column stiffness in SAC structures (pre-northridge desgin)

0

0.25

0.5

0.75

1

0 0.5 1 1.5 2

z/H

LA 3-story LA 9-story LA 20-storySE 3-story SE 9-story SE 20-story

Fig. A.4 Variation along the height of column stiffness ratio in SAC structures (pre-

Northridge design)

Fig. A.5 Schematic representation of three variations along the height of beam moment of

inertia (“Shear”, “Unif”, “Int.”)

z/H

Variation along the height of beam moment of inertia in base case generic moment resisting frames

, ,1/b i bEI EIL L

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Alt.1: Shear

Alt. 2: Unif.

Alt.3: Int.

1

- 251 -

Effect of variation along the height of ρ i on 1st Mode shapeN = 9, ρ avg = var., (Ic ) I = (I b ) i

0123456789

0.0 0.2 0.4 0.6 0.8 1.0Mode Shape

Floo

r Num

ber

Series1 Series3Series2 Series4“Shear”, ρ avg = 0.3

, ρ avg = 0.6

, ρ avg = 0.3

, ρ avg = 0.6“Shear”

“Unif.”

“Unif”

Effect of variation along the height of ρ i on 1st Mode shapeN = 9, ρ avg = var., (Ic ) I = (I b ) i

0123456789

0.0 0.2 0.4 0.6 0.8 1.0Mode Shape

Floo

r Num

ber

Series1 Series3Series2 Series4“Shear”, ρ avg = 0.3

, ρ avg = 0.6

, ρ avg = 0.3

, ρ avg = 0.6“Shear”

“Unif.”

“Unif”

Fig. A.6 Effect of different ρavg and variation along the height of beam moment of inertia on

first mode period of a 9-story moment-resisting frame structure

Effect of variation along the height of I c on 1st Mode shapeN = 9, ρ avg = 0.3, (I c ) i = var.

0123456789

0.0 0.2 0.4 0.6 0.8 1.0Mode Shape

Floo

r Num

ber

Series1 Series3Series2 Series4“Shear”( I c ) i = (I b ) i

“Shear” ( I c ) i = constant

“Unif.” (I c ) i = (I b ) i

“Unif.” (I c ) i =constant

Effect of variation along the height of I c on 1st Mode shapeN = 9, ρ avg = 0.3, (I c ) i = var.

0123456789

0.0 0.2 0.4 0.6 0.8 1.0Mode Shape

Floo

r Num

ber

Series1 Series3Series2 Series4“Shear”( I c ) i = (I b ) i

“Shear” ( I c ) i = constant

“Unif.” (I c ) i = (I b ) i

“Unif.” (I c ) i =constant

Fig. A.7 Effect of different variation along the height of beam and column moment of

inertia on first mode period of a 9-story moment-resisting frame

- 252 -

My My My

My My My

1.2My

1.2My

1.2My 0.6My

0.6My

0.6My

My My My

My My My

1.2My

1.2My

1.2My 1.2My

1.2My

1.2My

My My My

My My My

2.4My

2.4My

2.4My 1.2My

1.2My

1.2My

SCB = 1.2-1.2

SCB = 2.4-1.2

SCB = 2.4-2.4







Fig. A.8 Schematic representation of three variations of column strength in generic

moment-resisting frames (SCB = 1.2-1.2, 2.4-1.2, 2.4-2.4)

- 253 -

APPENDIX B PROPERTIES OF BASE CASE GENERIC FRAMES AND WALLS

B.1 Properties of the base case generic moment-resisting frame models

The general characteristics of the base case generic moment-resisting frame models used in this

study were discussed in Chapter 4. Modal and structural properties are summarized in Tables B.1

to B.18. The Tables are divided into 2 major sections:

Table of modal properties (Tables B.1 to B.12)

Modal properties (first five modes only, where i denotes mode number)

• Period ratios, Ti / T1

• Participation factors, PFi

• Mass participation, MPi (as a fraction of the total mass)

• Modal damping, ξi

• Normalized mode shapes, φi

Table of structural properties (Tables B.13 to B.48)

Stiffness and strength properties (i denotes floor number)

• Weight ratio, Wi / W (W = total weight)

• Beam moment of inertia ratio, Ib,i / Ib,1

• Story shear ratio, Vb,i / Vb,1

• Beam strength ratio, Mb,i / Mb,1

- 254 -

Properties of the base case generic shear wall models

The general characteristics of the base case generic shear wall models used in this study were

discussed in Chapter 4. Modal and structural properties are summarized in Tables B.49 to B.52.

The Tables are divided into 2 major sections:

Table of modal properties (Tables B.49 to B.52)

Modal properties (first five modes only, where i denotes mode number)

• Period ratios, Ti / T1

• Participation factors, PFi

• Mass participation, MPi (as a fraction of the total mass)

• Modal damping, ξi

• Normalized mode shapes, φi

Table of structural properties (Tables B.53 to B.56)

Mass and Strength properties (i denotes floor number)

• Weight ratio, Wi / W (W = total weight)

• Shear wall bending moment strength ratio, Mi / Mbase

-255-

Table B.1 Modal properties of generic moment-resisting frames: N = 4, T1 = var., Stiff. &

Str. = Shear

Model Properties Floor Mode ShapesMode Ti/T1 PFi MPi ξi φ1 φ2 φ3 φ4

1 1.000 -1.607 0.831 0.051 1 0.000 0.000 0.000 0.0002 0.345 0.603 0.117 0.040 2 -0.239 0.704 0.900 0.9183 0.180 0.347 0.039 0.059 3 -0.512 1.000 0.218 -1.0004 0.111 0.200 0.013 0.089 4 -0.775 0.406 -1.000 0.548

Roof -1.000 -0.995 0.448 -0.132


Str. = Int.


1 1.000 -1.631 0.856 0.051 1 0.000 0.000 0.000 0.0002 0.318 0.568 0.104 0.040 2 -0.274 0.801 0.988 -0.8003 0.167 0.311 0.031 0.060 3 -0.573 1.000 -0.022 1.0004 0.108 -0.163 0.009 0.083 4 -0.828 0.216 -1.000 -0.689

Roof -1.000 -0.971 0.570 0.217


Str. = Unif.


1 1.000 -1.645 0.871 0.051 1 0.000 0.000 0.000 0.0002 0.303 0.546 0.096 0.039 2 -0.298 0.878 1.000 -0.7163 0.158 0.291 0.027 0.061 3 -0.611 1.000 -0.194 1.0004 0.104 -0.135 0.006 0.080 4 -0.855 0.112 -0.913 -0.805

Roof -1.000 -0.968 0.601 0.290

-256-


Str. = Shear

Model Properties Floor Mode ShapesMode Ti/T1 PFi MPi ξi φ1 φ2 φ3 φ4 φ5

1 1.000 -2.225 0.797 0.050 1 0.000 0.000 0.000 0.000 0.0002 0.373 -0.836 0.113 0.041 2 -0.117 -0.273 0.551 -0.711 0.8613 0.216 0.521 0.044 0.054 3 -0.254 -0.542 0.900 -0.802 0.4414 0.143 -0.370 0.022 0.077 4 -0.393 -0.711 0.760 -0.049 -0.7195 0.102 0.277 0.012 0.105 5 -0.528 -0.726 0.154 0.790 -0.645

6 -0.660 -0.552 -0.607 0.713 0.6627 -0.785 -0.177 -1.000 -0.370 0.6668 -0.901 0.372 -0.503 -1.000 -1.000

Roof -1.000 1.000 0.966 0.610 0.351


Str. = Int.


1 1.000 -2.264 0.825 0.050 1 0.000 0.000 0.000 0.000 0.0002 0.338 -0.794 0.102 0.040 2 -0.140 -0.389 0.684 -0.880 0.9533 0.192 0.480 0.037 0.055 3 -0.302 -0.743 1.000 -0.755 0.1634 0.127 -0.335 0.018 0.079 4 -0.461 -0.911 0.616 0.350 -0.9455 0.091 0.244 0.010 0.108 5 -0.611 -0.826 -0.231 1.000 -0.139

6 -0.745 -0.490 -0.929 0.268 0.9777 -0.858 0.024 -0.899 -0.885 -0.0208 -0.946 0.579 -0.062 -0.695 -1.000

Roof -1.000 1.000 0.982 0.796 0.574


Str. = Unif.


1 1.000 -2.285 0.841 0.050 1 0.000 0.000 0.000 0.000 0.0002 0.322 -0.767 0.095 0.039 2 -0.155 -0.458 0.737 -0.957 1.0003 0.181 0.459 0.034 0.055 3 -0.332 -0.855 1.000 -0.665 -0.0394 0.120 -0.317 0.016 0.080 4 -0.503 -1.000 0.474 0.577 -0.9795 0.086 0.228 0.008 0.110 5 -0.657 -0.837 -0.432 0.977 0.214

6 -0.787 -0.415 -0.988 -0.046 0.9417 -0.889 0.136 -0.746 -1.000 -0.3808 -0.960 0.650 0.107 -0.487 -0.864

Roof -1.000 0.989 0.937 0.831 0.637

-257-


Str. = Shear


1 1.000 -2.711 0.789 0.050 1 0.000 0.000 0.000 0.000 0.0002 0.380 -1.003 0.108 0.041 2 -0.079 -0.167 -0.307 0.504 -0.6203 0.227 -0.628 0.042 0.052 3 -0.172 -0.349 -0.593 0.851 -0.8504 0.156 0.457 0.022 0.071 4 -0.267 -0.505 -0.739 0.796 -0.4155 0.115 -0.355 0.014 0.094 5 -0.360 -0.615 -0.697 0.341 0.365

6 -0.452 -0.665 -0.464 -0.303 0.8567 -0.541 -0.646 -0.088 -0.809 0.6118 -0.628 -0.548 0.340 -0.874 -0.2169 -0.712 -0.369 0.692 -0.390 -0.87910 -0.792 -0.108 0.822 0.423 -0.61411 -0.869 0.226 0.597 1.000 0.48012 -0.940 0.612 -0.048 0.640 1.000

Roof -1.000 1.000 -1.000 -0.926 -0.651


Str. = Int.


1 1.000 -2.757 0.816 0.050 1 0.000 0.000 0.000 0.000 0.0002 0.344 -0.961 0.099 0.040 2 -0.095 -0.257 -0.440 0.654 -0.7803 0.201 -0.584 0.037 0.053 3 -0.206 -0.528 -0.809 0.999 -0.8884 0.137 0.418 0.019 0.073 4 -0.317 -0.742 -0.918 0.718 -0.1025 0.101 -0.322 0.011 0.097 5 -0.426 -0.865 -0.714 -0.032 0.793

6 -0.530 -0.878 -0.258 -0.769 0.8657 -0.627 -0.775 0.299 -1.000 0.0138 -0.718 -0.560 0.762 -0.541 -0.8719 -0.799 -0.256 0.950 0.322 -0.78910 -0.871 0.103 0.762 0.975 0.22711 -0.930 0.471 0.234 0.880 1.00012 -0.973 0.788 -0.446 0.016 0.477

Roof -1.000 1.000 -1.000 -0.998 -0.850

-258-


Str. = Unif.


1 1.000 -2.781 0.830 0.050 1 0.000 0.000 0.000 0.000 0.0002 0.327 -0.931 0.093 0.039 2 -0.104 -0.313 -0.523 0.699 -0.8553 0.190 -0.561 0.034 0.053 3 -0.226 -0.636 -0.933 1.000 -0.8584 0.130 0.400 0.017 0.074 4 -0.348 -0.878 -0.994 0.585 0.1115 0.096 -0.306 0.010 0.099 5 -0.464 -0.994 -0.670 -0.256 0.950

6 -0.574 -0.967 -0.085 -0.909 0.6927 -0.673 -0.800 0.533 -0.895 -0.3648 -0.762 -0.517 0.943 -0.224 -1.0009 -0.839 -0.160 0.985 0.611 -0.48210 -0.901 0.221 0.643 0.998 0.59211 -0.949 0.571 0.049 0.651 0.98112 -0.982 0.840 -0.569 -0.182 0.228

Roof -1.000 1.000 -1.000 -0.940 -0.870


& Str. = Shear


1 1.000 -3.128 0.787 0.050 1 0.000 0.000 0.000 0.000 0.0002 0.381 -1.140 0.105 0.041 2 -0.061 -0.121 -0.208 0.340 0.5003 0.231 -0.715 0.041 0.052 3 -0.132 -0.258 -0.422 0.644 0.8554 0.162 0.521 0.022 0.068 4 -0.205 -0.385 -0.583 0.778 0.8285 0.122 0.409 0.013 0.089 5 -0.277 -0.493 -0.660 0.693 0.415

6 -0.348 -0.573 -0.639 0.406 -0.1997 -0.417 -0.622 -0.519 -0.005 -0.7278 -0.485 -0.635 -0.314 -0.427 -0.9069 -0.552 -0.607 -0.050 -0.728 -0.624

10 -0.617 -0.538 0.237 -0.801 0.00711 -0.679 -0.426 0.498 -0.596 0.66712 -0.740 -0.269 0.681 -0.147 0.95913 -0.799 -0.071 0.729 0.408 0.62414 -0.855 0.167 0.593 0.835 -0.22815 -0.909 0.437 0.241 0.848 -1.00016 -0.958 0.726 -0.321 0.219 -0.782

Roof -1.000 1.000 -1.000 -1.000 0.917

-259-


& Str. = Int.


1 1.000 -3.176 0.812 0.050 1 0.000 0.000 0.000 0.000 0.0002 0.346 -1.101 0.098 0.040 2 -0.072 -0.192 -0.324 0.470 -0.6363 0.205 -0.669 0.036 0.052 3 -0.156 -0.406 -0.642 0.844 -0.9914 0.142 0.481 0.019 0.070 4 -0.242 -0.597 -0.845 0.913 -0.7565 0.107 -0.374 0.011 0.091 5 -0.327 -0.747 -0.883 0.637 -0.058

6 -0.409 -0.843 -0.746 0.116 0.6797 -0.489 -0.877 -0.458 -0.454 0.9988 -0.565 -0.846 -0.072 -0.853 0.6899 -0.637 -0.750 0.336 -0.919 -0.067

10 -0.705 -0.592 0.681 -0.611 -0.79211 -0.768 -0.381 0.886 -0.040 -1.00012 -0.826 -0.130 0.897 0.566 -0.51913 -0.877 0.143 0.695 0.938 0.35114 -0.922 0.419 0.310 0.882 0.98915 -0.958 0.671 -0.182 0.380 0.86816 -0.984 0.872 -0.661 -0.365 -0.004

Roof -1.000 1.000 -1.000 -1.000 -1.000


& Str. = Unif.


1 1.000 -3.202 0.825 0.050 1 0.000 0.000 0.000 0.000 0.0002 0.330 -1.071 0.092 0.039 2 -0.079 -0.236 -0.392 0.544 -0.6813 0.194 -0.645 0.033 0.052 3 -0.171 -0.495 -0.764 0.946 -1.0004 0.134 0.462 0.017 0.071 4 -0.265 -0.721 -0.975 0.952 -0.6405 0.101 -0.359 0.010 0.093 5 -0.356 -0.889 -0.967 0.548 0.163

6 -0.445 -0.983 -0.741 -0.094 0.8527 -0.529 -0.995 -0.346 -0.695 0.9478 -0.608 -0.925 0.128 -0.994 0.3829 -0.682 -0.778 0.572 -0.862 -0.450

10 -0.749 -0.566 0.887 -0.355 -0.96811 -0.809 -0.308 1.000 0.306 -0.81212 -0.862 -0.023 0.887 0.834 -0.09013 -0.907 0.263 0.572 1.000 0.69514 -0.943 0.527 0.128 0.731 0.99515 -0.971 0.748 -0.346 0.144 0.60016 -0.990 0.908 -0.745 -0.511 -0.221

Roof -1.000 1.000 -0.995 -0.982 -0.943

-260-

Table B.13 Weight, stiffness and strength properties of generic moment-resisting frames: N

= 4, T1 = 0.4, Stiff. & Str. = Shear.

Floor Floor Stiffness and Strength PropertiesWi/W Ib,i/Ib,1 Vi/V1 Mb,i/Mb,1

1 0.250 1.000 #REF! #REF!2 0.250 1.000 1.000 1.0003 0.250 0.900 0.900 0.9004 0.250 0.700 0.700 0.700

Roof 0.250 0.400 0.400 0.400W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)1200 60889 1200 53200


= 4, T1 = 0.4, Stiff. & Str. = Int.


1 0.250 1.000 #REF! #REF!2 0.250 1.000 1.000 1.0003 0.250 0.950 0.900 0.9504 0.250 0.850 0.700 0.850

Roof 0.250 0.700 0.400 0.700W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)1200 55765 1200 53200


= 4, T1 = 0.4, Stiff. & Str. = Unif.


1 0.250 1.000 #REF! #REF!2 0.250 1.000 1.000 1.0003 0.250 1.000 0.900 1.0004 0.250 1.000 0.700 1.000

Roof 0.250 1.000 0.400 1.000W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)1200 52535 1200 53200

-261-


= 4, T1 = 0.6, Stiff. & Str. = Shear.


1 0.250 1.000 #REF! #REF!2 0.250 1.000 1.000 1.0003 0.250 0.905 0.905 0.9054 0.250 0.708 0.708 0.708

Roof 0.250 0.407 0.407 0.407W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)1200 26919 1200 53300


= 4, T1 = 0.6, Stiff. & Str. = Int.


1 0.250 1.000 #REF! #REF!2 0.250 1.000 1.000 1.0003 0.250 0.953 0.905 0.9534 0.250 0.854 0.708 0.854

Roof 0.250 0.704 0.407 0.704W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)1200 24737 1200 53300


= 4, T1 = 0.6, Stiff. & Str. = Unif.


1 0.250 1.000 #REF! #REF!2 0.250 1.000 1.000 1.0003 0.250 1.000 0.905 1.0004 0.250 1.000 0.708 1.000

Roof 0.250 1.000 0.407 1.000W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)1200 23349 1200 53300

-262-


= 4, T1 = 0.8, Stiff. & Str. = Shear.


1 0.250 1.000 #REF! #REF!2 0.250 1.000 1.000 1.0003 0.250 0.914 0.914 0.9144 0.250 0.724 0.724 0.724

Roof 0.250 0.422 0.422 0.422W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)1200 14991 1200 53500


= 4, T1 = 0.8, Stiff. & Str. = Int.


1 0.250 1.000 #REF! #REF!2 0.250 1.000 1.000 1.0003 0.250 0.957 0.914 0.9574 0.250 0.862 0.724 0.862

Roof 0.250 0.711 0.422 0.711W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)1200 13864 1200 53500


= 4, T1 = 0.8, Stiff. & Str. = Unif.


1 0.250 1.000 #REF! #REF!2 0.250 1.000 1.000 1.0003 0.250 1.000 0.914 1.0004 0.250 1.000 0.724 1.000

Roof 0.250 1.000 0.422 1.000W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)1200 13134 1200 53500

-263-


= 8, T1 = 0.8, Stiff. & Str. = Shear.


1 0.125 1.000 #REF! #REF!2 0.125 1.000 1.000 1.0003 0.125 0.978 0.978 0.9784 0.125 0.930 0.930 0.9305 0.125 0.854 0.854 0.8546 0.125 0.747 0.747 0.7477 0.125 0.609 0.609 0.6098 0.125 0.439 0.439 0.439

Roof 0.125 0.237 0.237 0.237W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)2400 58348 2400 111000


= 8, T1 = 0.8, Stiff. & Str. = Int.


1 0.125 1.000 #REF! #REF!2 0.125 1.000 1.000 1.0003 0.125 0.989 0.978 0.9894 0.125 0.965 0.930 0.9655 0.125 0.927 0.854 0.9276 0.125 0.874 0.747 0.8747 0.125 0.805 0.609 0.8058 0.125 0.720 0.439 0.720

Roof 0.125 0.618 0.237 0.618W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)2400 53159 2400 111000

-264-


= 8, T1 = 0.8, Stiff. & Str. = Unif.


1 0.125 1.000 #REF! #REF!2 0.125 1.000 1.000 1.0003 0.125 1.000 0.978 1.0004 0.125 1.000 0.930 1.0005 0.125 1.000 0.854 1.0006 0.125 1.000 0.747 1.0007 0.125 1.000 0.609 1.0008 0.125 1.000 0.439 1.000

Roof 0.125 1.000 0.237 1.000W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)2400 50055 2400 111000


= 8, T1 = 1.2, Stiff. & Str. = Shear.


1 0.125 1.000 #REF! #REF!2 0.125 1.000 1.000 1.0003 0.125 0.985 0.985 0.9854 0.125 0.945 0.945 0.9455 0.125 0.877 0.877 0.8776 0.125 0.777 0.777 0.7777 0.125 0.642 0.642 0.6428 0.125 0.469 0.469 0.469

Roof 0.125 0.255 0.255 0.255W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)2400 25378 2400 111000

-265-


= 8, T1 = 1.2, Stiff. & Str. = Int.


1 0.125 1.000 #REF! #REF!2 0.125 1.000 1.000 1.0003 0.125 0.992 0.985 0.9924 0.125 0.973 0.945 0.9735 0.125 0.939 0.877 0.9396 0.125 0.889 0.777 0.8897 0.125 0.821 0.642 0.8218 0.125 0.734 0.469 0.734

Roof 0.125 0.628 0.255 0.628W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)2400 23441 2400 111000


= 8, T1 = 1.2, Stiff. & Str. = Unif.


1 0.125 1.000 #REF! #REF!2 0.125 1.000 1.000 1.0003 0.125 1.000 0.985 1.0004 0.125 1.000 0.945 1.0005 0.125 1.000 0.877 1.0006 0.125 1.000 0.777 1.0007 0.125 1.000 0.642 1.0008 0.125 1.000 0.469 1.000

Roof 0.125 1.000 0.255 1.000W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)2400 22245 2400 111000

-266-


= 8, T1 = 1.6, Stiff. & Str. = Shear.


1 0.125 1.000 #REF! #REF!2 0.125 1.000 1.000 1.0003 0.125 0.989 0.989 0.9894 0.125 0.957 0.957 0.9575 0.125 0.897 0.897 0.8976 0.125 0.804 0.804 0.8047 0.125 0.672 0.672 0.6728 0.125 0.496 0.496 0.496

Roof 0.125 0.274 0.274 0.274W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)2400 14025 2400 112000


= 8, T1 = 1.6, Stiff. & Str. = Int.


1 0.125 1.000 #REF! #REF!2 0.125 1.000 1.000 1.0003 0.125 0.995 0.989 0.9954 0.125 0.979 0.957 0.9795 0.125 0.949 0.897 0.9496 0.125 0.902 0.804 0.9027 0.125 0.836 0.672 0.8368 0.125 0.748 0.496 0.748

Roof 0.125 0.637 0.274 0.637W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)2400 13101 2400 112000

-267-


= 8, T1 = 1.6, Stiff. & Str. = Unif.


1 0.125 1.000 #REF! #REF!2 0.125 1.000 1.000 1.0003 0.125 1.000 0.989 1.0004 0.125 1.000 0.957 1.0005 0.125 1.000 0.897 1.0006 0.125 1.000 0.804 1.0007 0.125 1.000 0.672 1.0008 0.125 1.000 0.496 1.000

Roof 0.125 1.000 0.274 1.000W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)2400 12514 2400 112000


= 12, T1 = 1.2, Stiff. & Str. = Shear.


1 0.083 1.000 #REF! #REF!2 0.083 1.000 1.000 1.0003 0.083 0.994 0.994 0.9944 0.083 0.978 0.978 0.9785 0.083 0.951 0.951 0.9516 0.083 0.910 0.910 0.9107 0.083 0.855 0.855 0.8558 0.083 0.786 0.786 0.7869 0.083 0.700 0.700 0.700

10 0.083 0.597 0.597 0.59711 0.083 0.476 0.476 0.47612 0.083 0.336 0.336 0.336

Roof 0.083 0.178 0.178 0.178W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)3600 56513 3600 169000

-268-


= 12, T1 = 1.2, Stiff. & Str. = Int.


1 0.083 1.000 #REF! #REF!2 0.083 1.000 1.000 1.0003 0.083 0.997 0.994 0.9974 0.083 0.989 0.978 0.9895 0.083 0.975 0.951 0.9756 0.083 0.955 0.910 0.9557 0.083 0.928 0.855 0.9288 0.083 0.893 0.786 0.8939 0.083 0.850 0.700 0.850

10 0.083 0.798 0.597 0.79811 0.083 0.738 0.476 0.73812 0.083 0.668 0.336 0.668

Roof 0.083 0.589 0.178 0.589W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)3600 51937 3600 169000


= 12, T1 = 1.2, Stiff. & Str. = Unif.


1 0.083 1.000 #REF! #REF!2 0.083 1.000 1.000 1.0003 0.083 1.000 0.994 1.0004 0.083 1.000 0.978 1.0005 0.083 1.000 0.951 1.0006 0.083 1.000 0.910 1.0007 0.083 1.000 0.855 1.0008 0.083 1.000 0.786 1.0009 0.083 1.000 0.700 1.000

10 0.083 1.000 0.597 1.00011 0.083 1.000 0.476 1.00012 0.083 1.000 0.336 1.000

Roof 0.083 1.000 0.178 1.000W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)3600 49185 3600 169000

-269-


= 12, T1 = 1.8, Stiff. & Str. = Shear.


1 0.083 1.000 #REF! #REF!2 0.083 1.000 1.000 1.0003 0.083 0.997 0.997 0.9974 0.083 0.986 0.986 0.9865 0.083 0.966 0.966 0.9666 0.083 0.934 0.934 0.9347 0.083 0.887 0.887 0.8878 0.083 0.824 0.824 0.8249 0.083 0.742 0.742 0.742

10 0.083 0.641 0.641 0.64111 0.083 0.517 0.517 0.51712 0.083 0.370 0.370 0.370

Roof 0.083 0.198 0.198 0.198W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)3600 24444 3600 170000


= 12, T1 = 1.8, Stiff. & Str. = Int.


1 0.083 1.000 #REF! #REF!2 0.083 1.000 1.000 1.0003 0.083 0.998 0.997 0.9984 0.083 0.993 0.986 0.9935 0.083 0.983 0.966 0.9836 0.083 0.967 0.934 0.9677 0.083 0.944 0.887 0.9448 0.083 0.912 0.824 0.9129 0.083 0.871 0.742 0.871

10 0.083 0.820 0.641 0.82011 0.083 0.759 0.517 0.75912 0.083 0.685 0.370 0.685

Roof 0.083 0.599 0.198 0.599W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)3600 22854 3600 170000

-270-


= 12, T1 = 1.8, Stiff. & Str. = Unif.


1 0.083 1.000 #REF! #REF!2 0.083 1.000 1.000 1.0003 0.083 1.000 0.997 1.0004 0.083 1.000 0.986 1.0005 0.083 1.000 0.966 1.0006 0.083 1.000 0.934 1.0007 0.083 1.000 0.887 1.0008 0.083 1.000 0.824 1.0009 0.083 1.000 0.742 1.000

10 0.083 1.000 0.641 1.00011 0.083 1.000 0.517 1.00012 0.083 1.000 0.370 1.000

Roof 0.083 1.000 0.198 1.000W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)3600 21860 3600 170000


= 12, T1 = 2.4, Stiff. & Str. = Shear.


1 0.083 1.000 #REF! #REF!2 0.083 1.000 1.000 1.0003 0.083 0.998 0.998 0.9984 0.083 0.992 0.992 0.9925 0.083 0.977 0.977 0.9776 0.083 0.951 0.951 0.9517 0.083 0.912 0.912 0.9128 0.083 0.855 0.855 0.8559 0.083 0.779 0.779 0.779

10 0.083 0.680 0.680 0.68011 0.083 0.555 0.555 0.55512 0.083 0.402 0.402 0.402

Roof 0.083 0.218 0.218 0.218W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)3600 13474 3600 169000

-271-


= 12, T1 = 2.4, Stiff. & Str. = Int.


1 0.083 1.000 #REF! #REF!2 0.083 1.000 1.000 1.0003 0.083 0.999 0.998 0.9994 0.083 0.996 0.992 0.9965 0.083 0.989 0.977 0.9896 0.083 0.976 0.951 0.9767 0.083 0.956 0.912 0.9568 0.083 0.928 0.855 0.9289 0.083 0.890 0.779 0.890

10 0.083 0.840 0.680 0.84011 0.083 0.778 0.555 0.77812 0.083 0.701 0.402 0.701

Roof 0.083 0.609 0.218 0.609W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)3600 12758 3600 169000


= 12, T1 = 2.4, Stiff. & Str. = Unif.


1 0.083 1.000 #REF! #REF!2 0.083 1.000 1.000 1.0003 0.083 1.000 0.998 1.0004 0.083 1.000 0.992 1.0005 0.083 1.000 0.977 1.0006 0.083 1.000 0.951 1.0007 0.083 1.000 0.912 1.0008 0.083 1.000 0.855 1.0009 0.083 1.000 0.779 1.000

10 0.083 1.000 0.680 1.00011 0.083 1.000 0.555 1.00012 0.083 1.000 0.402 1.000

Roof 0.083 1.000 0.218 1.000W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)3600 12297 3600 169000

-272-


= 16, T1 = 1.6, Stiff. & Str. = Shear.


1 0.063 1.000 #REF! #REF!2 0.063 1.000 1.000 1.0003 0.063 0.998 0.998 0.9984 0.063 0.992 0.992 0.9925 0.063 0.981 0.981 0.9816 0.063 0.964 0.964 0.9647 0.063 0.940 0.940 0.9408 0.063 0.907 0.907 0.9079 0.063 0.866 0.866 0.86610 0.063 0.816 0.816 0.81611 0.063 0.756 0.756 0.75612 0.063 0.684 0.684 0.68413 0.063 0.602 0.602 0.60214 0.063 0.508 0.508 0.50815 0.063 0.401 0.401 0.40116 0.063 0.281 0.281 0.281

Roof 0.063 0.147 0.147 0.147W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)4800 55154 4800 226000


= 16, T1 = 1.6, Stiff. & Str. = Int.


1 0.063 1.000 #REF! #REF!2 0.063 1.000 1.000 1.0003 0.063 0.999 0.998 0.9994 0.063 0.996 0.992 0.9965 0.063 0.991 0.981 0.9916 0.063 0.982 0.964 0.9827 0.063 0.970 0.940 0.9708 0.063 0.954 0.907 0.9549 0.063 0.933 0.866 0.93310 0.063 0.908 0.816 0.90811 0.063 0.878 0.756 0.87812 0.063 0.842 0.684 0.84213 0.063 0.801 0.602 0.80114 0.063 0.754 0.508 0.75415 0.063 0.700 0.401 0.70016 0.063 0.640 0.281 0.640

Roof 0.063 0.574 0.147 0.574W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)4800 51155 4800 226000

-273-


= 16, T1 = 1.6, Stiff. & Str. = Unif.


1 0.063 1.000 #REF! #REF!2 0.063 1.000 1.000 1.0003 0.063 1.000 0.998 1.0004 0.063 1.000 0.992 1.0005 0.063 1.000 0.981 1.0006 0.063 1.000 0.964 1.0007 0.063 1.000 0.940 1.0008 0.063 1.000 0.907 1.0009 0.063 1.000 0.866 1.00010 0.063 1.000 0.816 1.00011 0.063 1.000 0.756 1.00012 0.063 1.000 0.684 1.00013 0.063 1.000 0.602 1.00014 0.063 1.000 0.508 1.00015 0.063 1.000 0.401 1.00016 0.063 1.000 0.281 1.000

Roof 0.063 1.000 0.147 1.000W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)4800 48728 4800 226000


= 16, T1 = 2.4, Stiff. & Str. = Shear.


1 0.063 1.000 #REF! #REF!2 0.063 1.000 1.000 1.0003 0.063 0.999 0.999 0.9994 0.063 0.996 0.996 0.9965 0.063 0.990 0.990 0.9906 0.063 0.979 0.979 0.9797 0.063 0.961 0.961 0.9618 0.063 0.936 0.936 0.9369 0.063 0.903 0.903 0.903

10 0.063 0.859 0.859 0.85911 0.063 0.804 0.804 0.80412 0.063 0.737 0.737 0.73713 0.063 0.656 0.656 0.65614 0.063 0.559 0.559 0.55915 0.063 0.447 0.447 0.44716 0.063 0.317 0.317 0.317

Roof 0.063 0.169 0.169 0.169W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)4800 23798 4800 226000

-274-


= 16, T1 = 2.4, Stiff. & Str. = Int.


1 0.063 1.000 #REF! #REF!2 0.063 1.000 1.000 1.0003 0.063 1.000 0.999 1.0004 0.063 0.998 0.996 0.9985 0.063 0.995 0.990 0.9956 0.063 0.989 0.979 0.9897 0.063 0.981 0.961 0.9818 0.063 0.968 0.936 0.9689 0.063 0.951 0.903 0.951

10 0.063 0.930 0.859 0.93011 0.063 0.902 0.804 0.90212 0.063 0.868 0.737 0.86813 0.063 0.828 0.656 0.82814 0.063 0.780 0.559 0.78015 0.063 0.724 0.447 0.72416 0.063 0.659 0.317 0.659

Roof 0.063 0.584 0.169 0.584W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)4800 22490 4800 226000


= 16, T1 = 2.4, Stiff. & Str. = Unif.


1 0.063 1.000 #REF! #REF!2 0.063 1.000 1.000 1.0003 0.063 1.000 0.999 1.0004 0.063 1.000 0.996 1.0005 0.063 1.000 0.990 1.0006 0.063 1.000 0.979 1.0007 0.063 1.000 0.961 1.0008 0.063 1.000 0.936 1.0009 0.063 1.000 0.903 1.000

10 0.063 1.000 0.859 1.00011 0.063 1.000 0.804 1.00012 0.063 1.000 0.737 1.00013 0.063 1.000 0.656 1.00014 0.063 1.000 0.559 1.00015 0.063 1.000 0.447 1.00016 0.063 1.000 0.317 1.000

Roof 0.063 1.000 0.169 1.000W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)4800 21657 4800 226000

-275-


= 16, T1 = 3.2, Stiff. & Str. = Shear.


1 0.063 1.000 #REF! #REF!2 0.063 1.000 1.000 1.0003 0.063 0.999 0.999 0.9994 0.063 0.997 0.997 0.9975 0.063 0.991 0.991 0.9916 0.063 0.980 0.980 0.9807 0.063 0.963 0.963 0.9638 0.063 0.939 0.939 0.9399 0.063 0.906 0.906 0.906

10 0.063 0.864 0.864 0.86411 0.063 0.809 0.809 0.80912 0.063 0.743 0.743 0.74313 0.063 0.662 0.662 0.66214 0.063 0.566 0.566 0.56615 0.063 0.453 0.453 0.45316 0.063 0.322 0.322 0.322

Roof 0.063 0.171 0.171 0.171W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)4800 13346 4800 226000


= 16, T1 = 3.2, Stiff. & Str. = Int.


1 0.063 1.000 #REF! #REF!2 0.063 1.000 1.000 1.0003 0.063 1.000 0.999 1.0004 0.063 0.998 0.997 0.9985 0.063 0.995 0.991 0.9956 0.063 0.990 0.980 0.9907 0.063 0.982 0.963 0.9828 0.063 0.970 0.939 0.9709 0.063 0.953 0.906 0.953

10 0.063 0.932 0.864 0.93211 0.063 0.905 0.809 0.90512 0.063 0.871 0.743 0.87113 0.063 0.831 0.662 0.83114 0.063 0.783 0.566 0.78315 0.063 0.726 0.453 0.72616 0.063 0.661 0.322 0.661

Roof 0.063 0.586 0.171 0.586W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)4800 12636 4800 226000

-276-


= 16, T1 = 3.2, Stiff. & Str. = Unif.


1 0.063 1.000 #REF! #REF!2 0.063 1.000 1.000 1.0003 0.063 1.000 0.999 1.0004 0.063 1.000 0.997 1.0005 0.063 1.000 0.991 1.0006 0.063 1.000 0.980 1.0007 0.063 1.000 0.963 1.0008 0.063 1.000 0.939 1.0009 0.063 1.000 0.906 1.000

10 0.063 1.000 0.864 1.00011 0.063 1.000 0.809 1.00012 0.063 1.000 0.743 1.00013 0.063 1.000 0.662 1.00014 0.063 1.000 0.566 1.00015 0.063 1.000 0.453 1.00016 0.063 1.000 0.322 1.000

Roof 0.063 1.000 0.171 1.000W Ib,1 V1 Mb,1

γ = 1.0 (k) (in4) (k) (k.in)4800 12183 4800 226000

Table B.49 Modal properties of generic shear wall: N = 4, T1 = var.


1 1.000 -1.201 0.696 0.051 1 0.000 0.000 0.000 0.0002 0.155 0.660 0.210 0.027 2 -0.093 0.505 1.000 -1.0003 0.055 0.379 0.069 0.050 3 -0.328 1.000 0.334 0.9694 0.031 -0.223 0.024 0.071 4 -0.647 0.544 -0.972 -0.619

Roof -1.000 -0.727 0.427 0.175



1 1.000 -1.645 0.653 0.050 1 0.000 0.000 0.000 0.000 0.0002 0.158 -0.910 0.200 0.025 2 -0.025 -0.171 0.429 -0.683 0.9373 0.056 0.533 0.069 0.050 3 -0.095 -0.527 1.000 -1.000 0.5404 0.029 -0.380 0.035 0.091 4 -0.200 -0.853 0.958 -0.007 -0.9235 0.017 0.292 0.021 0.138 5 -0.333 -0.983 0.225 0.951 -0.295

6 -0.486 -0.829 -0.642 0.491 1.0007 -0.652 -0.391 -0.931 -0.677 0.0708 -0.825 0.255 -0.342 -0.752 -0.959

Roof -1.000 1.000 0.829 0.619 0.467

-277-



1 1.000 -1.993 0.639 0.050 1 0.000 0.000 0.000 0.000 0.0002 0.159 -1.104 0.196 0.025 2 -0.076 -0.076 -0.213 0.389 -0.5373 0.057 -0.647 0.067 0.050 3 -0.259 -0.259 -0.631 0.948 -1.0004 0.029 0.462 0.034 0.093 4 -0.485 -0.485 -0.959 1.000 -0.4765 0.017 -0.359 0.021 0.149 5 -0.695 -0.695 -1.000 0.396 0.546

6 -0.840 -0.840 -0.702 -0.467 0.9147 -0.883 -0.883 -0.161 -0.976 0.2128 -0.805 -0.805 0.421 -0.767 -0.7399 -0.606 -0.606 0.817 -0.002 -0.83410 -0.297 -0.297 0.864 0.745 0.03211 0.094 0.094 0.517 0.896 0.82912 0.536 0.536 -0.141 0.271 0.563

Roof 1.000 1.000 -0.954 -0.845 -0.667



1 1.000 -2.289 0.633 0.050 1 0.000 0.000 0.000 0.000 0.0002 0.159 -1.268 0.194 0.024 2 -0.007 -0.043 -0.124 0.238 0.3643 0.057 -0.743 0.067 0.050 3 -0.025 -0.153 -0.403 0.686 0.9064 0.029 0.531 0.034 0.093 4 -0.056 -0.302 -0.706 1.000 1.0005 0.017 0.413 0.021 0.151 5 -0.096 -0.467 -0.922 0.965 0.475

6 -0.145 -0.621 -0.976 0.557 -0.3487 -0.202 -0.744 -0.841 -0.070 -0.9158 -0.266 -0.820 -0.536 -0.656 -0.8399 -0.336 -0.837 -0.126 -0.955 -0.177

10 -0.411 -0.787 0.302 -0.842 0.61011 -0.489 -0.670 0.649 -0.368 0.97512 -0.571 -0.490 0.836 0.257 0.66513 -0.655 -0.254 0.813 0.756 -0.09914 -0.740 0.026 0.573 0.900 -0.77215 -0.826 0.336 0.149 0.597 -0.85916 -0.913 0.664 -0.398 -0.077 -0.229

Roof -1.000 1.000 -1.000 -0.942 0.839

Table B.53 Generic shear wall weight and strength properties: N = 4, T1 = var., Str. = var.

Mi/Mbase = 1.0 Mi/Mbase = -0.05/floorT = 0.2 T = 0.3 T = 0.4 T = 0.2 T = 0.3 T = 0.4

Floor Wi/W Mi/Mbase Mi/Mbase Mi/Mbase Mi/Mbase Mi/Mbase Mi/Mbase

1 0.000 1.000 1.000 1.000 1.000 1.000 1.0002 0.250 1.000 1.000 1.000 0.950 0.950 0.9503 0.250 1.000 1.000 1.000 0.900 0.900 0.9004 0.250 1.000 1.000 1.000 0.850 0.850 0.850

Roof 0.250 1.000 1.000 1.000 0.800 0.800 0.800W Mbase Mbase Mbase Mbase Mbase Mbase

γ = 1.0 (k) (k.in) (k.in) (k.in) (k.in) (k.in) (k.in)800 345600 345600 345600 345600 345600 345600

-278-

Table B.54 Generic shear wall weight and strength properties: N = 8, T1 = var., Str. = var.



1 0.000 1.000 1.000 1.000 1.000 1.000 1.0002 0.125 1.000 1.000 1.000 0.950 0.950 0.9503 0.125 1.000 1.000 1.000 0.900 0.900 0.9004 0.125 1.000 1.000 1.000 0.850 0.850 0.8505 0.125 1.000 1.000 1.000 0.800 0.800 0.8006 0.125 1.000 1.000 1.000 0.750 0.750 0.7507 0.125 1.000 1.000 1.000 0.700 0.700 0.7008 0.125 1.000 1.000 1.000 0.650 0.650 0.650




var.



1 0.000 1.000 1.000 1.000 1.000 1.000 1.0002 0.083 1.000 1.000 1.000 0.950 0.950 0.9503 0.083 1.000 1.000 1.000 0.900 0.900 0.9004 0.083 1.000 1.000 1.000 0.850 0.850 0.8505 0.083 1.000 1.000 1.000 0.800 0.800 0.8006 0.083 1.000 1.000 1.000 0.750 0.750 0.7507 0.083 1.000 1.000 1.000 0.700 0.700 0.7008 0.083 1.000 1.000 1.000 0.650 0.650 0.6509 0.083 1.000 1.000 1.000 0.600 0.600 0.600

10 0.083 1.000 1.000 1.000 0.550 0.550 0.55011 0.083 1.000 1.000 1.000 0.500 0.500 0.50012 0.083 1.000 1.000 1.000 0.450 0.450 0.450



-279-


var.



1 0.000 1.000 1.000 1.000 1.000 1.000 1.0002 0.063 1.000 1.000 1.000 0.950 0.950 0.9503 0.063 1.000 1.000 1.000 0.900 0.900 0.9004 0.063 1.000 1.000 1.000 0.850 0.850 0.8505 0.063 1.000 1.000 1.000 0.800 0.800 0.8006 0.063 1.000 1.000 1.000 0.750 0.750 0.7507 0.063 1.000 1.000 1.000 0.700 0.700 0.7008 0.063 1.000 1.000 1.000 0.650 0.650 0.6509 0.063 1.000 1.000 1.000 0.600 0.600 0.60010 0.063 1.000 1.000 1.000 0.550 0.550 0.55011 0.063 1.000 1.000 1.000 0.500 0.500 0.50012 0.063 1.000 1.000 1.000 0.450 0.450 0.45013 0.063 1.000 1.000 1.000 0.400 0.400 0.40014 0.063 1.000 1.000 1.000 0.350 0.350 0.35015 0.063 1.000 1.000 1.000 0.300 0.300 0.30016 0.063 1.000 1.000 1.000 0.250 0.250 0.250



- 281 -

APPENDIX C LINEAR MULTIVARIATE REGRESSION ANALYSIS

Throughout this study, we tried to develop simple linear relations between a response parameter

(e.g., median of collapse capacity ηc) and basic structural parameters (e.g., component plastic

hinge rotation capacity θp). Although the decision on the form of the linear multivariate

regression model is tied to the choice of predictor variables (Neter et. al., 1996), but the general

approach for finding the associated regression coefficients and tests of hypotheses is standard.

In a multiple regression model, we assume that the relationship between response

variable Y and p predictor variables X1 … Xp can be shown in the form of Equation C.1.

0 1 1 1 1p pY X Xβ β β ε− −= + + + + (C.1)

In Equation C.1, β0, β1, …, βp are constants known as regression coefficients and ε is a

random variable representing the error in the regression estimates. Using the regression model

presented in Equation C.1, each ith realization out of n number of realizations could be written as

shown in Equation C.2 or in matrix format as shown in Equation C.3.

0 1 1 1 , 1i p i p iY X Xβ β β ε− −= + + + + (C.2)

1 1Y = X β εn n p nn p× × ××

⋅ + (C.3)

In Equation C.3, Y is the vector of responses, X is the matrix of predictor variables, β is

the vector of regression coefficients, and ε is the vector of regression errors (i.e., residuals).

Using the least square method, the vector of regression coefficients b and sum of squares of

estimation residuals SSE are obtained using Equation C.4 and C.5, respectively.

( ) ( )1

11

b = X X X Yp

p p p

−

×× ×

′ ′ (C.4)

- 282 -

2ˆ=e e=YY -b X Y= ( )SSE y yi i′ ′ ′ ′ −∑ (C.5)

SSE is obtained using vector b that has p parameters. This means that SSE has n-p

degrees of freedom. The mean of SSE, denoted as MSE, is obtained by dividing SSE by the

associated degrees of freedom as shown in Equation C.6. The expectation of MSE is σ2 that is

the dispersion in of random variable ε, i.e., Equation C.7.

= SSEMSE

n p− (C.6)

2 E MSEσ = ⎡ ⎤⎣ ⎦ (C.7)

To test the adequacy of the fit, “coefficient of multiple determination”, denoted as R2, is

defined as shown in Equation C.8.

2

2

2

ˆ( )1 1

( )

y y SSEi iRSSTy yi

−= − = −

−∑∑ (C.8)

where SST is the total sum of squared deviations in y from its mean value, Equation C.9.

2( )SST y yi −= ∑ (C.8)

R2 measures the proportion of the total variability in Y due to use of the set of variables

X. R2 varies between 0 and 1. When a model fits well with the data, R2 is close to unity.

However, R2 is not good measure for comparing two regression models because adding more

variables to X only increases the value of R2. For this reason, it is suggested to use an adjusted

form of R2 where considers the number of predictors in the model. This measure is called

“adjusted coefficient of multiple determination”, denoted by 2aR and defined as shown in

Equation C.9.

2 1

1

SSEn pR SSTn

−= −−

(C.9)

- 283 -

2aR can become smaller by adding variables to X. Such case happen when the gain in

addition of a variable to X (i.e., reduction in SSE and SST) is compensated with the loss of

degree of freedom (i.e., increase in n-p).

In order to assess the adequacy of a regression model with less number of variables we

have used the F Test. For this reason the model with all variables (i.e., p variables), denoted as

“Full Model” (FM) is compared to a model with less number of variables (i.e., k variables and k

< p) denoted as “Reduced Model” (RM). Hypothesis H0 and H1 are defined as shown in Equation

C.10.

0 0 1 1 1 , 1

1 0 1 1 1 , 1

: :: :

i k i k RM

i p i p FM

RM H Y X Xp k

FM H Y X Xβ β β εβ β β ε

− −

− −

= + + + +⎧⎪ >⎨ = + + + +⎪⎩ (C.10)

If the goodness of fit of the reduced model RM is as good as the full model FM, the null

hypothesis H0 is not rejected. For this reason the variable F is defined as shown in Equation

C.11.

RM FM

FM

SSE SSE p kF

SSE n p− −⎡ ⎤⎣ ⎦=

− (C.11)

In Equation C.11, SSERM, and SSERM are the SSE associated with the reduced model RM

and full model FM, respectively. It is assumed that the variable F defined in Equation C.11 has F

distribution with (p-k) and (n-p) degrees of freedom (i.e., Figure C.1). If the value of F obtained

from Equation C.11 is larger than the tabulated value shown in Figure C.1 for the associated

degrees of freedom (i.e., Equation C.12) and significance level α, then the reduced model RM is

rejected.

( ), ;p k n pF F α− −≥ (C.12)

- 284 -

1 2 4 6 8 10 12 24 2000001 161.45 199.50 224.58 233.99 238.88 241.88 243.90 249.05 254.322 18.51 19.00 19.25 19.33 19.37 19.40 19.41 19.45 19.503 10.13 9.55 9.12 8.94 8.85 8.79 8.74 8.64 8.534 7.71 6.94 6.39 6.16 6.04 5.96 5.91 5.77 5.635 6.61 5.79 5.19 4.95 4.82 4.74 4.68 4.53 4.376 5.99 5.14 4.53 4.28 4.15 4.06 4.00 3.84 3.677 5.59 4.74 4.12 3.87 3.73 3.64 3.57 3.41 3.238 5.32 4.46 3.84 3.58 3.44 3.35 3.28 3.12 2.939 5.12 4.26 3.63 3.37 3.23 3.14 3.07 2.90 2.7110 4.96 4.10 3.48 3.22 3.07 2.98 2.91 2.74 2.5411 4.84 3.98 3.36 3.09 2.95 2.85 2.79 2.61 2.4012 4.75 3.89 3.26 3.00 2.85 2.75 2.69 2.51 2.3013 4.67 3.81 3.18 2.92 2.77 2.67 2.60 2.42 2.2114 4.60 3.74 3.11 2.85 2.70 2.60 2.53 2.35 2.1315 4.54 3.68 3.06 2.79 2.64 2.54 2.48 2.29 2.0720 4.35 3.49 2.87 2.60 2.45 2.35 2.28 2.08 1.8425 4.24 3.39 2.76 2.49 2.34 2.24 2.16 1.96 1.7130 4.17 3.32 2.69 2.42 2.27 2.16 2.09 1.89 1.6240 4.08 3.23 2.61 2.34 2.18 2.08 2.00 1.79 1.5160 4.00 3.15 2.53 2.25 2.10 1.99 1.92 1.70 1.39

120 3.92 3.07 2.45 2.18 2.02 1.91 1.83 1.61 1.25200000 3.84 3.00 2.37 2.10 1.94 1.83 1.75 1.52 1.01

n 1

n 2

( )1 2 1 2, , ;0.05 0.05n n n nP F f≥ =

1 2 4 6 8 10 12 24 2000001 161.45 199.50 224.58 233.99 238.88 241.88 243.90 249.05 254.322 18.51 19.00 19.25 19.33 19.37 19.40 19.41 19.45 19.503 10.13 9.55 9.12 8.94 8.85 8.79 8.74 8.64 8.534 7.71 6.94 6.39 6.16 6.04 5.96 5.91 5.77 5.635 6.61 5.79 5.19 4.95 4.82 4.74 4.68 4.53 4.376 5.99 5.14 4.53 4.28 4.15 4.06 4.00 3.84 3.677 5.59 4.74 4.12 3.87 3.73 3.64 3.57 3.41 3.238 5.32 4.46 3.84 3.58 3.44 3.35 3.28 3.12 2.939 5.12 4.26 3.63 3.37 3.23 3.14 3.07 2.90 2.7110 4.96 4.10 3.48 3.22 3.07 2.98 2.91 2.74 2.5411 4.84 3.98 3.36 3.09 2.95 2.85 2.79 2.61 2.4012 4.75 3.89 3.26 3.00 2.85 2.75 2.69 2.51 2.3013 4.67 3.81 3.18 2.92 2.77 2.67 2.60 2.42 2.2114 4.60 3.74 3.11 2.85 2.70 2.60 2.53 2.35 2.1315 4.54 3.68 3.06 2.79 2.64 2.54 2.48 2.29 2.0720 4.35 3.49 2.87 2.60 2.45 2.35 2.28 2.08 1.8425 4.24 3.39 2.76 2.49 2.34 2.24 2.16 1.96 1.7130 4.17 3.32 2.69 2.42 2.27 2.16 2.09 1.89 1.6240 4.08 3.23 2.61 2.34 2.18 2.08 2.00 1.79 1.5160 4.00 3.15 2.53 2.25 2.10 1.99 1.92 1.70 1.39

120 3.92 3.07 2.45 2.18 2.02 1.91 1.83 1.61 1.25200000 3.84 3.00 2.37 2.10 1.94 1.83 1.75 1.52 1.01

n 1

n 2

( )1 2 1 2, , ;0.05 0.05n n n nP F f≥ =

(a)

1 2 4 6 8 10 12 24 2000001 4052.18 4999.34 5624.26 5858.95 5980.95 6055.93 6106.68 6234.27 6365.592 98.50 99.00 99.25 99.33 99.38 99.40 99.42 99.46 99.503 34.12 30.82 28.71 27.91 27.49 27.23 27.05 26.60 26.134 21.20 18.00 15.98 15.21 14.80 14.55 14.37 13.93 13.465 16.26 13.27 11.39 10.67 10.29 10.05 9.89 9.47 9.026 13.75 10.92 9.15 8.47 8.10 7.87 7.72 7.31 6.887 12.25 9.55 7.85 7.19 6.84 6.62 6.47 6.07 5.658 11.26 8.65 7.01 6.37 6.03 5.81 5.67 5.28 4.869 10.56 8.02 6.42 5.80 5.47 5.26 5.11 4.73 4.3110 10.04 7.56 5.99 5.39 5.06 4.85 4.71 4.33 3.9111 9.65 7.21 5.67 5.07 4.74 4.54 4.40 4.02 3.6012 9.33 6.93 5.41 4.82 4.50 4.30 4.16 3.78 3.3613 9.07 6.70 5.21 4.62 4.30 4.10 3.96 3.59 3.1714 8.86 6.51 5.04 4.46 4.14 3.94 3.80 3.43 3.0015 8.68 6.36 4.89 4.32 4.00 3.80 3.67 3.29 2.8720 8.10 5.85 4.43 3.87 3.56 3.37 3.23 2.86 2.4225 7.77 5.57 4.18 3.63 3.32 3.13 2.99 2.62 2.1730 7.56 5.39 4.02 3.47 3.17 2.98 2.84 2.47 2.0140 7.31 5.18 3.83 3.29 2.99 2.80 2.66 2.29 1.8060 7.08 4.98 3.65 3.12 2.82 2.63 2.50 2.12 1.60

120 6.85 4.79 3.48 2.96 2.66 2.47 2.34 1.95 1.38200000 6.63 4.61 3.32 2.80 2.51 2.32 2.18 1.79 1.01

n 1

n 2

( )1 2 1 2, , ;0.01 0.01n n n nP F f≥ =

1 2 4 6 8 10 12 24 2000001 4052.18 4999.34 5624.26 5858.95 5980.95 6055.93 6106.68 6234.27 6365.592 98.50 99.00 99.25 99.33 99.38 99.40 99.42 99.46 99.503 34.12 30.82 28.71 27.91 27.49 27.23 27.05 26.60 26.134 21.20 18.00 15.98 15.21 14.80 14.55 14.37 13.93 13.465 16.26 13.27 11.39 10.67 10.29 10.05 9.89 9.47 9.026 13.75 10.92 9.15 8.47 8.10 7.87 7.72 7.31 6.887 12.25 9.55 7.85 7.19 6.84 6.62 6.47 6.07 5.658 11.26 8.65 7.01 6.37 6.03 5.81 5.67 5.28 4.869 10.56 8.02 6.42 5.80 5.47 5.26 5.11 4.73 4.3110 10.04 7.56 5.99 5.39 5.06 4.85 4.71 4.33 3.9111 9.65 7.21 5.67 5.07 4.74 4.54 4.40 4.02 3.6012 9.33 6.93 5.41 4.82 4.50 4.30 4.16 3.78 3.3613 9.07 6.70 5.21 4.62 4.30 4.10 3.96 3.59 3.1714 8.86 6.51 5.04 4.46 4.14 3.94 3.80 3.43 3.0015 8.68 6.36 4.89 4.32 4.00 3.80 3.67 3.29 2.8720 8.10 5.85 4.43 3.87 3.56 3.37 3.23 2.86 2.4225 7.77 5.57 4.18 3.63 3.32 3.13 2.99 2.62 2.1730 7.56 5.39 4.02 3.47 3.17 2.98 2.84 2.47 2.0140 7.31 5.18 3.83 3.29 2.99 2.80 2.66 2.29 1.8060 7.08 4.98 3.65 3.12 2.82 2.63 2.50 2.12 1.60

120 6.85 4.79 3.48 2.96 2.66 2.47 2.34 1.95 1.38200000 6.63 4.61 3.32 2.80 2.51 2.32 2.18 1.79 1.01

n 1

n 2

( )1 2 1 2, , ;0.01 0.01n n n nP F f≥ =

(b)

Fig. C.1 α Critical values for F Distribution with n1 and n2 degrees of freedom , where

( )1 2 1 2, , ;n n n nP F f α α≥ = . (a) α = 0.05, (b) α = 0.01

- 285 -

APPENDIX D MOELING PLASTIC HINGES AND STRUCTURAL DAMPING

D.1 Introduction

Modeling nonlinear behavior using plastic hinges (i.e., concentrated plasticity) in combination

with Rayleigh damping need special considerations. Without proper modeling of structural

elements, damping forces become unrealistically large at plastic hinge locations (Hall, 2006;

Medina and Krawinkler, 2003; Bernal, 1994). In this appendix, a simple methodology for

modeling concentrated plasticity along with Rayleigh damping is provided. In this method, a

beam element (i.e., denoted as “original” beam element) is modeled as a combination of an

elastic beam element with one/two end spring(s). It is assumed that all the inelastic deformation

is concentrated in the end spring(s) with no contribution to damping forces. The stiffness

proportional damping (i.e., modeling structural damping using the Rayleigh method involves

decomposition of the dynamic damping matrix to a mass proportional and stiffness proportional

damping matrices) is concentrated in the elastic element. This method for modeling Rayleigh

damping and nonlinear action was first proposed by Medina & Krawinkler (2003). This work is a

continuation and enhancement of their work. Other studies by Bernal (1994) and Hall (2006)

have pointed out same problems involving misuse of Rayleigh damping and remedies are

provided.

D.2 Rayleigh Damping

During the response history analysis, the equation of motion in the form of Equation D.1 is

solved at each time step.

[ ] ( ){ } [ ] ( ){ } [ ] ( ){ } [ ] ( ){ }gM x t C x t K x t M x t+ + = − (D.1)

- 286 -

where [ ]M is the mass matrix, [ ]C is the damping matrix, [ ]K is the tangent stiffness

matrix, ( ){ }x t is a vector of deformation response at degrees of freedom, ( ){ }gx t is the ground

motion acceleration vector at degrees of freedom, and dots represent a derivation to time. One

method for solving this equation is to decompose [ ]C in the form of Equation D.2 (i.e., Rayleigh

damping):

[ ] [ ] [ ]0C M Kα β= + (D.2)

where α and β are denoted as the mass proportional, and stiffness proportional, damping

coefficients respectively. [ ]0K is the initial (elastic) stiffness matrix. These coefficients are

obtained using Equation D.3 and Equation D.4, respectively (Hart, 1975).

2 22 j i i ji j

i j

ξ ω ξωα ωω

ω ω−

=−

(D.3)

2 22 j i i j

i j

ξ ω ξωβ

ω ω−

=−

(D.4)

In Equation D.3 and Equation D.4, ωi and ωj are the two circular frequencies that we

want to set their critical damping ratio to ξi and ξj, respectively.

D.3 Implementation of Rayleigh Damping in DRAIN-2DX and State of the Problem

DRAIN-2DX program is a tool that we have used to perform nonlinear response history analysis.

At each time step, the program solves the Newmark equation using average acceleration method

as shown in Equation D.5.

( )

[ ] ( ) [ ] { } ( )[ ]{ } [ ] ( )( )

( ) { } { }2 24 2 4

g damp elasticM C K x x t t M M x t x t F Ftt t

⎡ ⎤ ⎧ ⎫⎪ ⎪+ + Δ = − +Δ + + + −⎢ ⎥ ⎨ ⎬ΔΔ Δ⎢ ⎥ ⎪ ⎪⎣ ⎦ ⎩ ⎭

(D.5)

where { }elasticF and { }dampF are vectors of damping and elastic forces, respectively, at

degrees of freedom. As explained by Median & Krawinkler (2003), damping forces are

calculated as the product of the stiffness proportional factor, β, initial stiffness matrix, and

- 287 -

current velocity of degrees of freedom. The left hand side of Equation D.5 includes the damping

matrix and tangent stiffness matrix. With this setup, velocities used to calculate { }dampF will be

“correct” if the damping matrix is decomposed using the tangent stiffness matrix which is not the

case if Equation D.2 is used for decomposition of the damping matrix. For this reason, excessive

forces appear in those degrees of freedom that their stiffness is not equal to their initial stiffness

D.4 Solution to the Damping Problem

Several solutions to the damping problem have been proposed by Hall (2006), Medina &

Krawinkler (2003), and Bernal (1994). These solutions are:

• Using only mass proportional damping. This solution is not an attractive one because the

higher modes will be under-damped. Usually the critical damping ratios of higher modes

are not much smaller than critical damping ratios of lower modes (Hart, 1975).

• Using tangent stiffness matrix for decomposing the damping matrix instead of initial

stiffness matrix. This solution not only increases the run-time, but also become

ambiguous to implement for cases where stiffness becomes negative (i.e., monotonic

deterioration).

• Modeling each beam element with a combination of an elastic beam element and

rotational end spring(s). Plastic hinging occurs in these zero-length rotational spring

elements with zero damping. As the initial stiffness of the zero-length rotational end

springs are set to be much larger than the stiffness of the elastic beam, most of the elastic

deformation occurs in the elastic beam. We assign all the damping, as an equivalent

stiffness proportional damping coefficient, to the elastic beam element.

• Using a modified damping formulation in which stiffness-proportional contribution is

bounded (Hall 2006)

• Using static condensation to eliminate massless degrees of freedom (Bernal, 1994).

- 288 -

Among solutions provided above, the third method was pursed by Medina & Krawinkler

(2003). In their study, plastic hinging is modeled by using rotational spring at both ends of elastic

beam elements. Figure D.1 shows this type of modeling. The upper portion of Figure D.1 shows

an ordinary beam element and the lower portion of this figure shows the equivalent elastic beam

element with end springs. For each case, the left hand side shows the nodes needed for modeling

these elements and the right hand side shows the associated degrees of freedom. End springs are

zero-length elements (i.e., hollow circles depict end springs) so nodes i, is1, and js1 are essentially

overlapping. The same overlapping exists for j, is2, and js2. If we assume that moments at two

ends of the original beam element are equal in sign and value (i.e., double curvature moment

gradient), then the characteristics of the equivalent element model (i.e., elastic beam with end

springs) is found. For this reason, we assume that the stiffness of rotational springs are equal to n

times the rotational stiffness of the elastic beam element. If the original beam element has a

moment of inertia equal to Ii, modulus of elasticity equal to E, and length of L, then the moment

of inertia of the elastic beam (i.e., Ie) and the stiffness of the end springs (i.e., Ks) are calculated

using Equation D.6 and Equation D.7, respectively (i.e., n = 10 in Medina & Krawinkler 2003).

1e i

nI In+

= (D.6)

( ) 61 is

EIK nL

= + (D.7)

In order to apply all the stiffness proportional damping to the elastic beam element, we

need to find the equivalent stiffness proportional damping factor, β’, for the elastic beam

element. β’ is found by equating the damping work of the original beam element and the

damping work of the elastic beam element in the equivalent model. Equation D.8 shows the

equivalent stiffness proportional damping factor β’.

11n

β β⎛ ⎞′ = +⎜ ⎟⎝ ⎠

(D.8)

An essential assumption in deriving Equation D.6 to Equation D.8 is that the moments at

two ends of the original beam element are equal in sign and value. This assumption has been

valid for the research conducted by Medina & Krawinkler (2003) because they have used one-

bay generic frames where nonlinear behavior only occurs in beams (i.e., no nonlinearity in

- 289 -

columns). In such one-bay generic frames, essentially the moments at two ends of the original

beam are equal in sign and value. Unfortunately, this is not the case for elements used in the

generic structures used in this study. Generic moment-resisting frames of this study are three-bay

frames and inelastic deformation in allowed in columns. Generic shear walls are modeled as a

series of cantilever elements. In both structural systems, moment gradient does not conform to

the assumptions made for finding the properties of the equivalent elastic beam and corresponding

end springs mentioned above.

Another problem that we encountered is that by assigning n = 10, the initial stiffness of

the spring becomes very large. This stiffness compared to the stiffness of the spring after

yielding is significantly large. Therefore, in order to obtain convergence in the step-by-step

dynamic solution algorithm, the time step should be extremely small (i.e., Δt = 0.0001 for

generic moment-resisting frames and no convergence for generic shear walls). At such small

time steps, accuracy of results is a matter of faith. For example, when very small time step is

used, the acceleration response is polluted with many high amplitude spikes which are the

consequence of the Newton-Raphson method for finding the response point in conjunction with

very small time step.

In order to obviate from such problems, we introduce an enhancement to the solution

proposed by Krawinkler & Medina (2003). This enhancement is in the method we obtain the

characteristics of the elastic beam and end springs. These characteristics are obtained regardless

of the moment gradient. Also, using the new method, we can assign smaller values to n (e.g., n =

1 for generic moment-resisting frames and n = 5 for generic shear walls). A smaller value for n

reduces the difference between the elastic stiffness and inelastic stiffness of the spring elements,

therefore the time step could be a reasonable value (e.g., Δt = 0.005 for generic moment-resisting

frames and Δt = 0.0005 for generic shear walls).

D.5 Development of an Equivalent Element Model for Beam Element

D.5.1 Development of an equivalent element model for beam element (two-end-spring)

- 290 -

In Section D.4, characteristics of the elastic beam element and the associated end springs were

found assuming that end moments are equal in sign and value (i.e., double curvature moment

gradient). In this section we want to do the same exercise for a general moment gradient. The

approach for this purpose is to perform a static condensation on the stiffness matrix of the

combination of elastic beam element and end springs (i.e., lower portion of Figure D.1) and

eliminate extra degrees of freedom (i.e., degrees of freedom 3 and 6 in the lower portion of

Figure D.1) such that the condensed stiffness matrix becomes comparable with the stiffness

matrix of the original beam element. Figure D.2 shows the stiffness matrix of the beam element

(i.e., this is the stiffness matrix of a general beam element). The stiffness matrix of the original

beam is shown in Figure D.3. Properties of the elastic beam element are obtained using the

aforementioned method for a given value of n and end spring stiffness. (i.e., Equation D.9 to

D.12.)

6 is

EIK nL

= (D.9)

1i

nI In+

= (D.10)

6(1 )2 3ij ji

nS Sn

+= =

+ (D.11)

6ii jj ijS S S= = − (D.12)

This means that the original beam element could be modeled with an elastic beam

element with moment of inertia obtained from Equation D.10, stiffness coefficients obtained

from Equation D.11 and Equation D.12, and two end springs with initial stiffness obtained from

Equation D.9. In Equation D.7 and Equation D.8, Ii is the stiffness of the original beam element.

Variation of Sii and Sij with n is plotted in Figure D.4. As seen, values of Sii and Sij asymptotically

reach 4.0 and 2.0 for large values of n.

The equivalent stiffness proportional damping coefficient, β’, for the elastic beam

element is found in the same way that was described in Section D.4 by equalizing the damping

work done by the original beam element and the damping work done by the elastic beam

element. Calculations show that Equation D.8 holds for finding β’in this case.

- 291 -

D.5.2 Development of an equivalent element model for beam element (one-end-spring)

Similar to the approach outlined above, equivalent element model with a single spring at end i of

an elastic beam is developed. Such model is useful for cases where only one end of the original

beam element yields. In this study, as long as generic moment-resisting frames are symmetric,

each three-bay frame is modeled as a one-bay-and-a-half frame (i.e., Figure D.5). The interior

beam is modeled with half of it original length with a pin at its far end. Also, each generic shear

wall used in this study is molded as cantilever beam with springs at each floor level. If the initial

stiffness shown in Equation D.9 is used for the end spring, and if the moment of inertia of the

elastic element is obtained using Equation D.10, then Equation D.13, Equation D.14, and

Equation D.15 show the stiffness coefficients for the elastic element. Figure D.5 shows the

variation of these stiffness coefficients for various n values. As seen, values of Sii, Sii, and Sij

asymptotically reach 4.0, 4.0, and 2.0, respectively, for large values of n.

213

ij jinS Sn

= =+

(D.13)

2ii ijS S= (D.14)

1 21jj ij

nS Sn

+=

+ (D.15)

The equivalent stiffness proportional damping coefficient, β’, is found as shown in

Equation D.16. The method to find this coefficient is similar to previous two cases (i.e., by

equalizing the damping work done by the beam element and the damping work done by the

elastic beam element).

112n

β β′ = + (D.16)

-292-


springs at both ends

3 2 3 2

2

3 2

0 0 0 0

( 2 ) ( ) ( 2 ) ( )0

( )0

0 0

( 2 ) ( ).

⎡ ⎤−⎢ ⎥⎢ ⎥

+ + + + + +⎢ ⎥− − −⎢ ⎥⎢ ⎥

+⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

+ + +⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

ii jj ij e ii ji e ii jj ij e jj ij e

ii ji e ji eii e

ii jj ij e jj ij e

jj e

AE AEL L

S S S EI S S EI S S S EI S S EIL L L L

S S EI S EIS EIL L L

AEL

S S S EI S S EISym

L LS EI

L

Fig. D.2 Stiffness matrix of the elastic beam element

1

23

4

5 67 8

1

23

4

5 6

Ordinary beam element

Equivalent elastic beam element with end springs

Modeling Nodes Degrees of freedom


i j

i j

is1 js1 is2 js2

- 293 -

3 2 3 2

2

3 2

0 0 0 0

12 6 12 60

4 6 20

0 0

12 6.

4

⎡ ⎤−⎢ ⎥⎢ ⎥⎢ ⎥− − −⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

i i i i

i i i

i i

i

AE AEL L

EI EI EI EIL L L L

EI EI EIL L L

AEL

EI EISymL L

EIL

Fig. D.3 Stiffness matrix of the original beam element

Stiffness coefficients for equivalent elastic beamsprings at both ends of elastic element

2

2.5

3

3.5

4

0 5 10 15 20 25n

stiff

ness

coe

ffici

ent

S ii = S jj

S ij = S ji

Fig. D.4 Stiffness coefficients for elastic beam in the equivalent model, springs at both ends

of elastic element

- 294 -


spring at one end

Stiffness coefficients for equivalent elastic beamspring at end i of elastic element

0

0.5

1

1.5

2

2.5

3

3.5

4

0 5 10 15 20 25n

stiff

ness

coe

ffici

ent

S ii

S jj

S ij = S ji

Fig. D.6 Stiffness coefficients for elastic beam in the equivalent model, spring at end i of

elastic element

1

23

4

5 67

1

23

4

5 6

Ordinary beam element

Equivalent elastic beam element with end springs



i j

i j

is1 js1

- 295 -

APPENDIX E LIST OF EDPS FOR GENERIC STRUCTURAL SYSTEMS

The EDPs obtained from Incremental Dynamic Analysis of generic structural systems are listed

in this appendix. First, the EDPs recorded and storied for generic moment-resisting frames are

presented. Afterwards, the list of EDPs recorded for generic shear walls are listed.

E.1 List of EDPs for generic moment resisting frames

Global displacement-based EDPs

• Maximum displacement at floor i: maxiδ

• Maximum absolute velocity at floor i: maxiv

• Maximum absolute acceleration at floor i (i.e., this parameter was named as peak floor

acceleration in the text of this dissertation ): maxia

• Maximum drift angle in story i (i.e., this parameter was named as maximum interstory drift

ratio in the text of this dissertation ):

1

max

max

i ii

hδ δ

θ−⎛ ⎞−

= ⎜ ⎟⎝ ⎠

• Residual displacement at floor i: resiδ

• Residual drift angle in story i:

1res res

res

max

i ii

hδ δ

θ−⎛ ⎞−

= ⎜ ⎟⎝ ⎠

Global force-based EDPs

• Maximum story-shear in story i: maxiV

• Maximum floor-overturning moment at floor i: (Column axial load only): max

iaxialOTM

• Maximum floor-overturning moment at floor i: (Column axial load and moment): max

itotOTM

Global energy-based EDPs

- 296 -

• Total damping energy: maxDE

• Total input energy: maxIE

• Total hysteretic energy: maxHE

Local displacement-based EDPs

• Maximum rotation of exterior edge of exterior beam at floor i: ext. beam max

iθ

• Maximum rotation of interior beam at floor i. int. beam max

iθ

• Maximum plastic rotation of exterior edge of exterior beam at floor i. ext._ext. beam max

ipθ

• Maximum plastic rotation of interior edge of exterior beam at floor i. ext._int. beam max

ipθ

• Maximum plastic rotation of interior beam at floor i. int. beam max

ipθ

• Maximum accumulated positive plastic rotation of exterior edge of exterior beam at floor i: ext._ext. beam

.maxi

accpθ +

• Maximum accumulated negative plastic rotation of exterior edge of exterior beam at floor i: ext._ext. beam

.maxi

accpθ −

• Maximum accumulated positive plastic rotation of exterior edge of exterior beam at floor i

(pre-peak): ext._ext. beam

. .maxi

acc ppkpθ +

• Maximum accumulated negative plastic rotation of exterior edge of exterior beam at floor i

(pre-peak): ext._ext. beam

. .maxi

acc ppkpθ −

• Maximum accumulated positive plastic rotation of interior edge of exterior beam at floor i: ext._int. beam

.maxi

accpθ +

• Maximum accumulated negative plastic rotation of interior edge of exterior beam at floor i: ext._int. beam

.maxi

accpθ −

• Maximum accumulated positive plastic rotation of interior edge of exterior beam at floor i

(pre-peak): ext._int. beam

. .maxi

acc ppkpθ +

- 297 -

• Maximum accumulated negative plastic rotation of interior edge of exterior beam at floor i

(pre-peak): ext._int. beam

. .maxi

acc ppkpθ −

• Maximum accumulated positive plastic rotation of interior beam at floor i: int. beam

.maxi

accpθ +

• Maximum accumulated negative plastic rotation of interior beam at floor i: int. beam

.maxi

accpθ −

• Maximum accumulated positive plastic rotation of interior beam at floor i (pre-peak):: int. beam

. .maxi

acc ppkpθ +

• Maximum accumulated negative plastic rotation of interior beam at floor i (pre-peak):: int. beam

. .maxi

acc ppkpθ −

• Maximum residual rotation of exterior edge of exterior beam at floor i: ext. beam res

iθ

• Maximum residual rotation of interior beam at floor i: int. beam res

iθ

• Maximum rotation at top of the exterior column in story i: ext._top column max

iθ

• Maximum rotation at bottom of the exterior column in story i: ext._bot. column max

iθ

• Maximum rotation at top of the interior column in story i: int._top. column max

iθ

• Maximum rotation at bottom of the interior column in story i: int._bot. column max

iθ

• Maximum plastic rotation at top of the exterior column in story i: ext._top column max

ipθ

• Maximum plastic rotation at bottom of the exterior column in story i: ext._bot. column max

ipθ

• Maximum plastic rotation at top of the of interior column in story i:. int._top column max

ipθ

• Maximum plastic rotation at bottom of the interior column in story i: int._bot. column max

ipθ

• Maximum accumulated positive plastic rotation at top of the exterior column in story i: ext._top column

.maxi

accpθ +

• Maximum accumulated negative plastic rotation at top of the exterior column in story i: ext._top column

.maxi

accpθ −

• Maximum accumulated positive plastic rotation at bottom of the exterior column at story i: ext._bot. column

.maxi

accpθ +

- 298 -

• Maximum accumulated negative plastic rotation at bottom of the exterior column at story i: ext._bot. column

.maxi

accpθ −

• Maximum accumulated positive plastic rotation at top of the interior column at story i: int._top column

.maxi

accpθ +

• Maximum accumulated negative plastic rotation at top of the interior column at story i: int._top column

.maxi

accpθ −

• Maximum accumulated positive plastic rotation at bottom of the interior column at story i: int._bot. column

.maxaccpθ +

• Maximum accumulated negative plastic rotation at bottom of the interior column at story i: int._bot. column i

.maxaccpθ −

• Maximum accumulated positive plastic rotation at top of the exterior column at story i (pre-

peak): ext._top column

. .maxi

acc ppkpθ +

• Maximum accumulated negative plastic rotation at top of the exterior column at story i (pre-

peak): ext._top column

. .maxi

acc ppkpθ −

• Maximum accumulated positive plastic rotation at bottom of the exterior column at story i

(pre-peak): ext._bot. column

. .maxi

acc ppkpθ +

• Maximum accumulated negative plastic rotation at bottom of the exterior column at story i

(pre-peak):ext._bot. column

. .maxi

acc ppkpθ −

• Maximum accumulated positive plastic rotation at top of the interior column at story i (pre-

peak): int._top column

. .maxacc ppkpθ +

• Maximum accumulated negative plastic rotation at top of the interior column at story i (pre-

peak): int._top column

. .maxi

acc ppkpθ −

• Maximum accumulated positive plastic rotation at bottom of the interior column at story i

(pre-peak): int._bot. column

. .maxi

acc ppkpθ +

• Maximum accumulated negative plastic rotation at bottom of the interior column at story i

(pre-peak): int._bot. column

. .maxi

acc ppkpθ −

- 299 -

• Maximum residual rotation at top of the exterior column at story i: ext._top column iresθ

• Maximum residual rotation at bottom of the exterior column at story i: int._bot. column iresθ

• Maximum residual rotation at top of the interior column at story i: int._top column iresθ

• Maximum residual rotation at bottom of the interior column at story i: int._bot. column iresθ

Local forced-based EDPs

• Maximum moment at exterior edge of exterior beam at floor i: ext._ext. beam max

iM

• Maximum moment at interior edge of exterior beam at floor i: ext._int. beam max

iM

• Maximum moment of interior beam at floor i: int. beam max

iM

• Maximum moment at top of the exterior column in story i: ext._top column max

iM

• Maximum moment at top of the interior column in story i: int._top column max

iM

• Maximum moment at bottom of the exterior column in story i: ext._bot. column max

iM

• Maximum moment at bottom of the interior column in story i: int._bot. column max

iM

• Maximum moment at middle of the exterior column in story i: ext._mid. column max

iM

• Maximum moment at middle of the interior column in story i: int._mid. column max

iM

• Maximum shear force in exterior column in story i: ext. column

maxiV

• Maximum shear force in interior column in story i: int. column

maxiV

• Maximum axial compression force in exterior column at story i: ext. column max

icP

• Maximum axial compression force in interior column at story i: int. column max

icP

• Maximum axial tension force in exterior column at story i: ext. column max

itP

• Maximum axial tension force in interior column at story i: int. column max

itP

- 300 -

Local energy-based EDPs

• Maximum hysteretic energy dissipated in exterior edge of exterior beam at floor i: ext._ext. beam max

iHE

• Maximum hysteretic energy dissipated in interior edge of exterior beam at floor i: ext._int. beam max

iHE

• Maximum hysteretic energy dissipated in interior beam at floor i: int. beam max

iHE

• Maximum hysteretic energy dissipated at top of the exterior column in story i: ext._top column max

iHE

• Maximum hysteretic energy dissipated at bottom of the exterior column in story i: ext._bot. column max

iHE

• Maximum hysteretic energy dissipated at top of the interior column in story i: int._top column max

iHE

• Maximum hysteretic energy dissipated at bottom of the interior column in story i: int._bot. column max

iHE

Minimum and maximum response history of displacement-based EDPs

• Drift angle of story i:

1i iit

th

δ δθ−⎛ ⎞−

= ⎜ ⎟⎝ ⎠

• Rotation of exterior edge of exterior beam at floor i: ext. beam t

iθ

• Rotation of interior beam at floor i. int. beam t

iθ

• Rotation at top of the exterior column at story i: ext._top column t

iθ

• Rotation at bottom of the exterior column at story i: ext._bot. column t

iθ

• Rotation at top of the interior column at story i: int._top. column t

iθ

• Rotation at bottom of the interior column at story i: int._bot. column t

iθ

E.2 List of EDPs for generic shear walls

- 301 -

Global displacement-based EDP’s

• Maximum displacement at floor i: maxiδ

• Maximum rotation at floor i: maxiθ

• Maximum absolute velocity at floor i: maxiv

• Maximum absolute acceleration at floor i: maxia

• Maximum drift angle in story i:

1

max

max

i ii

dh

δ δθ

−⎛ ⎞−= ⎜ ⎟

⎝ ⎠

• Maximum rotation in story i: ( )1max max

i i irθ θ θ −= −

• Residual displacement at floor i: resiδ

• Residual rotation at floor i: iresθ

• Residual drift angle at story i:

1res res

res

i ii

hδ δ

θ−⎛ ⎞−

= ⎜ ⎟⎝ ⎠

• Residual rotation in story i: ( )1i i ires res resrθ θ θ −= −

Global force-based EDPs

• Maximum story-shear in story i: maxiV

• Maximum floor-overturning moment at floor i: max

itotOTM

Global energy-based EDPs

• Total damping energy: maxDE

• Total input energy: maxIE

• Total hysteretic energy: maxHE

Local displacement-based EDPs

• Maximum plastic rotation at floor i: imaxpθ

- 302 -

• Maximum accumulated positive plastic rotation at floor i: .maxi

accpθ +

• Maximum accumulated negative plastic rotation at floor i: .maxi

accpθ −

• Maximum accumulated positive plastic rotation at floor i (pre peak): . .maxi

acc ppkpθ +

• Maximum accumulated negative plastic rotation at floor i (pre peak): . .maxi

acc ppkpθ −

Minimum and maximum response history of displacement-based EDPs

• Drift angle of story i:

1i iit

th

δ δθ−⎛ ⎞−

= ⎜ ⎟⎝ ⎠

• Rotation of exterior edge of exterior beam at floor i: ext. beam t

iθ

• Rotation of interior beam at floor i. int. beam t

iθ

• Rotation at top of the exterior column at story i: ext._top column t

iθ

• Rotation at bottom of the exterior column at story i: ext._bot. column t

iθ

• Rotation at top of the interior column at story i: int._top. column t

iθ

Rotation at bottom of the interior column at story i: int._bot. column t

iθ

simplified performance based earthquake engineering

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