erformance-based assessment of existing …tz123dt3431/tr153_ruizgarcia_0.pdfearthquake ground...

Department of Civil and Environmental Engineering

Stanford University

PERFORMANCE-BASED ASSESSMENT OF EXISTING STRUCTURES ACCOUNTING FOR RESIDUAL DISPLACEMENTS

by

Jorge Ruiz-Garcia and

E. Miranda

Report No. 153

August 2005

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

©2005 The John A. Blume Earthquake Engineering Center

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Abstract

Recent seismic events have highlighted the necessity of demolishing damaged structures due

to excessive permanent deformations, even though they did not experience collapse.

Moreover, the evaluation of permanent (residual) lateral displacements plays an important

role for determining the technical feasibility of retrofitting structures that have been damaged

due to earthquake excitation. In addition, an adequate estimation of permanent lateral

displacements is important to assess seismic risks from aftershocks events. Thus, an adequate

estimation of residual displacement demands that existing structures may experience after

earthquake ground shaking should be of primary importance in modern performance-based

assessment procedures. For that purpose, this dissertation is aimed to provide further

information about residual displacement demands and their dependence on several ground

motion and structural features of single-degree-of-freedom (SDOF) and multi-degree-of-

freedom (MDOF) systems. In particular, this study proposes simplified probabilistic

approaches to estimate residual displacement demands for different seismic hazard levels in

the context of recently introduced performance-based methodologies.

In this investigation a special emphasis is given to the evaluation of residual displacement

demands, although information about maximum (transient) displacement demands is also

provided for reference purposes. Thus, the first part of this investigation reports

comprehensive statistical studies to quantify residual and maximum displacement demands of

inelastic SDOF systems considering a relatively large earthquake ground motion database,

and considering a large number of structural parameters. The second part of this study focuses

on the evaluation of permanent (residual) and maximum (transient) drift demands of multi-

story framed building models under different levels of ground motion intensity. Both parts

include the formulation and implementation of simplified probabilistic approaches to estimate

maximum and residual displacement demands accounting for the uncertainty in the structural

response and the ground motion hazard.

It is believed that this study provides further information towards incorporating explicitly

the evaluation of residual displacement demands for assessing the seismic performance of

existing structures or, even, for the preliminary design phase of new structures where

structural damage control is achieved through control of lateral deformation demands.

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Acknowledgments

This report was originally published as the Ph.D. dissertation of the first author. The

authors would like to express gratitude to the Consejo Nacional de Ciencia y

Tecnologia ( CONACYT) in Mexico for providing financial support.

vi

Table of Contents

Abstract iv

Acknowledgments v

List of Figures xv

List of Tables xvi

1. Introduction

1.1 Motivation..................................................................................................... 1

1.2 Overview of Probabilistic Seismic Demand Analysis (PSDA)........................ 2

1.3 Brief Review of PSDA Applications.............................................................. 3

1.4 Some Needs of Current PSDA Methodologies ............................................... 5

1.4.1 Improved Intensity Measures..................................................................... 5

1.4.2 Characterization of Central Tendency and Dispersion of EDP’s................. 6

1.5 Objectives ..................................................................................................... 7

1.6 Outline .......................................................................................................... 7

2. Maximum Inelastic Displacement Demands of SDOF systems: Firm Soil Sites

2.1 Introduction................................................................................................... 11

2.2 Definition of Inelastic Displacement Ratios ................................................... 12

2.3 Review of Previous Studies on Inelastic Displacement Ratios ........................ 15

2.3.1 Inelastic Displacement Ratios for Elastoplastic and Bilinear Systems........... 15

2.3.2 Inelastic Displacement Ratios for Degrading-systems .................................. 18

2.4 Earthquake Ground Motions Used in This Study............................................ 18

2.5 Hysteretic models Considered in This Study ................................................. 19

2.5.1 Stiffness-Degrading Hysteretic Models........................................................ 20

2.5.2 Stiffness-and-Strength Hysteretic Models .................................................... 22

2.5.3 Structural-Degrading Hysteretic Models ...................................................... 23

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2.6 Statistical results............................................................................................ 25

2.6.1 Central Tendency for CR.............................................................................. 25

2.6.1.1 Different firm local soil conditions.............................................. 26

2.6.1.2 All firm site classes..................................................................... 28

2.6.2 Dispersion................................................................................................... 29

2.6.2.1 Different firm local soil conditions.............................................. 30

2.6.2.2 All firm site classes..................................................................... 31

2.6.3 Effect of Firm Soil Conditions ..................................................................... 32

2.6.4 Effect of Hysteretic Behavior ...................................................................... 34

2.6.4.1 Effect of post-yield stiffness ....................................................... 34

2.6.4.2 Effect of unloading stiffness ....................................................... 38

2.6.4.3 Effect of stiffness-and-strength degradation ................................ 41

2.6.4.4 Effect of structural degradation ................................................... 43

2.7 Functional Models to Estimate CR.................................................................. 45

2.7.1 Review of Functional Models to Estimate CR............................................... 46

2.7.2 Proposed Simplified Functional Model to Estimate CR................................. 49

2.7.3 Evaluation of Proposed Functional Model to Estimate CR ............................ 52

2.8 Summary....................................................................................................... 55

3. Maximum Inelastic Displacement Demands of SDOF systems: Soft soil site and

Near-Fault Effects

3.1 Introduction................................................................................................... 57

3.2 Review of Previous Studies ........................................................................... 58

3.2.1 Inelastic Displacement Ratios for Soft-Soil Sites ......................................... 59

3.2.2 Inelastic Displacement Ratios for Near-Fault Ground Motions..................... 60

3.3 Earthquake Ground Motions Used in This Study............................................ 61

3.3.1 Soft Soil Site Records.................................................................................. 61

3.3.2 Near-Fault Ground Motions......................................................................... 62

3.4 Ground Motion Characterization on CR.......................................................... 63

3.4.1 Soft-Soil Site Records ................................................................................. 63

3.4.2 Near-Fault Ground Motions......................................................................... 66

3.5 Statistical results for soft-soil sites ................................................................. 69

3.5.1 Central Tendency of CR ............................................................................... 69

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3.5.2 Dispersion of CR.......................................................................................... 70

3.5.3 Effect of Lateral Strength Ratio ................................................................... 72

3.5.4 Effect of Earthquake Magnitude and Distance to the Source ........................ 74

3.5.5 Effect of Hysteretic Behavior ..................................................................... 74

3.5.5.1 Effect of post-yield stiffness ...................................................... 75

3.5.5.2 Effect of stiffness degradation .................................................... 76


3.6 Statistical Results for Near-Fault Ground Motions ........................................ 79

3.6.1 Central tendency of CR ................................................................................ 80

3.6.2 Dispersion of CR.......................................................................................... 82

3.6.3 Effect of Lateral Strength Ratio .................................................................. 84

3.6.4 Effect of Earthquake Magnitude ................................................................. 85

3.6.5 Effect of Distance to the Source................................................................... 87

3.6.6 Effect of Peak Ground Velocity................................................................... 88

3.6.7 Effect of Pulse Period.................................................................................. 89

3.6.8 Effect of Hysteretic Behavior ..................................................................... 90

3.6.8.1 Effect of post-yield stiffness ratio ............................................... 90


3.7 Simplified Equation to Estimate CR .............................................................. 94

3.8 Summary ...................................................................................................... 96

4. Probabilistic Evaluation of Maximum Inelastic Displacement Demand of SDOF

Systems

4.1 Introduction................................................................................................... 100

4.2 Formulation of Proposed Simplified Approach to Estimate ( )iδλ .................. 102

4.3 Evaluation of Simplified Assumptions ........................................................... 106

4.3.1 Effect of Earthquake Magnitude on CR ........................................................ 107

4.3.2 Effect of the Distance to the Rupture on CR ................................................. 110

4.3.3 Effect of the Duration of the Ground Motion on CR...................................... 112

4.3.4 Statistical Dependence Between CR and Sd ................................................... 115

4.3.4.1 Relative error by neglecting correlation between CR and Sd ......... 115

4.3.4.2 Dispersion of Sd and Correlation between CR and Sd .................... 117

4.3.4.3 Evaluation of relative error ......................................................... 120

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4.3.5 Cumulative Distribution Function of CR ...................................................... 121

4.3.5.1 Empirical distribution of CR ........................................................ 122

4.3.5.2 Parametric distribution of CR....................................................... 124

4.3.5.3 Selection of statistical parameters for the lognormal probability

distribution of CR ........................................................................ 127

4.4 Proposed Statistical Models to Estimate the Conditional Distribution of CR.... 130

4.4.1 Central Tendency Functional ....................................................................... 131

4.4.2 Dispersion Functional.................................................................................. 132

4.4.3 Evaluation of Proposed Functional Models to Estimate Conditional

Probability of CR ......................................................................................... 134

4.5 Evaluation of the Proposed Approach to Compute ( )iδλ ............................... 136

4.5.1 Spectral Displacement Seismic Hazard Curves ............................................ 137

4.5.2 Maximum Inelastic Displacement Demand Hazard Curves .......................... 138

4.5.3 Uniform Hazard Spectra of Maximum Inelastic Displacement Demand ...... 142

4.6 Summary....................................................................................................... 144

5. Residual Displacement Demands of SDOF systems

5.1 Introduction................................................................................................... 146

5.2 Review of Previous Studies on Residual Displacement Demands of

SDOF systems............................................................................................... 147

5.3 Evaluation of Residual Displacement Demands ............................................. 149

5.3.1 Direct Approach.......................................................................................... 149

5.3.2 Indirect Approach........................................................................................ 150

5.4 Hysteretic Models Considered in This Study.................................................. 151

5.5 Earthquake Ground Motions Used in this study ............................................. 152

5.6 Statistical Results Using Direct Approach...................................................... 152

5.6.1 Central Tendencies for Different Soil Conditions ........................................ 152

5.6.2 Central Tendencies for All Site Classes ....................................................... 154

5.6.3 Dispersion of Cr ......................................................................................... 155

5.6.4 Effect of Firm Soil Conditions ..................................................................... 158

5.6.5 Effect of Lateral Strength Ratio .................................................................. 160

5.6.6 Effect of the Frequency Content (Type of Ground Motion) .......................... 164

5.6.6.1 Effect of pulse period on Cr ........................................................ 164

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5.6.7.1 Effect of positive post-yield stiffness ratio .................................. 166


5.7 Statistical Results Using the Indirect Approach ............................................. 172

5.7.1 Central Tendencies for Different Soil conditions ......................................... 172

5.7.2 Central Tendencies for All Site Classes ....................................................... 173

5.7.3 Dispersion of Residual Ratio ...................................................................... 174

5.7.4 Effect of the Frequency Content (Type of Ground Motion) .......................... 177


5.7.5.1 Effect of post-yielding stiffness................................................... 180


5.8 Summary ...................................................................................................... 183

6. Probabilistic Evaluation of Residual Displacement Demands of SDOF systems

6.1 Introduction................................................................................................... 186

6.2 Proposed Approach to Estimate ( )rδλ .......................................................... 188

6.3 Evaluation of Simplified Assumptions ........................................................... 189

6.3.1 Effect of Earthquake Magnitude on Cr ......................................................... 192

6.3.2 Effect of the Distance to the Rupture on Cr .................................................. 198

6.3.3 Effect of the Duration of the Ground Motion on Cr ..................................... 203

6.3.4 Statistical Dependence Between Cr and Sd ................................................... 205

6.3.5 Cumulative Distribution of Cr ..................................................................... 209

6.3.5.1 Empirical distribution of Cr......................................................... 209

6.3.5.2 Parametric distribution of Cr ....................................................... 212

6.4 Statistical Models to Estimate the Conditional Distribution of Cr ................... 215

6.4.1 Central Tendency Functional ....................................................................... 215

6.4.2 Dispersion Functional.................................................................................. 216

6.4.3 Evaluation of Proposed Functional Models to Estimate the Cumulative

Conditional Distribution of Cr ..................................................................... 219

6.5 Evaluation of the Proposed Approach to Compute ( )rδλ .............................. 221

6.5.1 Residual Displacement Demand Hazard Curves .......................................... 222

6.5.2 Uniform Hazard Spectra of Residual Displacement ..................................... 226

6.5.3 Comparison of Displacement Demand Hazard Curves ................................. 227

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6.6 Summary....................................................................................................... 229

7. Statistical Studies of Deformation Demands of MDOF systems

7.1 Introduction................................................................................................... 231

7.2 Previous Studies on Residual Deformation Demands of MDOF Systems ..... 232

7.3 Generic Frame Building Models Used in This Study ..................................... 233

7.4 Deformation Demand Measures..................................................................... 234

7.5 Ground Motion Characterization ................................................................... 234

7.5.1 Ensemble of Earthquake Ground Motions.................................................... 235

7.5.2 Definition of Relative Intensity Measure...................................................... 236

7.5.3 Advantages of An Inelastic Intensity Measure ............................................. 238

7.6 Statistical Evaluation of Maximum and Residual Deformation Demands

for One-Bay Generic Framed Building Models ............................................... 242

7.6.1 Effect of Number of Stories ......................................................................... 242

7.6.2 Effect of Period of Vibration ....................................................................... 252

7.6.3 Effect of Frame Mechanism......................................................................... 257

7.6.4 Effect of Member Hysteretic Behavior......................................................... 265

7.6.4.1 Effect of positive strain-hardening .............................................. 266

7.6.4.2 Effect of strength deterioration.................................................... 270

7.6.4.3 Effect of stiffness degradation..................................................... 274

7.6.5 Effect of Structural Overstrength ................................................................. 277

7.6.6 Effect of Ground Motion Duration .............................................................. 280

7.7 Summary....................................................................................................... 288

8. Statistical Evaluation of Deformation Demand Ratios for MDOF Systems

8.1 Introduction................................................................................................... 292

8.2 Previous Findings on Deformation Demand Ratios ........................................ 293

8.3 Deformation Demand Ratios ......................................................................... 294

8.4 Evaluation of Maximum Deformation Demand Ratios for MDOF systems .... 295




8.4.4 Effect of Member Hysteretic Behavior......................................................... 300

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8.4.5 Effect of Ground Motion Duration............................................................... 302

8.5 Evaluation of Residual Deformation Ratios for MDOF Systems .................... 302




8.5.4 Effect of Member Hysteretic Behavior ........................................................ 309

8.5.4.1 Effect of positive strain-hardening .............................................. 309

8.5.4.2 Effect of member degrading behavior ........................................ 310

8.5.5 Effect of Ground Motion Duration .............................................................. 313

8.6 Summary ...................................................................................................... 315

9. Probabilistic Evaluation of Maximum Residual Displacement Demand of MDOF

Systems

9.1 Introduction................................................................................................... 318

9.2 Formulation of Probabilistic Estimation of Residual Deformation Demands .. 320

9.2.1 Proposed Approach to Estimate r∆λ .......................................................... 321

9.3 Probabilistic Distribution of Residual Deformation Demands........................ 323

9.3.1 Empirical Distribution of Residual Deformation Demands........................... 323

9.3.2 Parametric Distribution of Residual Deformation Demands ......................... 327

9.3.3 Selection of Statistical Parameters for the Lognormal Probability

Distribution of Residual Deformation .......................................................... 328

9.3.4 Variation of Statistical Parameters of Residual Deformation Demands

as a Function of the Ground Motion Intensity .............................................. 331

9.4 Evaluation of the Proposed Approach............................................................ 336

9.4.1 Elastic and Inelastic Displacement Demand Hazard Curves ......................... 337

9.4.2 Residual and Maximum Deformation Demand Hazard Curves..................... 339

9.4.3 Comparison of Residual and Maximum Deformation Hazard Curves ........... 343

9.4.4 Summary .................................................................................................... 345

10. Summary and Conclusions

10.1 Overview....................................................................................................... 346

10.2 Summary and Main Findings ......................................................................... 347

10.2.1 Evaluation of Maximum Inelastic Displacement Demands of

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SDOF Systems......................................................................................................... 347

10.2.1.1 Statistical studies ........................................................................ 347

10.2.1.2 Simplified probabilistic approach to estimate i∆ ........................ 352

10.2.2 Evaluation of Residual Displacement Demands of SDOF Systems............... 354

10.2.2.1 Statistical studies ........................................................................ 354

10.2.2.2 Simplified probabilistic approach to estimate r∆ ........................ 357

10.2.3 Evaluation of Maximum and Residual Deformation Demands of MDOF

Systems....................................................................................................... 357

10.2.4 Probabilistic Estimation of Maximum and Residual Deformation

Demands of MDOF Systems ....................................................................... 361

10.3 Limitations .................................................................................................... 361

10.3.1 Hysteretic Modeling .................................................................................... 362

10.3.2 Probabilistic Approach ................................................................................ 362

10.3.3 Multi-Story Building Models....................................................................... 362

10.3.4 Range of Ground Motion Intensity .............................................................. 363

10.4 Suggested Research....................................................................................... 364

A. Earthquake Ground Motion Records

A.1 Introduction....................................................................................................366

A.2 Suites of Earthquake Ground Motions ........................................................... 366

A.2.1 Firm Soil Site Set .......................................................................................... 366

A.2.2 Soft Soil Site Set ............................................................................................372

A.2.3 Fault-Normal Near-Fault Set ......................................................................... 376

A.2.4 Long- and Short-Duration Set ........................................................................ 377

B. Ground Motion Characterization

B.1 Introduction....................................................................................................380

B.2 Definition of Frequency Content Parameters...................................................380

B.2.1 Predominant Period of the Ground Motion......................................................380

B.2.1 Bandwidth......................................................................................................383

C. Sample Statistical Measures

C.1 Measures of central Tendency ........................................................................384

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C.1.1 Sample Mean .................................................................................................384

C.1.2 Counted Median.............................................................................................384

C.1.3 Geometric mean .............................................................................................385

C.2 Measures of Dispersion ..................................................................................385

C.2.1 Standard Deviation.........................................................................................385

C.2.2. Coefficient of Variation..................................................................................385

C.2.3 Standard Deviation of the Natural Logarithm of the Data................................385

C.3 Relationship among Statistical Parameters......................................................386

D. Generic Frame Models

D.1 Introduction....................................................................................................387

D.2 Modeling of Generic Framed Building Models Used in This Study.................388

D.2.1 Fundamental Period of Vibration....................................................................388

D.2.2 Height-Wise Stiffness Distribution .................................................................389

D.2.3 Fundamental Mode Shape ..............................................................................392

D.2.4 Height-Wise Strength Distribution..................................................................393

D.2.5 Modeling Assumptions ...................................................................................395

D.2.5.1 Modeling of the generic frames .....................................................................395

D.2.5.2 Frame elements and hysteretic behavior..........................................................395

List of References .....................................................................................................398

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List of Tables

2.1 Parameter estimates a0 for equation. (2.8) ...................................................................47

2.2 Site-dependent parameter estimates for equation (2.10) ...............................................48

2.3 Parameter estimates for equation (2.12)...................................................................... 49

2.4 Parameter estimates and 95% confidence intervals for equation (2.15)

corresponding to three different central tendency measures computed from all 240

ground motions ...........................................................................................................50

2.5 Site-dependent parameter estimates and 95% confidence intervals for equation

(2.15) obtained from median CR corresponding to each site class.................................51

2.6 Measures of error using parameter estimates from Table 2.4 and equation (2.15).........54

2.7 Measures of error using parameter estimates from Table 2.5 and equation (2.15).........54

2.8 Parameter estimates summary for equation (3.2) .........................................................95

2.9 Parameter estimates summary for equation (3.3) .........................................................96

A.1 Earthquake ground motions recorded in the NEHRP Site Class AB...........................369

A.2 Earthquake ground motions recorded in the NEHRP Site Class C..............................370

A.3 Earthquake ground motions recorded in the NEHRP Site Class D .............................371

A.4 Earthquake ground motions recorded in the San Francisco Bay Area used in this

study.........................................................................................................................373

A.5 Earthquake ground motions recorded in Mexico City used in this study.....................374

A.6. Fault-normal near-fault earthquake ground motions used in this study........................377

A.7 Short-duration earthquake ground motions used in this study ....................................379

A.8 Short-duration earthquake ground motions used in this study ....................................379

D.1. Parameters used in equations (D.3) and (D.4).............................................................392

D.2 Parameters used to simulate strength deterioration in this investigation .......................397

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List of Figures

2.1 Relationship between normalized strength (1/R) and displacement ductility

demand: (a) For a T=2.0s system; (b) For a T=0.2s system (after Ruiz-Garcia and

Miranda). ....................................................................................................................13

2.2 Hysteretic models used in this investigation: (a) Modified-Clough (MC); (b)

Takeda model (TK); and (c) origin-Oriented model (O-O)...........................................21

2.3 Hysteretic models used in the second stage of this investigation: (a) SD

(stiffness-degrading); (b) MSD (moderately-degrading); and (c) SSD (severely-

degrading)...................................................................................................................22

2.4 Experimental and analytical response for typical composite beam-column

connections with inelastic action in the joint panel zone (after Kanno and

Deierlein, 1997) ..........................................................................................................23

2.5 Experimental and analytical response for a poorly-detailed reinforced concrete

beam-column connection (after Walker et al, 2000).....................................................24

2.6 Comparison of inelastic displacement ratios computed from different central

tendency measures (R=4) ............................................................................................26

2.7 Mean inelastic displacement ratios for different NEHRP site classes: (a) Site

class AB; (b) site class C; and (c) site class D..............................................................27

2.8 Central tendency of inelastic displacement ratios for all 240 ground motions

recorded in NEHRP site classes AB, C and D: (a) Mean; (b) median; and (c)

geometric mean...........................................................................................................29

2.9 Coefficients of variation of inelastic displacement ratios for each site condition:

(a) site classes AB; (b) site class C; and (c) site class D...............................................30

2.10 Dispersion of CR computed from all 240 ground motions recorded in NEHRP site

classes AB, C and D: (a) COV; (b) standard deviation of the natural logarithm of

CR ...............................................................................................................................31

2.11 Inelastic displacement ratios corresponding to different percentiles: (a) for R=2;

(b) for R =4; and (c) for R = 6 .....................................................................................32

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2.12 Ratio of mean CR computed from each site class normalized by mean CR

computed from all ground motions: (a) site class AB; (b) site class C; (c) site

class D........................................................................................................................33

2.13 Mean inelastic displacement ratios of bilinear systems: (a)? α=3%; (b) α=5%; (c)

α= 10%; and (d) α=20% .............................................................................................35

2.14 Median inelastic displacement ratios of bilinear systems: (a) for α=3%; and (b)

for α= 10% .................................................................................................................36

2.15 Mean ratios of maximum deformation of bilinear to elastoplastic systems: (a)? α =

3%; (b) α = 5%; (c) α = 10%; and (d) α = 20%...........................................................37

2.16 Coefficients of variation of inelastic displacement ratio computed with all 240

ground motions for bilinear systems with: (a) α=3%; (a) α=5%; (c) α=10%; and

(d) α=20% .................................................................................................................38

2.17 Inelastic displacement ratios computed for three types of stiffness-degrading

systems: (a) Modified-Clough model; (b) Takeda model; and (c) origin-oriented

model .........................................................................................................................39

2.18 Influence of hysteretic behavior for three types of stiffness-degrading systems:

(a) Modified-Clough model; (b) Takeda model; and (c) origin -oriented model ...........40

2.19 Mean inelastic displacement ratios computed for two types of stiffness- and

strength-degrading systems: (a) MSD model; and (b) SSD model................................41

2.20 Mean ratio for inelastic displacement demands in stiffness- and strength-

degrading systems: (a) MSD model; (b) SSD model. .................................................42

2.21 Mean inelastic displacement ratios computed for two types of structural

degrading systems: (a) SSD-1 model; and (b) SSD-2 model ........................................43

2.22 Mean ratio of inelastic displacement demands in structural degrading and bilinear

systems: (a) SSD-1 model; and (b) SSD-2 model.........................................................44

2.23 Mean inelastic displacement ratios computed with equation (2.15) and parameter

estimates given in Table 2.4 ........................................................................................51

2.24 Comparison of mean inelastic displacement ratios computed using all 240 firm

site records to those computed with equation (2.15).....................................................52

xviii

3.1 Estimation of the predominant period of the ground motion: (a) for Foster City

(recording station FC, 10/17/89, comp. 360); (b) for Villa del Mar (recording

station 09, sate 25/04/89, comp. 90). ...........................................................................64

3.2 Relationship between predominant period of the ground motion (Tg) and

bandwidth for earthquake ground motions recorded on soft soil sites...........................65

3.3 Mean inelastic displacement ratios computed for three ground motions recorded

in the San Francisco Bay Area: (a) as a function of T; (b) as a function of T/Tg. ...........66

3.4 Relationship between Tg and Tp for fault-normal near-fault ground motions; (b)

relationship between Tg and bandwidth........................................................................67

3.5 Mean inelastic displacement ratios computed from the earthquake ground motion

obtained from the Sylmar Converter Station (1994 Northridge earthquake): (a) as

a function of T; (b) as a function of T/Tg ..................................................................... 68

3.6 Estimation of the predominant period for the Sylmar Converter Station (1994

Northridge earthquake). ..............................................................................................68

3.7 Mean inelastic displacement ratios computed for: (a) the San Francisco Bay Area

ground motion set, and (b) Mexico City ground motion set..........................................70

3.8 Coefficient of variation of inelastic displacement ratios computed for: (a) the San

Francisco Bay Area ground motion set; and (b) Mexico City ground motion set. ........71

3.9 Inelastic displacement ratios corresponding to various counted percentiles

computed with the Mexico City set: (a) R=2; (b) R = 4; and (c) R =6. ..........................72

3.10 Effect of lateral strength ratio on inelastic displacement ratios: (a) T/Tg=0.5; (b)

T/Tg=1.0; and (c) T/Tg=2.0 ..........................................................................................73

3.11 Effect of earthquake magnitude and distance to the source on inelastic

displacement ratios computed from sot-soil records.....................................................74

3.12 Effect of post-yield stiffness ratio on inelastic displacement ratios: (a) Mexico

City set (α=3%); (b) San Francisco set (R = 4, α?= 1.5%, 3%, 5%, 10%)......................76

3.13 Mean inelastic displacement ratios for stiffness-degrading systems: (a) San

Francisco Bay Area set; (b) Mexico City set ................................................................77

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3.14 Degrading to non-degrading inelastic displacement ratios: (a) San Francisco Bay

area Set; (b) Mexico City set .......................................................................................78

3.15 Degrading to non-degrading inelastic displacement ratios computed from the San

Francisco Bay Area set: (a) R = 3; and (b) R=5............................................................79

3.16 Degrading to non-degrading inelastic displacement ratios computed from the

Mexico City set: (a) R = 3; and (b) R=5.......................................................................79

3.17 Central tendency of inelastic displacement ratios using all 40 fault-normal near-

fault ground motions computed without normalized periods: (a) Sample mean;

and (b) counted median...............................................................................................81


fault ground motions computed with normalized periods respect to Tp: (a) Sample

mean; and (b) counted median.....................................................................................81


fault ground motions computed with normalized periods respect to Tp: (a) Sample

mean; and (b) counted median.....................................................................................82

3.20 Coefficient of variation of inelastic displacement ratios computed for the near-

fault ground motion set: (a) As a function of T; (b) as a function of T/Tp; and (c)

as a function of T/Tg ....................................................................................................83

3.21 Inelastic displacement ratio corresponding to different counted percentiles

computed for the near-fault ground motion set: (a) R=2.0; (b) R=4.0; and (c)

R=6.0..........................................................................................................................84

3.22 Effect of lateral strength ratio on inelastic displacement ratios computed for the

near-fault ground motion set: (a) T/Tg=0.5; (b) T/Tg=1.0; and (c) T/Tg=2.0...................85

3.23 Effect of earthquake magnitude on inelastic displacement ratios computed for the

near-fault ground motion set: (a) R = 2; (b) R = 4; (c) R = 6.........................................86

3.24 Effect of distance [km] to the source on inelastic displacement ratios computed

for the near-fault ground motion set: (a) R = 2; (b) R = 4; (c) R = 6..............................87

3.25 Effect of peak ground velocity [cm/s] on inelastic displacement ratios computed

from the near-fault ground motion set: (a) R=2; (b) R=4; (c) R=6 ................................88

xx

3.26 Effect of pulse period on inelastic displacement ratios computed from the near-

fault ground motion set: (a) R = 2; (b) R = 4; (c) R = 6................................................89

3.27 Effect of post-yield stiffness ratio on inelastic displacement ratios computed from

the near-fault ground motion set: (a) α=3%; (b) α=5%; (c) α=10% .............................90

3.28 Coefficient of variation of inelastic displacement ratios computed from the near-

fault ground motion set: (a) α=0%; (b) α=3%; (c) α =5%; and (d)? α=10% ..................91

3.29 Degrading to non-degrading inelastic displacement ratios computed from the

near-fault ground motion set: (a) R = 2; (b) R=4; and (c) R=6 ......................................93

3.30 Mean inelastic displacement ratios for elastoplastic systems computed with

equations (3.2) and (3.3) .............................................................................................96

4.1 Effect of earthquake magnitude on mean inelastic displacement ratios: (a) R=2;

(b) R=4 .....................................................................................................................107

4.2 Correlation between mean inelastic displacement ratios and earthquake

magnitude for all 240 ground motions recorded in NEHRP site classes AB, C and

D ..............................................................................................................................108

4.3 CR fitted residuals versus earthquake magnitude for R = 4 and four periods of

vibration: (a) T=0.2s; (b) T=0.5s; (c) T=1.0s; and (d) T=3.0s. ...................................109

4.4 Effect of distance to the source on rupture on mean inelastic displacement ratios

for (a) for R=2; and (b) for R=4.................................................................................111

4.5 Correlation between mean inelastic displacement ratios and distance to the source

for all 240 ground motions recorded in NEHRP site classes AB, C and D..................111

4.6 CR fitted residuals versus distance to the source for R = 4: (a) T=0.2s; (b) T=0.5s;

(c) T=1.0s; and (d) T=3.0s.........................................................................................112

4.7 Effect of strong ground motion duration on mean CR: (a) for R=2; (b) for R=4;

and (c) for R=6 .........................................................................................................114

4.8 Mean CR of each strong motion duration set normalized by mean CR from all 40

ground motions: (a) Short–duration set; (b) long-duration set ....................................115

4.9 COV of Sd computed from all 240 ground motions recorded in NEHRP site

classes AB, C and D .................................................................................................118

xxi

4.10 Correlation between CR and Sd computed for ground motions corresponding to

NEHRP site classes: (a) Site class AB; (b) site class C; (c) site class D......................119

4.11 Correlation between CR and Sd computed for all 240 ground motions recorded in

NEHRP site classes AB, C and D..............................................................................119

4.12 Calculated absolute relative error by assuming lack of correlation between CR

and Sd for different site classes: (a) Site classes AB; (b) site class C; (c) site class

D ..............................................................................................................................120

4.13 Calculated absolute relative error by assuming lack of correlation between CR

and Sd for all NERP site classes AB, C, D .................................................................121

4.14 Empirical cumulative distribution of CR as a function of period of vibration for:

(a) for R= 2.0; (b) for R = 4.0; and (c) for R = 6.0......................................................122

4.15 Empirical cumulative distribution of CR as a function of lateral strength ratio for:

(a) T= 0.5 s.; and (b) T = 2.0 s ...................................................................................123

4.16 Effect of site conditions on empirical cumulative distribution of CR (R=4.): (a)

T=0.5s, (b) T=2.0s.....................................................................................................124

4.17 Effect of hysteretic behavior on empirical cumulative distribution of CR (R=4):

(a) T=0.5s; (b) T=2.0s ...............................................................................................124

4.18 Comparison of parametric CDF with respect to empirical distribution of CR for a

short-period system (T=0.5 s) and three levels of lateral strength...............................126

4.19 Comparison of three parametric CDF with respect to empirical distribution of CR

for a long-period system (T=2.0 s) and three levels of lateral strength........................127

4.20 Fitting of the empirical distribution of CR with different combination of statistical

parameters of the lognormal distribution for a system with T=0.2s and R=2.0............128

4.21 Fitting of the empirical cumulative distribution of CR with different parameters of

the lognormal distribution for a short-period weak system (T=0.2 s, R=2.0)...............129

4.22 Comparison of counted percentiles and percentiles of CR assuming lognormal

CDF (using statistical parameters from sample data) for: (a) R = 2; (b) R = 4; (c)

R = 6.........................................................................................................................130

4.23 Mean of CR for elastoplastic SDOF systems estimated with equation (4.22)...............132

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4.24 Dispersion of CR for elastoplastic SDOF systems: (a) Dispersion computed from

equation (4.20) and σ~ sample data; (b) dispersion estimated with equations

(4.23) and (4.24) .......................................................................................................133

4.25 Dispersion of CR for elastoplastic SDOF systems estimated with equation (4.23)

and parameter estimates obtained from σ~ obtained from equation (4.20)...................133

4.26 Lognormal CDF: (a) using parameter from sample data; (b) using parameters

from equations (4.20)-(4.24) .....................................................................................135

4.27 Percentiles of CR computed assuming lognormality of CR and statistical

parameters estimated from equations (4.22)-(4.24) for: (a) R=2; (b) R=4; (c) R= 6.....136

4.28 Maximum inelastic displacement demand hazard curve corresponding to T=0.2 s .....138

4.29 Maximum inelastic displacement demand hazard curve corresponding to T=0.3s ......139




4.33 Maximum inelastic displacement demand hazard curve as a function of period of

vibration for: (a) Cy = 0.1; and (b) Cy = 0.4 ................................................................141

4.34 Uniform hazard spectra of maximum inelastic displacement demand

corresponding to: (a) 10% in 50 years; and (b) 2% in 50 years...................................143

5.1 Example of direct approach computation displacement time-history computed for

the NS component of the 1940 El Centro record........................................................150

5.2 Example of indirect approach computation using displacement time-history

computed for the NS component of the 1940 El Centro record ..................................151

5.3 Mean residual displacement ratios for NEHRP site classes AB, C and D ...................153

5.4 Mean residual displacement ratios for all 240 earthquake ground motions

recorded in NEHRP site classes B, C and D: (a) Mean; (b) median............................154

5.5 Coefficient of variation of residual displacement ratios corresponding to: (a) Site

classes AB, (b) site classes C, and (c) site class D .....................................................155

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5.6 Coefficient of variation of residual displacement ratios for all 240 ground

motions recorded in NEHRP site classes B, C and D.................................................156

5.7 Residual inelastic displacement ratios corresponding to different percentiles for:

(a) R =2; (b) R=4 and (c) R = 6..................................................................................157

5.8 Mean residual displacement ratios on each group normalized by mean ratios from

all ground motions: (a) site Classes AB; (b) site class C; (c) site class D....................159

5.9 Effect of lateral strength ratio on residual displacement ratios: (a) T=0.5s; (b)

T=1.0s; and (c) T=2.0s ..............................................................................................161

5.10 Mean residual displacement ratios computed from: (a) San Francisco Area set;

(b) Mexico City set; (c) near-fault set; and (d) firm soil set (site classes AB,C,D)......162

5.11 Coefficients of variation of Cr corresponding to: (a) Soft-soil records in the San

Francisco Bay Area; (b) soft soil records in Mexico City; (c) near-fault records;

and (d) firm-soil records............................................................................................164

5.12 Effect of pulse period of near-fault ground motions on median Cr for: (a) R=2.0;

(b) R=4.0; and (c) R=6.0 ...........................................................................................165

5.13 Effect of post-yield stiffness ratio, α, on mean residual displacement ratios: (a)

α=1%; and (b) α=5%................................................................................................167

5.14 Coefficients of variation of Cr computed from bilinear systems with: (a) α = 1%,

(b) α = 3%. ...............................................................................................................168

5.15 Mean residual displacement ratios computed from stiffness of stiffness-degrading

systems with different unloading stiffness: (a) HC ∞→ ; (b) HC=2.5; (c)

HC=1.5; (d) HC=0.1. ................................................................................................169

5.16 Comparison of hysteretic response and displacement time-history from a SDOF

system (T=1.0s, ζ =5%, Cy=0.015) subjected to the records collected in station

24399 comp. 360 (1994 Northridge earthquake) considering three types of

hysteretic behavior ....................................................................................................170

5.17 Coefficients of variation of residual displacement ratios corresponding to

different stiffness-degrading systems: (a) HC ∞→ ; (b) HC=2.5; (c) HC=1.5; (d)

HC=0.1.....................................................................................................................171

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5.18 Mean ratios of residual displacement demand to maximum inelastic displacement

demand corresponding to NEHRP site classes AB, C and D......................................173

5.19 Ratios of residual displacement demand to maximum inelastic displacement

demand for all 240 ground motions recorded in NEHRP site classes B, C and D:

(a) Mean; and (b) median. .........................................................................................174

5.20 Coefficient of variation of residual ratios corresponding to: (a) Site Classes AB;

(b) site Class C; (c) site Class D................................................................................175

5.21 Coefficient of variation of residual ratios for all 240 ground motions recorded in

NEHRP site classes B, C and D ................................................................................176

5.22 Residual ratios corresponding to different percentiles for: (a) R =2; (b) R=4 and

(c) R = 6 ...................................................................................................................176


demand computed for: (a) San Francisco Bay Area set; (b) Mexico City set; (c)

near-fault set, and (d) firm soil set. ............................................................................178

5.24 Coefficient of variation of residual ratios corresponding to: (a) San Francisco

Bay Area set; (b) Mexico City set; (c) near-fault set; and (d) Firm soil site set ...........179

5.25 Effect of pulse period on median residual ratios: (a) Tp < 1.0s; (b) 1.0s < Tp <

2.0s; and (c) Tp >2.0s ................................................................................................180

5.26 Effect of post-yield stiffness ratio on mean ratios of residual displacement

demands of bilinear to elastoplastic systems: (a) α=3%; (b) α=5%; and (c)

α=10% .....................................................................................................................181


demand computed for different stiffness-degrading systems: (a) HC ∞→ ; (b)

HC=2.5; (c) HC=1.5; (d) HC=0.1..............................................................................182

6.1 Effect of earthquake magnitude on residual displacement ratios for: (a) R = 2, (b)

R = 4, and (c) R = 6...................................................................................................193

6.2 Mean Cr computed from each set of ground motions corresponding to each

magnitude range normalized with respect to mean Cr from all ground motions:

(a) 5.7 < Ms < 6.2; (b) 6.3 < Ms < 6.9; and (c) 7.0 < Ms < 7.8.....................................194

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6.3 Coefficient of correlation between Cr and earthquake magnitude for each site

condition: (a) Site classes AB; (b) site class C; (c) site class D ................................. .195

6.4 Coefficient of correlation between Cr and earthquake magnitude for all 240

earthquake ground motions recoded in site classes AB, C, and D...............................196

6.5 Cr fitted residuals versus earthquake magnitude for R = 4: (a) T=0.2 s; (b) T = 0.5

s; (c) T = 1.0 s; and (d) T = 2.0 s................................................................................198

6.6 Effect of nearest distance to the horizontal projection of rupture, D [kms], on

residual inelastic displacement ratios for (a) R = 2; (b) R = 4; and (c) R = 6 ...............199

6.7 Mean Cr computed from each set of ground motions corresponding to each

distance range [D in Km] normalized with respect to mean Cr from all ground

motion: (a) 10.1 < D < 20.0; (b) 20.1 < D < 45.0; and (c) 45.1 < D < 160.0 ...............200

6.8 Coefficient of correlation between Cr and distance to the rupture for each site

class: (a) Site classes AB; (b) site class C; and (c) site class D...................................201

6.9 Coefficient of correlation between Cr and distance to the rupture for all 240

earthquake ground motions recoded in NEHRP site classes AB, C, and D.................202

6.10 Cr fitted residuals versus earthquake magnitude for R = 4: (a) T=0.2 s; (b) T =

0.5 s; (c) T = 1.0 s; and (d) T = 2.0 s..........................................................................203

6.11 Effect of ground motion duration on residual displacement ratios for: (a) R=2; (b)

R=4, and (c) R=6.......................................................................................................204

6.12 Mean Cr computed from each ground motion duration set normalized with

respect to mean Cr from all 40 ground motions: (a) SD records; (b) LD records .........205

6.13 Coefficient of correlation between Cr and Sd computed from all 240 earthquake

ground motions from NEHRP site classes AB, C, and D ...........................................207

6.14 Absolute relative error when computing ∆r with approximate (6.9) and exact

(6.10) formulation.....................................................................................................208

6.15 Empirical cumulative distribution of Cr as a function of period of vibration for:

(a) R= 2.0; (b) R = 4.0; and (c) R = 6.0 ......................................................................210

6.16 Empirical cumulative distribution of Cr as a function of lateral strength ratio for:

(a) T= 0.5 s.; and (b) T = 2.0 s ...................................................................................211

xxvi

6.17 Effect of hysteretic behavior on empirical cumulative distribution of Cr (T=0.5s):

(a) R= 2; (b) R=4; and (c) R = 6.................................................................................211

6.18 Comparison of three parametric CDF with empirical distribution of Cr for a

short-period system (T=0.5 s)....................................................................................213

6.19 Comparison of three parametric CDF with respect to empirical distribution of Cr

for a long-period system (T=2.0 s).............................................................................213

6.20 Comparison of counted percentiles and percentiles of Cr computed assuming

lognormal CDF (using statistical parameters from sample data) for: (a) R = 2; (b)

R = 4; (c) R = 6 .........................................................................................................214

6.21 Median residual displacement ratio computed with equations (6.16) and (6.17)

and parameter estimates given in Table 6.1 for site classes AB, C, D.........................217

6.22 Coefficient of variation of residual displacement ratio computed with equations

(6.18) and (6.19) and parameter estimates given in Table 6.1 for site classes

AB,C, D ...................................................................................................................218

6.23 Comparison of empirical and lognormal cumulative distribution functions of Cr

(using proposed models) for two periods of vibration: (a) T=0.5s; and (b) T=2.0s......220

6.24 Comparison of counted percentiles and percentiles of Cr computed assuming

lognormal CDF (using statistical parameters from proposed models) for: (a) R =

2; (b) R = 4; (c) R = 6................................................................................................221

6.25 Residual displacement demand hazard curve corresponding to five different yield

strength coefficient and T=0.2s .................................................................................222


strength coefficient and T=0.3s .................................................................................223


strength coefficient T=0.5s........................................................................................223





xxvii

6.30 Residual displacement demand hazard curve corresponding to different yield

strength coefficients: (a) Cy=0.1; (b) Cy=0.2 ..............................................................225

6.31 Uniform hazard spectra of residual displacement demand corresponding to: (a)

10% in 50 years; (b) 2% in 50 years..........................................................................227

6.32 Comparison of residual and maximum displacement demand hazard curves for a

short-period system (T=0.5s) with two levels of yield strength: (a) Cy = 0.2; (b)

Cy = 0.8.....................................................................................................................228

7.1 Pushover curves for generic frames: (a) Rigid models; and (b) flexible models..........237

7.2 Variation of θroof with changes in IM: (a) Sd (T1); and (b) ∆i (T1) ...............................239

7.3 Variation of IDRmax with changes in IM: (a) Sd (T1); and (b) ∆i (T1)...........................239

7.4 Influence of IM in the variation of dispersion: (a) roof drift ratio; and (b)

maximum inter-story drift ratio .................................................................................240

7.5 Variation of θr,roof with changes in IM: (a) Sd (T1); and (b) ∆i (T1) .............................241

7.6 Variation of RIDRmax with changes in IM: (a) Sd (T1); and (b) ∆i (T1) .........................241

7.7 Influence of IM in the variation of dispersion for: (a) residual roof drift ratio; and

(b) maximum residual inter-story drift ratio...............................................................241

7.8 Effect of number of stories on height-wise distribution of median IDR for three

rigid building models: (a) GF-3R (T1=0.5 s); (b) GF-9R (T1=1.185 s); and (c)

GF-18R (T1=2.0 s) ....................................................................................................242

7.9 Effect of number of stories on height-wise distribution of median IDR for three

flexible building models: (a) GF-3F (T1=0.79 s); (b) GF-9F (T1=1.902 s); and (c)

GF-18F (T1=3.311 s).................................................................................................243

7.10 Effect of number of stories on the height-wise dispersion distribution of IDR for

three rigid building models: (a) GF-3R (T1=0.5 s); (b) GF-9R (T1=1.185 s); and

(c) GF-18R (T1=2.0 s)...............................................................................................244

7.11 Effect of number of stories on height-wise dispersion distribution of IDR for

three flexible building models: (a) GF-3F (T1=0.79 s); (b) GF-9F (T1=1.902 s);

and (c) GF-18F (T1=3.311 s).....................................................................................244

xxviii

7.12 Effect of number of stories on height-wise distribution of median RIDR for three

rigid building models: (a) GF-3R (T1=0.5 s); (b) GF-9R (T1=1.185 s); and (c)

GF-18R (T1=2.0 s) ....................................................................................................245

7.13 Effect of number of stories on height-wise distribution of median RIDR for three

flexible building models: (a) GF-3F (T1=0.79 s); (b) GF-9F (T1=1.902 s); and (c)

GF-18F (T1=3.311 s).................................................................................................245

7.14 Effect of number of stories on the height-wise dispersion distribution of RIDR for

three rigid building models: (a) GF-3R (T1=0.5 s); (b) GF-9R (T1=1.185 s); and

(c) GF-18R (T1=2.0 s)...............................................................................................246

7.15 Effect of number of stories on height-wise dispersion distribution of IDR for

three flexible building models: (a) GF-3F (T1=0.79 s); (b) GF-9F (T1=1.902 s);

and (c) GF-18F (T1=3.311 s).....................................................................................247

7.16 Effect of number of stories on generic rigid frames: (a) Median θroof; and (b)

median θr,roof .............................................................................................................247

7.17 Effect of number of stories on generic flexible frames: (a) Median θroof; and (b)

median θr,roof .............................................................................................................248

7.18 Effect of number of stories on dispersion of rigid frames: (a) Dispersion of θroof;

and (b) dispersion of θr,roof.........................................................................................249

7.19 Effect of number of stories on dispersion of flexible frames: (a) Dispersion of

θroof; and (b) dispersion of θr,roof.................................................................................249

7.20 Effect of number of stories and ground motion for intensity of rigid frames on:

(a) Median IDRmax; and (b) median RIDRmax ..............................................................250

7.21 Effect of number of stories and ground motion for intensity of flexible frames on:


7.22 Effect of number of stories on dispersion of rigid frames: (a) Dispersion of

IDRmax; and (b) dispersion of RIDRmax .......................................................................251

7.23 Effect of number of stories on dispersion of flexible frames: (a)Dispersion

IDRmax; and (b) dispersion RIDRmax ...........................................................................251

xxix

7.24 Effect of period of vibration on generic rigid framed building models: (a)

Median θroof; and (b) median θr,roof.............................................................................253

7.25 Effect of period of vibration on generic flexible framed building models: (a)

Median θroof; and (b) median θr,roof.............................................................................253

7.26 Effect of period of vibration on dispersion for generic rigid frames: (a)

Dispersion of θroof; and (b) dispersion of θr,roof ...........................................................254


Dispersion of θroof; and (b) dispersion of θr,roof ...........................................................254

7.28 Effect of period of vibration for generic rigid frames: (a) Median IDRmax, and

(b) median RIDRmax..................................................................................................255

7.29 Effect of period of vibration for generic flexible frames: (a) Median IDRmax, and

(b) median RIDRmax...................................................................................................255


Dispersion of IDRmax; and (b) dispersion of RIDRmax .................................................256

7.31 Effect of period of vibration on dispersion of IDRmax and RIDRmax for generic

flexible frames ..........................................................................................................256

7.32 Effect of the type of mechanism on the height-wise distribution of median IDR

for GF-9R (T1=1.185 s) building model: (a) FH mechanism; (b) BH mechanism;

and (c) CH mechanism..............................................................................................259

7.33 Effect of the type of mechanism on the height-wise distribution of median IDR

for GF-9F (T1=1.902 s) building model: (a) FH mechanism; (b) BH mechanism;

and (c) CH mechanism..............................................................................................259

7.34 Height-wise dispersion distribution of IDR for GF-9R (T1=1.185 s) model: (a)

FH mechanism; (b) BH mechanism; and (c) CH mechanism .................................... 260

7.35 Height-wise dispersion distribution of IDR for GF-9F (T1=1.902 s) building

model: (a) FH mechanism; (b) BH mechanism; and (c) CH mechanism.....................260

7.36 Effect of the type of mechanism on the height-wise distribution of RIDR for GF-

9R (T1=1.185 s) building model: (a) FH mechanism; (b) BH mechanism; and (c)

CH mechanism .........................................................................................................261

xxx

7.37 Effect of the type of mechanism on the height-wise distribution of RIDR for GF-

9F (T1=1.902s.) building model: (a) FH mechanism; (b) BH mechanism; and (c)

CH mechanism .........................................................................................................261

7.38 Height-wise dispersion distribution of RIDR for GF-9F (T1=1.902 s) building


7.39 Height-wise dispersion distribution of RIDR for GF-9F (T1=1.902 s) building


7.40 Effect of the type of mechanism for GF-9R (T1 = 1.185 s) building model on:(a)

Median IDRmax; and (b) median RIDRmax ...................................................................263

7.41 Effect of the type of mechanism for GF-9R (T1 = 1.185 s) building model on: (a)

Dispersion of IDRmax; and (b) dispersion of RIDRmax.................................................264

7.42 Effect of the type of mechanism for GF-9F (T1 = 1.902 s) building model on: (a)

Median IDRmax ; and (b) RIDRmax..............................................................................264

7.43 Effect of the type of mechanism for GF-9F (T1 = 1.902 s) building model on: (a)

Dispersion of IDRmax; and (b) dispersion of RIDRmax.................................................265

7.44 Effect of positive strain-hardening on the height-wise distribution of IDR for GF-

9R (T1=1.195 s) building model: (a) η?=2.0; (b) η?= 4.0; and (c) η?= 6.0 ......................267

7.45 Height-wise dispersion distribution of IDR for GF-9R (T1=1.185 s) building

model: (a)? α= 0.1%; (b) α= 2%; and (c) α = 5% .......................................................267

7.46 Effect of positive strain-hardening on the height-wise distribution of RIDR for

GF-9R (T1=1.185 s) building model: (a) η=2.0; (b) η= 4.0; and (c) η= 6.0................268

7.47 Height-wise dispersion distribution of RIDR for GF-9R (T1=1.185 s) building

model: (a)? α = 0.1%; (b) α= 2%; (c) α = 5%.............................................................268

7.48 Effect of positive strain-hardening on median IDRmax and RIDRmax for GF-9R

(T1=1.185 s) building model on: a) Median IDRmax ; and (b) RIDRmax ......................269

7.49 Effect of member strain hardening for GF-9F (T1 = 1.902 s) building model on:

(a) Dispersion of IDRmax; and (b) dispersion of RIDRmax ...........................................270

7.50 Height-wise distribution of IDR for GF-3R (T1=0.5 s): (a) NSD model; (b) MSD

model; and (c) SSD model ........................................................................................271

xxxi

7.51 Height-wise distribution of dispersion of IDR for GF-3R (T1=0.5 s) building

model: (a) NSD model; (b) MSD model; and (c) SSD model.....................................271

7.52 Height-wise distribution of RIDR for GF-3R (T1=0.5 s) building model: (a) NSD

model; (b) MSD model; and (c) SSD model ..............................................................272


model: (a) NSD model; (b) MSD model; and (c) SSD model.....................................272

7.54 Effect of strength deterioration on deformation demands of GF-3R (T1 = 0.5 s):


7.55 Effect of strength deterioration on deformation demands of GF-3R (T1 = 0.5 s):


7.56 Effect of stiffness degradation on the height-wise distribution of median IDR for

GF-18R (T1=2.0 s.): (a) η=2.0; (b) η= 4.0; and (c) η= 6.0 .........................................275

7.57 Effect of stiffness degradation on the height-wise distribution of median RIDR

for GF-18R (T1=2.0 s.): (a) η =2.0; (b) η = 4.0; and (c) η?= 6.0..................................275

7.58 Effect of member stiffness degradation on GF-3R (T1 = 0.5 s): (a) Median

IDRmax; and (b) median RIDRmax................................................................................276

7.59 Effect of member stiffness degradation on GF-18R (T1 = 2.0 s): (a) Median

IDRmax; and (b) median RIDRmax................................................................................276

7.60 Height-wise strength distribution: (a) Parabolic variation per NEHRP provisions;

(b) overstrength distribution along the height ............................................................278

7.61 Height-wise distribution of IDR for GF-9R (T1=1.185 s) building model: (a)

Without overstrength; (b) uniform overstrength; and (c) non-uniform

overstrength..............................................................................................................279


model: (a) without over strength; (b) uniform over strength; (c) non-uniform

over strength.............................................................................................................280

7.63 Height-wise distribution of IDR for GF-18R (T1=2.0 s) building model obtained

from three suites of ground motions: (a) s40-LMSR-N; (b) s20-SD; and (c) s20-

LD............................................................................................................................282

xxxii

7.64 Height-wise dispersion distribution of IDR for GF-18R (T1=2.0 s) building model

obtained from three suites of ground motions: (a) s40-LMSR-N; (b) s20-SD; and

(c) s20-LD ................................................................................................................283

7.65 Height-wise distribution of RIDR for GF-18R (T1=2.0 s) building model obtained

from three suites of ground motions: (a) s40-LMSR-N; (b) s20-SD; and (c) s20-

LD............................................................................................................................283


model obtained from three suites of ground motions: (a) s40-LMSR-N; (b) s20-

SD; and (c) s20-LD...................................................................................................284

7.67 Effect of ground motion duration on GF-3R (T1 = 0.5 s): (a) Median IDRmax; and

(b) median RIDRmax...................................................................................................285

7.68 Effect of the ground motion duration on GF-3R (T1 = 0.5 s): (a) Dispersion of

IDRmax; and (b) dispersion of RIDRmax .....................................................................285

7.69 Effect of ground motion duration on GF-18R (T1 = 2.0 s): (a) Median IDRmax; and

(b) median RIDRmax...................................................................................................286


IDRmax; and (b) dispersion of RIDRmax ......................................................................286

7.71 Effect of ground motion duration on GF-18R (T1 = 3.31 s): (a) Median IDRmax;

and (b) median RIDRmax. ...........................................................................................286


IDRmax; and (b) dispersion of RIDRmax ......................................................................287

7.73 Effect of ground motion duration on GF-3R (T1 = 0.5 s) considering moderate

member strength deterioration: (a) Median IDRmax; and (b) median RIDRmax..............287

7.74 Effect of the ground motion duration on GF-3R (T1 = 0.5 s) considering moderate

member strength deterioration: (a) Dispersion of IDRmax; and (b) dispersion of

RIDRmax ....................................................................................................................288

8.1 Effect of number of stories on median IDRmax /θroof ratio: (a) Generic rigid

frames; and (b) generic flexible frames .....................................................................296

xxxiii

8.2 Effect of number of stories on dispersion of IDRmax /θroof ratio: (a) Generic rigid


8.3 Effect of period of vibration on median ratio of IDRmax to θroof: (a) Generic rigid


8.4 Variation of median IDRmax /θroof ratio for both generic rigid and flexible frame

models as a function of the period of vibration and the ground motion intensity ........298

8.5 Effect of period of vibration on dispersion of IDRmax /θroof ratio: (a) Generic rigid

frames; and (b) Generic flexible frames.....................................................................299

8.6 Effect of the frame mechanism on IDRmax /θroof ratio for two building models: (a)

GF-9R (T1=1.185 s); and (b) GF-9F (T1=1.902 s)......................................................300

8.7 Effect of the frame mechanism on dispersion of IDRmax/θroof ratio for two

building models: (a) GF-9R (T1=1.185 s); and (b) GF-9F (T1=1.902 s)......................300

8.8 Effect of member strain hardening on IDRmax/θroof ratio for GF-9R model:(a)

median; and (b) dispersion ........................................................................................301

8.9 Effect of member stiffness-degrading hysteretic behavior on median IDRmax/θroof

ratio: (a) GF-3R (T1=0.5s); and (b) GF-18R (T1=2.0s)...............................................301

8.10 Effect of ground motion duration on median IDRmax/θroof ratio: (a) GF-

3R(T1=0.5s); and (b) GF-18R(T1=2.0s)....................................................................302

8.11 Height-wise distribution of residual deformation ratio for three rigid frame

models: (a) GF-3R(T1=0.5s); (b) GF-9R(T1=1.185s); and (c) GF-18R(T1=2.0s).........303

8.12 Height-wise dispersion distribution of residual deformation ratio for GF-9R

(T1=1.185s)...............................................................................................................304

8.13 Effect of number of stories on γroof: (a) Generic rigid frames; and (b) generic

flexible frames ..........................................................................................................305

8.14 Effect of number of stories on γmax: (a) Generic rigid frames; and (b) generic

flexible frames ..........................................................................................................305

8.15 Effect of period of vibration on γroof: (a) Generic rigid frames; and (b) generic

flexible frames ..........................................................................................................306

xxxiv

8.16 Effect of period of vibration on γmax: (a) Generic rigid frames; and (b) generic

flexible frames ..........................................................................................................306

8.17 Height-wise distribution of median residual ratio for three frame mechanism

models: (a) FH mechanism; (b) BH mechanism; and (c) CH mechanism...................307

8.18 Height-wise distribution of median residual ratio for three frame mechanism

models: (a) FH mechanism; (b) BH mechanism; and (c) CH mechanism.................. .307

8.19 Effect of type of mechanism on GF-9R (T1?=1.185s): (a) Median ??γmax; and (b)

median γroof ...............................................................................................................308

8.20 Effect of type of mechanism on GF-9F (T1=1.902s): (a) Median ??γmax; and (b)

median γroof ...............................................................................................................308

8.21 Height-wise distribution of median residual deformation ratio along the height

for GF-9R (T1=1.185 s) considering three levels of member strain hardening: (a)

α=0.1%; (b) α=2%; and (c) α=5% ............................................................................309

8.22 Effect of member strain hardening on residual deformation ratios for GF-9F (T1 =

1.902 s): (a) Median γmax; and (b) median γroof ...........................................................310

8.23 Distribution of residual deformation ratio along the height for GF-9R

(T1=1.185s) considering three member hysteretic behaviors: (a) EPP; (b) MC;

and (c) TK ................................................................................................................311

8.24 Effect of stiffness degradation on median γmax for two building models: (a) GF-

3R (T1=0.5s); (b) GF-18R (T1=2.0s)..........................................................................312

8.25 Effect of member strength deterioration on median γmax for two building models:

(a) without strain hardening in column elements; and (b) with strain hardening in

column elements .......................................................................................................312

8.26 Distribution of median residual deformation ratio along the height for GF-9R

(T1=1.185 s) considering three sets of ground motions: (a) s40-LMSR-N; and (b)

s20-SD; (c) s20-LD...................................................................................................313

8.27 Effect of ground motion duration on residual deformation ratio for GF-

3R(T1=0.5s) : (a) Median γmax; and (b) median γroof....................................................314

xxxv

8.28 Effect of ground motion duration on residual deformation ratio for GF-

18R(T1=2.0s): (a) Median γmax; and (b) median γroof ...................................................314

9.1 Empirical cumulative distribution of residual deformation demands obtained

from the GF-3R (T1=0.5 s) building model for different relative intensities: (a)

θr,roof; and (b) RIDRmax...............................................................................................325

9.2 Empirical distribution of residual deformation demands obtained from the GF-

18R (T1=2.0 s) building model for different relative intensities: (a) θr,roof; and (b)

RIDRmax ....................................................................................................................325

9.3 Empirical distribution of RIDRmax for different relative intensities obtained for

building models with same number of stories but different period of vibration: (a)

GF-9R (T1=1.185 s); and (b) GF-9F(T1=1.902 s).......................................................326

9.4 Empirical cumulative distribution of RIDRmax obtained for GF-9R building model

exhibiting two types of building mechanism: (a) BH mechanism; and (b) CH

mechanism................................................................................................................326

9.5 Comparison of three parametric cumulative distribution functions with the

empirical distribution of RIDRmax obtained from the GF-9R (T1=1.185 s) ..................328

9.6 Fitting of the parametric lognormal CDF of RIDRmax for: (a) GF-9R (T1=1.185s);

and (b) GF-9F (T1=1.902s)........................................................................................330

9.7 Fitting of the parametric lognormal CDF of RIDRmax for the GF-18R building

model for two relative intensities: (a) η= 4; and (b) η= 6...........................................330

9.8 Variation of central tendency and dispersion of RIDR computed for five stories of

the GF-9R building model: (a) Median RIDR; (b) dispersion of RIDR ( RIDRlnσ ) ......331

9.9 Evaluation of equations (9.15) and (9.16) to estimate θr,roof: (a) GF-3R (T1=0.5 s);

and (b) GF-18R (T1=2.0 s) ........................................................................................334

9.10 Evaluation of equations (9.15) and (9.16) to estimate RIDRmax: (a) GF-3R (T1=0.5

s); and (b) GF-18R (T1=2.0 s) ...................................................................................335

9.11 Evaluation of equations (9.15) and (9.16) to estimate dispersion of θr,roof: (a) GF-

3R(T1=0.5 s); and (b) GF-18R (T1=2.0 s) ..................................................................335

xxxvi

9.12 Evaluation of equations (9.15) and (9.16) to estimate dispersion of RIDRmax: (a)

GF-3R (T1=0.5 s); and (b) GF-18R (T1=2.0 s) ...........................................................336

9.13 Evaluation of equation (9.16) to estimate dispersion of RIDRmax: (a) GF-9R; and

(b) GF-9F .................................................................................................................336

9.14 Comparison of elastic and inelastic displacement demand hazard curve (for a

system with T=2.0 s and Cy=0.2)...............................................................................339

9.15 Residual drift hazard curves for GF-3R building model: (a)θr, roof; and (b)

RIDRmax ....................................................................................................................340

9.16 Residual drift hazard curves for GF-18R building model: (a) θr, roof; and (b)

RIDRmax ....................................................................................................................340

9.17 Comparison of residual deformation demand hazard curves considering variable

and constant variation of dispersion with changes in ground motion intensity: (a)

)( ,roofrθλ ; and (b) )( maxRIDRλ .............................................................................341

9.18 Maximum drift hazard curves for GF-3R building model: (a) θroof; and (b) IDRmax.....342

9.19 Maximum drift hazard curves for GF-18R building model: (a) θroof; and (b)

IDRmax.......................................................................................................................342

9.20 Comparison of θroof and θr,roof hazard curves obtained from two building model:

(a) GF-3R (T1=0.5 s); and (b) GF-18R (T1=2.0 s)......................................................343

9.21 Comparison of IDRmax and RIDRmax hazard curves obtained from two building

model: (a) GF-3R(T1=0.5 s); and (b) GF-18R(T1=2.0 s) ............................................344

A.1 Magnitude versus distance to horizontal projection of rupture for earthquake

ground motions considered in this study....................................................................367

A.2 Peak ground acceleration (PGA) versus distance to horizontal projection of

rupture for earthquake ground motions considered in this study.................................368

A.3 Median spectral acceleration response, Sa, spectra computed for ζ = 5%: a) Scaled

at T=0.5 s; b) scaled at T = 1.0 s of Sa: (a) from each site class; (b) from all 240

earthquake ground motions .......................................................................................372

xxxvii

A.4 Dispersion of Sa: (a) from each site class; (b) from all 240 earthquake ground

motions.....................................................................................................................372

A.5 Location of ground motion accelerographic stations in Mexico City where

records used in this study were obtained....................................................................375

A.6. (a) Mean and median Sa spectra; (b) Dispersion (COV) ..............................................376

A.7. (a) Peak ground acceleration (PGV) versus earthquake magnitude; (b) PGV

versus closest distance to the rupture.........................................................................376

A.8 Median elastic acceleration spectra for s20-LD and s20-SD ground motion sets:

(a) Scaled to T=0.5s; b) scaled to T=1.0s ...................................................................378

D.1. Fundamental periods of vibration of generic frames considered in this study. .............389

D.2. Height-wise stiffness variation using equation (D.4) ..................................................391

D.3. Height-wise stiffness variation for two building models: (a) 9-story; (b) 18-story.......391

D.4. Fundamental mode shapes computed for two generic building models: (a) 3-story;

(b) 18-story...............................................................................................................393

D.5. Design spectrum used for the design of generic frames ..............................................395

D.6. Flexural strength degradation model implemented in RUAUMOKO (Carr, 2003) ......396

D.6. Calibration of bilinear strength-degrading model using RUAUMOKO (Carr,

2003) ........................................................................................................................396

___________________________________________________________________________________ Chapter 1 Introduction

1

Chapter 1

Introduction

1.1 Motivation

There is a consensus among the earthquake engineering community that both structural and

nonstructural damage are primarily the result of lateral deformation demands in the structure

induced by earthquake ground shaking. In recently proposed performance-based assessment

methodologies structural performance is based on maximum inter-story drift demands. In

addition to maximum deformation demands , the evaluation of residual deformation demands

should play an important role in performance-based design. Some recommended seismic

assessment provisions specify some limiting values on residual deformation (e.g., FEMA,

2000), but they do not include specific procedures to estimate residual deformation demands.

The amplitude of residual deformations is particularly important in determining the technical

or economical feasibility of repairing damaged structures. For example , several dozen

damaged reinforced concrete (RC) buildings in Mexico City had to be demolished after the

1985 because of the technical difficulties to repair structures with large permanent drifts

(Rosenblueth and Meli, 1986). Another example of the consequence of significant permanent

lateral displacements is a two story steel office building that was severely damaged during the

1994 Northridge earthquake (SAC, 1995). The excessive permanent displacements were

consequence of concentrated structural damage in the first story, with yielding noted at the

base plate connections, and a number of through column fractures at the second floor moment

connections. The extent of this damage was such that the building owner decided to demolish

the structure above the foundation level. Similarly, many RC bridge piers were demolished in

Kobe after the 1995 Hyogo-Ken-Nambu earthquake for the elevated cost that would be

required to repair piers with large permanent drifts (Kawashima, 2000). Furthermore, they

also represent one of the most important response parameters in the evaluation of the residual

capacity of damaged structures to sustain aftershocks (Luco et al., 2004; Bazzurro et al.,

2004; Mackie and Stojadinovic, 2004).


2

Recognizing the importance of residual deformation demands in the decision-making

process for retrofitting existing structures and in mitigating potential permanent deformations

in the design of new structures, researchers have suggested innovative structural systems such

as unbonded post-tensioned bridge RC bridge columns (Kwan and Billington, 2003; Sakai

and Mahin, 2004) or RC walls (Kurama et al., 1999) and post-tensioned steel moment

connections (Ricles et al., 2001, Christopoulos et al., 2002,). However, existing RC or steel

structural systems still are prone to experiencing excessive permanent deformations under

seismic loading and its performance can not be fully characterized with only taking into

account the maximum lateral deformations and neglecting possible permanent (residual)

deformation demands. Furthermore, conceptual global predesign approaches for design of

new structures based on controlling maximum deformation demands (e.g., Bertero and

Bertero, 2002; Teran-Gilmore, 2004) are not complete if permanent deformation demand

evaluation and constraint are not included.

Therefore, a reliable evaluation of global and local residual deformation demands should

be explicitly incorporated in seismic performance-based methodologies for the assessment of

existing structures or the design of new structures.

1.2 Overview of Probabilistic Seismic Demand Analysis (PSDA)

In this dissertation, a rational procedure to estimate the mean annual frequency of exceeding a

specified seismic demand for a specific structure at a given seismic environment (i.e., site soil

condition as well as specific magnitude and distance to the source conditions), known as

Probabilistic Seismic Demand Analysis (Cornell, 1996), is used as a framework to account for

residual deformation demands into performance-based assessment of existing structures.

In essence, Probabilistic Seismic Demand Analysis (PSDA) can be viewed as an extension

of a Probabilistic Seismic Hazard Analysis (PSHA) to estimate mean annual frequencies of

exceedance of ground motion intensity parameters to computing mean annual frequencies of

exceedance of structural response parameter. In a PSDA mean annual frequency of exceeding

seismic response parameter of a mathematical model representative of a civil engineering

structure (e.g., bridges, buildings, offshore platform, etc.) are obtained from rigorous nonlinear

dynamic time-history analyses when subjected to a suite of earthquake ground motions scaled


3

to intensities in accordance with the seismic hazard curve at the site, obtained from

conventional PSHA.

It should be mentioned that, similarly to PSHA, the PSDA approach is an application of

the total probability theorem, which is mathematically expressed as follows:

( ) )()(

)(|)(

0imd

imdimd

imIMedpEDPPedp IMEDP

λλ ⋅=>= ∫

∞ (1.1)

where, consistent with the nomenclature suggested in current seismic performance-based

assessment methodologies (Cornell and Krawinkler, 2000; Moehle and Deierlein 2004), EDP

refers to a structural engineering demand parameter (e.g., peak roof displacement, peak inter-

story drift, peak floor acceleration, etc.) and IM denotes the ground motion intensity measure

(e.g., spectral elastic acceleration at the first-mode period of vibration). The mean annual

frequency of exceeding a predefined engineering demand parameter, edp, is denoted as

)(edpEDPλ while )(imIMλ refers to the seismic hazard at the site, measured in terms of mean

annual frequency of a ground motion intensity parameter IM, exceeding a level of intensity im.

In addition, the term )|( imIMedpEDPP => expresses the conditional probability of

exceeding a specific edp given that the ground motion intensity parameter IM is equal to im.

Information of )|( imIMedpEDPP => is obtained from non-linear dynamic analyses

performed for a specific structure subjected to a set of ground motions scaled to various levels

of intensity that represent the local seismicity.

1.3 Brief Review of PSDA Applications

Very recently, there have been an increased number of investigations that have made use of

PSDA framework in order to reach different objectives. In general, PSDA has served to

estimate the mean annual probability of exceeding a given target level of inelastic response in

a specific structure located at a given site with potential seismic hazards. In this section, a

brief overview of the PSDA applications in earthquake engineering problems is presented.

Conceptually, one of the first applications using PSDA framework was done by Esteva

and Ruiz (1989) to compute the mean annual frequency of collapse of regular frame buildings

designed according to the 1987 Mexico City Building Code. In their study, building collapse


4

was defined as the limit state when any story level exceeded a given story displacement

ductility capacity. This study provided mean annual probabilit ies of exceedance of story

displacement ductility capacities.

After being instrumental in the conceptual development of PSHA, Cornell and his co-

workers at Stanford University have devised the use of the PSDA framework for different

challenging structural engineering applications (e.g., Sewell and Cornell, 1987; Bazzurro and

Cornell, 1994; Cornell, 1996; Shome and Cornell, 1999; Luco and Cornell, 1998; 2000). Their

studies have had an important impact in the development of a new generation of earthquake

engineering provisions. As an example, Luco and Cornell (1998, 2000) developed a PSDA

methodology to evaluate the mean annual frequency of exceeding maximum inter-story drift

ratios (IDRmax) for the seismic assessment of steel frame buildings with welded connections

prone to suffer fracture damage. The methodology suggested the inclusion of simplifying

assumptions that allows a closed-form solution in equation (1.1). The reader is referred to

Cornell (1996) and McGuire (2004) for details in the derivation of the closed-form solution

Among the simplifying assumptions , the closed form solution assumes a linear relationship, in

the log-log space, of IDRmax and the spectral acceleration corresponding to the building’s

fundamental period of vibration, Sa (T1), which was selected as IM, and that dispersion of

IDRmax for different levels of intensity is constant. Their approach was more recently modified

to a Load and Resistance Factor Design Format type format (Cornell et al., 2002), more

commonly used in structural engineering practice, and incorporated in recently released

seismic performance-based guidelines for the assessment of existing steel structures

designated as FEMA 350-353 (FEMA, 2001).

In addition, a further extension of the PSDA approach has been recently adopted for the

Pacific Earthquake Engineering Center (PEER) to evaluate seismic building performance

primarily in terms of economic losses (Cornell and Krawinkler, 2000; Moehle and Deierlein,

2004). As part of the PEER vision, Krawinkler and his co-workers (Ibarra et al., 2002; Medina

and Krawinkler, 2003; Krawinkler et al., 2004) have evaluate the mean annual probability of

collapse capacity of regular frame buildings, where building collapse capacity is associated to

the loss of lateral strength capacity in the onset of collapse under earthquake loading rather

than deformation capacity associated to collapse.

Miranda and his collaborators (Miranda and Aslani, 2003; Miranda et al., 2004) have

employed PSDA framework to estimate the building-specific expected annual losses and mean

annual frequency of exceedance of economic losses. Recognizing the important contribution


5

of acceleration-dependent non-structural components in the total economic losses produced in

buildings by earthquakes, the authors focused not only on peak inter-story drifts at each floor

but also on peak floor acceleration at each floor as key structural demand parameters.

Finally, it should be mentioned that interesting applications of PSDA has been done

recently for the design of new frame steel buildings incorporating structural control devices

such as friction-pendulum type base isolation or viscous dampers (Barroso and Winterstein,

2002) as well as for evaluating retrofit strategies of existing reinforced concrete frame

buildings incorporating energy dissipation devices (Torres and Ruiz, 2004).

1.4 Some Needs of Current PSDA Methodologies

1.4.1 Improved Intensity Measures

An important component in PSDA is the selection of an appropriate parameter to characterize

the intensity of earthquake ground motions. This parameter is commonly referred to as ground

motion intensity measure (IM) in conventional PSDA and its selection depends on several

aspects. For example, Luco and Cornell (2004) have suggested three criteria for selecting an

appropriate IM for PSDA: a) Efficiency; b) sufficiency; and c) computability. The term

efficient is used to designate an IM that produces a small record-to-record variability of the

EDP of interest conditioned on the ground motion intensity. An IM is designated as sufficient

when its use allows computation of the EDP conditionally independent of the earthquake

magnitude , distance to the earthquake source, and other source parameters that influence IM.

Finally, it is desirable that any pre-selected IM, on the basis of efficiency and sufficiency,

should be readily available in the form of )(imIMλ to be used in PSDA methodologies. As

mentioned by the authors, there is a trade-off between the efficiency and sufficiency of the IM

and the computability of )(imIMλ . It should be noted that an IM can be a scalar (e.g., spectral

acceleration corresponding to the first-mode structure period of vibration, Sa (T1)) or a vector-

valued IM which couples two, or more, basic IM’s (e.g., Sa (T1) and the elastic spectral

acceleration corresponding to the second-mode period of the structure, Sa (T2))(e.g., Shome

and Cornell, 1999; Luco and Cornell, 2004).


6

The limitations of originally proposed Sa (T1) as IM has been highlighted by several

researchers (Cordova et al., 2001; Luco and Cornell, 2004; Miranda et al., 2004). Among

others, Sa (T1) is not an efficient IM when evaluating the seismic performance of buildings

subjected to near-fault ground motions, to account for higher-mode effects of tall flexible

structures or to evaluate acceleration demands in buildings. Several alternative IM’s have been

proposed and evaluated recently (Giovenale et al., 2003; Conte et al., 2003; Giovenale et al.,

2004; Luco and Cornell, 2004) showing more robustness, in terms of efficiency and

sufficiency, than Sa (T1) but, up to date, most of identified promising IMs lack of the

computability requirement. In particular, scaling the ground motion records to produce the

same maximum inelastic displacement of a SDOF system having the same structure first-

mode period of vibration, ∆i(T1), and yield displacement has shown to be a more robust IM

than Sa (T1) (Luco and Cornell, 2004; Miranda et al., 2004). However, seismic hazard curves

of maximum inelastic displacement, λ(∆i), have not been yet developed. In order to develop

such an inelastic seismic hazard curves an attenuation relationship of maximum inelastic

displacement should be developed. In order to partially overcome this critical component in

PSDA, Luco et al. (2002) proposed a simulation-based approach to obtain the ground motion

hazard at a given site in terms of any IM. This simulation approach seems appealing, but still

computationally intensive for practical implementation.

1.4.2 Characterization of Central Tendency and Dispersion of EDPs

Another key assumption in simplified PSDA methodologies which incorporates the closed-

form solution to estimate the mean annual probability of exceeding seismic demands as

proposed by Cornell and his collaborators (e.g., Cornell, 1996; Luco and Cornell, 1998, 2000)

is that dispersion of central tendency is assumed to be constant regardless the level of intensity

of the ground motion. However, statistical results have showed that the record-to-record

variability of EDPs depends on the level of ground motion intensity. Furthermore, the

relationship between the EDP and the intensity of the ground motion even in the log-log space

might not be linear. Then, improved functional forms that consider an improved

characterization of the variation of probability parameters (e.g., central tendency and

dispersion of the EDP) with changes in the level of ground motion intensity should be

included in the estimation of the conditional probability of exceeding a specified EDP for a

given IM. The latter observation has been overcome by Miranda and Aslani (2003) for the


7

probabilistic evaluation of IDRmax and PFA through improved, still simple, functional forms,

but their applicability to other EDPs of interest such as residual deformation demands remains

an open question.

1.5 Objectives

The main goal of this dissertation consist of developing an improved PSDA approach for

assessing the seismic performance of existing standard regular buildings which explicitly takes

into account both maximum and residual deformation demands. In order to develop such a

methodology, the following tasks were carried out in this dissertation:

a) Comprehensive statistical studies to evaluate maximum inelastic displacement

demands, ∆i, and residual displacement demands, ∆r, of single-degree-of-freedom

(SDOF) systems considering a relatively large earthquake ground motion database,

and considering a large number of structural parameters.

b) Development of both maximum and residual displacement demand hazard curves

and, in consequence, uniform hazard spectra of maximum and residual

displacement demand.

c) Detailed statistical studies to evaluate lateral deformation, both maximum and

residual, demands of multi-story frame building models under different levels of

ground motion intensity.

d) To explore an alternative ground motion intensity measure based on maximum

inelastic displacement demand of an equivalent SDOF system to reduce the

variability in the estimation of deformation demands.

e) Coupling of maximum inelastic displacement demand hazard curves with

deformation demands of MDOF systems to estimate mean annual frequency of

exceeding maximum and residual deformation demands.

1.6 Outline

An improved approach for the performance-based assessment of existing structures or the

conceptual global preliminary design of new structures is introduced in this dissertation. This

approach accounts for the estimation of both global (e.g., residual roof drift ratio) and local


8

(e.g., maximum residual inter-story drift ratio) residual deformation demands along with

commonly accepted maximum deformation demands (e.g., peak roof drift ratio and peak inter-

story drift ratio) to establish the seismic performance of site-specific existing structures. Both

maximum and residual deformation demands are expressed in terms of the mean annual

frequency of exceeding a certain level of deformation demand, which can be related to

deformation-based performance limit states linked to discrete performance levels of both

individual components as well as complete systems. An important component of the improved

methodology is the use of maximum inelastic displacement demand of SDOF systems,

corresponding to the period of vibration and relative lateral strength of interest, as intensity

measure instead of traditional elastic intensity measures (e.g., elastic spectral displacement

associated to the first-mode period of vibration of the structure). Then, maximum inelastic

displacement demand hazard curves need to be developed. The basis for fast and novel

approach to develop inelastic seismic hazard curves which uses readily-available elastic

seismic hazard curves is described in Chapter 4. The procedure is based on information

developed from comprehensive statistical studies of inelastic deformation demands for SDOF

systems subjected to earthquake ground motions recorded on different spatial and site

environment conditions.

To achieve the objectives described in Section 1.5, this dissertation is organized in two

parts. The first part reports statistical studies and probabilistic estimation of maximum and

residual displacement demands of SDOF systems, which are reported from Chapters 2 to 6.

The second part includes statistical studies and probabilistic estimation of maximum drift

demands for MDOF systems, which are reported from Chapter 7 to 9. An outline of the

dissertation is described as follows:

The statistical evaluation of inelastic deformation demands for SDOF systems subjected to

acceleration time histories recorded on rock or firm site conditions at distances to the fault

larger than 15 km is the subject of Chapter 2. In this chapter, maximum inelastic

displacement demands are expressed in terms of inelastic displacement ratios, CR, which

allows the estimation of maximum inelastic displacement demands from maximum elastic

displacement demands. Inelastic displacement ratios and their dependence on period of

vibration, level of lateral strength ratio, soil site conditions and hysteretic behavior of the

inelastic system are studied in detail. Dispersion associated to the estimation of central

tendencies is also carefully evaluated.


9

Chapter 3 reports the results of a parallel study on inelastic displacement ratios

considering earthquake ground motions recorded on soft-soil site conditions and acceleration

time-histories collected from recording stations close to the fault (e.g., distance to the fault

rupture smaller than 15 km.) with forward directivity effects in the normal component. It is

shown that in spite of their different nature (e.g., different frequency content, ground motion

duration, shear wave velocity, etc.) inelastic displacement ratios for this type of seismic

environment exhibit some common features when they are normalized with respect to the

predominant period of the ground motion or to the pulse period.

Chapter 4 introduces and applies the methodology to estimate inelastic displacement

seismic hazard curves which provide information about the mean annual frequency of

exceeding various levels of inelastic deformation demand for SDOF systems having different

periods of vibration and levels of relative lateral strength for a specific site. These curves are

also used to develop uniform hazard spectra of maximum inelastic displacement

corresponding to different return periods, which per se are very useful tools to establish

adequate deformation-based performance limit states. Included in this chapter is an evaluation

of the simplifying assumptions inherent in the proposed methodology as well as its limitations

and possible future improvements.

Another important piece of the methodology suggested in this dissertation is the

estimation of residual deformation demands. Chapter 5 reports the results of a comprehensive

statistical study aimed at evaluating residual deformation demands of inelastic SDOF systems.

For this purpose, two approaches are introduced: a) A direct approach, that makes use of

residual displacement ratios, Cr, which allows the estimation of residual displacement

demands of SDOF systems at the end of the earthquake ground shaking from maximum elastic

spectral displacement; and (b) An indirect approach, which estimates residual displacement

demands from maximum inelastic displacement demand through a residual ratio. Parametric

studies are conducted that identify the influence of the period of vibration, the level of lateral

strength ratio, type of firm soil site conditions, type of ground motion, and the hysteretic

behavior on central tendencies, as well as on the dispersion associated.

In Chapter 6, the methodology introduced in Chapter 4 is adapted to compute site-

specific residual displacement seismic hazard curves for inelastic SDOF systems with

different periods of vibration and different levels of relative lateral strength. In addition,

uniform hazard spectra of residual displacement demand are developed for two return periods.


10

Reported in Chapter 7 are statistical studies on the seismic response of generic one-bay

moment-resisting frames performed to evaluate both maximum and residual deformation

demands of MDOF systems under different levels of intensity of the earthquake ground

motion. For this purpose, both maximum elastic spectral displacements and maximum

inelastic displacement are used as intensity measures. Derived from the former study,

statistical results of deformation ratios that can be implemented in simplified methods for

rapidly assessing the seismic performance of existing structures, or during the global

predesign phase of new structures, are reported in Chapter 8.

The proposed approach to assess the seismic performance of existing structures

accounting for residual deformation demands is formulated and illustrated in Chapter 9. The

suggested approach combines the results obtained in previous chapters (Chapters 2, 4 and 7).

Thus, maximum and residual deformation fragility curves are combined with maximum

displacement seismic hazard curves to obtain mean annual frequency of exceeding maximum

and residual deformation demands for site-specific structures. It is shown that the proposed

methodology can be very helpful for the performance-based assessment of existing structures

and even for the conceptual global preliminary design of new structures.

Finally, Chapter 10 summarizes relevant conclusions derived from this investigation. In

addition, it discusses the limitations of the proposed approach and further research needed for

enhancing and extending the methodology.

In order to supply additional information, this dissertation includes four appendices.

Appendix A lists the suites of earthquake ground motions used throughout this research.

Appendix B discusses ground motion characterization issues. Appendix C defines the sample

statistical measures. Finally, Appendix D describes the design process of the generic frame

buildings models as well as the modeling assumptions.

___________________________________________________________________________________ Chapter 2 Maximum Inelastic Displacement Demands of SDOF Systems: Firm Sites

11

Chapter 2

Maximum Inelastic Displacement Demands of SDOF

Systems: Firm Soil Sites

2.1 Introduction

Recently introduced displacement-based seismic design criteria use displacements rather than

forces as basic demand parameters for the design, evaluation and rehabilitation of structures.

However, implementation of displacement-based seismic design criteria into structural

engineering practice requires simplified analysis procedures to estimate inelastic displacement

demands on structures for ground motions in which the structure is expected to behave

nonlinearly. This is particularly true when the seismic hazard at the site for design is specified

as design spectra rather than as acceleration time histories. Recent recommendations for the

evaluation and rehabilitation of existing structures have introduced simplified analysis

methods in which single -degree-of-freedom (SDOF) systems are used to estimate global

inelastic displacement demands on structures. Examples of those recommendations are the

ATC-40 guidelines (ATC, 1996), FEMA-273 (FEMA, 1997a) and FEMA 356 (FEMA,

2000). Furthermore, in these new resource documents global inelastic displacement demands

of structures are computed taking into account the relationship between the maximum

inelastic displacement demands of nonlinear SDOF systems and the maximum elastic

displacement demands of linear elastic SDOF systems. Thus, recently there is a renewed

interest on approximate methods to estimate maximum displacement demands of inelastic

SDOF systems.

The objective of this chapter is to present the results of a statistical study of the ratio of

maximum inelastic displacement demand to maximum elastic displacement demand for SDOF

systems on firm sites with known relative strength. The effects of period of vibration, level of

lateral strength, and local site conditions are investigated. In addition, the effect of post-yield

stiffness and hysteretic behavior is specially addressed. In particular, the dispersion of


12

constant relative strength inelastic displacement ratios is assessed. This study makes use of

improved information that has been recently made available on the geological characteristics

at accelerographic recording stations in California. The investigation is limited to rock and

relatively firm soil sites with shear wave velocities higher than 180 m/s in the upper 30m of

the site profile. Note that several written portions of this chapter (mainly related to the results

for elastoplastic SDOF systems) were already published in Earthquake Engineering and

Structural Dynamics (Ruiz-Garcia and Miranda, 2003), although related to a different ground

motion database.

2.2 Definition of Inelastic Displacement Ratios

The inelastic displacement ratio, CR, is defined as the maximum lateral inelastic displacement

demand, ∆i, divided by the maximum lateral elastic displacement demand, Sd, on systems with

the same mass and initial stiffness (i.e., same period of vibration) when subjected to the same

earthquake ground motion. In both cases displacements are relative to the ground.

Mathematically this is expressed as:

d

iR S

C∆= (2.1)

In equation (2.1), ∆i is computed in systems with constant yielding strength relative to the

strength required to maintain the system elastic (i.e., constant relative strength). Here, the

relative lateral strength is measured by the strength ratio R, which is defined as:

y

a

FSm

R

= (2.2)

where m is the mass of the system, Sa is the acceleration spectral ordinate and Fy is the lateral

yielding strength of the system. The numerator in equation (2.2) represents the lateral strength

required to maintain the system elastic, which sometimes is also referred to as the elastic

strength demand.

Nomenclature in equation (2.1) is meant to be consistent with the nomenclature used in

NEHRP publications (FEMA, 1997a; 1997b; FEMA, 2000) in which the letter C is used as a

factor modifying elastic displacements and is also consistent with the nomenclature previously


13

used by Miranda (2000) in which the subscript in the inelastic displacement ratio represents

the parameter that remains constant. Thus, constant ductility inelastic displacement ratios are

represented by Cµ, and constant relative strength (or constant strength ratio) inelastic

displacement ratios are represented by CR. Both types of inelastic displacement ratios permit

the estimation of maximum inelastic displacement demands from maximum elastic

displacement demands.

T=2.0s

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 2.0 4.0 6.0 8.0Ductility Demand µ

1/R

Using E[C R ] and eq. (2.3)

Using E[C µ ] and eq. (2.4)

T=0.2s

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 2.0 4.0 6.0 8.0Ductility Demand µ

1/R

Using E[CR ] and eq. (2.3)

Using E[Cµ ] and eq. (2.4)

Figure 2.1. Relationship between normalized strength (1/R) and displacement ductility demand: (a) For

a T=2.0 s system; (b) For a T=0.2s system (after Ruiz-Garcia and Miranda, 2003).

Inelastic displacement ratios were computed for SDOF systems having a viscous damping

ratio of 5%, a nonlinear elasto-plastic hysteretic behavior, and with the following strength

ratios R=1, 1.5, 2, 3, 4, 5 and 6. For each earthquake record and each relative strength ratio,

inelastic displacement ratios were computed for a set of 50 periods of vibration between 0.05

and 3.0 s. Unlike the constant ductility inelastic displacement ratio Cµ that has to be computed

through iteration on the lateral strength until the computed displacement ductility demand is

within a certain tolerance equal to the target ductility ratio, the constant relative strength

inelastic displacement ratio CR can be computed without any iteration and thus, for a given

acceleration time history it is significantly faster to compute.

Figure 2.1 shows the relationship between the lateral strength of SDOF systems with a

period of vibration of 1.0 s, and the maximum displacement when subjected to 264 earthquake

ground motions recorded on firm sites (Miranda, 2000). In this figure, the lateral strength is

normalized by the lateral strength required to maintain the system elastic (i.e., Fy/(mSa)=1/R),

and the maximum deformation is normalized by the yield displacement. The continuous line


14

represents the rela tionship between the lateral strength and the expected value of the ductility

demand computed using the expected value of CR as follows:

][][ RCERE ⋅=µ (2.3)

where µ is the displacement ductility ratio and E[ ] denotes expectation. The dotted line

represents the relationship between the displacement ductility ratio and the expected value of

the normalized lateral strength computed for R=1, 1.5, 2, 3, 4, 5 and 6 using the expected

value of Cµ as follows:

µµ ][1 CE

RE =

(2.4)

It can be seen that, in this case, the use of E[Cµ ] using equation (2.4) leads to very similar

results than when using E[CR] in equation (2.3). However, it can be noticed that for a given R

the ductility demand computed with constant ductility inelastic displacement ratios is always

smaller than the one computed with constant relative strength inelastic displacement ratios.

Furthermore, the difference increases as R and µ increase. The practical implication of this

observation is that the use of constant ductility inelastic displacement ratios like those reported

by Miranda (2000), if used for the evaluation of existing structures, in which R is known for a

given ground motion, can lead to underestimations of the maximum inelastic displacement

demands. The underestimation increases as the dispersion on Cµ and CR increases. For short

period structures, where the dispersion on CR is large, the underestimation can be much larger.

An example similar to that shown in figure 2.1a but for a period of 0.2 s is shown in figure

2.1b. In this case, for an R=3 the underestimation in maximum displacement can be larger than

40%. Hence, it is clear that for the evaluation of existing structures there is a need for

statistical studies on CR. Only when there is no dispersion on Cµ and CR, as for example in the

case of R=1 or for a single ground motion (for any level of R) is that equations 2.3 and 2.4

lead to the same displacement. For a further discussion on the reason for this difference, the

reader is referred to Miranda (2001).


15

2.3 Review of Previous Studies on Inelastic Displacement Ratios

In this section, a brief review of prior published studies aimed to explicitly evaluate inelastic

displacement ratios, either from a constant-ductility or constant-strength perspective, is

presented. The section is divided in two parts. The first part introduces main findings from

previous investigations that considered elastoplastic and bilinear SDOF systems, whereas the

second part presents relevant observations from studies that employed degrading SDOF

systems.

2.3.1 Inelastic Displacement Ratios for Elastoplastic and Bilinear systems

The first study that investigated the relationship between the maximum deformations of

inelastic and of elastic systems was conducted by Veletsos and Newmark (1960, 1965) who

studied SDOF systems with an elasto-plastic hysteretic behavior subjected to simple pulses

and to three recorded earthquake ground motions. They observed that in the low frequency

region the maximum deformation of the inelastic and elastic systems was approximately the

same. This observation gave rise to the well-known equal displacement rule (i.e., maximum

inelastic displacement is equal to maximum elastic displacement). The study also concluded

that, in the high frequency region, the inelastic displacements are significantly higher than

their elastic counterparts. In a later study (Veletsos and Vann, 1971) , the equal displacement

rule was also recommended for the medium frequency region. They also noted that the width

of each frequency region is generally different for different ground motions. Shimazaki and

Sozen (1984) noticed that, in the short period region the ratio of maximum inelastic

displacement demand to maximum elastic displacement demand depended critically on the

lateral strength of the structure relative to the elastic strength demand and that the estimate of

the inelastic displacement demand was beyond a simple procedure. Later, these observations

were confirmed by Qi and Moehle (1991) using short-period equivalent SDOF systems

subjected to three recorded ground motions.

Miranda (1993) studied ratios of maximum inelastic displacement to maximum elastic

displacement of SDOF systems undergoing specific levels of displacement ductility when

subjected to 124 earthquake ground motions recorded on different site conditions. Mean

constant-ductility ratios of maximum inelastic to maximum elastic response for three types of


16

soil conditions were computed as part of the investigation. The study gave a special insight to

this ratio in the short period range and to the limiting periods of the spectral regions where the

equal displacement rule is applicable. The results of Miranda (1993) were confirmed by

Krawinkler and his co-workers at Stanford University (1993, 1997) using smaller sets of

ground motions.

In order to evaluate the nonlinear static procedure of the FEMA guidelines for the seismic

rehabilitation of buildings (FEMA-273), Whittaker et al. (1998) also conducted a study of the

ratio of inelastic to elastic displacements. Results of the ratio of mean inelastic displacements

to mean elastic displacements were presented corresponding to 20 horizontal components of

10 ground motions recorded on either stiff soil or soft rock sites. The study concluded that for

periods smaller than about 1.0s, mean inelastic displacements exceed mean elastic

displacements. Furthermore, it was concluded that for systems with lateral strengths smaller

than 20% of the strength required to maintain the system elastic, mean inelastic displacements

systematically exceed mean elastic displacements, suggesting that the equal displacement rule

may not be applicable in these situations.

More recently, Miranda (2000) presented a comprehensive statistical study of constant

ductility inelastic displacement ratios for the design of structures on firm sites. This study

provided new information regarding the dispersion of this ratio and regarding the influence of

period, level of inelastic deformation, magnitude, distance to the source and local site

conditions. Miranda concluded that for SDOF systems undergoing the same displacement

ductility ratio, inelastic displacement ratios were not affected by magnitude or by distance to

the source. Furthermore, the study concluded that for sites with average shear wave velocities

higher than 180 m/s (600 ft/s) in the upper 30 m (100 ft) of the site profile local, site

conditions do not affect significantly constant-ductility inelastic displacement ratios. As part

of the study, a simplified equation to estimate ratios of maximum inelastic to maximum elastic

displacement as a function of period of vibration and of displacement ductility ratio was also

developed. The simplified equation to estimate constant-ductility inelastic displacement ratios

developed by Miranda (2000) is very useful in preliminary design of new structures where

control of maximum inelastic deformations is desired for structures where an estimate of the

global displacement ductility capacity is known. However, in the evaluation of existing

structures the main interest is to determine the global and local deformations that a structure

with known lateral strength may undergo when subjected to earthquakes of different


17

intensities. As showed in the previous section, the use of constant-ductility inelastic

displacement ratios underestimates the expected value of the maximum deformations in

systems with known lateral strength. Hence, inelastic displacement ratios corresponding to

systems with equal relative lateral strength (lateral yielding strength relative to the lateral

strength required to maintain the system elastic) are particularly useful when evaluating

existing structures.

Motivated for the improvement of current seismic guidelines for the design of buildings

with damping-type passive energy dissipating devices, Ramirez et al. (2002) conducted an

analytical study of bilinear (i.e., considering four levels of positive post-yield stiffness)

inelastic SDOF systems having three different viscous damping ratios (ζ = 0.05, 0.20 and

0.35), which account for additional linear viscous damping. The systems where subjected to a

suite of 20 earthquake ground motions that matched, on average, the 2000 NEHRP spectrum

for a region of high seismicity located on stiff soil site. The authors found that, in the range of

5 to 30%, the effect of viscous damping ratio on inelastic displacement ratios computed for

bilinear systems is not significant.

Very recently, Ruiz-Garcia and Miranda (2003) published a comprehensive statistical

study on constant-strength inelastic displacement ratios of elastoplastic SDOF systems

subjected to an relatively large ensemble of 216 ground motions. Among other conclusions,

they concluded that magnitude and distance to the source does not have a significant influence

on inelastic displacement ratios for systems with periods of vibration in the medium- and

long-period region and with R < 4. The authors proposed an equation to obtain estimates of

mean inelastic displacement ratios. Their results have been recently confirmed by Chopra and

Chintanapakdee (2004).

While prior studies have provided valuable information regarding central tendency of

constant relative strength inelastic displacement ratios of SDOF systems, few of them

provided reliable information on the dispersion of this ratio, mainly due to the small sample of

ground motions.


18

2.3.2 Inelastic Displacement Ratios for Degrading Systems

The effect of hysteretic behavior on the seismic response of inelastic SDOF systems has

deserved much attention from researches in the last two decades. In particular, the influence of

the hysteretic behavior has been studied on lateral strength demands for systems undergoing

specific levels of ductility (i.e., on constant-ductility inelastic strength demand spectra).

However, very few investigations have studied its effect directly on displacement demands

and even fewer have investigated this effect on constant-ductility (Gupta and Kunnath, 1998)

or constant-relative strength inelastic displacement ratios (Gupta and Krawinkler, 1999; Song

and Pincheira, 2000; Pekoz and Pincheira, 2004). From these previous studies, Gupta and

Kunnath (1998) noted that inelastic displacement ratios are greatly affected by the type of

hysteretic behavior adopted to compute the response of nonlinear SDOF oscillators.

Particularly, Gupta and Krawinkler (1999) showed that nonlinear SDOF oscillators having

pinched stiffness-degrading hysteretic behavior leads to larger maximum inelastic

displacements than pure stiffness-degrading systems. One of the most complete studies on the

effect of structural deterioration on displacement demands on inelastic displacement demands

is to the one by Pincheira and his co-workers (Song and Pincheira, 2000; Pekoz and Pincheira,

2004). They reported that maximum inelastic displacement for degrading systems are larger

than those of non-degrading systems when the period of vibration is shorter than the

predominant period of the ground motion (defined as the peak in the input energy spectra of

an elastic SDOF system).

2.4 Earthquake Ground Motions Used in This Study

A total of 240 earthquake acceleration time histories recorded in the state of California in 12

different earthquakes with magnitude ranging from 5.8 to 7.7 were used in this study. A

particularly large number of earthquake ground motions were selected in order to carefully

assess the dispersion of the inelastic displacement ratios and in order to be able to obtain

inelastic displacement ratios corresponding to different percentiles. All the ground motions

selected have the following characteristics: (1) recorded on accelerographic stations where

enough information exists on the geological and geotechnical conditions at the site that

enables the classification of the recording site in accordance to recent code recommendations

(FEMA, 1997a; 1997b; FEMA, 2000); (2) recorded on rock or firm sites with average shear


19

wave velocities higher than 180 m/s (600 ft/s) in the upper 30 m (100 ft) of the site profile; (3)

recorded on free field stations or in the first floor of low-rise buildings with negligible soil-

structure interaction effects; (4) recorded in earthquakes with surface wave magnitudes (Ms)

larger than 5.7; and (5) records in which at least one of the two horizontal components had a

peak ground acceleration larger than 40 cm/s2.

The earthquake ground motions were divided into three groups according to the local site

conditions at the recording station. The first group consisted of 80 ground motions recorded

on stations located on rock with average shear wave velocities between 760 m/s (2,500 ft/s)

and 1,525 m/s (5,000 ft/s). The second group consisted of 80 records obtained on stations on

very dense soil or soft rock with average shear wave velocities between 360 m/s (1,200 ft/s)

and 760 m/s while the third group consisted of 80 ground motions recorded on stations on stiff

soil with average shear wave velocities between 180 m/s (600 ft/s) and 360 m/s. Recording

stations in the first group correspond to site class A and B according to recent design

provisions while recording stations in the second and third groups correspond to site classes C

and D, respectively. A complete list of all ground motions including peak ground

accelerations, earthquake magnitude, site class at the recording station, and distance to the

horizontal projection of the fault rupture is reported in Appendix A of this dissertation.

2.5 Hysteretic Models Considered in this Study

Most previous studies that have investigated the inelastic response of structures when

subjected to earthquake ground motions have used the elastic -perfectly plastic (EPP) model.

However, it is clear that this hysteretic model does not adequately represents the hysteretic

behavior of structural elements or of structures that exhibit structural degradation (i.e.,

combination of stiffness degradation, strength deterioration, pinching, etc.) such as that

occurring in most reinforced concrete (RC) structures under seismic excitations. It should be

noticed that the type and amount of structural degradation depends on several factors such as

loading history, the level of shear stress acting on the element, bar anchorage length, amount

of transverse reinforcement, level of axial load, etc. It should be noted that some degree of

structural degradation is expected to occur even in RC elements with adequate detailing.

Similarly, some structural degradation is also expected to occur in steel or composite

structural elements and connections due to local web and/or flange local buckling, among


20

other phenomena. Many hysteretic models have been proposed in the literature to represent

the load-deformation characteristics of steel or RC structures when subjected to alternate

cyclic loading. A comprehensive review of early RC hysteretic models can be found in Otani

(1981) while more recent models are proposed in (e.g., Kunnath et al., 1992; Cheok et al.,

1998; Song and Pincheira, 2000; Ibarra, 2004).

In this investigation, an enhanced version of the original three-parameter model (Kunnath

et al., 1992) was used to reproduce the global nonlinear response of RC structural systems

when subjected to cyclic lateral loading (Cheok et al., 1998). Initially, this analytical model

was proposed to simulate several types of hysteretic behavior features of RC members through

an adequate selection of four parameters. Each parameter control the level of stiffness

degradation (parameter HC), the level of strength degradation (parameters HBD and HBE)

and pinching (parameter HS). In particular, the enhanced version allows strength degradation

under cycling loading as a function of the cumulative hysteretic energy (parameter HBE) and

the displacement ductility (parameter HBD) reached at each previous half cycle. Further

information about the analytical model can be found in (Cheok et al., 1998). However, it

should be noted, that the type of strength degradation that is captured by this model only

includes cyclic degradation in which the peak deformation at a certain point in the loading

history will only affect the strength and stiffness in future cycles or in future half-cycles of

deformation, but will not decrease the lateral strength during the current half-cycle of

deformation. Reduction in lateral strength during half cycles has more recently been defined

as post-capping behavior (Ibarra, 2004).

In this chapter, the effect of hysteretic behavior on CR was studied in three stages. In the

first stage, the effect of stiffness degradation and, in particular, the effect of unloading

stiffness was addressed. Next, the combined effect of stiffness- and strength-degradation was

investigated in the second stage while the effect of structural degradation was studied in the

last stage.

2.5.1 Stiffness-Degrading Hysteretic Models

In order to address the effect of stiffness degradation and, in particular, the influence of

unloading stiffness three different values of parameter HC of the modified three-parameter

model (Cheok et al., 1998), were considered: ∞ , 2.5, and 0.1. In this analytical model, a very


21

large value of HC (HC? ∞ ) reproduces the well-known modified-Clough model (Mahin and

Bertero, 1976) where the unloading stiffness is equal to the initial stiffness, while a value of

HC = 2.5 simulates the Takeda model (Takeda et al., 1970). The two latter models have been

successfully used in the past to analytically reproduce the hysteretic behavior of well-detailed

flexurally-controlled reinforced concrete elements with negligible strength deterioration under

cyclic loading. A value of HC = 0.1 reproduces the force-deformation relationship of RC

elements with unloading branches towards the origin (origin-centered capability) under

repeated loading, which is aimed to reproduce the hysteretic behavior of unbonded post-

tensioned bridge piers without significant axial load (Kwan and Billington, 2003). The

hysteretic behaviors considered in this stage are illustrated in figure 2.2

(c) O-O

-1.2-1.0-0.8-0.6-0.4-0.20.0

0.20.40.60.81.01.2

-15 -10 -5 0 5 10 15Displacement Ductility

Nor

mal

ized

For

ce

(b) TK

-1.2-1.0

-0.8-0.6

-0.4-0.2

0.00.20.40.6

0.81.01.2

-15 -12 -9 -6 -3 0 3 6 9 12 15Displacement Ductility

Nor

mal

ized

For

ce

(a) MC

-1.2-1.0

-0.8-0.6

-0.4-0.2

0.00.2

0.40.60.8

1.01.2

-15 -10 -5 0 5 10 15Displacement Ductility

Nor

mal

ized

For

ce

Figure 2.2. Hysteretic models used in this investigation: (a) Modified-Clough (MC); (b) Takeda model (TK); and (c) Origin-Oriented model (O-O).


22

2.5.2 Stiffness- and Strength-Degrading Hysteretic Models

The second stage on the effect of hysteric behavior on inelastic displacement ratios considers

stiffness- and-strength degrading analytical models. Thus, the modified three parameter model

was calibrated to represent the following global behavior of RC structural systems under

cyclic loading: (a) Stiffness-degrading (SD), to represent flexural behavior that only exhibit

stiffness degradation; (b) moderately-degrading (MSD), to simulate systems that experience

stiffness degradation in conjunction with moderate strength deterioration; and (c) Severely-

degrading (SSD), to emulate systems that suffer both stiffness degradation and severe strength

deterioration. In the aforementioned analytical models, the level of strength deterioration

corresponding to a displacement ductility of 6 is approximately 55% for the SSD model and

21% for the MSD model. A graphical representation of the simulated hysteretic behaviors is

showed in figure 2.3.

(a) SD

-1.2

-1.0-0.8

-0.6

-0.4-0.2

0.0

0.20.4

0.6

0.81.0

1.2

-15 -12 -9 -6 -3 0 3 6 9 12 15

Displacement Ductility

Nor

mal

ized

For

ce

(b) MSD

-1.2

-1.0-0.8

-0.6

-0.4-0.2

0.00.2

0.40.6

0.81.0

1.2

-15 -12 -9 -6 -3 0 3 6 9 12 15


Noo

rmal

ized

For

ce

(c) SSD

-1.2

-1.0-0.8

-0.6-0.4

-0.20.0

0.20.4

0.60.8

1.01.2

-15 -12 -9 -6 -3 0 3 6 9 12 15


Nor

mal

ized

For

ce

Figure 2.3. Hysteretic models used in the second stage of this investigation: (a) SD (stiffness-

degrading); (b) MSD (Moderately-degrading); and (c) SSD (severely-degrading).


23

As can be noted from the figure, strength degradation depends on the maximum

deformation as well as on the cumulative dissipated hysteretic energy reached in the previous

cycles, which has been referred as in-cycle strength deterioration, but it does not degrades in

the same cycle . It should be mentioned that pinching behavior was not included in this stage.

2.5.3 Structural-Degrading Hysteretic Models

Finally, the last stage investigates the influence of structural degradation (i.e., combination of

stiffness degradation, strength deterioration, pinching, etc.) on inelastic displacement ratios.

For that purpose, two hysteretic behaviors are considered in this stage to characterize the

global response of structural-degrading SDOF systems.

OJS4-1Experimental

-400

-300

-200

-100

0

100

200

300

400

-0.10 -0.05 0.00 0.05 0.10

Drift Angle (rad)

For

ce (B

eam

She

ar)(

kN)

OJS4-1Analytical

-400

-300

-200

-100

0

100

200

300

400

-0.10 -0.06 -0.02 0.02 0.06 0.10

Drift Angle (rad)

For

ce (B

eam

She

ar)(

kN)

OJS1-1Experimental

-250

-200

-150

-100

-50

0

50

100

150

200

250

-0.08 -0.04 0 0.04 0.08Drift Angle (rad)

For

ce (

Bea

m S

hear

)(kN

)

OJS1-1Analytical

-250

-200

-150

-100

-50

0

50

100

150

200

250

-0.08 -0.04 0.00 0.04 0.08

Drift Angle (rad)

For

ce (

Bea

m S

hear

)(kN

)

Figure 2.4. Experimental and analytical response for typical composite beam-column connections with

inelastic action in the joint panel zone (after Kanno and Deierlein, 1997).

First, the modified version of the three-parameter model (Cheok et al., 1998) was used to

reproduce and to carefully calibrate the load-deformation response of two half-scale composite

steel beam-reinforced concrete column connections (specimens OJS1-1 and OJS4-1)


24

previously tested at Cornell University by Kanno and Deierlein (1992). The parameters that

control the analytical response were carefully selected for each specimen to minimize the

difference, through an iterative process, between the experimental and analytical response. A

comparison of the experimental and analytical (simulated) load-deformation response of

specimens OJS4-1 and OJS4-1 is shown in figure 2.4. It is important that note that these

specimens represent composite subassemblies where the steel beam was designed to remain

elastic or practically elastic in order to concentrate most inelastic deformation and structural

damage in the composite joint. For example, it can be seen that the specimen OJS4-1 exhibits

a relatively good hysteretic behavior showing some stiffness degradation for cycles with drifts

larger than 1% and some strength deterioration for cycles with drift larger than 4%. A post-

yield stiffness ratio of 9% was identified and used when computing the analytical response.

SDOF systems having hysteretic behavior calibrated to specimen OJS4-1 will be referred to as

hysteretic behavior SSD-1, which is used in this investigation to represent the global behavior

of structures with stiffness degradation but that exhibit only a small level of strength

degradation.

Another comparison between experimental and analytical response is shown in figure 2.5,

which corresponds to a poorly-detailed beam-column joint specimen (PEER-14) tested at the

University of Washington (Walker et al., 2000). It can be seen that despite the severe strength

degradation, the three-parameter model is capable of reproducing relatively well the

experimental results. Systems with hysteric parameters calibrated to this specimen will be

referred to as hysteretic behavior SSD-2, which is used in this investigation to represent the

global behave ior of structures with severe strength and stiffness degradation.

Specimen PEER-14Walker et al. (2001)

-80

-60

-40

-20

0

20

40

60

80

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0

Drift [in]

Forc

e [k

ips]

Specimen PEER-14Walker et al., (2001)

-80

-60

-40

-20

0

20

40

60

80

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0

Drift [in]

Fo

rce

[kip

s]

Figure 2.5. Experimental and analytical response for a poorly-detailed reinforced concrete beam-

column connection (after Walker et al, 2001).


25

2.6 Statistical Results

A first round of results of this chapter includes a total of 72,000 inelastic displacement ratios

(corresponding to 240 ground motions, 50 periods of vibration and 6 levels of relative

strength) computed from the nonlinear response of SDOF systems having elastic -perfectly-

plastic hysteretic behavior. The main purpose of the first set of results was to examine the

effects of period of vibration, lateral strength ratio and firm site conditions. The same set of

results is employed to investigate the effect of earthquake magnitude and distance to the

source on inelastic displacement ratios, which will be reported separately in Chapter 5. The

influence of hysteretic behavior in bilinear and degrading systems comprises a new set of

results which are described in Section 2.6.4.

2.6.1 Central Tendency of CR

Prior statistical studies on inelastic displacement demands have traditionally used sample

mean as a measure of central tendency. However, alternative statistical measures can be used

to characterize the central tendency of inelastic displacement ratios. For example, counted

median and geometric mean (i.e., expected value of the log of the data) are additional

statistical measures that also provide information about central values (Benjamin and Cornell,

1970). Particularly, counted median has the advantage of being less sensitive to outliers (i.e.,

individual observations that are significantly larger than the rest of the sample) than sample

mean. On the other hand, geometric mean is a logical estimator of central tendency when

sample data approximately follows a lognormal distribution, but it could be also influenced by

outliers. Recognizing the importance of providing information of each statistical measure, in

this section, and throughout this dissertation otherwise noted, three different statistical

measures of central tendency were computed: sample mean, counted median and geometric

mean. Definition of this central tendency measures is provided in Appendix C. Thus, each

central tendency measure of CR was computed for each period and each lateral strength ratio.

A comparison of central tendency measures of CR computed for a lateral strength ratio

equal to four and for all 240 ground motions is shown in figure 2.6. It can be seen that

inelastic displacement ratios follows a similar trend regardless of the central tendency

measure. It should be noted that, for a given period of vibration, CR ordinates are larger when


26

sample mean is used as statistical measure than geometric mean and counted median, mainly

for periods of vibration shorter than about 1.0 s A first interesting observation is that

maximum inelastic displacement becomes smaller than maximum elastic displacements for

periods longer than about 1.0 s when counted median is used as central tendency measure,

which means that the equal displacement approximation might be only adequate for mean

values. Therefore, next sections will report sample mean CR but relevant observations derived

from the other central tendency measures will be highlighted.

R = 4 SITE CLASSES AB, C, D

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

sample mean

geometric mean

counted median

Figure 2.6. Comparison of inelastic displacement ratios computed from different central tendency

measures (R=4).

2.6.1.1 Different firm local site conditions

For the site classes considered here current seismic design provisions in the U.S. (FEMA,

2000) specify linear elastic design spectra that are significantly different from each other.

Thus, it is particularly important to know if inelastic displacement ratios to be used for

estimating maximum inelastic displacements from maximum elastic displacements are

affected by local site conditions and, if so, to quantify these differences.

Figure 2.7 shows mean inelastic displacement ratios corresponding to the three groups of

site conditions considered here. It can be seen that, in general, constant relative strength

inelastic displacement ratios exhibit the same trend regardless of the local site condition.

These ratios are characterized by being larger than one in the short period spectral region


27

(maximum inelastic displacements larger than maximum elastic displacements) and relatively

close to one (maximum inelastic displacements on average approximately equal to maximum

elastic displacement) for long periods. For periods smaller than 1.0 s inelastic displacement

ratios are strongly dependent on the period of vibration and on the lateral strength ratio. In

general, in this spectral region maximum inelastic displacements become much larger than

maximum elastic displacements as the lateral strength ratio increases (i.e., as the lateral

strength decreases with respect to the lateral strength required to maintain the system elastic)

and as the period of vibration decreases. Furthermore, for elasto-plastic systems constant

relative strength inelastic displacement ratios tend towards ∞ (become unbounded) as the

period of vibration tends to zero, which means that existing structures with very short periods

may undergo very large inelastic displacement demands relative to their elastic counterparts

unless they have lateral strengths that allow them to remain elastic or nearly elastic.

(a) SITE CLASS AB(mean of 80 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(b) SITE CLASS C(mean of 80 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(c) SITE CLASS D(mean of 80 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 2.7. Mean inelastic displacement ratios for different NEHRP site classes: (a) Site classes AB; (b) site class C; and (c) site class D.


28

It is important to notice that the limiting period dividing spectral regions where the equal

displacement rule is applicable from those where this rule is not applicable and is on average

unconservative (e.g., produces an underestimation of the maximum lateral displacement

demand) depends primarily on the lateral strength ratio, although it is also influenced by local

site conditions. In general, this limiting period increases as lateral strength ratio increases and

as the average shear wave velocity in the upper 30 m of the site profile decreases (i.e., as site

conditions become softer). For example, for structures on a site class C the equal displacement

rule is applicable on average for periods longer than about 0.4s for a lateral strength ratio of

1.5, but the rule is only approximately correct for periods longer than about 1.0s for lateral

strength ratios of 6. Similarly, for a lateral strength ratio of 2, the equal displacement rule is

approximately correct for periods longer than about 0.45s, 0.65s and 0.80s for structures on

site classes AB, C and D, respectively. It can be seen that in the short period spectral region CR

increases as the average shear wave velocity in the upper 30 m of the site profile decreases.

2.6.1.2 All firm site classes

Figure 2.8a shows mean constant-strength inelastic displacement ratios corresponding to

all 240 ground motions, regardless of the site conditions at the recording station. Ratios shown

in this figure represent what, on average, can be expected for existing structures built on firm

sites. Again, it can be seen that in the short period region the ratio of inelastic to elastic

displacement demand is strongly dependent on the relative strength of the system.

Furthermore, in this period region the equal displacement rule can result in significant

underestimations of the maximum inelastic displacement demand. In addition, counted median

inelastic displacement ratios are shown in figure 2.8b. From the figure, it is interesting to note

that the equal displacement approximation does not hold for systems in the medium and long

period region. In these regions, maximum inelastic displacements are slightly smaller than

maximum elastic displacements. Thus, the equal approximation as suggested originally by

Veletsos and Newmark (1960) and adopted in many seismic design codes worldwide only

applies when evaluating mean (average) results. Finally, geometric mean of CR is shown in

figure 2.8c. It can be seen that for periods of vibration longer than about 1.0s CR tends towards

1, regardless of the lateral strength ratio, which mean that the equal displacement rule might

hold when evaluating geometric mean of CR.


29

(a) mean of 240 ground motionsSITE CLASSES AB,C,D

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(b) median of 240 ground motionsSITE CLASSES AB,C,D

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(c) geometric mean of 240 ground motionsSITE CLASSES AB, C, D

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 2.8. Central tendency of inelastic displacement ratios for all 240 ground motions recorded in NEHRP site classes AB, C and D: (a) Mean; (b) median; and (c) geometric mean.

2.6.2 Dispersion

While mean inelastic displacement ratios are very important, as they represent the first

moment of the probability distribution and what can be expected on average to occur, it is also

very important to quantify the level of dispersion that exists around mean values of CR. A

common and effective way to quantify the dispersion is through the coefficient of variation

(COV), which is defined as the ratio of the standard deviation to the mean and represents a

normalized (nondimensional) measure of dispersion. In addition, the standard deviation of the

natural log of CR,RClnσ , also provides information about the variability in the estimation of

CR, specially when the data is lognormally distributed. Information about the variability of CR

is of high importance in the implementation of probabilistic -based methods. Therefore, it is

reported in this section.


30

2.6.2.1 Different firm local site conditions

Coefficient of varia tion of inelastic displacement ratios corresponding to each site class is

shown in figure 2.9. It can be seen that, in general, COV increases as the level of lateral

strength ratio increases over the whole period region, regardless of the firm site condition. In

spite of differences in local firm site conditions (e.g., shear wave velocity) particular, two

trends of COV can be identified depending on the spectral region. In the first spectral region,

COV significantly increases as the period of vibration shortens and as the lateral strength ratio

increases. In the second spectral region, COV increases as the lateral strength ratio increases,

but it does not strongly changes with variation in the period of vibration. The level of COV

and the limiting period that divides these spectral regions depends on the local firm site

conditions. For example, for site class C the limiting period that divides both spectral regions

is about 0.6 s In particular, it can be seen that COV of CR is slightly higher for site classes AB

and C than that computed site class D for very short periods of vibration, but the opposite

occurs for periods of vibration longer than about 0.5 s.

(a) SITE CLASSES AB

0.0

0.3

0.6

0.9

1.2

1.5

1.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV CR

R = 6.0

R = 5.0R = 4.0

R = 3.0R = 2.0

R = 1.5

(b) SITE CLASSES C

0.0

0.3

0.6

0.9

1.2

1.5

1.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV C R

R = 6.0R = 5.0

R = 4.0R = 3.0

R = 2.0R = 1.5

(c) SITE CLASSES D

0.0

0.3

0.6

0.9

1.2

1.5

1.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV CR

R = 6.0

R = 5.0R = 4.0

R = 3.0R = 2.0

R = 1.5

Figure 2.9. Coefficients of variation of inelastic displacement ratios for each site condition: (a) Site

classes AB; (b) site class C; and (c) site class D.


31

2.6.2.2 All firm site classes

Figure 2.10 shows COV’s of inelastic displacement ratios corresponding to ground motions

from all site classes considered in this study (i.e., from all 240 ground motions). From the

figure, it can be confirmed that COV increases as the level of relative lateral strength increases

over the period region considered in this study. COV becomes particularly high for periods of

vibration smaller than about 0.4 sec and as the lateral strength ratio increases. With the

exception of very short periods (smaller than about 0.15 sec), for a given level of lateral

strength ratio COV tends to decrease with increasing periods. This trend is more noticeable for

weak systems relative to the ground motion intensity (i.e., with higher values of R). It should

be noted that COV of CR is larger than the COV reported by Miranda (2000) for Cµ,

particularly for periods smaller than 1.0 s.

(a)SITE CLASSES B,C,D

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV CR

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

(b) SITE CLASSES B,C,D

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

σ ln CR

R = 6.0

R = 5.0

R = 4.0R = 3.0

R = 2.0

R = 1.5

Figure 2.10. Dispersion of CR computed from all 240 ground motions recorded in NEHRP site classes

AB, C and D: (a) COV; (b) standard deviation of the natural logarithm of CR.

A better way to provide information about the dispersion on CR is to compute inelastic

displacement ratios corresponding to different percentiles. Inelastic displacement ratios

corresponding to sample percentiles of 10%, 30%, 50%, 70% and 90% are shown in figure

2.10 for R= 2, 4 and 6. Sample percentiles were computed by counting data previously sorted

in ascending order as described in Benjamin and Cornell (1970). For example, for R=4, it can

be seen that although median inelastic displacement ratios (i.e., p=50%) are just slightly

smaller than one for periods longer than about 1.0s, there is an 80% probability that CR will be

between the curves associated to percentiles 10% and 90%, which in this spectral region


32

implies that in 80% of the cases CR varies approximately between 0.62 and 1.67. Similarly, in

this spectral region, in 40% of the cases (between p=30% and 70%) CR is larger than 0.78 and

smaller than 1.19. A further example is provided for inelastic displacement ratios

corresponding to the same percentiles but for R=6 (figure 2.10c). In this case, it can be seen

that for periods of vibration longer than about 1.0 s there is an 80% probability that maximum

inelastic displacement demand will be approximately between 0.62 and 1.97 times the

maximum elastic displacement demand. A comparison of counted percentiles and those

computed assuming a lognormal distribution of CR will be shown in Section 5.22 (Chapter 5).

(a) R = 2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

p = 90%

p = 50%

p = 50%

p = 30%

p = 10%

(b) R = 4

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

p = 90%

p = 50%

p = 50%

p = 30%

p = 10%

(c) R = 6

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

p = 90%

p = 50%

p = 50%

p = 30%

p = 10%

Figure 2.11. Inelastic displacement ratios corresponding to different percentiles: (a) for R=2; (b) for R =4; and (c) for R = 6.

2.6.3 Effect of Firm Soil Conditions

Most structures are built on sites that are classified as firm sites (site classes B, C and D).

Thus, it is important to quantify the differences on inelastic displacement ratios computed

from ground motions recorded in these site classes. In order to assess the effect of local site


33

conditions, that is to evaluate the differences in CR for different site conditions within firms

sites (site classes B, C, and D), ratios of mean CR on each group to mean CR computed from all

240 ground motions were computed.

(a) SITE CLASS AB

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR,AB/CR,ABCD

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(b) SITE CLASS C

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR,C/CR,ABCD

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(c) SITE CLASS D

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR,D/CR,ABCD

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 2.12. Ratio of mean CR computed from each site class normalized by mean CR computed from all

ground motions: (a) Site class AB; (b) site class C; (c) site class D.

Figures 2.12a, 2.12b and 2.12c show mean inelastic displacement ratios for site class B,

site class C and site class D normalized by mean inelastic displacement ratios computed from

all three site classes, respectively. It can be seen that if one neglects the effects of local site

condit ions for structures built on rock (site class AB) and uses mean values from all 240

ground motions one would, in general, tend to slightly overestimate maximum inelastic

displacement demands. For periods between 0.2 and 1.2 s the overestimation is less than 15%,

while for periods longer than 1.2 s the overestimation is less than 5%. For structures on site

class C the use of mean inelastic displacement ratios from all site classes considered here

would produce small underestimations of maximum displacements for T<0.2 s, small


34

overestimations for 0.2 s <T< 0.9 s and on average practically no errors for T>1.0 s.

Meanwhile , for structures on sites classified as D ignoring the effects of site conditions in the

estimation of CR could, in general, result on small underestimations of maximum displacement

for structures with periods smaller than 1.4s. It can be seen that the difference in inelastic

displacement ratio produced by local site conditions increases with increasing lateral strength

ratio. Thus, for lateral strength ratios smaller than 3 the errors produced by neglecting the

effect of local site conditions on CR are typically smaller than 10%, while for strength ratios of

4 and 5 site conditions are typically smaller than 20%.

2.6.4 Effect of Hysteretic Behavior

In this section, the effect of hysteretic behavior on CR was studied in four parts. The first part

considered the effect of post-yield stiffness on CR using non-degrading bilinear systems

having four different positive post-yielding stiffness ratios (i.e., post-yield stiffness

normalized by initial stiffness). The second part explores the influence of unloading stiffness

on the seismic response of three stiffness-degrading SDOF systems while the third part

considers three types of stiffness- and-strength degrading SDOF systems. Finally, the last part

includes the effect of two types of structural-degrading systems described in subsection

2.6.4.4.

2.6.4.1 Effect of post-yield stiffness

Mean inelastic displacement ratios were computed for bilinear SDOF systems having four

different values of positive post-yield stiffness ratio : α = 3%, 5%, 10% and 20%. The bilinear

SDOF systems were subjected to all 240 earthquake ground motions included in this

investigation. Mean CR spectra corresponding to each bilinear SDOF system is showed in

figure 2.13. It can be seen that, for a given lateral strength ratio, CR tends to decrease as α

increases, which means that maximum inelastic displacement demands decreases as the post-

yielding stiffness ratio increases. The former observation is consistent with previous

investigations. In addition, it is interesting to note that CR becomes smaller than 1 (i.e.,

maximum inelastic displacements becomes smaller than maximum elastic displacements) as


35

the α increases in the medium- and long-period spectral range, which mean that the equal

displacement rule is no longer a good average approximation for bilinear systems. The corner

period where maximum inelastic displacements are smaller than maximum elastic

displacements shortens as α increases, without significant influence on the lateral strength

ratio. In addition, to point out that the equal displacement approximation does not hold for

bilinear SDOF systems, median CR spectra for two bilinear systems is showed in figure 2.14.

It can be seen that median maximum inelastic displacement demands becomes smaller than

median maximum elastic displacement demand with increments of α.

(a) α = 3%

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0

R = 5.0R = 4.0R = 3.0R = 2.0

R = 1.5

(c) α = 10%

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0

R = 5.0R = 4.0R = 3.0R = 2.0

R = 1.5

(b) α = 5%

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(d) α = 20%

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 2.13. Mean inelastic displacement ratios of bilinear systems : (a) α = 3%; (b) α = 5%; (c) α = 10%; and (d) α = 20%.


36

(a) α = 3%

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(b) α = 10%

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 2.14. Median inelastic displacement ratios of bilinear systems :

(a) For α = 3%; and (b) for α = 10%.

In order to study further the effect of positive post-yield stiffness on maximum inelastic

displacement demands, maximum deformations of bilinear systems were computed for SDOF

systems with different values of post-yield stiffness ratios , α, (post-yield stiffness normalized

by initial stiffness). Maximum inelastic deformations were computed of systems with α = 3, 5,

10 and 10% when subjected to all 240 ground motions. For each ground motion and for each

different system, the maximum inelastic deformation of systems with positive post-yield

stiffness ratios were normalized by the maximum deformation of the elastoplastic system

(α=0%). A total of 288,000 different ratios of bilinear to elastoplastic peak inelastic

deformation were computed.

Figure 2.15 shows mean ratios of maximum inelastic displacement demand of bilinear

systems having different levels of post-yield stiffness ratio to the maximum inelastic

displacement demand of elastoplastic systems. It can be seem that the maximum inelastic

deformation of the bilinear systems becomes smaller than the one of elastoplastic systems as

the strength ratio increases. It can be observed that periods of vibrations larger than about 1.0s

the effect of post-yield stiffness is relatively small and approximately period independent.

However, for periods smaller than about 0.5s the maximum deformation of systems with

positive post-yield stiffness can be significantly smaller than that of elastoplastic systems.


37

(a) α = 3%

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆ i,α=3%/∆i,α=0

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(b) α = 5%

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆ i,α=5/∆ i,α=0

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(c) α = 10%

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆ i,α=10/∆ i,α=0

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(d) α = 20%

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆i,α=20/∆i,α=0

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 2.15. Mean ratios of maximum deformation of bilinear to elastoplastic systems :

(a) α = 3%; (b) α = 5%; (c) α = 10%; and (d) α = 20%.

Besides studying how much smaller the displacement demands of bilinear systems are

with respect to those of elastoplastic systems it is important to examine if having a positive

post-yield stiffness also helps in reducing the dispersion or record-to-record variability of the

inelastic displacement ratios. This issue is examined in figure 2.16 where the coefficient of

variation of inelastic displacement ratio of bilinear systems (with α>0) are shown. Comparing

figures 2.10a and 2.16 it can be seen that the dispersion of inelastic displacement ratios of

systems with positive post-yield stiffness is essentially the same as that with elastoplastic

systems except for periods smaller than about 0.25s where important reductions in dispersion

are observed as α increases.


38

(a) α = 3%

0.0

0.4

0.8

1.2

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(b) α = 5%

0.0

0.4

0.8

1.2

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(c) α = 10%

0.0

0.4

0.8

1.2

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(d) α = 20%

0.0

0.4

0.8

1.2

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 2.16. Coefficients of variation of inelastic displacement ratio computed with all 240 ground motions for bilinear systems with: (a) α=3%; (a) α=5%; (c) α=10%; and (d) α=20%

2.6.4.2 Effect of unloading stiffness

Early studies on the effect of the unloading stiffness on the maximum displacement of

inelastic systems concluded that unloading stiffness does not have a significant effect on peak

deformation demands. However, a later study by Otani (1980) concluded that unloading

stiffness was an important parameter for the response of reinforced concrete structures. In

order to provide additional information on the influence of unloading stiffness of stiffness-

degrading systems in the maximum inelastic displacement demands, the ratio of CR for

stiffness-degrading systems to CR for elastoplastic systems was computed. As was discussed in

Section 2.5.1, three different values of unloading stiffness, measured by the parameter HC of

the modified three-parameter model (Cheok et al., 1998), were considered: ∞ , 2.5, and 0.1. , a

very large value of HC (HC? ∞ ) reproduces the well-known modified-Clough model (Mahin


39

and Bertero, 1976) where the unloading stiffness is equal to the initial stiffness, while a value

of HC = 2.5 simulates the Takeda model (Takeda et al., 1970). A value of HC = 0.1

reproduces the force-deformation relationship of RC elements with unloading branches

towards the origin (origin-oriented capability) under repeated loading.

(b) HC=2.5 (Takeda model)SITE CLASS D

(mean of 80 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(c) HC=0.1(origin-oriented model)SITE CLASS D


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(a) HC (modified-Clough model)SITE CLASS D


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 2.17. Inelastic displacement ratios computed for three types of stiffness-degrading systems: (a)

Modified-Clough model; (b) Takeda model; and (c) origin-oriented model.

Inelastic displacement ratios corresponding to each of the three stiffness-degrading models

are shown in figure 2.17. It can be seen that mean inelastic displacement ratios of the Takeda

model are practically the same as those of the modified Clough model, indicating that for

moderate levels of degradation of the unloading stiffness, as those suggested by Takeda et al.

(1970), the unloading stiffness does not have a significant influence in the maximum inelastic

displacement demand of stiffness-degrading systems. However, it can be seen that for systems

that unload toward the origin (i.e., models with HC=0.1) maximum inelastic displacements are

larger than those experienced by the other two stiffness-degrading systems. Differences

increase as the level of nonlinearity increase. In particular, in can be seen that maximum


40

inelastic displacements that are, on average, larger than maximum elastic displacements (CR >

1) for the whole range of periods of practical interest. Therefore, for structures exhibiting this

type of hysteretic behavior the equal displacement approximation is not longer valid and can

yield underestimation of mean inelastic displacement demands.

(b) HC = 2.5 (Takeda model)SITE CLASS D


0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆i,SD /∆i,EP

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(c) HC = 0.1 (origin-oriented model)SITE CLASS D


0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆i,SD /∆i,EP

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(a) HC (modified-Clough model)SITE CLASS D


0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆i,SD / ∆i,EP

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 2.18. Influence of hysteretic behavior for three types of stiffness-degrading systems: (a)

Modified-Clough model; (b) Takeda model; and (c) origin-oriented model.

In order to further study the effect of stiffness degradation on the lateral displacement

demands, ratios of maximum inelastic deformations of stiffness-degrading systems to

maximum inelastic deformations of elastoplastic systems were computed. These ratios

represent a measure of how larger or smaller the inelastic displacement demands are in

systems with stiffness-degradation, ∆i,SD, compared to those in elastoplastic systems, ∆i,EP.

These ratios were obtained for each period of vibration, each level of relative strength, each

record, and each type of hysteretic behavior. Mean stiffness-degrading to elastoplastic

inelastic displacement ratios computed for the three types of stiffness-degrading hysteretic

behaviors are shown in figure 2.18. It can be seen that for periods of vibration smaller than


41

0.7s stiffness-degradation leads to peak displacement demands that are, on average, larger than

those of elasto-plastic systems. For periods between 0.15s and 0.7s, the increase in

displacement demand caused by stiffness degradation increases as the lateral strength ratio R

increases, regardless of the unloading stiffness of the system. For systems, with periods longer

than 0.7s, peak deformation demands of SDOF systems with modified-Clough or Takeda

hysteretic behavior are, on average slightly smaller than those experienced by elastoplastic

systems. This observation agrees well with those available in the literature (e.g., Riddell and

Newmark, 1979; Nassar and Krawinkler, 1991; Rahnama and Krawinkler, 1993). As

previously noted, for periods longer than 0.7s, the maximum inelastic deformations of systems

that unload toward the origin are, on average, larger than those of elastic systems and larger

than those of elastoplastic systems.

2.6.4.3 Effect of stiffness- and strength-degradation

In the third stage studying the effect of hysteretic behavior on CR special emphasis was placed

on the response of stiffness- and strength-degrading systems. Thus, inelastic displacement

ratios corresponding to two stiffness- and strength-degrading models, described in subsection

2.5.3, are shown in figure 2.19.

(b) SSD modelSITE CLASSES AB,C,D


0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(a) MSD modelSITE CLASSES AB,C,D


0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 2.19. Mean inelastic displacement ratios computed for two types of stiffness- and strength-

degrading systems: (a) MSD model; and (b) SSD model.

It can be observed that mean inelastic displacement ratios of MSD model are slightly

smaller than those of the SSD model meaning that the level of strength degradation (both


42

hysteretic behaviors have the same unloading stiffness) has an influence in the maximum

inelastic displacement demand of stiffness- and strength-degrading systems.

In order to further study the effect of combined stiffness and strength degradation on the

lateral displacement demands, ratios of maximum inelastic deformations of stiffness- and

strength-degrading systems to maximum inelastic deformations of elastoplastic systems were

computed. Simila rly, these ratios provides a quantitative measure of how larger or smaller the

inelastic displacement demands are in systems with both stiffness and strength degradation,

∆i,MSD or ∆i,SD, compared to those in elastoplastic systems, ∆i,EP. These ratios were obtained for

each period of vibration, each level of relative strength, each record, and each type of

hysteretic behavior. Mean stiffness- and strength-degrading to elastoplastic inelastic

displacement ratios computed for the two types of stiffness- and strength-degrading hysteretic

behaviors are shown in figure 2.19. Again, it can be observed that for periods of vibration

smaller than 0.7s both hysteretic behaviors yields maximum displacement demands that are,

on average, larger than those of elasto-plastic systems. For periods between 0.15 and 0.7s, the

increase in displacement demand caused by combined stiffness and strength degradation

increases as the lateral strength ratio R increases, regardless of the level of strength

deterioration of the system. However, it is evident that maximum inelastic displacement

becomes larger as the level of strength deterioration increases. For systems, with periods

longer than 0.7s, peak deformation demands of SDOF systems with MSD and SSD hysteretic

behavior are, on average similar to those experienced by elastoplastic systems.

(b) SSD modelSITE CLASSES AB,C,D


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆i,SSD / ∆i,EP

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(a) MSD modelSITE CLASSES AB,C,D


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆i,MSD / ∆i,EP

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 2.20. Mean ratio for inelastic displacement demands in stiffness- and strength-degrading

systems.


43

2.6.4.4 Effect of structural degradation

Figure 2.21 shows mean constant-strength inelastic displacement ratios, CR?, as a function of

period of vibration and strength ratio for two types of structural-degrading systems. As

mentioned in Section 2.5.3 the SSD-1 model is representative of hysteretic behaviors that

exhibit small to moderate level of structural degradation (both strength and stiffness

degradation) whereas the SSD-2 model is representative of hysteretic behaviors that exhibit

severe levels of structural degradation. Similarly to hysteretic behaviors previously discussed

in previous sections, inelastic displacement ratios strongly depend on the period of vibration

and on the lateral strength ratio. Mean inelastic displacement ratios computed for these

structural degrading systems show, in general, a very similar trend to hysteretic behaviors

previously discussed which is characterized by being larger than one in the short period

spectral region (i.e., maximum inelastic displacements are larger than maximum elastic

displacements) and approximately equal or slightly smaller than one for medium and long

periods. It should be noticed that regardless of the hysteretic behavior, the limiting periods that

divides the region where maximum inelastic displacements are greater than their elastic

counterparts depends on the lateral strength ratio. For example, the limiting period for systems

with SSD-1 hysteretic behavior and R=2 is 0.65 s while for R=4 is 0.90 s. In the same figure, it

can also be observed that mean inelastic displacement ratios become independent of the lateral

strength ratio for periods of vibration of longer than about 1.4 s.

(a) SSD-2 modelSITE CLASS D


0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5



0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 2.21. Mean inelastic displacement ratios computed for two types of structural degrading systems: (a) SSD-1 model; and (b) SSD-2 model.


44

In order to further study the effect of structural degradation on the lateral displacement

demands, the ratio of CR for structural-degrading systems to CR for non-degrading bilinear

systems was computed. Bilinear systems had the same post-yielding stiffness of their

structural-degrading counterparts. This ratio was obtained for each period of vibration, each

level of relative strength, each record, and each type of hysteretic behavior. Means of these

ratios provide a measure of how la rger or smaller the inelastic displacement demands are in

systems with structural-degradation, ∆i,SSD, compared to those in non-degrading (bilinear)

systems, ∆i,BI.

(b) SSD-2 modelSITE CLASS D


0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆i,SSD / ∆i ,EP

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5



0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆i,SSD / ∆i,EP

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 2.22. Mean ratio of inelastic displacement demands in structural degrading and bilinear systems:

(a) SSD-1 model; and (b) SSD-2 model.

Mean structural-degrading to non-degrading inelastic displacement ratios computed for

the two types of hysteretic behavior are shown in figure 2.22. It can be been that, in general,

inelastic displacement demands of short-period SDOF systems exhibiting this type of

structural-degrading hysteretic behavior may be significantly larger than those with bilinear

hysteretic behavior depending on the level of lateral strength. Limiting periods where this ratio

is, on average, larger than one from those in which is , on average, similar or smaller than one

are not constant and vary with the level of relative strength. These limiting periods that divide

the region where it is unconservative to neglect the effects of structural degradation (i.e.,

structural-degradation produces larger inelastic displacement demands than those of non-

degrading systems) from spectral regions where it is conservative to neglect the effects of

structural degradation (i.e., regions in which structural degradation leads to smaller inelastic

displacement demands than those of non-degrading systems) tends to increase with increasing


45

the level of structural degradation. As seen in this figure these limiting periods are

approximately 0.7 s for systems with the SSD-1 hysteretic model and 0.8 s for SSD-2

hysteretic models.

2.7 Functional Models to Estimate CR

For displacement-based design and, in general, in earthquake resistant design it is desirable to

have a simplified expression of CR’s central tendency in order to estimate maximum inelastic

displacement demands from maximum elastic displacement demands for structures where the

lateral strength is approximately known. In a recent study, Miranda (2001) concluded that

expressions derived directly from statistical analyses of mean inelastic displacement ratios

produce better results than using expressions derived from mean strength reduction factors

(i.e., Rµ - µ - T relationships). For example, Chopra and Goel (1999, 2000) and Fajfar (1999,

2000) have proposed to estimate the maximum inelastic displacement of existing structures

with known lateral strength, by multiplying the yielding displacement by a displacement

ductility ratio, µ, computed from existing Rµ – µ – T relations, which typically provide an

estimate of the mean strength reduction factor Rµ as a function of the displacement ductility

ratio and the period of vibration T. A common choice in the aforementioned studies was the Rµ

– µ – T relationship reported by Nassar and Krawinkler (1991), mainly because its functional

form rather than its accuracy with respect to other existing relationships. While this approach

is very simple and provides a way to estimate the maximum inelastic displacement from the

ratio of the lateral strength required to maintain elastic to the lateral strength in the system,

Miranda (2001) showed that the ductility demand computed from Rµ – µ – T relations does not

correspond to the mean displacement ductility demand of system with relative strength equal

to Rµ , and that the computed displacement ductility ratio is the first approximation of the

mean displacement ductility demand, hence this procedure introduces a systematic error that

will tend to underestimate the maximum inelastic displacement. Although the actual size of

the error depends on the particular Rµ – µ – T relation that is used, the error will typically

increase with increasing ductility. For the evaluation of existing structures more accurate

estimates of maximum inelastic displacements of systems where the relative strength is known

can be achieved by using results directly derived from statistical studies of systems with

known relative strength as opposed to those based on systems undergoing specific levels of


46

ductility. Then, a simplified equation to estimate constant-strength inelastic displacement

ratios is desired for earthquake engineering purposes with parameter estimates based directly

on statistical results such as those reported in Section 2.6.

In this section, a review of several equations that has been recently proposed in the

literature is offered. Next, a simplified equation to estimate CR is proposed followed by an

error analysis to evaluate the adequacy of the proposed simplified equation.

2.7.1 Review of Functional Models to Estimate CR

Nassar and Krawinkler (1991). As previously mentioned, simplified expressions to estimate

the central tendency of CR has been proposed from available Rµ – µ – T relationships. The

main example of such equations is the one derived from Nassar and Krawinkler (1991)

simplified equation to originally estimate Rµ for structures built on firm soil conditions. This

derived equation adopts the following functional form

−+=

cR

RC

c

R1

11~ (2.5)

where c is a period-dependent parameter given by

Tb

T

Tc

a

a+

+=

1 (2.6)

and coefficients a and b depends on the strain-hardening level. Ramirez, Constantinou, Whittaker, Kircher, Chrysostomou (2002). More recently,

motivated by the improvement of current seismic guidelines for the design of buildings with

damping-type passive energy dissipating devices, Ramirez et al. (2002) conducted an

analytical study of bilinear inelastic SDOF systems having three different elastic viscous

damping ratios (βv = 0.05, 0.20 and 0.35), which account for the additional damping, when

subjected to 20 earthquake ground motions that matched, on average, the 2000 NEHRP


47

spectrum for a region of region of high seismicity located on stiff soil site with a characteristic

soil period, Ts, equal to 0.6 s Based on their statistical results, the ratio of the average

maximum inelastic displacement to average maximum elastic displacement demand was used

for the authors to develop an equation to estimate mean CR to be used in the design of new

buildings with damping systems, which is given as follows

bs

R TT

aR

RC

−−+=

)1)(1(1

~ α (2.7)

where

)( 210 Raa θθ += (2.8)

vRb βθθθ 543 −−= (2.9) where Ts is the characteristic soil period, parameter βv is the linear viscous damping under

elastic conditions, α is the post-elastic to elastic stiffness ratio, and coefficients θ1 through θ4

were estimated from non-linear regression analysis from their statistical results. Estimates of

parameter a0 depend on both βv and α, which are given in Table 2.1. In addition, coefficient

estimates 0.21 =θ , 45.02 =θ , 24.33 =θ , 1.04 =θ , 5.44 =θ were suggested.

Table 2.1. Parameter estimates a0 for equation. (2.8).

βv α = 0.05 α = 0.15 α = 0.25 α = 0.50

0.05 0.116 0.100 0.093 0.071

0.30 0.195 0.160 0.143 0.111

Ruiz-Garcia and Miranda (2003). Motivated by providing a simplified expression to be used

during the evaluation of existing structures, Ruiz-Garcia and Miranda (2003) suggested the

following expression to estimate mean CR


48

( )( )1

111

~

312

−

−

⋅+= R

TTC

sR θθ θ

(2.10)

where Ts was defined as the characteristic soil period while coefficients θ1, θ2, θ3 depend on the

soil conditions (Table 2.2).

Table 2.2. Site-dependent parameter estimates for equation (2.10).

Site class 1θ 2θ 3θ Ts

AB 42.0 1.60 45.0 0.75

C 48.0 1.80 50.0 0.85

D 57.0 1.85 60.0 1.05

All sites 50.0 1.80 55.0

Erdik and Aydinoglu 2003). An equation to estimate the ratio of maximum inelastic

displacement to maximum inelastic displacement was suggested by Aydinoglu (Erdik and

Aydinoglu 2003) as part of a vulnerability study of buildings in Turkey which is given as

follows

⋅+

−+=

23221

2 4

exp1)1(

1~

R

T

T

RCR

θ

θθθ

(2.11)

where parameter estimates 3001 =θ , 102 =θ , 203 −=θ , 5.04 =θ were obtained through

nonlinear regression analysis.

Chopra and Chintanapakdee (2004). Aimed to estimate maximum inelastic displacement

demands from spectral elastic displacement demands to be employed in the recently

introduced modal pushover analysis , Chopra and Chintanapakdee (2004) suggested the

following equation


49

1

311

4

2)1(1

~−

−

++−+=

θ

θθ

θ

cRR T

T

RLC (2.12)

where

−

+=α

11

1 RR

LR (2.13)

In the above equations, Tc is the corner period that divides the so-called acceleration- and

velocity-sensitive region in a tripartite spectral representation of the elastic spectral

acceleration, velocity and displacement response of a SDOF system and α is the post-yielding

stiffness ratio. Coefficients 1θ , 2θ , 3θ , 4θ were obtained through nonlinear regression

analysis (Table 2.3) which depended on the ground-motion ensemble .

Table 2.3. Parameter estimates for equation (2.12).

Ground-motion ensemble 1θ 2θ 3θ 4θ

LMSR 63 2.3 1.7 2.3

LMSR-LMLR-SMSR-SMLR

61 2.4 1.5 2.4

2.7.2 Proposed Simplified Functional Model to Estimate CR

As was noted in the previous section, several equations to estimate central tendency of CR

have been recently published in the literature. However, further simplified but still adequate

equations to capture main trends of inelastic displacement ratios are needed in order to be

implemented in seismic code regulations. Therefore, based on the observations from previous

sections a simple nonlinear equation to estimate central tendency of inelastic displacement

ratios, RC~ , should adopt a functional form as follows:


50

( )θ,,~

RTfCR = (2.14)

where the response variables are the period of vibration, T , and the level of lateral

strength ratio, R, as well as a set of unknown parameters, θ . Therefore, the proposed

simplified equation is given by:

21

11

~θθ T

RCR

⋅

−+= (2.15)

where θ1 and θ2 are unknown parameters. In this investigation, a nonlinear regression analysis

was conducted using the Gauss-Newton algorithm with the Levenberg-Marquardt method

modifications for global convergence (Bates and Watts, 1988) to compute parameter estimates

1θ and 2θ . It should be noted that parameter estimates can be found to fit any statistical central

tendency measure (e.g., sample mean, counted median, geometric mean) of CR. For instance,

Table 2.4 reports the resulting parameter estimates of θ1 and θ2 as well as their 95%

confidence interval corresponding to the three central tendency measures of CR computed

from all 240 ground motions, regardless of the soil condition.

Table 2.4. Parameter estimates and 95% confidence intervals for equation (2.15) corresponding to three different central tendency measures computed from all 240 ground motions.

Central Tendency 1θ 2θ )ˆ.(. 1θic )ˆ.(. 2θic

Sample mean 35.79 2.12 32.40, 38.19 2.07, 2.17

Median 79.12 1.98 65.60, 91.68 1.89, 2.07

Geometric mean 49.03 1.87 44.73, 53.33 1.82, 1.92

In addition, parameter estimates of θ1 and θ2 and their confidence intervals obtained from

median inelastic displacement ratios corresponding to each of the ground motion ensembles

are reported in Table 2.5. From the table, it is evident that parameter estimates depends on the

site class.


51

Table 2.5. Site-dependent parameter estimates and 95% confidence intervals for equation (2.15) obtained from median CR corresponding to each site class.

Site Class 1θ 2θ )ˆ.(. 1θic )ˆ.(. 2θic

AB 88.10 1.74 65.75, 110.45 1.59, 1.89

C 131.10 2.15 91.75, 170.44 1.98, 2.32

D 50.17 1.95 42.98, 58.06 1.86, 2.04

It should be noted that equation (2.15) corresponds to a surface in the CR – R – T space

and, as illustrated figure 2.23, it can provide central tendency estimates of inelastic

displacements ratios as a function of R and T. It should be mentioned that the functional form

of equation 2.15 was recently chosen as an improvement to coefficient C1 in the so-called

displacement coefficient method included in FEMA 356 (FEMA, 2000), but with parameter

estimates 901 =θ and 21 =θ (Comartin et al., 2004).

1.0

2.0

3.0

4.05.0

6.0

0.10.5

0.91.3

1.72.2

3.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0C R

R

PERIOD [s]

Figure 2.23. Mean inelastic displacement ratios computed with equation (2.15) and parameter estimates given in Table 2.4.


52

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0 (Eqn. 2.15)R = 6.0 (data)R = 3.0 (Eqn. 2.15)R = 3.0 (data)

Figure 2.24. Comparison of mean inelastic displacement ratios computed using all 240 firm site records

to those computed with equation (2.15).

Finally, figure 2.24 shows a comparison of mean inelastic displacement ratios for R = 3

and R = 6 computed for all 240 firm site records to CR values computed with equation (2.15).

It can be seen that, in general, the analytical expression (equation 2.15) provides very good

results and is able to capture the effects of both R and T on inelastic displacements ratios.

.

2.7.3 Evaluation of Proposed Functional Models to Estimate CR

There are two kinds of errors associated with the use of equation (2.15) as an estimator of the

population central tendency of CR. The first is related to the error in using a sample central

tendency (e.g., sample mean obtained from a finite number of observations) as the basis of the

nonlinear regression analyses. The second is related to the fact that regression analyses do not

yield a perfect match of the sample central tendency (i.e., goodness-of-fit error). The first kind

of error is called estimation error (Benjamin and Cornell, 1970) and, for example, it can be

shown that the coefficient of variation of the sample mean is directly proportional to the

coefficient of variation of the random variable and inversely proportional to the square root of

the number of observations used to estimate the sample mean. Hence, the estimation error

will, in general, increase as the ductility ratio increases and will decrease as the number of

records considered in computing the mean sample increases. The sample sizes in this study

have been selected to be quite large, so for periods of vibration larger than 0.8s, where

coefficients of variation are smaller then 0.5, the estimation error on the mean will be at most


53

5% when 80 ground motions are considered and at most 3% when 240 ground motions are

considered. In the short period region these errors could double in size. The second kind of

error is the one associated with differences between the smooth fitted surfaces to the computed

sample mean. This second source of error is discussed in the following paragraphs. Both types

of errors represent epistemic uncertainties in the estimation of maximum inelastic

displacements.

In this study, the standard error of the residuals and the coefficient of correlation were

used as measures of “goodness-of-fit” of equation (2.15) with respect to the results of

nonlinear regression analyses for each level of lateral strength. The standard error of the

residuals, SE, is mathematically defined as follows:

p

n

iiRiR

nn

CC

RSE−

−

≡∑

=1

2,, )

~(

)( (2.16)

where iRC , is the mean inelastic displacement ratio computed from nonlinear response history

analysis for the i-th period of vibration, iRC ,~

is the inelastic displacement ratio estimated from

the nonlinear regression model (i.e., with equation 2.15) for the i-th period of vibration, n is

the number of data points which corresponds to the number of periods (i.e., 50) and np is the

number of parameters in the model which in the proposed model is equal to 2. SE is a

measure of the average error in the model. The standard error of the residuals was computed

using parameter estimates of Table 2.4 for each level of lateral and for the range of periods

considered in this investigation (i.e. between 0.05s and 3.0s). The other parameter that was

computed was the correlation coefficient, ρ (Benjamin and Cornell, 1970) which is defined as:

)~

()(

)~

,cov(

RR

RR

CC

CC

σσρ ≡ (2.17)

where )~

,cov( RR CC is the covariance between RC and RC~ , )( RCσ and )

~( RCσ are the

standard deviation of RC and RC~ , respectively. It should be noted that that as the nonlinear

regression model describes better the data as ρ approaches one. Table 2.6 presents a summary


54

of the two parameters of “goodness-of-fit” for each measure of central tendency and each

level of lateral strength. It can be seen that the correlation between observed inelastic

displacement ratios and those computed with equation (2.15) is very good and that standard

errors of the residuals in most cases are relatively small. Although these standard errors of the

residuals can be reduced further at the expense of using a more complicated function with

more parameters, equation (2.15) is deemed appropriate for engineering purposes.

Table 2.6. Measures of error using parameter estimates from Table 2.4 and equation (2.15).

Central

Tendency

Parameter

R=1.5

R=2.0

R=3.0

R=4.0

R=5.0

R=6.0

SE 0.1252 0.1908 0.1799 0.1576 0.1789 0.2421 Sample mean

ρ 0.9944 0.9968 0.9993 0.9998 0.9998 0.9998

SE 0.0757 0.1342 0.2309 0.2942 0.3546 0.4241 Median

ρ 0.9765 0.9816 0.9864 0.9902 0.9920 0.9927

SE 0.0643 0.1315 0.2280 0.3090 0.3635 0.4169 Geometric

mean ρ 0.9873 0.9867 0.9900 0.9919 0.9937 0.9947

Table 2.7. Measures of error using parameter estimates from Table 2.5 and equation (2.15).

Site Class Parameter

R=1.5

R=2.0

R=3.0

R=4.0

R=5.0

R=6.0

SE 0.0375 0.0697 0.1219 0.1811 0.2487 0.2988 AB

ρ 0.9701 0.9743 0.9804 0.9807 0.9796 0.9811

SE 0.0582 0.0987 0.1643 0.1789 0.1931 0.2285 C

ρ 0.9860 0.9899 0.9930 0.9963 0.9976 0.9978

SE 0.1153 0.1966 0.3458 0.5054 0.5278 0.6575 D

ρ 0.9734 0.9808 0.9852 0.9859 0.9914 0.9914

Equation (2.15) can be used to estimate central tendency of maximum deformations of

SDOF systems. Several studies have shown that maximum deformations of SDOF systems

can be used to compute approximations of global displacement demands of low and mid-rise

buildings (Miranda, 1991; Qi and Moehle, 1991; Collins et al., 1994). However, it should be


55

noted that such approximations are only acceptable for buildings whose response is primarily

dominated by the first mode, and that considerable errors may be introduced when the

contribution of higher modes to displacement response parameters is significant. In addition,

it is important to point out that the proposed equation and the parameter estimates adequately

represents the trend of inelastic displacement ratios computed for a suit of ground motions

recorded on site classes B, C and D and is not applicable to very soft soils, nor to sites that

may be affected by near-fault site affected by forward directivity effects. Inelastic

displacement ratios for these other conditions are discussed in Chapter 3.

2.8 Summary

The purpose of this chapter was to assess inelastic displacement ratios that permit the

estimation of maximum inelastic displacements from maximum elastic displacements for

structures built on firm sites. A statistical study has been presented of inelastic displacement

ratios computed for SDOF systems with an elasto-plastic and structural-degrading hysteretic

behaviors with different levels of lateral strengths relative to the strength required to maintain

the system elastic when subjected to a large number of earthquake ground motions. The

following conclusions are drawn from the results of this investigation.

1. In the short period spectral region, maximum inelastic displacements demands of systems

with constant relative strength are on average much larger than maximum elastic

demands. In this spectral region the ratio of maximum inelastic to maximum elastic

displacement demand is strongly dependent on the period of vibration, on the lateral

strength ratio and on the type of hysteretic behavior. Constant relative strength inelastic

displacement ratios increase as the lateral strength of the system decreases with respect to

the lateral strength required to maintain the system elastic. For periods longer than 1.2s

and lateral strength ratios smaller than six maximum inelastic displacement demands are

approximately equal to maximum elastic demands.

2. Coefficients of variation of inelastic displacement ratios increase with increasing lateral

strength ratios. Dispersion is relatively large for lateral strength ratios higher than 4 and

periods of vibration smaller than 1.5s. With exception of periods shorter than 0.15s,

coefficients of variation decrease with increasing period of vibration.


56

3. It was found that the effects of local site conditions on constant relative strength inelastic

displacement ratios are slightly larger than those of constant ductility inelastic

displacement ratios, however, in general the effect are still relatively small, particularly

for periods longer than 1.2 s. Neglecting the effect of site conditions for structures with

periods smaller than 1.5s built on firm sites will typically result in errors less than 20% in

the estimation of mean inelastic displacement ratios, whereas for periods longer than 1.5s

the errors are smalle r than 10%. Differences are even smaller if the lateral strength ratio is

equal or smaller than 3.

4. Limiting periods dividing regions where the equal displacement rule is applicable from

those where this rule is not applicable depend primarily on the lateral strength ratio,

although they are also influenced by local site conditions and type of hysteretic behavior.

In general these limiting periods increase primarily as the lateral strength ratio increases

and to a lesser degree as the average shear wave velocity in the upper 30 m of the site

profile decreases or as the level of structural degradation increases.

5. Maximum deformation demands of short period structures are on average larger than

those on non-degrading systems. In general, the increment in displacement produced by

degradation effects increases as the strength ratio increases (i.e., as the system becomes

weaker relative to the lateral strength required to maintain the system elastic). For

structures with periods longer than about 0.7 s, maximum deformation of degrading

systems are on average either similar or slightly smaller than those of non-degrading

system.

6. Maximum inelastic displacement demands of stiffness-degrading systems are not

significantly affected by the unloading stiffness provided that the reduction in unloading

stiffness is small or moderate. However, for systems that unload toward the origin (origin-

oriented systems) maximum inelastic displacement are on average larger than maximum

deformation demands and therefore the equal displacement rule should not be used for

these systems.

7. A simplified equation was derived using nonlinear regression analyses to estimate

inelastic displacement demands of structures on firm sites exhibiting elastoplastic

behavior. The functional form of the proposed equation has been recently adopted to

improve coefficient C1 in FEMA 356 (Comartin et al., 2004).

_____________________________________________________________________Chapter 3 Maximum Inelastic Displacement Demand: Soft Soil Sites and Near-Fault Effects

57

Chapter 3

Maximum Inelastic Displacement Demand of SDOF

Systems: Soft Soil Sites and Near Fault Effects

3.1 Introduction

It is widely recognized that local site conditions can have an important influence on the

ground motion intensity at a given site. This influence is particularly noticeable in the case of

soft soil deposits that give rise to ground motions characterized by narrow band spectra that

can produce very large lateral displacement demands. Evidence of inadequate seismic

performance of buildings and other types of structures located in soft soil sites have been

documented during various earthquakes (e.g., 1967 Caracas earthquake, 1977 Vrancea,

Romania earthquake, 1985 Michoacan earthquake, 1989 Loma Prieta earthquake). For

example, as a result of the earthquakes that struck Mexico in 1985, more than 300 buildings

built on very soft cohesive soil deposits in Mexico City collapsed or suffered severe structural

damage (Rosenblueth and Meli, 1986). In the United States, soft soil deposits in the San

Francisco Bay Area were a key factor in the observed damage during the 1989 Loma Prieta

earthquake, including the collapse of a one-mile segment of the Cypress Street Viaduct

(Miranda and Bertero, 1991).

On the other hand, an examination of accelerograms indicates that horizontal ground

motion components oriented perpendicular to the fault strike (fault normal) recorded at

stations near the causative fault are typically, but not always, characterized for having a

noticeable large velocity pulse (i.e., pulse period). This large velocity pulse is a consequence

of forward rupture directivity effects when the earthquake rupture moves towards the site.

Structures experiencing this type of ground motions (i.e., fault-normal near-fault ground

motions) are particularly susceptible of suffering structural damage as was observed after the

1994 Northridge and the 1995 Hyogo-Ken Nambu (Kobe) earthquakes.


58

Field post-earthquake evidence and subsequent research have showed that the nonlinear

seismic response of structures is greatly affected when they are located in soft soil sites or near

the causative fault. Therefore, implementation of performance-based methodologies for the

design of new structures and the evaluation of existing structures requires further

understanding about the behavior of structures located in this seismic environment.

The main objective of this Chapter is to present the results of a comprehensive statistical

study of inelastic displacement ratios, as defined in Chapter 2, computed from the inelastic

response of single degree-of-freedom (SDOF) systems subjected to earthquake ground

motions compiled in recording stations located in very soft soil conditions and recorded near

the fault rupture with forward directivity effects. Common goals in the investigation of soft-

soils site and near-fault effects on inelastic displacement ratios reported here were: (a) To

characterize the influence of normalized period of vibrations (i.e., periods of vibration

normalized with respect to the predominant period of the ground motion or the pulse period)

and relative lateral strength ratio on inelastic displacement ratios; (b) to obtain central

tendency and dispersion of inelastic displacement ratios; (c) to evaluate the influence of

ground motion characteristics (e.g., earthquake magnitude, distance to the source, peak ground

velocity, pulse period); (d) to investigate the effect of hysteretic features (e.g., positive post-

yield stiffness, stiffness degradation and stiffness-and-strength degradation) on inelastic

displacement ratios; and (d) to propose a simplified equation to estimate central tendency of

inelastic displacement ratios for the evaluation of existing structures located on very soft-soil

sites or subjected to near-fault ground motions to be implemented in performance-based

assessment methodologies.

3.2 Review of Previous Studies

In this section, a brief review of prior published studies aimed to explicitly evaluate inelastic

displacement ratios, either from a constant-ductility or constant-strength perspective,

computed from SDOF systems subjected to earthquake ground motions recorded on soft soil

conditions or near the causative fault are presented.


59

3.2.1 Inelastic Displacement Ratios for Soft-Soil Sites

There are only a small number of studies that have considered earthquake ground motions

recorded in very soft soil sites (i.e., soft-soil records) in the estimation of maximum inelastic

displacement demands and, particularly, in the evaluation of inelastic displacement ratios

(Miranda, 1993; Rahnama and Krawinkler, 1993; Song and Pincheira, 2000; Ruiz-Garcia and

Miranda, 2004). One of the first statistical studies that considered a relatively large sample of

soft-soil records (24 acceleration time histories) to estimate seismic demands for structures

located in this soil conditions was performed by Miranda (1993). As part of his study, the

author evaluated constant-ductility inelastic displacement ratios from elastoplastic and bilinear

SDOF systems. That study concluded that the inelastic displacement ratios for structures

located in soft soil conditions are very different than those computed for structures placed on

rock or firm soil sites. He pointed out that seismic demands of structures built on soft soil

deposits are strongly influenced by the predominant period of the ground motion. Thus, it was

suggested that this ground motion parameter should be explicitly considered while computing

inelastic displacement ratios for this soil conditions. This conclusion was independently

confirmed by Rahnama and Krawinkler (1993), who used 10 soft-soil records collected in bay

mud sites of the San Francisco Bay Area to study seismic demands on structures.

More recently, Song and Pincheira (2000) conducted an analytical study of constant-

strength inelastic displacement ratios computed from the nonlinear response of SDOF systems

having strength and stiffness degrading hysteretic behavior when subjected to a small set of 5

ground motions recorded on soft-soil conditions. From their study, the authors noted that

nonlinear systems exhibiting strength-and-stiffness degrading characteristics experience, on

average, larger maximum inelastic deformation demands than systems with stiffness-

degrading hysteretic behavior (e.g., modified-Clough model). In general, they found that

inelastic displacement amplification depends on the period of vibration, the level of lateral

strength, and the level of degrading characteristics. However, in spite of their effort the

authors did not observe any significant trend of the central tendency and they did not provide

information about the dispersion of these ratios, which is of very important for performance-

based assessment of existing structures.

Finally, Ruiz-Garcia and Miranda (2004) conducted a statistical study of constant-ductility

inelastic displacement ratios using a 100 ground motions recorded in the former lake-bed of

Mecxico City and 16 records collected in the San Francisco Bay Area. In particular, they


60

concluded that maximum inelastic displacements of stiffness-degrading (i.e., modified-Clough

model) systems are, in general, larger than those of elastoplastic systems for systems with

small normalized periods of vibration (i.e., period of vibration smaller than the predominant

period of the ground motion). In this spectral region, for certain normalized periods, inelastic

displacements of stiffness-degrading systems can be on average 25% larger than those of

elastoplastic systems.

3.2.2 Inelastic Displacement Ratios for Near-fault Ground Motions

Several researchers have focused their attention on the estimation of maximum inelastic

displacement demands , through inelastic displacement ratios, for structures subjected to

ground motions recorded at stations near the fault rupture (e.g., Baez and Miranda, 2000;

Alavi and Krawinkler, 2001; Pincheira and Song, 2000; Chopra and Chintapakdee, 2003,

2004; Akkar et al., 2004). In particular, Baez and Miranda (2000) used a relatively large

sample of 85 near-fault ground motions, including fault normal and fault parallel components,

recorded at distances shorter than 15 km to compute constant-ductility inelastic displacement

ratios. The authors observed that, on average, near-fault ground motions lead to larger

inelastic displacement ratios than those computed using far-field ground motions for periods

of vibration shorter than 1.3 s. Among near-fault ground motions, records collected in the

fault-normal direction, which exhibit forward directivity effects, can particularly increase the

ordinates of inelastic displacement ratios with respect to those computed from the fault-

parallel component. In addition, they noted that peak ground velocity and maximum

incremental velocity are the ground motion parameters that most influence inelastic

displacement ratios. On the other hand, based on a family of pulse-type input motions, Alavi

and Krawinkler (2000) observed that the pulse period plays an important role for

characterizing inelastic displacement demands for systems subjected to near-fault ground

motions.

Very recently, Chopra and Chintapakdee (2003) computed constant-ductility inelastic

displacement ratios for elastoplastic SDOF systems using a small set of 14 near-fault

earthquake ground motions. They found that inelastic displacement ratios show a very similar

trend than those computed from far-field ground motions when inelastic displacement ratios

are computed for normalized periods of vibration with respect to the corner period, Tc (i.e., the

period that divides the so-called acceleration- and velocity-sensitive region in a tripartite


61

spectral representation of the elastic acceleration, velocity and displacement response of a

SDOF system (Newmark and Hall, 1969)).

It should be noted that very few studies have studied the nonlinear response of SDOF

degrading-systems when subjected to near-fault ground motions (Pincheira and Song, 2000;

Akkar et al., 2004), and only the study by Pincheira and Song (2000) evaluated nonlinear

SDOF systems having combined stiffness and strength degradation hysteretic features. In

particular, the former authors analyzed SDOF systems having three different levels of

stiffness- and strength-degradation when subjected to an ensemble of 5 near-fault ground

motions. They found that constant-strength inelastic displacement ratios are higher for

moderate and high stiffness- and strength-degrading systems than that of purely stiffness-

degrading (i.e., modified-Clough model) systems, depending on the period region and the

level of lateral strength. However, the authors did not identify clear tendencies, mainly due to

the small sample ground motion set, and they did not provide information about dispersion of

CR for degrading-systems. Very recently, Akkar et al. (2004) reported results of constant-

strength inelastic displacement ratios from non-degrading and stiffness-degrading SDOF

systems considering 56 records showing pulse signal. He found that stiffness-degrading

systems leads to larger inelastic displacement demands than those from non-degrading

(elastoplastic) systems for periods of vibration shorter than about 1.5 times the pulse period,

but he could not identify a noticeable trend with respect to the lateral strength ratio.


3.3.1 Soft Soil Site Records

Two sets of ground motions with a total of 118 records were considered in this investigation.

The first set includes 18 acceleration time histories recorded in accelerographic stations

located on bay mud sites in the San Francisco Bay Area during the 1984 Morgan Hill

earthquake (Ms=6.1) and the 1989 Loma Prieta earthquake (Ms=7.1). The San Francisco Bay

is located in a basin about 15 km wide, bounded by the active San Andreas and Hayward fault

zones. This region is characterized by a wide variety of geologic deposits from rocks sites in

the hill area to estuarine mud and clay deposits in the flatlands along the margins of the bay.

The bay mud area is comprised of unconsolidated, water-saturated, dark plastic clay and silty


62

clay with well-sorted silt and sand dunes in some areas. It may contain more than 50% of

water content and low shear-wave velocities in the range of 67 to 116 m/s (Borchedt, 1996). A

detailed list of all grounds motions is presented in Appendix A.

The second set of earthquake ground motions consists of 100 acceleration time histories

captured in five recent earthquakes in recording stations located in the soft soil zone of

Mexico City. Mexico City is built partly on an old lake bed that was formed by the Texcoco,

the Chalco and the Xochimilco lakes. Relatively thick deposits of lacustrine clay form the soft

zone. In the former lake bed zone of Mexico City the depth of the soft clay deposits varies

from 10 m to 60 m. These clay deposits are very deformable and are characterized by very

high water contents that reach more than 400%, shear wave velocities as low as 40 m/s, and

high plasticity indexes. All ground motions in this second set were obtained from the region

that experienced more damage during the 1985 earthquake (Rosenblueth and Meli, 1986). A

complete list of all grounds motions can be found in Appendix A.

3.3.2 Near-Fault Ground Motions

In this study, a suite of 40 fault-normal near-fault ground motions was assembled to evaluate

inelastic displacement ratios for this seismic environment. Selected acceleration time histories

have the following characteristics: (1) recorded at horizontal distances to the surface

projection of the rupture not larger than 20 km; (2) recorded in earthquakes with strike-slip or

dip-slip faulting mechanisms with moment magnitudes (Mw) equal or larger than 6.0; (3)

records with peak ground velocity (PGV) larger than 20 cm/s. In addition, the period

associated to the velocity pulse (i.e., pulse period, Tp) was available for all 40 near-fault

ground motions. The pulse period for each ground motion was identified by Fu and Menun

(2004) using a velocity pulse model fitted to match each of the fault-normal near-fault ground

motion components. A detailed list of all grounds motions including earthquake name, station

name, the horizontal distance from the recording site to the surface projection of the rupture,

PGV, velocity pulse and Tp can be found in Appendix A.


63

3.4 Ground Motion Characterization on CR

A key aspect in performance-based earthquake engineering and, particularly, in site response

analysis is the adequate selection of a suite of ground motions. Some important features of a

ground motion are the intensity, duration of the ground shaking, and frequency content.

Several parameters have been proposed in the literature to characterize the latter features. For

example, predominant period of the ground motion and bandwidth have been suggested as

measures of the ground motion frequency content, while peak ground acceleration has been

commonly used as a measure of the intensity of a ground motion (Kramer, 1996). Then, it is

desirable to consider these ground motion characteristics in the evaluation of inelastic seismic

demands. In this study, the frequency content of ground motions collected from the soft soil

deposits and recorded from stations near the causative fault was measured by the predominant

period, Tg, and the bandwidth, Ω, of the ground motion as well as the pulse period, Tp. of the

velocity pulse.

3.4.1 Soft Soil Sites Records

Several definitions have been proposed to characterize the predominant period of the ground

motion (Rathje et al., 1998). In this investigation, Tg was computed as the period at which the

maximum ordinate of a five percent elastic damped relative velocity spectrum occurs

(Miranda, 1993). For the soft soil deposits of Mexico City, Tg has been found closely related

with the predominant period of the soil deposit computed from elastic one-dimensional theory

(Reinoso and Ordaz, 1998). Examples of the computation of the predominant period of the

ground motion for records obtained in the San Francisco Bay Area and in Mexico City are

shown in figure 3.1.


64

(a) Tg = 1.14 s

0.0

20.0

40.0

60.0

80.0

100.0

120.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIODO [s]

Sv [cm/s]

(b) Tg = 2.96 s

0.0

20.0

40.0

60.0

80.0

100.0

120.0

0.0 1.0 2.0 3.0 4.0 5.0PERIODO [s]

Sv [cm/s]

Figure 3.1. Estimation of the predominant period of the ground motion: (a) for Foster City (recording station FC, 10/17/89, comp. 360); (b) for Villa del Mar (recording station 09, sate

25/04/89, comp. 90).

In addition, in this investigation an analog definition of bandwidth, Ω, from that originally

formulated by Vanmarcke (1972) was used to measure the dispersion, or spread, of the ground

motion with respect to its central frequency (i.e., predominant period of the ground motion).

Details of the definition employed to measure Ω are given in Appendix A. This analog

definition allows differentiating earthquake ground motions as narrowband or broadband The

relationship between Tg and Ω for all 118 ground motions recorded in soft-soil conditions is

showed in figure 3.2. From this figure, it can be observed that Tg follows a linear trend with

respect to its bandwidth (e.g., Tg decreases as bandwidth increases). According with the

original definition postulated by Vanmarcke (1972), smaller values of the bandwidth are

associated with narrow band signals (e.g., Ω < 0.5 is considered a narrow-band ground

motion). It can be seen that bandwidth of ground motions recorded on soft soil deposits of

Mexico City ranges from 0.18 to 0.47 while those recorded on the bay mud area of San

Francisco varies from 0.29 to 0.58. Then, it can be seen that, for the suite of records

considered in this study, the ground motions recorded in soft soil deposits of the old lake bed

zone of Mexico City have narrower bandwidth than the ground motions collected in the San

Francisco Bay Area.


65

0.0

1.0

2.0

3.0

4.0

0.0 0.2 0.4 0.6 0.8Bandwidth, Ω

T g

Lake bed Mexico City

San Francisco Bay mud

Figure 3.2. Relationship between predominant period of the ground motion (Tg) and bandwidth for earthquake ground motions recorded on soft soil sites.

In recent studies, Miranda (1993) showed that estimation of the predominant frequency of

the ground motion was very important in order to adequately assess seismic demands on both

linear elastic and nonlinear structures built on soft soils. For example, inelastic displacement

ratios computed for a relative strength ratio of 6 and for three different ground motions

recorded in the San Francisco Bay Area are shown in figure 3.3a. It can be seen that inelastic

displacement ratios of ground motions recorded on soft soils follow a similar trend but for

different spectral regions. Shown in the same figure is the average of the inelastic

displacement ratios of these three ground motions (in the black line). It can be seen that

despite containing information of all three ground motions, this average does not provide an

adequate characterization of the inelastic displacement ratios of any of the ground motions. To

overcome this problem Miranda (1993) suggested to normalize the period of vibration, T, of

the SDOF systems by the predominant period of vibration of the ground motion, Tg. Inelastic

displacement ratios for these 3 ground motions but with normalized periods, T/Tg, are shown

in figure 3.3b. Also shown in the figure is the average of the three normalized CR spectra. It

can be seen that this period normalization leads to a good characterization of the inelastic

deformation demands of simple structures built on soft soils.


66

(b)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

Foster City-360Redwood City-233

Larkspur Ferry-360mean of 3 records

(a)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

Foster City-360

Redwood City-233Larkspur Ferry-360

mean of 3 records

Figure 3.3. Mean inelastic displacement ratios computed for three ground motions recorded in the San Francisco Bay Area: (a) as a function of T; (b) as a function of T/Tg.

3.4.2 Near-Fault Ground Motions

For fault-normal near-fault ground motions, the velocity pulse and the period of the velocity

pulse (i.e., pulse period, Tp) have been identified as important parameters that characterize this

type of ground motions. In particular, Tp appears strongly correlated to earthquake magnitude

and empirical predictive relationships have been proposed to estimate Tp as a function of

moment magnitude, Mw (Alavi and Krawinkler, 2000; Fu and Menun, 2004; Mavroeidis et al.,

2004). Furthermore, employing a small sample of near-fault ground motions, Menun and Fu

(2002) noted that Tp appears to be also in good correlation to Tg. This relationship was verified

in this investigation and it is illustrated in figure 3.4a. It can be seen that Tp seems linearly

related to Tg with a correlation coefficient of 0.83. Actually, larger correlation exists between

Tp and Tg when Tp is shorter than about 1.5s. Then, it is suggested that Tg might also provide a

good characterization of the frequency content for both soft-soil site records and near-fault

ground motions. In addition, the relationship between Tg and Ω for the suite of 40 fault-normal

near-fault ground motions is shown in figure 3.4b. It can also be also seen that a strong

correlation exists between Tg and Ω for fault-normal near-fault ground motions considered in

this study. Bandwidths for this type of ground motions range from 0.22 s to 0.60.


67

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.0 0.2 0.4 0.6 0.8 1.0

Bandwidth, Ω

Tg [s]

Near-fault ground motions

ρ = 0.83

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

Tp [s]

Tg [s]

Near-Fault Ground Motions

Figure 3.4. (a) Relationship between Tg and Tp for fault-normal near-fault ground motions; (b) relationship between Tg and bandwidth.

As mentioned previously, Chopra and Chintapakdee (2003) computed inelastic

displacement ratios using a small set of earthquake ground motions recorded near the

causative fault (near-fault ground motions, for short). They suggested to compute CR for this

type of ground motions by normalize the period of vibration with respect to the corner period,

Tc (i.e., the period that divides the so-called acceleration- and velocity-sensitive region in a

tripartite spectral representation of the elastic acceleration, velocity and displacement response

of a SDOF system (Newmark and Hall, 1982)). However, as recognized by the authors, Tc,

varies with the excitation and its definition mainly has a judgmental basis. Moreover, Jain and

Pal (1991) showed that the estimation of Tc involves large variability even from earthquake

ground motions with very similar frequency content characteristics. Recognizing the key role

of Tp in adequately characterizing the response of elastic and inelastic SDOF systems

subjected to near fault excitations, several researchers (Alavi and Krawinkler, 2000; Fu and

Menun, 2004; Mavroe idis et al., 2004) have recently suggested obtaining constant-ductility

response spectra for normalized periods of vibration with respect to the pulse period, T/Tp, to

facilitate incorporation of design spectra for near-fault ground motions of different magnitudes

ranges. Examples of constant-strength inelastic displacement ratios computed from the record

obtained in the Sylmar Converter station (SYL) during the Northridge 1994 earthquake

without and with normalized periods of vibration, T/Tg, are shown in figures 3.5a and 3.5b.

When CR is computed with respect to the period of vibration, a large amplification of CR can

be seen for periods between 0.5 and 1.0s. However, when CR is computed as a function of T/Tp


68

inelastic displacement ratios seems to follow a similar trend than that observed for soft soil

sites.

(a)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(b)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 3.5. Mean inelastic displacement ratios computed from the earthquake ground motion obtained

from the Sylmar Converter Station: (a) as a function of T; (b) as a function of T/Tg.

In addition, figure 3.6 illustrates the computation of Tg according to the previous definition

for the same record. The predominant period of this record is 2.58 s, which is very close to the

pulse period of 2.53 s reported by Menun and Fu (2002). Menun and Fu estimated Tp by fitting

the recorded ground motion with a stochastic velocity-pulse model.

Northridge 1994 EarthquakeStation SYL-Fault Normal

0

50

100

150

200

250

300

350

0.0 1.0 2.0 3.0 4.0 5.0PERIOD [s]

Sv [cm/s2]

Tg = 2.58 s

Tp = 2.53 s

Sv,max=262 cm/s

Tg

Figure 3.6. Estimation of the predominant period of the ground motion for a near-fault ground

motion.


69

3.5 Statistical Results for Soft Soil Sites

In the first stage of this study, a total of 35,400 inelastic displacement ratios for elastoplastic

SDOF systems were computed (corresponding to 118 ground motions, 50 normalized periods

of vibration and 6 levels of relative strength). Later, inelastic displacement ratios were

statistically processed (i.e., obtaining central tendencies and dispersion) by each normalized

period, each level of relative strength and each ground motion set. The main purpose of the

first set of results was to examine the effects of period of vibration, lateral strength ratio,

earthquake magnitude and distance to the source on inelastic displacement ratios. The

influence of hysteretic behavior in bilinear and structural-degrading systems comprises a new

set of results which are described in Section 3.5.5.


Figures 3.7a and 3.7b show mean constant-relative strength inelastic displacement ratios as a

function of normalized periods, T/Tg, computed for elastoplastic systems subjected to

acceleration time histories recorded in the San Francisco Bay Area and in the former bed-lake

of Mexico City. In spite of the differences in frequency content of each ground motion set and

geologic characteristics (e.g., Mexico City soft soil deposits have much smaller shear wave

velocities and much higher water content than soft soils in the Bay Area), it can be seen that

inelastic displacement ratios follow a similar trend. From this trend, three spectral regions can

be identified. In the first region, maximum inelastic displacements are larger than maximum

elastic displacements, increasing as the lateral strength ratio increases and as normalized

periods T/Tg decreases. The limiting T/Tg ratios for this region are 0.75 and 0.85 for the San

Francisco Bay Area set and the Mexico City set, respectively. In the second region, maximum

inelastic displacements are smaller than maximum elastic displacement, which mean that

structures with period of vibration close to the predominant period of the ground motion

experience, on average, inelastic deformation demands smaller than their elastic counterparts.

In this spectral region, mean CR decreases as the lateral strength ratio increases. The limiting

normalized periods for this region are approximately 1.55 for both ground motion sets. After

this region, maximum inelastic displacement demands are on average equal to maximum

elastic displacement demands regardless of the level of lateral strength for the San Francisco


70

Bay Area set. On the other hand, inelastic displacements are slightly greater than maximum

inelastic displacement demands for the Mexico City set. In general, a similar trend for

constant-ductility inelastic displacement ratios, using a similar ground motion database, was

reported by the authors (Ruiz-Garcia and Miranda, 2004).

(b) Mexico City Set


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(a) San Francisco Bay Area set(mean of 18 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tg

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 3.7. Mean inelastic displacement ratios computed for: (a) the San Francisco Bay Area ground motion set, and (b) Mexico City ground motion set.

From the former observations and those reported in Chapter 2, it should be pointed out

that mean constant-strength inelastic displacement ratios for structures built on firm soil sites

are different from those of structures built on soft soil sites and hence, maximum inelastic

deformation demands can also be very different. It should also be noted that the well-known

equal displacement approximation (i.e., maximum inelastic displacements are, on average,

equal to maximum elastic displacements) originally suggested by Veletsos and Newmark

(1960) and often used in current seismic design codes worldwide to estimate inelastic

deformation demands is not applicable for structures built on soft soil sites in a wide range of

periods.

3.5.2 Dispersion of CR

Dispersion on inelastic displacement ratios was quantified by computing coefficients of

variation (COV). This parameter was computed for each normalized period of vibration and

for each level of displacement ductility demand. Figure 3.8a shows COV of inelastic

displacement ratios corresponding to the ground motions recorded in the San Francisco Bay


71

Area. A similar plot corresponding to the Mexico City ground motion set is showed in figure

3.8b. COV for the Mexico City ground motion set are smoother as a result of the significantly

larger sample size.

Dispersion of inelastic displacement ratios of structures on soft soil deposits are

characterized by: (1) an increase in dispersion as the level of inelastic deformation increases;

(2) larger dispersion for systems with periods of vibration longer than the predominant period

of the ground motion; and (3) increases in dispersion for periods of vibration equal to those

corresponding to the first and second modes of vibration of the soil deposit. It should also be

noted that, with exception of periods of vibration close to the predominant period of the

ground motion, levels of dispersion computed from ground motions recorded on very soft soil

sites are smaller than those reported by Miranda (2000) computed from ground motions

recorded on rock or firm soil sites.

(b) Mexico City Set

0.0

0.3

0.6

0.9

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

COV CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(a)San Francisco Bay Area Set

0.0

0.3

0.6

0.9

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

COV CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 3.8. Coefficient of variation of inelastic displacement ratios computed for: (a) the San Francisco

Bay Area ground motion set; and (b) Mexico City ground motion set.

Figure 3.9 shows inelastic displacement ratios corresponding to various percentiles

computed from systems undergoing displacement ductility ratios of three and five when

subjected to ground motions recorded in Mexico City. Here it can be seen that the same

general trend is observed at different probability levels. For systems with periods close to the

predominant period of the ground motion the equal displacement approximation would result

in a significant overestimation of the inelastic displacement even for a conditional probability

level of 90%. Furthermore, the effect of the second mode of vibration of the soil deposit is

much more pronounced for low levels of conditional probability of occurrence.


72

(a) R = 2Mexico City Set

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

p = 90%

p = 70%

p = 50%

p = 30%

p = 10%

(b) R = 4Mexico City Set

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

p = 90%

p = 70%

p = 50%

p = 30%

p = 10%

(c) R = 6 Mexico City Set

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

p = 90%p = 70%

p = 50%

p = 30%

p = 10%

Figure 3.9. Inelastic displacement ratios corresponding to various counted percentiles computed with the Mexico City set: (a) R=2; (b) R = 4; and (c) R =6.

3.5.3 Effect of Lateral Strength Ratio

In order to further study the effects of the level of inelastic behavior on inelastic displacement

ratio ,s all of the individual results were plotted for selected T/Tg ratios as a function of

displacement ductility ratio. Results computed with the Mexico City ground motion set for

elastoplastic systems with T/Tg = 0.5, 1.0 and 2.5 are shown in figure 3.10. It can be seen that

for T/Tg=0.5 inelastic displacement ratios increase almost linearly with increases in ductility

demand. This linear trend is evident not only for median values (p=50%), but also for high and

low percentiles. For periods of vibration equal to the predominant period of the ground motion

the effect of the level of inelastic behavior is significantly different. As shown in this figure,

inelastic displacement ratios decrease as the level of inelastic behavior increases, however, the

observed reduction is not linearly proportional to the increase in ductility. In particular, the

rate of reduction is larger for small levels of ductility than for larger levels of ductility

demand. This means that for structures on very soft soil deposits, whose periods of vibration


73

are equal or close to the predominant period of the ground motion, even a small level of

inelastic behavior is very effective in reducing lateral displacement demands. These suggests

that, as previously noted by Miranda (1991), the use of energy dissipation devices for

structures in this situation could be particularly beneficial by not only significantly reducing

lateral strength demands, but also by significantly reducing lateral displacement demands. For

systems with T/Tg=2.5 median inelastic displacement ratios increase only a little bit with

increases in inelastic behavior, however, for p=90% the increase in inelastic displacement with

respect to elastic displacement is more important, showing that even though the equal

displacement approximation is, on average, approximately valid in this spectral region, for

systems undergoing large levels of inelastic behavior for some ground motions the inelastic

displacement could be as large as two times the elastic displacement. It is also important to

notice that a stronger asymmetry of the probability distribution is observed in this spectral

region as the level of inelastic behavior increases.

(a) T/Tg = 0.5

0.0

1.0

2.0

3.0

4.0

1.0 2.0 3.0 4.0 5.0 6.0 7.0Lateral Strength Ratio, R

CR

p = 90%p = 50%p = 10%

(b) T/Tg = 1.0

0.0

1.0

2.0

3.0

4.0


CR

p = 90%p = 50%p = 10%

(c) T/Tg = 2.0

0.0

1.0

2.0

3.0

4.0


CR

p = 90%p = 50%p = 10%

Figure 3.10. Effect of lateral strength ratio on inelastic displacement ratios: (a) T/Tg=0.5; (b)

T/Tg=1.0; and (c) T/Tg=2.0.


74

3.5.4 Effect of Earthquake Magnitude and Distance to the Source

Previous studies reported in the literature have noted that both the amplitude of elastic spectral

ordinates and frequency content of the ground motion depend on the magnitude of the

earthquake and distance to the source. Thus, it is important to study if inelastic displacement

ratios are also modified by earthquake magnitude and/or epicentral distance. For this purpose,

mean inelastic displacement ratios for elastoplastic systems were computed using four groups

of 18 ground motions from the Mexico City acceleration time-history ensemble corresponding

to four ranges of surface-wave magnitude (Ms). Figure 3.11a shows a comparison of mean CR

computed for 4 subsets of ground motions corresponding to four different magnitude ranges. It

can be seen that the spectral shape and amplitude of inelastic displacement ratios are not

significantly affected by earthquake magnitude. Similarly, the effect of distance to the source

on CR can be seen in figure 3.11b by comparing mean inelastic displacement ratios computed

from two subsets of 8 ground motions recorded in stations located between 60-80 km and

between 80 and 120 km with respect to mean CR computed for all 18 ground motions using

the San Francisco Bay Area set. It can be seen that, for these ranges of distances, distance to

the source has not strongly influence on the CR ordinates.

(a)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

Ms = 6.3

Ms = 6.6

Ms = 6.9

Ms = 7.1

(b)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

60 < D < 80 km80 < D < 120 kmmean all records

Figure 3.11. Effect of earthquake magnitude and distance to the source on inelastic displacement ratios

computed from sot-soil records.


In this section, the effect of hysteretic behavior on CR was studied in three parts. The first part

considered the effect of post-yield stiffness on CR using non-degrading bilinear systems


75

having four different positive post-yielding stiffness ratios (i.e., post-yield stiffness

normalized by initial stiffness). The second part explores the influence of unloading stiffness

on the seismic response of three stiffness-degrading SDOF systems while the third part

considers three types of stiffness- and-strength degrading SDOF systems.

3.5.5.1 Effect of post-yield stiffness

In order to examine the effect of positive post-yield stiffness on the maximum inelastic

displacement demands, maximum deformations of bilinear systems with post-yield stiffness

ratios (i.e., post-yield stiffness normalized by initial stiffness) of 3%, typical of steel moment

resisting frame connections, were computed when subjected to all 100 ground motions

recorded in Mexico City. Then, ratios of the maximum inelastic deformation of bilinear

systems to maximum deformation of the elastoplastic system were obtained for each record,

each normalized period of vibration, and each lateral strength ratio. Figure 3.12 shows mean

ratios of maximum inelastic displacement demand of bilinear systems, ∆i( α), to the maximum

inelastic displacement demand of elastoplastic systems, ∆i,EP, for each level of lateral strength.

It can be seen that, with exception of very short normalized periods, the maximum inelastic

deformation of the bilinear systems becomes smaller with respect to the one of the

elastoplastic system as the lateral strength ratio increases. It should be noted that for periods of

vibration equal or closer to the predominant period of the ground motion, maximum inelastic

deformation demands for bilinear are similar to those of elastoplastic systems. For T/Tg ratios

smaller than about 0.2 the maximum deformation of systems with positive post-yield stiffness

can be significantly smaller than that of elastoplastic systems. In addition, mean ∆i( α) /∆i,EP

ratios computed for four levels of post-yielding stiffness of bilinear systems with R= 4 and

subjected to the San Francisco Bay Area set are shown in figure 3.12b. The figure illustrates

that, for a given lateral strength ratio, maximum inelastic deformation demands for bilinear

systems becomes smaller than that of elastoplastic systems as the level of positive post-yield

stiffness ratio increases depending on the spectral region.


76

(a)Mexico City Soft Soil Set

(mean of 100 ground motions)α = 3%

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tg

∆ i(α) / ∆ i, EP

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

(b) R = 4San Francisco Bay Area Set(mean of 18 ground motions)

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tg

∆ i(α) / ∆ i, EP

α = 1.5%

α = 3.0%

α = 5.0%

α = 10.0%

Figure 3.12. Effect of post-yield stiffness ratio on inelastic displacement ratios: (a) Mexico City set (α =

3%); (b) San Francisco set (R = 4, α = 1.5%, 3%, 5%, 10%).

3.5.5.2 Effect of stiffness degradation

In this section, the effect of stiffness degradation on constant-strength CR spectra is further

studied considering the well-known Modified-Clough model described in Section 2.5.1 of

Chapter 2. Mean inelastic displacement ratios corresponding to each of soft soil ground

motion ensemble are shown in figure 3.13. Again, it can be seen that, in spite of the difference

in frequency content between both ensembles, mean CR spectra show a similar trend. In

general, CR ordinates change with changes in the T/Tg ratio and the level of lateral strength.

For the case of stiffness-degrading systems, two regions can be identified. In the first region,

maximum inelastic displacement demands from stiffness-degrading systems becomes larger

than maximum elastic displacement demands as the T/Tg ratio decreases and as the level of

lateral strength increases. In the second region, maximum elastic displacement demands are

expected to be larger than maximum inelastic displacement of stiffness-degrading systems. It

should be noted that the limiting T/Tg ratio that divide CR ordinates smaller than one than those

larger than one is around 0.75 for both soft soil sites, which means that the equal displacement

approximation does not hold for stiffness-degrading systems having periods of vibration

larger than 0.75 times the predominant period of the ground motion.


77

(a) San Francisco Bay Area Set (mean of 18 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

R = 6.0

R = 5.0R = 4.0

R = 3.0R = 2.0R = 1.5

(b) Mexico City Set (mean of 100 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0

R = 1.5

Figure 3.13. Mean inelastic displacement ratios for stiffness-degrading systems : (a) San Francisco Bay

Area set; (b) Mexico City set.

In order to further study the effect of stiffness degradation on the inelastic deformation

demands, the ratio of CR for stiffness-degrading systems to CR for non-degrading (i.e.,

elastoplastic) systems was computed. This relationship also represents a measure of how

larger or smaller the inelastic displacement demands are in systems with stiffness-degradation,

∆i,MC, compared to those in non-degrading systems, ∆i,EP. This ratio was obtained for each

normalized period of vibration, each level of relative strength, and each record for both ground

motion ensembles. Figures 3.14a and 3.14b show the ratio ∆i,MC /∆i,EP computed for both the

San Francisco Bay Area and the Mexico City ground motions set. It can be seen that, with

exception of short T/Tg ratios, for both soft soil sites the effect of stiffness degradation is not

significant over a wide spectral region. It is interesting to note that over a large T/Tg region

systems having stiffness-degrading characteristics will have inelastic displacement demands

smaller than the structures that do not exhibit stiffness degradation regardless of the level of

lateral strength ratio. T/Tg ratios where this ratio is, on average, larger than one from those in

which is, on average, smaller than one are not constant and vary with the level of relative

strength. These limiting periods that divide the region where it is unconservative to neglect the

effects of stiffness degradation (i.e., stiffness degradation produces larger inelastic

displacement demands than those of non-degrading systems) from spectral regions where it is

conservative to neglect the effects of structural degradation (i.e., regions in which stiffness

degradation leads to smaller inelastic displacement demands than those of non-degrading


78

systems) increase, in general, with decreasing level of lateral strength ratio. As seen in this

figure, these limiting normalized periods range from 0.95 for R=2.0 to 0.35 for R=6.

(b) Mexico City Set(mean of 100 ground motions)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

∆ i,MC/∆ i,EP

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

∆ i, MC/∆ i,EP

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 3.14. Degrading to non-degrading inelastic displacement ratios: (a) San Francisco Bay area Set;

(b) Mexico City set.

3.5.5.3 Effect of stiffness- and strength-degradation

In the third stage studying the effect of hysteretic behavior on CR special emphasis was placed

on the response of stiffness- and strength-degrading systems. Thus, inelastic displacement

ratios corresponding to stiffness- and strength-degrading models described in subsection 2.5.3

of Chapter 2 were computed. Similarly to previous section, the ratio of maximum inelastic

displacement demands in systems with stiffness- and strength-degradation, ∆i,D, compared to

those in non-degrading systems, ∆i,EP, was obtained and statistically processed. The ratio

∆i,D/∆i,EP corresponding to two levels of lateral strength and four hysteric behaviors is showed

in figure 3.15 for the San Francisco Bay Area and in figure 3.16 for the Mexico City ground

motion set. It can be seen that the ordinates of ∆i,D/∆i,EP depends on the level of deterioration

(i.e., severe stiffness- and strength-degrading, SSD model, systems experiences larger inelastic

displacements than systems having MC model hysteretic behavior), the level of lateral

strength ratio and the T/Tg ratio. In particular, it can be seen that it is expected that degrading-

systems sustain maximum inelastic displacements larger than those of non-degrading

(elastoplastic) systems for periods of vibration equal to or smaller than Tg, depending on the

level of lateral strength. Outside of this spectral region, degrading-systems would experience


79

maximum inelastic displacements smaller than non-degrading systems, regardless of the level

of lateral strength.

(b) R = 5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tg

∆ i,D / ∆ i,EP

SSD

MSD

SD

MC

(a) R = 3

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

∆ i,D / ∆ i,EP

SSD

MSD

SD

MC

Figure 3.15. Degrading to non-degrading inelastic displacement ratios computed from the San

Francisco Bay Area set: (a) R = 3; and (b) R=5.

(b) R = 5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / T g

∆ i,D / ∆i,EP

SSD

MSD

SD

MC

(a) R = 3

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

∆ i,D / ∆ i,EP

SSD

MSD

SD

MC

Figure 3.16. Degrading to non-degrading inelastic displacement ratios computed from the Mexico City

set: (a) R = 3; and (b) R=5.

3.6 Statistical Results for Near Fault Ground Motions

Initially, a total of 12,000 inelastic displacement ratios for elastoplastic SDOF systems were

computed (corresponding to 40 ground motions, 50 normalized periods of vibration and 6

levels of relative strength) and they were statistically processed (i.e., obtaining central

tendencies and dispersion) by each normalized period and each level of relative strength. The


80

main purpose of the first round of results was to examine the effects of normalized period of

vibration, lateral strength ratio, earthquake magnitude, distance to the source, peak ground

velocity, and pulse period on inelastic displacement ratios. The influence of hysteretic

behavior in bilinear and structural-degrading systems comprises a new set of results which are

described in Section 3.6.5.


Inelastic displacement ratios were computed using the suite of near-fault earthquake ground

motions described in Section 3.2.2. Central tendency measures (i.e., sample mean and counted

median) as a function of the period of vibration and the lateral strength ratio are shown in

figure 3.17. It can be seen that these statistical parameters provides different estimates of the

central tendency of CR. For example, mean CR ordinates are larger than one for periods of

vibration longer than about 1.5 s while median CR are slightly smaller than one in the same

spectral region. Besides this observation, a local amplification in CR ordinates can be seen for

periods of vibration around 0.6 s when median values are reported, but this feature is loss

when mean values are reported. Local amplification seems to increase as the lateral strength

ratio increases. On the other hand, mean and median inelastic displacement ratios computed

when the period of vibration is normalized with respect of the pulse period, Tp, are shown in

figure 3.18. It can be seen that CR computed from both statistical measures follow a more

similar trend than that without period normalization. It should be noted that the overall trend is

very similar to that observed for inelastic displacement ratios computed with soft soil ground

motion records. Then, three regions can be identified: (a) For T/Tp ratios smaller than 0.85,

inelastic displacement ratios increases as the period of vibration decreases with respect of Tp

and as the lateral strength ratio increases; (b) for T/Tp between 0.85 and 1.5, inelastic

displacement ratios are smaller than 1 which means that, on average, maximum inelastic

displacements are smaller that their maximum elastic counterparts, and; (c) for T/Tp ratios

greater than 2.0, inelastic displacement ratios are approximately equal to one which means that

the equal displacement approximation is only valid for this spectral region. It should be noted

that the peak where CR is smaller than 1 approximately correspond to periods of vibration

equal or close to the pulse period, but with a smaller values of CR than that expected for soft

soil site conditions.


81

(a) Near-Fault Set(mean of 40 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0R = 5.0R = 4.0

R = 3.0R = 2.0R = 1.5

(b) Near-fault Set(median of 40 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

R = 6.0R = 5.0R = 4.0

R = 3.0R = 2.0R = 1.5

Figure 3.17. Central tendency of inelastic displacement ratios using all 40 fault-normal near-fault ground motions computed without normalized periods: (a) Sample mean; and (b) counted median.

(a) Near-Fault Set (mean of 40 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tp

CR

R = 6.0

R = 5.0R = 4.0

R = 3.0R = 2.0R = 1.5

(b) Near-Fault (median of 40 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tp

CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0

R = 1.5

Figure 3.18. Central tendency of inelastic displacement ratios using all 40 fault-normal near-fault

ground motions computed with normalized periods respect to Tp: (a) Sample mean; and (b) counted median.

Since a good correlation between Tp and Tg, was observed it is interesting to investigate if

inelastic displacement ratios computed for periods of vibration normalized with respect to the

predominant period of the ground motions, T/Tg, follows a similar trend than that observed

when CR was computed for T/Tp ratios. Mean and median CR ratios as a function of T/Tg are

shown in figure 3.19. It can be seen that, in general, mean CR follows a similar trend than that

described in the last section. However, it should be noted that the region where the equal

approximation is valid begins for T/Tg ratios larger than 1.5.


82

(a) Near-fault Set(mean of 40 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tp

CR

R = 6.0R = 5.0R = 4.0

R = 3.0R = 2.0R = 1.5

(b) Near-Fault Set (median of 40 ground motions)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tp

CR

R = 6.0

R = 5.0R = 4.0R = 3.0R = 2.0

R = 1.5

Figure 3.19. Central tendency of inelastic displacement ratios using all 40 fault-normal near-fault

ground motions computed with normalized periods respect to Tp: (a) Sample mean; and (b) counted median.

3.6.2 Dispersion of CR

The coefficient of variation of inelastic displacement ratios computed without normalizing

and normalizing the period of vibration with respect to Tp and Tg is shown in figure 3.20a,

3.20b and 3.20c, respectively. In all cases, dispersion is not constant and it depends on the

spectral region and the lateral strength ratio. In particular, it can be seen that dispersion is

considerably reduced for T/Tp larger than 1, but it becomes much larger than that computed

without normalized periods. Also dispersion is significantly reduced when for T/Tg larger than

1 and it has similar ordinates than that when CR is computed without normalized periods.

Therefore, inelastic displacement ratios computed with respect to T/Tg ratios allows to better

characterize the frequency content of the ground motion and to reduce the variability in the

estimation of CR.


83

(a)

0.0

0.4

0.8

1.2

1.6

2.0

2.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV C R

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(b)

0.0

0.4

0.8

1.2

1.6

2.0

2.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tp

COV C R

R = 6.0

R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(c)

0.0

0.4

0.8

1.2

1.6

2.0

2.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

COV CR

R = 6.0R = 5.0R = 4.0

R = 3.0R = 2.0R = 1.5

Figure 3.20. Coefficient of variation of inelastic displacement ratios computed for the near-fault ground

motion set: (a) As a function of T; (b) as a function of T/Tp; and (c) as a function of T/Tg.

As an alternative way of measuring dispersion, counted percentiles corresponding to different

probabilities of exceeding inelastic displacement ratios were computed from the statistical

data. For example, figure 3.21 shows inelastic displacement ratios as a function of T/Tg

computed for systems having relative strength ratio of 4 and corresponding to five percentiles.

It can be seen that for pe riods of vibration close to Tg there is 80% of probability that CR

would be between 0.59 and 1.16, which mean that the equal displacement approximation

would result in an important overestimation of the maximum inelastic deformation demand.

On the other hand, for systems in the short normalized period range, for example T/Tg = 0.5,

CR would vary between 1.0 and 4.0 in 80% of the cases, which means that the equal

displacement rule would considerably underestimate maximum inelastic displacement

demands, which reflects a large variability in the estimation of CR.


84

(a) R = 2.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

p = 90%

p = 70%

p = 50%

p = 30%

p = 10%

(b) R = 4.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

p = 90%

p = 70%

p = 50%

p = 30%

p = 10%

(c) R = 6.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

p = 90%

p = 70%

p = 50%

p = 30%

p = 10%

Figure 3.21. Inelastic displacement ratio corresponding to different counted percentiles computed for

the near-fault ground motion set: (a) R=2; (b) R=4; and (c) R=6.


From figures 3.6 and 3.7, it was observed that the influence of the lateral strength ratio, R, on

CR changes depending of the spectral region. Therefore, a further look at this influence for

specific normalized period T/Tg is desirable. CR ratios computed with the near-fault ground

motion set for systems having six levels of lateral strength ratio are shown in figure 3.22 for

T/Tg ratios equal to 0.5, 1.0 and 2.0.


85

(c) T /Tg = 2.0

0.0

1.0

2.0

3.0

4.0

1 2 3 4 5 6 7Lateral Strength Ratio, R

CR

p = 90%

p = 50%

p = 10%

(a) T /Tg = 0.5

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

1 2 3 4 5 6

Lateral Strength Ratio, R

CR

p = 90%

p = 50%p = 10%

(b) T /Tg = 1.0

0.0

1.0

2.0

3.0

4.0

1 2 3 4 5 6 7Lateral Strength Ratio, R

CR

p = 90%p = 50%p = 10%

Figure 3.22. Effect of lateral strength ratio on inelastic displacement ratios computed for the near-fault

ground motion set: (a) T/Tg =0.5; (b) T/Tg = 1.0; and (c) T/Tg=2.0.

It can be seen that for T/Tg=0.5 inelastic displacement ratios increase nonlinearly as R

increases. The latter observation is true also for low and high probabilities of exceedance

(p=10% and 90%). However, T/Tg = 1.0 and 2.0 inelastic displacement ratios decrease with a

smooth nonlinear trend as the level of lateral strength increases. In addition, it can be seen that

for T/Tg = 0.5 dispersion increases as the lateral strength ratio increases, which means that

stronger asymmetry of the probability distribution is observed in this spectral region as the

level of lateral strength ratios increases.

3.6.4 Effect of Earthquake Magnitude

The effect of earthquake magnitude on CR for near-fault ground motions was evaluated by

computing CR spectra using three bins of 14 records corresponding to three earthquake


86

magnitude range. Figure 3.23 compares CR spectra computed from each magnitude bin for

three levels of lateral strength.

(a) R = 2.0

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

mean

Series4

Series3

Series2 5.61.6 ≤≤ wM

7.66.6 ≤≤ wM

3.79.6 ≤≤ wM

(b) R = 4.0

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

mean

Series4

Series3


7.66.6 ≤≤ wM

3.79.6 ≤≤ wM

(c) R = 6.0

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

mean

Series4

Series3


7.66.6 ≤≤ wM

3.79.6 ≤≤ wM

Figure 3.23. Effect of earthquake magnitude on inelastic displacement ratios computed for the near-

fault ground motion set: (a ) R = 2; (b) R = 4; (c) R = 6.

From the figure, it can be seen that earthquake magnitude has a moderate effect on CR

ordinates when the period of vibration is shorter than the predominant period of the ground

motion. In this region, earthquakes with large magnitude size would lead to larger CR

ordinates than earthquakes with moderate magnitude size. For periods of vibration longer than

Tg, earthquake magnitude does not have a significant effect on CR ordinates. These

observations are in good agreement with recent findings from Mavroeidis et al. (2004) using

constant-ductility inelastic spectra. In addition, it was observed that earthquake magnitude

influences CR ordinates when the system has lateral strength ratio larger than three and for T/Tg

ratios equal to or smaller than one.


87

3.6.5 Effect of Distance to the Source

A comparison of mean inelastic displacement ratios computed for three bins of 14 records

representative of three distance ranges of the near-fault rupture is shown in figure 3.24.

(a) R = 2.0

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tg

CR

mean

0.50.0 ≤≤ D0.100.5 ≤< D

0.150.10 ≤< D

(b) R = 4.0

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tg

CR

mean

0.100.5 ≤< D

0.150.10 ≤< D

0.50.0 ≤≤ D

(c) R = 6.0

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

mean

0.150.10 ≤< D

0.100.5 ≤< D

0.50.0 ≤≤ D

Figure 3.24. Effect of distance [km] to the source on inelastic displacement ratios computed for the

near-fault ground motion set: (a) R = 2; (b) R = 4; (c) R = 6.

It can be observed that, with exception of systems with T/Tg ratios smaller than 0.5,

distance to the fault rupture does no have strong influence on CR ordinates regardless of the

level of lateral strength. For systems with T/Tg ratios smaller than 0.5, structures very close to

the causative fault would cause largest CR ordinates for the three levels of lateral strength ratio

illustrated.


88

3.6.6 Effect of Peak Ground Velocity

The effect of peak ground velocity (PGV) on mean inelastic displacement ratios with

normalized periods of vibration with respect to Tg of near-fault ground motions was studied

using three bins of 14 records representative of three ranges of PGV. The influence of PGV on

mean CR for three levels of lateral strength is shown in figure 3.25.

(a) R = 2.0

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

mean

0.550.0 ≤≤ PGV0.1100.55 ≤< PGV0.1800.110 ≤< PGV

(b) R = 4.0

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

mean

0.1800.110 ≤< PGV0.1100.55 ≤< PGV

0.550.0 ≤≤ PGV

(c) R = 6.0

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

mean

0.1800.110 ≤< PGV

0.1100.55 ≤< PGV

0.550.0 ≤≤ PGV

Figure 3.25. Effect of peak ground velocity [cm/s] on inelastic displacement ratios computed from the

near-fault ground motion set: (a) R = 2; (b) R = 4; (c) R = 6.

In general, it can be seen that PGV does not have a strong influence on the mean CR

ordinates, once T is normalized with respect to Tg, for the three levels of lateral strength and

for the spectral region considered here. However, it should be mentioned that Baez and

Miranda (2000) found that near fault records with PGV larger than 40 cm/s leads to slightly

larger constant-ductility inelastic displacement ratios for periods of vibration between 0.1 and

0.9s and for ductility demands in excess of 3.


89

3.6.7 Effect of Pulse Period

The influence of the pulse period, Tp, of the velocity pulse on the amplitude of median CR

ratios was studied using three bins of 12 records representative of three ranges of Tp. The

effect of Tp on mean CR for three levels of lateral strength is illustrated in figure 3.26. In

general, it can be observed that the range of Tp have an important influence on both the shape

and the mean CR ordinates for the three levels of lateral strength considered, particularly true

for periods of vibration smaller than the pulse period. In this spectral region, near-fault ground

motions with Tp shorter than 1.0 s leads to larger ordinates than those records having longer

Tp, regardless of the levels of relative lateral strength. Moreover, for near-fault records with Tp

shorter than 2.0 s, an amplification of mean CR can be observed for periods of vibration near

0.5 times the pulse period. This local amplification seems to increase as the lateral strength

ratio increases and as the pulse period shortens, which is more noticeable when Tp is shorter

than 1.0 s.

(a) R = 2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tp

CR

Tp < 1.0 sec.1.0 sec. < Tp < 2.0 sec.Tp > 2.0 sec. median

(b) R = 4

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tp

CR

Tp < 1.0 sec.1.0 sec. < Tp < 2.0 sec.Tp > 2.0 sec. median

(c) R = 6

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / TP

CR

Tp < 1.0 sec1.0 sec < Tp < 2.0 sec.Tp > 2.0 sec.

median

Figure 3.26. Effect of pulse period on inelastic displacement ratios computed from the near-fault ground

motion set: (a) R = 2; (b) R = 4; (c) R = 6.


90


3.6.8.1 Effect of post-yield stiffness ratio

In order to examine the effect of positive post-yield stiffness on the maximum inelastic

displacement demands, maximum deformations of bilinear systems with post-yield stiffness

ratio (i.e., post-yield stiffness normalized by initial stiffness) of 3%, 5% and 10% were

computed when subjected to all 40 fault-normal near-fault ground motions. Therefore, ratios

of the maximum inelastic deformation of bilinear systems to maximum deformation of the

elastoplastic system were obtained for each record, each normalized period of vibration (i.e.,

T/Tg), and each lateral strength ratio.

(a) α = 3%

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / T g

∆i, α

=3%

/ ∆i,

α=0

%

R = 6.0R = 5.0

R = 4.0R = 3.0R = 2.0

R = 1.5

(b) α = 5%

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / T g

∆i, α

=5%

/ ∆i, α

=0%

R = 6.0R = 5.0R = 4.0

R = 3.0R = 2.0

R = 1.5

(c) α = 10%

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / T g

∆i, α

=10

%/ ∆

i,α

=0%

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

Figure 3.27. Effect of post-yield stiffness ratio on inelastic displacement ratios computed from the

near-fault ground motion set: (a) α = 3%; (b) α = 5%; (c) α = 10%;


91

Mean ratios of maximum inelastic displacement demand of bilinear systems to the

maximum inelastic displacement demand of elastoplastic systems are shown in figure 3.27. In

general, two spectral regions can be identified depending if T is smaller or larger than Tg. In

the first region (T < Tg), maximum inelastic deformation demands of bilinear systems becomes

smaller than that of elastoplastic system as the lateral strength ratio increases and as T/Tg ratio

decreases. In the second region (T > Tg), inelastic deformation demands computed from

bilinear systems becomes slightly smaller than those computed for elastoplastic systems as the

level of lateral strength ratio increases, but with significant changes as the T/Tg ratio increases.

In general comparing figure 2.13 (Chapter 2) with the results presented in this section, it can

be concluded that the effect of positive post-yielding stiffness is more beneficial in decreasing

inelastic displacement demands for systems subjected to far-field ground motions than fault-

normal near-fault ground motions.

(b) α = 3%

0.0

0.3

0.6

0.9

1.2

1.5

1.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

COV CR

R = 6.0R = 5.0

R = 4.0R = 3.0

R = 2.0

R = 1.5

(c) α = 5%

0.0

0.3

0.6

0.9

1.2

1.5

1.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

COV CR

R = 6.0R = 5.0R = 4.0R = 3.0

R = 2.0R = 1.5

(d) α = 10%

0.0

0.3

0.6

0.9

1.2

1.5

1.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

COV CR

R = 6.0R = 5.0R = 4.0R = 3.0

R = 2.0R = 1.5

(a) α = 0%

0.0

0.3

0.6

0.9

1.2

1.5

1.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

COV CR

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 3.28. Coefficient of variation of inelastic displacement ratios computed from the near-fault

ground motion set: (a) α = 0%; (b) α = 3%; (c) α = 5%; and (d) α =10%.


92

In addition of studying the effect of positive post-yield stiffness in reducing maximum

inelastic displacement demands of bilinear systems with respect to elastoplastic systems, it is

interesting of quantifying if an increment of post-yielding stiffness leads to smaller dispersion.

For this purpose, COV of CR of bilinear systems (with α > 0) are shown figure 3.28.

Comparing figures 3.28a with figures 3.28b, 3.28c, 3.28c it can be seen that the dispersion of

CR of systems with positive post-yield stiffness is very similar than that with elastoplastic

systems with exception of normalized periods smaller than about 0.25 where some reductions

in dispersion are observed as α increases. In addition, it should be noted that, when computed

with normalized periods, dispersion of CR for bilinear systems subjected to near-fault ground

motions is smaller than that of bilinear systems subjected to far-field ground motions (see

figure 2.9, Chapter 2).

3.6.8.2 Effect of strength-and-stiffness degradation

As part of this investigation, inelastic displacement ratios were computed from the nonlinear

response of SDOF systems having only stiffness-degrading feature (i.e., modified-Clough,

MC, model) and SDOF systems having three different degrading characteristics (i.e., stiffness-

degrading, SD, moderately-degrading, MSD, and severely-degrading, SSD) when subjected to

each of 40 fault-normal near-fault ground motions and later statistically processed according

to normalized period of vibration, T/Tg , level of lateral strength ratio , and level of

deterioration.

In order to illustrate the effect of combined stiffness and strength degradation on the

inelastic deformation demands the ratio of CR for degrading systems (e.g., MC, SD, MSD and

SSD) to CR for non-degrading (i.e., elastic-perfectly plastic) systems was computed. This

relationship also represents a measure of how larger or smaller the inelastic displacement

demands are in systems with degrading characteristics, ∆i,D, compared to those in non-

degrading systems, ∆i,EP. The ratio ∆i,D / ∆i,EP was obtained for each normalized period of

vibration, each level of relative strength and each record. Mean ∆i,D / ∆i,EP ratios

corresponding to three levels of lateral strength are shown in figures 3.29. In general, two

regions can be identified regardless of the level of lateral strength. In the first region, ∆i,D

becomes larger than ∆i,EP as the T/Tg ratio decreases and as the level of degrading features

increases. In the second region, ∆i,D becomes smaller than ∆i,EP as the T/Tg ratio increases


93

without significant influence on level of degrading characteristics. The corner normalized

period that divides both regions depends on the level of lateral strength and of the degrading

features. For example, the limiting normalized period systems with R=2 and MC behavior is

1.0, while for MSD behavior is 1.3. It should be noted that, for a given R, the limiting

normalized period does not significantly change for combined stiffness and strength degrading

systems. In addition, it should be noted that, for a given degrading-system, an increment in the

level of lateral strength seems to decrease inelastic deformation demands. Although this

observation seems counterintuitive it is consistent with previous findings reported by

Pincheira and Song (2000). In general, it can be concluded that increments in the level of

cyclic degradation increases inelastic deformation demands for systems with T shorter than Tg.

(b) R = 4

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / T g

∆ i,D / ∆ i,EP

SSD

MSD

SD

MC

(a) R = 2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

∆ i,D / ∆i,EP

SSD

MSD

SD

MC

(c) R = 6

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

∆ i,D / ∆ i,EP

SSD

MSD

SD

MC

Figure 3.29. Degrading to non-degrading inelastic displacement ratios computed from the near-fault

ground motion set: (a) R = 2; (b) R=4; and (c) R=6.


94

3.7 Simplified Equation to Estimate CR

For displacement-based design and, in general, in earthquake resistant design it is desirable to

have a simplified equation to estimate mean constant-strength inelastic displacement ratios

that can facilitate the estimation of maximum displacements in systems with inelastic behavior

from maximum displacements of systems with linear elastic behavior. In a recent study,

Miranda (2001) concluded that expressions derived directly from statistical analyses of mean

inelastic displacement ratios produce better results than using expressions derived from mean

strength reduction factors (i.e., Rµ - µ - T relationships). Thus, a functional form to estimate

mean inelastic displacement ratios, RC~ , must be a function of the T/Tg ratio, the level of

lateral strength ratio, R, and a set of parameters, θ , that minimize the difference between

mean inelastic displacement ratios, RC , and the computed values as follows

= θ,,ˆ R

TT

fCg

R (3.1)

Based on the observations described in earlier sections, the proposed generalized additive

model aimed to capture central tendency of inelastic displacement ratios for soft-soil site

conditions is given by:

( ) ( ) ( )[ ]+−⋅−⋅⋅−

⋅−+= 2

322

1 08.0ln3exp1

)1(~

ggg

R TTTTTT

RC θθ

θ

( ) ( ) [ ]24 67.0lnexp55.0 +⋅⋅⋅ gg TTTT θ (3.2)

where T is the period of vibration, Tg is the predominant period of the ground motion and

θ1, θ2, θ3,θ4 are constants which depend the type of ground motion ensemble. Equation (3.2)

corresponds to a surface in the CR – R – T/Tg space and provides estimates of mean inelastic

displacement ratios as a function of R and T/Tg. The first term in (3.2) keep the functional

form of the equation proposed in Section 2.7.2 (Chapter 2) to estimate mean inelastic


95

displacement ratios for firm sites while the second and third terms account for local reductions

in CR that take place for periods close to the fundamental period of vibration of the ground

motion (T/Tg ˜ 1) and at periods close to the second mode of vibration of the soil deposit (T/Tg

˜ 1/3). In this investigation, a nonlinear regression analysis was conducted using the

Levenberg-Marquardt method (Bates and Watts, 1988) from mean inelastic displacement

ratios corresponding to each set of ground motions to compute parameter estimates 1θ ?? 2θ ?? 3θ ,

and 4θ . As an example, the resulting values of these parameter estimates to fit mean CR

corresponding to the Mexico City Set are given in Table 3.1.

Table 3.1. Parameter estimates summary for equation (3.2)

Mexico City Set

R = 2.0

R = 3.0

R = 4.0

R = 5.0

R = 6.0

θ1 1.068 1.141 1.198 1.246 1.254 θ2 3.758 4.491 5.685 7.099 8.628 θ4 -0.555 -1.000 -1.228 -1.360 -1.401 θ8 -15.427 -44.868 -107.315 -147.789 -172.759

The functional form in equation (3.3) of the simplified model can be also used to

approximately estimate mean inelastic displacement ratios accounting for near-fault effects by

considering the first two terms as follows:

( ) ( ) ( )[ ]2322

1

08.0lnexp1

)1(1~

−⋅⋅⋅+

⋅−+= gg

gR TTTT

TTRC θθ

θ (3.3)

Similarly, nonlinear regression analysis was performed to obtain the best estimates of

parameters θ1, θ2, θ3 , which are reported in Table 3.2, to estimate mean CR. Finally, inelastic

displacement ratios predicted by using equations (3.3) and (3.4) are shown in figure 3.30.

From this figure, a good agreement between the predicted and computed inelastic

displacement ratios can be observed. Then, the functional models suggested in this study could

be very useful for estimating maximum inelastic displacement demands through inelastic

displacement ratios considering soft-soil and near-fault effects.


96

Table 3.2. Parameter estimates summary for equation (3.3)

Near-fault

set

R = 2.0

R = 3.0

R = 4.0

R = 5.0

R = 6.0

θ1 11.733 9.829 9.798 13.243 15.369 θ2 -0.072 -0.131 -0.344 -0.487 -0.518 θ3 -0.001 -0.291 -4.298 -5.805 -5.825

Mexico City SetNonlinear equation fit

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

Near-Fault SetNonlinear equation fit

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

CR

R = 6.0

R = 5.0R = 4.0

R = 3.0

R = 2.0R = 1.5

Figure 3.30. Mean inelastic displacement ratios for elastoplastic systems computed with equations (3.2)

and (3.3).

3.8 Summary

The main goal of this investigation was to assess maximum inelastic displacement demands of

single-degree-of-freedom (SDOF) systems subjected to ground motions recorded on soft soil

deposits and near the causative fault (near-fault). For this purpose, inelastic displacement

ratios, CR, were computed from the nonlinear response of SDOF systems having 6 different

levels of lateral strength when subjected to a total of 118 earthquake ground motions recorded

in the old lake-bed zone of Mexico City and on the bay-mud area of San Francisco as well as

40 fault-normal near-fault ground motions.

The following conclusions can be drawn from the results concerning soft-soil site effects

evaluated in this study:


97

1. Despite the important differences in the characteristics of soft soil deposits in Mexico City

and in the San Francisco Bay Area (e.g., shear wave velocities, water content, frequency

content, bandwidth) their constant-strength inelastic displacement ratios show a very

similar trend.

2. For systems with periods of vibration smaller than about 0.75 times the predominant

period of the ground motion (Tg), maximum inelastic displacements are larger than

maximum elastic displacements. In this spectral region mean CR ordinates increases

nonlinearly with increasing lateral strength ratio, R. For systems with periods of vibration

close to Tg, maximum inelastic displacements are, on average, significantly smaller than

maximum elastic displacements. In this spectral region the well-known equal

displacement approximation will lead to overestimations of lateral displacement demands

of inelastic systems. This was observed for practically all ground motion recorded on very

soft soil. For systems with periods of vibration that are more than 1.5 times Tg , maximum

inelastic displacements are, on average, close to the maximum elastic displacements.

3. Dispersion of CR is not constant over the whole normalized period range (T/Tg), tending to

increase for short T/Tg ratios. In general, the scatter of mean CR of soft soil sites is smaller

than that reported in Chapter 2 for firm sites.

4. For the range of earthquake magnitudes and epicentral distances considered here, it was

shown that CR ordinates are not significantly affected by earthquake magnitude or by

distance to the source.

The following conclusions are offered from the results involving fault-normal near-fault

effects evaluated in this study:

1. Constant-strength inelastic displacement ratios for near-fault ground motions depend

on the pulse period, Tp, of the velocity pulse. Therefore, an adequate characterization

of CR spectra for this type of ground motions should explicitly take into account Tp.

2. A good correlation between Tp and the predominant period of the ground motion, Tg,

as well as the ground motion bandwidth was found for the suite of fault-normal near-

fault ground motions considered in this investigation. This observation allows an

alternative way of computing CR for normalized periods since Tg can be computed

from the spectral velocity spectra.


98

3. For systems with periods of vibration smaller than about 0.85 times Tp, maximum

inelastic displacements are larger than maximum elastic displacements. In this spectral

region mean CR ordinates increases nonlinearly with increasing lateral strength ratio,

R. For systems with periods of vibration between 0.85 and 2.0 times Tp, maximum

inelastic displacements are, on average, significantly smaller than maximum elastic

displacements. In this spectral region the well-known equal displacement

approximation will lead to overestimations of lateral displacement demands of

inelastic systems. For systems with periods of vibration that are more than 2.0 times

Tp , maximum inelastic displacements are, on average, close to the maximum elastic

displacements.

4. Dispersion of CR is not constant over the whole normalized period range (T/Tp),

tending to significantly increase for short T/Tp ratios. In general, it was found that the

scatter of mean CR for near-fault ground motions is smaller when the period of

vibration is normalized with respect to Tg instead of Tp.

5. The most important ground motion characteristic that influences the shape and

amplitude of CR ordinates is the pulse period, which is particularly true for systems

with period of vibration shorter than the pulse period. Moreover, near-fault ground

motions with pulse period shorter than 1.0 s leads to a local amplification for systems

with period of vibration near 0.5 times the pulse period.

6. It was shown that CR ordinates computed for systems with T/Tg ratios larger than

about 0.5 are not significantly affected by earthquake magnitude or by distance to the

source (for the range of earthquake size and epicentral distances considered here).

Similarly, CR ordinates are not very sensitive to the peak ground velocity of the near-

fault records.

7. It was found that the effect of post-yielding stiffness in limiting maximum inelastic

displacements demands is less beneficial for near-fault ground motions than for far-

field ground motions.

8. An increment in the level of stiffness-and-strength degradation increases maximum

inelastic deformation demands for systems with T shorter than Tg. However, for

systems with T greater than Tg, maximum inelastic displacement demands are slightly

smaller than that of non-degrading systems.


99

Finally, a simplified equation to estimate mean inelastic displacement ratios accounting

for soft soil conditions was proposed. The proposed simplified expression provides relatively

good approximations of mean inelastic displacement ratios of SDOF systems. It can also be

used to obtain rough estimates of maximum inelastic displacements of MDOF structures built

on very soft soil deposits whose displacement response is dominated by the first mode. The

proposed equation might used for estimating mean inelastic displacement ratios considering

near-fault effects.

___________________________________________________________________________________ Chapter 4 Probabilistic Estimation of Maximum Inelastic Displacement Demands of SDOF Systems

100

Chapter 4

Probabilistic Estimation of Maximum Inelastic

Displacement Demand of SDOF Systems

4.1 Introduction

Current proposed performance-based procedures to estimate global (e.g., roof) lateral

deformation demands during the seismic evaluation of standard existing multi-story buildings

rely on the estimation of maximum displacement demands of inelastic single-degree-of

freedom (SDOF) systems. For example, in the Nonlinear Static Procedure (NLS) introduced

in the FEMA-356 document (FEMA, 2000) the target maximum inelastic displacement is

estimated from the so-called displacement coefficient method. In the displacement coefficient

method, coefficient C1 is defined as a “modification factor to relate expected maximum

inelastic displacements to displacement calculated from linear elastic analysis”. Another

example of a nonlinear static approach for earthquake-resistant evaluation of buildings that

make uses of maximum inelastic displacement demands of SDOF systems is the recently

introduced modal pushover analysis (e.g., Chintanapakdee and Chopra, 2003). A common and

key step in the aforementioned procedures is the estimation of maximum inelastic deformation

demands through the use of inelastic displacement ratios that allows the estimation of peak

inelastic displacement demands from maximum elastic displacement demands (through the

coefficient C1). Inelastic displacement ratios were first studied by Veletsos and Newmark

(1960) but for a limited range of earthquake ground motions, periods of vibration and levels of

inelastic deformation. Since then, several researchers have performed statistical studies aiming

to evaluate inelastic displacement ratios using larger sets of ground motions and for wider

range of periods of vibration than those pioneer studies (e.g., Shimazaki and Sozen, 1976; Qi

and Moehle, 1991; Whittaker et al., 1998; Song and Pincheira, 2000; Ramírez et al., 2001;

Ruiz-García and Miranda, 2003; Chopra and Chintanapakdee; 2004). Some, but not all, of the

published statistical studies have pointed out the large dispersion (i.e., record-to-record

variability) in the estimation of inelastic displacement ratios and, in consequence, in the


101

estimation of maximum inelastic displacement demands. Unfortunately, this, and other, source

of uncertainty have not been taken into account in the previously mentioned performance-

based procedures for the design of new structures nor for the evaluation of existing structures.

It is well known that the largest source of uncertainty in estimating the effects of

earthquakes on man-made structures lies on the estimation of the characteristics of future

earthquake ground motions that can occur at a specified site. A Probabilistic Seismic Hazard

Analysis (PSHA) represents a rational and quantitative procedure to estimate the hazard of

earthquake ground motions at this site (Cornell, 1968). Using the location and geometry of all

possible sources, the probability distribution of earthquake magnitudes at each source, and

attenuation relationships, a conventional PSHA permits the estimations of the mean annual

frequency of occurrence of a certain ground motion parameter (e.g., peak ground acceleration,

etc.) or elastic response parameters (e.g., pseudo-spectral acceleration) by integration over all

possible sources, earthquake magnitudes and distances. The first attempt to consider inelastic

response in PSHA was conducted by McGuire (1974) who developed the first attenuation

relationship for inelastic deformation of elastoplastic SDOF systems. However, this early

study only considered a small number of ground motions and the relationship was constrained

to inelastic systems having three levels of lateral strength and with 5 periods of vibration.

Derived from the same study, McGuire (1974) developed a procedure to obtain spectral elastic

ordinates having the same probability of exceeding maximum response of SDOF systems for a

range of periods of vibration. This type of spectrum is known as uniform hazard spectra

(UHS). Years later, Sewell and Cornell (1987) extended conventional PSHA analysis to

estimate the mean annual frequency that a specified nonlinear SDOF system exceeds a

threshold relative damage measure (e.g., a displacement ductility demand in excess of four).

In addition, their approach allowed computing UHS of nonlinear response parameters (e.g.,

the ratio of the absolute maximum linear base shear to the absolute maximum nonlinear base

shear).

Probabilistic Seismic Demand Analysis (PSDA) provides a rational way to evaluate the

seismic demand hazard of a specified structure built on a specific seismic environment

(Cornell, 1996). However, PSDA still requires performing a large number of nonlinear time-

history analyses on user-defined structural models whose implementation might be

questionable at an early stage in the evaluation of existing structures when limited information

of the structure is available. Then, approximate methods to estimate maximum inelastic

deformation demands of existing structures based on the estimation of maximum inelastic


102

displacement of SDOF systems which consider the variability in its estimation and,

furthermore, the probability of being exceeded seems a promising approach to be incorporated

in current performance-based assessment methodologies of existing structures.

The main objective of this study is to propose a simplified probabilistic approach to

estimate maximum inelastic displacement demands of SDOF systems and, specifically, the

site-specific mean annual frequency of exceeding a specified maximum inelastic displacement

demand. In particular, the study introduces: (1) a robust procedure for the evaluation of the

parameters to compute the probability distribution of inelastic displacement ratios; (2) a

careful evaluation of the influence of earthquake magnitude and distance to the source in the

estimation of maximum inelastic displacement demand; (3) an estimation of the error

introduced by simplifying assumptions; and (4) maximum inelastic displacement demand

hazard curves and nonlinear uniform hazard spectra to be used in performance-based

assessment of existing structures. This investigation makes use of statistical results of inelastic

displacement ratios computed from the dynamic response of inelastic SDOF systems having a

wide range of periods of vibration and lateral strength when subjected to a relatively large

suite of ground motions. However, in this study the implementation of the proposed procedure

is limited to estimate the maximum deformation demands of inelastic SDOF systems having

elastoplastic hysteretic behavior when subjected to far-field ground motions recorded on rock

or firm soil conditions.

4.2 Formulation of Proposed Simplified Approach to Estimate ( )iδλ

The main goal of the proposed approach consists on computing the mean annual frequency of

maximum inelastic displacement demand, i∆ , exceeding a certain lateral displacement,

iδ , )( iδλ , which can be expressed mathematically as follows:

[ ] dd

dyddiii ds

dSsd

CTsSP)(

,;|)(0

λδδλ ⋅=>∆= ∫

∞ (4.1)

In the above equation, )( dsλ is the site-specific mean annual frequency of exceedance

(MAF) of spectral displacement, Sd, evaluated at sd, which is also known as seismic hazard

curve. In this case, the ground motion intensity measure is Sd. Site-specific seismic hazard


103

curves are obtained from conventional PSHA and they are already available from the United

States Geological Survey (USGS) for any geographical location in the United States (Frankel

and Leyendecker, 2000). In addition, [ ]yddii CTsSP ,;| =>∆ δ is the probability of

i∆ exceeding iδ , conditioned on the system’s fundamental period of vibration, T, the yield

strength coefficient, Cy (defined as the lateral yielding strength of the system, Fy, normalized

by its weight), as well as the maximum spectral elastic displacement demand, Sd. A key

assumption in the proposed procedure is that i∆ just depends on the structural properties of the

system (i.e., T and Cy) and the level of intensity (measured by Sd) of the ground motion, and it

is not affected by other parameters such as earthquake magnitude nor distance to the source.

This core simplifying assumption allows us to separate the probabilistic estimation of the

seismic hazard at a given site (i.e., right-hand side in the integrand) from the probabilistic

estimation of i∆ (i.e., left-hand side in the integrand), which can be viewed as an extension of

conventional PSHA. It should be mentioned that the above formulation is conceptually similar

to that proposed by Cornell and his co-workers (e.g., Sewell and Cornell, 1987; Bazzurro and

Cornell, 1994) to estimate the annual seismic risk of exceeding a relative damage measure of

nonlinear SDOF oscillators.

By assuming that earthquakes occur according to a Poisson process (i.e., inter-arrival

times are time-independent and they are exponentially distributed) an that the annual rate of

exceeding Sd is relatively small, the mean (annual) probability of exceedance of sd,

][ dd sSP > is approximately )( dsλ , which also implies that the mean (annual) probability of

exceeding a certain level of maximum inelastic displacement demand is approximately )( iδλ ,

( )iiiP δλδ ≈>∆ ][ .

In order to evaluate the conditional probability in the right-hand side of equation (4.1), it

is convenient to make use of the constant-strength inelastic displacement ratio, CR, introduced

in Section 2.2 (Chapter2) which allows the estimation of maximum inelastic displacement

demands from the maximum elastic displacement demand. The inelastic displacement ratio is

a non-dimensional parameter defined as the maximum inelastic displacement demand, i∆ , of

an inelastic SDOF system with known lateral strength divided by the maximum lateral elastic

displacement demand, Sd, computed on a linear SDOF system with the same mass and initial

stiffness (i.e., same period of vibration) when subjected to the same earthquake ground


104

motion. In both cases displacements are relative to the ground. Mathematically this is

expressed as:

d

iR S

C∆

= (4.2)

In equation (4.2), maximum inelastic displacement demands can be computed for SDOF

systems with constant yield strength coefficient or, alternatively, with constant yield strength

relative to the strength required to maintain the system elastic (i.e., constant relative strength).

Here the relative lateral strength is measured by the lateral strength ratio, R, which is defined

as:

y

a

CgS

R/

= (4.3)

where Sa is the spectral acceleration ordinate and g is the acceleration of gravity. The

numerator in equation (4.3) represents the lateral strength required to maintain the system

elastic, which sometimes is also referred to as the elastic strength demand. For a given

structure with known lateral strength ratio and first-mode period of vibration, if information of

the expected CR and Sd ordinates is available, the maximum inelastic displacement demand

can be computed as follows:

dRi SC ⋅=∆ (4.4)

Therefore, substituting equation (4.4) in equation (4.1) yields:

( ) [ ] dd

dyddidRi ds

dSsd

CTsSSCP)(

,;|0

λδδλ ⋅=>⋅= ∫

∞ (4.5)

In the proposed procedure, it is assumed that the expected value of i∆ can be obtained as

the product of CR times Sd, which implies that CR and Sd are independent random variables and

that the is a lack of correlation between them. Furthermore, in computing the conditional

probability term in equation (4.5), it is assumed that is equivalent to


105

compute [ ]yRR CTcCP ,|> , where cR is defined as: diR sc δ= . Therefore, equation (4.5)

can be expressed as follows:

( ) [ ] dd

dyRRi ds

dSsd

CTcCP)(

,|0

λδλ ⋅>= ∫

∞ (4.6)

It should be mentioned that a core assumption in the above formulation is the lack of

dependence of CR on the intensity of the event and distance form the causative fault. This

assumption is based on a prior statistical study that showed little dependence of CR on

earthquake magnitude and distance to the source (Ruiz-Garcia and Miranda; 2003). The

former observation was recently verified by Chopra and Chintanapakdee (2004) using a

different ground motion set than that used by Ruiz-Garcia and Miranda (2003). This

simplified assumption will be further verified in Sections 4.3.1 and 4.3.2.

In addition, to evaluate the conditional probability inside the integrand, an examination of

the empirical cumulative distribution function of CR will be carry out in Section 4.4.3 and

candidate parametric cumulative distribution functions will be evaluated in Section 4.4.5.

In addition, in order to compute the seismic hazard curve of i∆ , the seismic hazard at the

designated site should be available. The site-specific seismic hazard information usually is

provided from seismologists in terms of the mean annual frequency of exceeding peak ground

acceleration (PGA) or spectral acceleration (Sa) at various periods, computed from the elastic

response of SDOF systems having 5% of damping ratio. For example, this information is

readily available from the United States Geological Survey (USGS) for any site in the U.S.

(Frankel and Leyendecker, 2001).

It should be mentioned that the lateral strength ratio has been a key parameter in

traditional force-based design. However, several researchers have promoted the yield

displacement of the structure (i.e., the displacement of the roof at yield) as a primary

parameter in their recently proposed procedures for the design of new structures, or for the

evaluation of existing structures, (Aschheim and Black, 2000; Paulay, 2002; Aschheim, 2004).

Particularly, those researchers have showed that the roof yield displacement is a very stable

parameter that can be estimated from nonlinear static analysis (pushover). Thus, it is

convenient to alternatively express equation (4.1) as a function of the yield displacement of

the SDOF system, ∆y, through the following relationship:


106

RSd

y =∆ (4.7)

which allow an alternative form of equation (4.1) as follows:

[ ] dd

dyddiii ds

dSsd

TsSP)(

,;|)(0

λδδλ ⋅∆=>∆= ∫

∞ (4.8)

In the above formulation, the capacity (i.e., yield strength coefficient, lateral strength ratio,

or yield (roof) displacement) and dynamic properties (i.e., period of vibration) of the existing

structure (e.g., bridge piers) are known, which mean that a reasonable estimate is available.

For example, nonlinear static analysis (pushover) of the structure of interest subjected to an

adequate lateral force profile provides information of the global capacity (Seneviratna and

Krawinkler, 1997). On the other hand, ambient vibration tests or empirical formulae (e.g.,

Chopra and Goel, 2000) can be used for estimating the period of vibration.

In addition, it should be mentioned that even though the proposed approach can be

implemented to estimate the maximum inelastic displacement demand of systems subjected to

different type of ground motions (e.g., far-field and near-fault ground motions), built on

different soil conditions (e.g., rock, firm soil or soft-soil sites), having different hysteretic

behavior and having different damping ratios, in this study the implementation of the proposed

approach is just illustrated for structures built on firm soil sites having elastoplastic hysteretic

behavior and with equivalent damping ratio of 5%.

4.3 Evaluation of Simplified Assumptions

In the formulation of the proposed probabilistic approach outlined in the previous section three

main simplifying assumptions have been made. The first simplified assumption implies that

the conditional expectation of CR on a given earthquake magnitude event (m) at a specified

distance to the source (d), [ ]dmCE R ,| , would be approximated by the unconditional expected

value of CR, [ ]RCE , which means that there is a lack of dependence of the expected CR on

earthquake magnitude and distance to the source. The second simplified assumption is that

there is a lack of correlation between CR and Sd (i.e., CR and Sd are statistically uncorrelated)


107

and they can be assumed as independent random variables. Finally, the third simplified

assumption implies that the empirical cumulative probability distribution of CR is

approximated by a parametric cumulative distribution function. Therefore, it is necessary to

verify the validity of these simplifying assumptions and to point out the limitations of the

suggested simplified probabilistic approach. For that purpose, statistical results of CR obtained

in the preceding chapter and corresponding to SDOF systems having elastoplastic hysteretic

behavior are used to examine the simplified assumption in the next sections.

4.3.1 Effect of Earthquake Magnitude on CR

Nowadays, it is well-known that elastic spectral ordinates (i.e., spectral displacement, Sd) are

dependent on the magnitude of the earthquake. Thus, it is important to know to what extent

earthquake magnitude affects inelastic displacement ratios. The influence of earthquake

magnitude on CR was studied by computing inelastic displacement ratios from ground motions

recorded on site class D and then grouped in three bins of 24 ground motions according to the

earthquake magnitude in which they were recorded. A comparison of mean inelastic

displacement ratios computed for the three bins and for lateral strength ratios equal to two and

four is shown in figure 4.1.

(a) SITE CLASS D, R = 2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

7.0< Ms < 7.8 (24 records)6.3< Ms < 6.9 (24 records)5.7< Ms < 6.2 (24 records)

(b) SITE CLASS D, R = 4

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

7.0< Ms < 7.8 (24 records)6.3< Ms < 6.9 (24 records)

5.7< Ms < 6.2 (24 records)

Figure 4.1. Effect of earthquake magnitude on mean inelastic displacement ratios: (a) for R=2; (b) for R=4.

It can be seen that for a lateral strength ratios equal to 2, earthquake magnitude has minor

effect on CR. However, earthquake magnitude can influence ordinates of CR for short-period

structures having larger lateral strength ratio (i.e., weaker structures relative to the intensity of


108

the ground motion). It can be also seen that for R=4 the inelastic displacement ratios of ground

motions recorded in earthquakes with surface-wave magnitudes between 5.7 and 6.2 tend to be

smaller than those of ground motions recorded in earthquakes with higher magnitudes. In

particular, for R=4 and T=0.5s, mean values of CR of ground motions recorded in earthquake

with magnitudes higher than 6.3 are approximately two times higher those from ground

motions recorded in earthquake with magnitudes between 5.7 and 6.2.

Another way to further investigate the effect of earthquake magnitude on inelastic

displacement ratios is through the coefficient of correlation, ρ. Figure 4.2 shows the

correlation between the inelastic displacement ratio and the earthquake magnitude for the

range of periods included in this study. It can be seen that, in general, CR has a positive

correlation with earthquake magnitude regardless of the period of vibration, meaning that an

increase in magnitude tends to produce an increase in CR. It can also be noticed that the

correlation increases as the level of lateral strength ratio increases (i.e. CR is more correlated

with earthquake magnitude for weak structures than for strong structures).

CR, Ms

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

ρ

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 4.2. Correlation between mean inelastic displacement ratios and earthquake magnitude for all 240 ground motions recorded in NEHRP site classes AB, C and D.

An approach commonly used in applied statistics to investigate the statistical significance

of adding an additional (prospective) explanatory variable (e.g., earthquake magnitude) on the

response variable (e.g., inelastic displacement ratio) consists on plotting the fitted residuals,

ie , computed by using baseline predictor variables (e.g., period of vibration, T, and lateral


109

strength ratio, R) versus the prospective predictor variable and then performing linear

regression of the ie on the prospective explanatory variable (Weisberg, 1985). The statistical

significance of the prospective variable is evaluated by performing the following test of

hypothesis: the parameter estimate of the slope is zero. The statistical significance is measured

by a quantity known as the p-value, assuming that the slope coefficient is a random variable

that follows a Student t-distribution (Weisberg, 1985). The hypothesis is rejected (i.e., the

slope parameter estimate is non-zero) if the p-value is smaller or equal than a threshold value

(e.g., 5% significance level). This statistical technique has been used by Cornell and his co-

workers (e.g., Shome and Cornell, 1999) to investigate the sufficiency of candidate intensity

measures with respect to earthquake magnitude. This statistical technique was used to evaluate

quantitatively the influence of earthquake magnitude on CR and it is illustrated next.

(a) T = 0.2 sp -value = 6.42e-06

-30.0

-20.0

-10.0

0.0

10.0

20.0

30.0

5.5 6.0 6.5 7.0 7.5 8.0Earthquake Magnitude

Residual

(b) T=0.5sp -value = 2.72e-06

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0


Residual

(c) T = 1.0sp -value = 0.0817

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0


Residual

(d) T = 3.0 sp -value = 0.657

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0


Residual

Figure 4.3. CR fitted residuals versus earthquake magnitude for R = 4 and four periods of vibration:

(a) T=0.2s; (b) T=0.5s; (c) T=1.0s; and (d) T=3.0s.

First, the fitted residuals, e , were computed for a given period of vibration and given

level of lateral strength ratio (i.e., ijRijR Cce ,,ˆ −= , where cR is the computed mean inelastic

displacement ratio and RC is the estimation of the mean CR obtained from equation (2.15) for

the i-th period of vibration and the j-th lateral strength ratio) using the whole set of results


110

(i.e., 240 fitted residuals). Next, 240 fitted residuals are plotted against their corresponding

earthquake magnitude and conventional linear regression of se'ˆ on earthquake magnitude is

performed. To illustrate this procedure, plots of pair-wise fitted residuals-earthquake

magnitude are showed in figure 4.3 for four periods of vibration (T = 0.2s, 0.5s, 1.0s, and 3.0s)

and for R = 4. The fitted line is also shown in the same figure as well as the corresponding p-

value. Similar plots were obtained for other T and R combinations. For example, it can be seen

that the p-value is smaller than 5% for short-period systems (T=0.2s and 0.5s), meaning that

the hypothesis test can be rejected (i.e., the slope parameter estimate is non-zero) and,

furthermore, meaning that there is some dependence of CR on earthquake magnitude. On the

other hand, the test of hypothesis can not be rejected (i.e., the slope parameter estimate is not

significantly different from zero) for long-period systems (T=1.0s and 3.0s), meaning that

there is evidence of lack of dependency of CR on earthquake magnitude.

In general, it was found that dependency of CR on earthquake magnitude is statistically

significant only for weaker systems (R > 4) in the short spectral period region (T < 0.5s). On

the other hand, the effect of earthquake magnitude on CR can be neglected for systems with

periods of vibration greater than 0.5s and lateral strength ratios smaller or equal than 4.

4.3.2 Effect of Distance to The Rupture on CR

It is recognized that elastic spectral ordinates attenuates as the distance to the earthquake

source decreases, for a given magnitude event. Therefore, it is of special interest to study if

ordinates of CR are also affected by the distance to the source. For that purpose, inelastic

displacement ratios were computed for three bins of 24 earthquake ground motions having

three different range of distances to the horizontal projection of the rupture (the so called

Joyner and Boore distance) (Boore et al., 1997). It should be mentioned that all 72 ground

motion were recorded in stations located in site class D. A comparison of mean inelastic

displacement ratios for three different distances to the rupture and for lateral strength ratios

equal to two and four are shown in figure 5.4. For the range of distances considered in their

study, changes in mean inelastic displacement ratios are relatively small. It should be noted,

however, that the ensemble of records considered did not include near-fault records.


111

(a) SITE CLASS D, R = 2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

45.1< D < 160.0 (24 records)20.1< D < 45.0 (24 records)

1.0< D < 20.0 (24 records)

(b) SITE CLASS D, R = 4

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

45.1< D < 160.0 (24 records)

20.1< D < 45.0 (24 records)

1.0< D < 20.0 (24 records)

Figure 4.4. Effect of distance to the source on mean inelastic displacement ratios for:

(a) R=2; and (b) R=4.

CR, D

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

ρ

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 4.5. Correlation between mean inelastic displacement ratios and distance to the source for all 240

ground motions recorded in NEHRP site classes AB, C and D.

The coefficient of correlation computed for the inelastic displacement ratio and the

distance to the rupture versus the period of vibration is showed in figure 4.5. It can be seen

that, in general, the inelastic displacement ratio is positively correlated with the distance to the

rupture over the period region and the correlation slightly increases as the lateral strength ratio

increases. It can also be observed that the correlation between CR and distance to the rupture is

smaller than 0.15 for periods of vibration longer than 0.5s. Following the approach suggested

in the preceding section, the residuals of the inelastic displacement ratio versus distance to the

rupture are shown in figure 4.6 corresponding to the same lateral strength ratio and periods of

vibration. It was also found that distance to the rupture has statistical significance on CR for


112

short period and weak systems, but its influence on CR is considered negligible for a wide

range of systems.

(a) T = 0.2 sp -value = 4.85e-06

-30.0

-20.0

-10.0

0.0

10.0

20.0

30.0

0 20 40 60 80 100 120Nearest Distance to the Source Projection [km]

Residual

(b) T=0.5sp -value = 0.000203

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0


Residual

(c) T = 1.0sp -value = 0.2426

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 20 40 60 80 100 120Nearest distance to the source projection [km]

Residual

(d) T = 3.0 sp -value = 0.406

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 20 40 60 80 100 120Nearest distance to the source projection [km]

Residual

Figure 4.6. CR fitted residuals versus nearest distance to the source for R = 4: (a) T=0.2s. (b) T=0.5s; (c) T=1.0s; (d) T=3.0s.

4.3.3 Effect of Duration of the Ground Motion on CR

Nowadays, still there is a controversy about the influence of earthquake ground motion

duration on the seismic response of structures. This controversy begins with the definition of

duration of strong ground shaking of acceleration time histories recorded from earthquake

events that adequately represent the time interval when the energy content of the earthquake

ground shaking produce damage to the excited structure. For instance, Bommer and Martinez-

Pereira (1999) published a comprehensive review about the merits and pitfalls of many

definitions proposed in the literature. This issue has gained attention since researchers have

found that strong motion duration is closely related to the number of inelastic cycles that

structural elements suffer during earthquake excitation and, thus, with cumulative damage due

to low-cycle fatigue phenomena (e.g., Mahin, 1980; Fajfar and Fischinger, 1990; Cosenza et


113

al., 1993). However, other researchers (e.g., Rahnama and Krawinkler, 1993; Bernal, 1992;

Ibarra, 2003; Cosenza et al., 2004) have found that td has small effect on the peak seismic

demand parameters (e.g., maximum inelastic displacement demands or even the collapse

capacity of SDOF systems).

In this section, the influence of strong motion duration on CR is investigated by using two

sets of earthquake ground motions containing short and long strong motion duration records.

Strong motion duration of each record was obtained from the definition proposed by Trifunac

and Brady (1975). They defined strong motion duration as the time interval between 5% and

95% of the Arias intensity. Even though this definition has been criticized for lacking of

correlation with the nature of the seismic event (e.g., duration of the rupture of the causative

fault), it has been widely used for many researchers while evaluating the effect of strong

motion duration. The strong motion duration of 40 earthquake ground motion was obtained

using the Trifunac-Brady definition and divided in two sets of 20 ground motions having short

strong motion duration (e.g., between 8.8 s. and 15.9 s) and long strong motion duration (e.g.,

between 25.7 s. and 51.7 s). The ensemble of short-duration records actually represent a subset

of the LMSR-N ground motion database compiled by Medina and Krawinkler (2003). A

complete list of both ground motions sets is given in Appendix A.

A comparison of mean inelastic displacement ratios computed from both sets of short

(SD) and long (LD) ground motion durations for three lateral strength ratios is shown in figure

4.7. It can be observed that for R=2, strong ground motion direction does not have a

significant effect on CR. However, as the system becomes weaker (i.e., relative lateral strength

ratio increases) strong motion duration can influence CR for systems with short and medium

period of vibration (e.g., until 2.0 s). For example, for R = 4 and T = 1.5 s., mean values of CR

computed from earthquake records having long strong motion duration are approximately 1.26

times higher than those computed from the short strong motion duration set. For a system with

the same period of vibration but R = 6, motions with long strong motion duration lead to mean

CR values approximately 1.36 higher than those estimated with the ground motion set

containing short strong motion duration. It should be mentioned that the long-duration ground

motions were obtained from seismic events having moment magnitude, Mw, between 7.6 and

8.0, which can be considered as large magnitude events, while short-duration records were

collected from events with Mw between 6.5 and 6.7. Thus, large-magnitude long-duration

records might have more influence in CR ordinates than moderate-magnitude short-duration

ground motions, which are typical of the seismic hazard in California.


114

(c) R = 6.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

LD

SD

(b) R = 4.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

LD

SD

(a) R = 2.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

LD

SD

Figure 4.7. Effect of strong ground motion duration on mean CR for: (a) R=2; (b) R=4; and (c) R=6.

In order to further study the effect of strong motion duration on mean CR values, the ratio

of mean CR computed from each strong motion duration set to the mean CR computed from all

40 earthquake ground motions is reported in figure 4.8. This ratio also represents the ratio

between mean maximum inelastic displacement demands computed from each set to mean

maximum inelastic displacement demands computed using all 40 ground motions. It can be

seen that, in general, if one neglects the effects of strong motion duration and uses mean CR

values from all 40 ground motions, it could lead to an underestimation of CR when systems are

subjected to short strong motion duration and to an overestimation when the systems are

excited from long duration motions. This underestimation or overestimation depends on

period of vibration and levels of lateral strength ratio, but it is more remarkable for systems in

the short period region.


115

Effect of duration of the ground motion(a) Short duration records

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR,SD / CR,SLD

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Effect of duration of the ground motion(b) Long duration records

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR,LD / CR,SLD

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 4.8. Mean CR of each strong motion duration set normalized by mean CR from all 40 ground

motions: (a) Short–duration set; (b) long-duration set.

4.3.4 Statistical Dependence Between CR and Sd

4.3.4.1 Relative error by neglecting correlation between CR and Sd

In the proposed procedure to estimate the mean annual frequency of exceeding a given

inelastic displacement threshold, it is assumed that i∆ can be computed as the product of CR

times Sd. This assumption implies that CR and Sd are uncorrelated random variables. However,

this assumption can lead to some systematic error in the estimation of i∆ in equation (4.2) if

CR and Sd are statistically correlated. Thus, it is important to have an estimation of the

statistical correlation between CR and Sd as well as the relative error in the estimation of i∆

over the over the whole range of periods of vibration and lateral strength ratios considered in

this study. Next, an examination of the relative error is presented.

First, the expected value of the maximum inelastic displacement can be written as follows:

[ ]dRi SCEE ⋅=∆ ][ (4.8)

where [ ]⋅E denotes expectation. If the CR and Sd are statistically correlated, the right hand side

in equation (4.8) can be written as:


116

dSCSCdRdR RdRSECESCE σσρ ⋅⋅+⋅=⋅ ,][][][ (4.9)

where [ ]RCE and RCσ are the expected value and standard deviation of CR, [ ]dSE and

dSσ are the expected value and standard deviation of Sd, while dSRC ,ρ is the correlation

coefficient between the CR and Sd.

In addition, the relationship between the expected value and the standard deviation of a

random variable is given by the coefficient of variation (COV). Therefore, the standard

deviation of CR and Sd can be defined as follows:

[ ]RCRRC COVCE ⋅=σ (4.10)

[ ]dSddS COVSE ⋅=σ (4.11)

where RCCOV and

dSCOV are the COV of CR and Sd, respectively. Thus, substituting (4.10)

and (4.11) into (4.9) yields:

[ ] ][][][][][, dRSCSCdRdRi SECECOVCOVSECESCEE

dRdR⋅⋅⋅⋅+⋅=⋅=∆ ρ

(4.12)

[ ] ( )dRdR SCSCdRdRi COVCOVSECESCEE ⋅⋅+⋅⋅=⋅=∆

,1][][][ ρ (4.12a)

However, if there is lack of correlation between CR and Sd (i.e., )0, =dR SCρ , the expected

value of i∆ can be simply estimated as the product of two independent random variables:

[ ] [ ]dRi SECEE ⋅=∆ ][ (4.13)

A common way to evaluate the error in the estimation of maximum inelastic displacement

is through the relative error between the exact and the approximate parameters. The relative

error in the evaluation of the expected value of the inelastic displacement produced by


117

neglecting the correlation between the inelastic displacement ratio and the spectral

displacement can be measured by:

1][][

−∆∆

=ei

ai

EE

ε (4.14)

where aiE ][∆ and eiE ][∆ are the approximate and exact expected value of ∆i, respectively

given by equations (4.12) and (4.13). Then, substituting equations (4.12) and (4.13) into (4.14)

yields:

dRdR

dRdR

SCSC

SCSC

COVCOV

COVCOV

⋅⋅+

⋅⋅−=

,

,

1 ρ

ρε (4.15)

4.3.4.2 Dispersion of Sd and Correlation Between CR and Sd.

To compute the relative error, ε, given in equation (4.15) it is necessary to obtain statistical

information about the correlation between CR and Sd, as well as information on the coefficient

of variation (COV) of Sd for the range of periods of vibration and lateral strength ratios of

interest. Information related to Sd should be obtained from the site-specific hazard

curve ( )dsλ . However, this information is not readily available from the USGS (e.g.,

correlation between CR and Sd or COV of Sd). To overcome this situation, it is assumed that

the acceleration time histories contained in the ground motion ensemble used in this

investigation are representative of the seismic hazard environment (i.e., earthquake magnitude

range, distance to the source range, and soil conditions) considered in the computation of

( )dsλ . Thus, in order to evaluate equation (4.15), dR SC ,ρ and COV of Sd are obtained from

the ground motion ensemble and statistical results generated in this study.

The coefficient of variation of Sd computed from ground motions collected in each firm

site condition is illustrated in figure 4.9. It can be seen that despite the differences in ground

motion characteristics, COV of Sd follows a similar trend regardless of local firm site


118

condition. It can be also observed that dispersion of Sd tends to increase as the period of

vibration increases.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV Sd

Site Class AB

Site Class C

Site Class D

Figure 4.9. COV of Sd computed from all 240 ground motions recorded in NEHRP site classes AB, C

and D.

In addition, dR SC ,ρ calculated for each site class is shown in figure 4.10. In general, it can

be seen that dR SC ,ρ is not constant over the whole spectral period and that CR is negatively

correlated with Sd, regardless of the firm soil condition. The level of correlation depends on

the period of vibration, level of lateral strength ratio and type of firm soil condition. In

particular, it can be observed that dSRC ,ρ is higher for site class D than that for site classes AB

and C. A similar plot of dSRC ,ρ computed from all 240 ground motions recorded in firm sites is

shown in figure 4.11. Similarly to the previous figure, it can be seen that CR and Sd are

negatively correlated (e.g., CR decreases as Sd increases) over the whole range of periods. In

general, for periods longer than 0.5s the correlation between CR and Sd is smaller than 25%,

which means that CR and Sd are weakly correlated. It should be noted that the level of

correlation is not significantly influenced by the level of lateral strength ratio for periods of

vibration shorter than 1.0 s. However, small influence of the lateral strength ratio on dSRC ,ρ is

observed for periods of vibration longer than 1.0 s, increasing as the lateral strength ratio

increases. It should be mentioned that since dSRC ,ρ is, in general, negative equation (4.13)


119

leads to larger estimates of i∆ than equation (4.12), which mean that the proposed approach is

conservative from an engineering point of view.

(b) SITE CLASS C

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

ρ (CR, Sd)

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(a) SITE CLASSES AB

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

ρ (CR, Sd)

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(c) SITE CLASS D

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

ρ (CR, Sd)

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 4.10. Correlation between CR and Sd computed for ground motions corresponding to NEHRP site

classes: (a) Site class AB; (b) site class C; (c) site class D.

CR, Sd

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

ρ

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 4.11. Correlation between CR and Sd computed from all 240 ground motions recorded in NEHRP

site classes AB, C and D.


120

4.3.4.3 Evaluation of relative error

The relative error evaluated for each site class is illustrated in figure 4.12. It should be noted

that the relative error provides information to evaluate how much larger is the approximate

estimation of ∆i compared to the exact estimation of ∆i. In general, it can be seen that the

relative error is not constant over the whole period region regardless of the firm soil condition.

The level of relative error for a given period of vibration and for each firm soil condition

depends on the lateral strength ratio. In particular, it can be observed that the relative error can

be very large for very short periods of vibration (e.g., T < 0.2 s) when earthquake ground

motions recorded in site classes AB and C are considered. Relative errors for periods of

vibration longer than about 0.2 s are smaller than 15%. However, the use of ground motions

collected in site class D leads to larger relative errors than those computed for site classes AB

and C for periods of vibration longer than about 1.0 s. It should be noted that larger relative

errors are computed for site class D since the COV of CR is larger when records captured in

this site class are employed.

(a) SITE CLASSES AB

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

|ε|

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(b) SITE CLASS C

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

|ε|

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(c) SITE CLASS D

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

|ε|

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 4.12. Calculated absolute relative error by assuming lack of correlation between CR and Sd for

different site classes: (a) Site classes AB; (b) site class C; (c) site class D.


121

The relative error computed using all 240 earthquake ground motions is showed in figure

4.13. As was observed before, the level of relative error depends on both the period of

vibration and the lateral strength ratio. For a given period of vibration, the relative error

decrease increases nonlinearly as the lateral strength ratio increases (e.g., is greater for weaker

structures than for strong structures relative to the ground motion intensity). It should be

mentioned that with exception of very short periods of vibration (e.g., T < 0.2 s), the relative

error is smaller than about 16%. Most structures have periods of vibration longer than 0.2 s,

therefore, it can be concluded that the relative error inherent when neglecting the correlation

between CR and Sd is small and, thus, CR and Sd can be treated as statistically independent

random variables.

SITE CLASSES AB, C, D

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

|ε|

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 4.13. Calculated absolute relative error by assuming lack of correlation between CR and Sd for all

NERP site classes AB, C, D.

4.3.5 Cumulative Distribution Function of CR

An explicit consideration of the uncertainty involved in the estimation of inelastic demands for

structures subjected to earthquake ground shaking through a probabilistic framework requires

the characterization of the observed conditional probability of exceeding a given demand level

of interest (i.e., empirical cumulative distribution) An early attempt to characterize the

empirical distribution of inelastic displacement demands of SDOF systems was conducted by

McGuire (1974). The author observed that the empirical cumulative distribution of inelastic


122

displacement ordinates corresponding to 0.9 and 0.99 fractiles is reasonably adjusted to a

parametric lognormal cumulative distribution function. Another example of characterizing

sample distribution of inelastic structural demands was conducted by Miranda (1993). The

author studied the empirical distribution of inelastic strength demands obtained from statistical

results of nonlinear SDOF systems undergoing constant ductility demands when subjected to a

set of 124 earthquake ground motions. The author found that parametric probability

distributions such as lognormal, gamma, Gumbel type I and Weibull were adequate to

represent the cumulative probability distribution function (CDF) of inelastic strength

demands.

4.3.5.1 Empirical distribution of CR

The empirical distribution of CR was obtained using the computed CR, for a given period of

vibration and lateral strength ratio, from all 240 earthquake ground motions as a random

sample and assuming each CR value as an independent outcome. Next, all 240 CR observations

were sorted in ascending order and each observation, i, was assigned a probability equal

to )1( +ni , where n corresponds to 240 observations.

(a)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0CR

P [

CR >

cR |

T=t

, R =

2]

T = 0.2 s

T = 0.5 s

T = 1.0 s

T = 2.0 s

T = 3.0 s

(b)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0CR

P [C

R >

cR |

T =

t, R

=4]

T = 0.2 s

T = 0.5 s

T = 1.0 s

T = 2.0 s

T = 3.0 s

(c)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0CR

P [

CR

> c

R |

T =

t, R

= 6

]

T = 0.2 s

T = 0.5 s

T = 1.0 s

T = 2.0 s

T = 3.0 s

Figure 4.14. Empirical cumulative distributions of CR as a function of period of vibration for:

(a) for R= 2.0; (b) for R = 4.0; and (c) for R = 6.0.


123

The empirical distribution of CR as a function of period of vibration is shown in figure

4.14 for three levels of relative lateral strength. In general, it can be observed that the

empirical distribution is not symmetric around median values and it has longer tails moving

towards upper exceedance probabilities, which is especially true for short-period systems.

Another general observation is that the empirical distribution of CR increases as the period of

vibration decreases and as the level of lateral strength ratio increases (as the ground motion

intensity increases relative to the lateral strength of the system). Particularly, it can be seen

that the empirical distribution of CR is very similar for SDOF systems with medium- and long-

period of vibration (e.g., T ≥ 1.0 s) regardless of the level of relative lateral strength, and

period of vibration.

Empirical probability distributions of CR for a short- and long-period system as a function

of three levels of lateral strength ratio is shown in figure 4.15. It can be seen that the empirical

distribution of CR for each level of lateral strength is very different for the short-period system,

even for very low exceedance probabilities, which means that for a given level of CR weaker

systems relative to the intensity of the ground motion (e.g., R=6) have higher exceedance

probability. Unlike the short-period system, the empirical distribution of the long-period

system, at least in the inter-quartile range, for the three levels of lateral strength. Finally, the

effects of site conditions and of hysteretic behavior on empirical probability distributions of

CR for short and long period system are shown in figures 4.16 and 4.17. It can be seen that

none of this two variables cause a significant change in the probability distribution.

(a) T = 0.5 s

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

CR

P [C

R >

cR |

R =

r]

R = 2.0

R = 4.0

R = 6.0

(b) T = 2.0 s

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0CR

P [C

R >

cR |

R =

r]

R = 2.0

R = 4.0

R = 6.0

Figure 4.15. Empirical cumulative distribution of CR as a function of lateral strength ratio for:

(a) T= 0.5s; and (b) T = 2.0 s.


124

(a) T = 0.2 s

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

CR

P[C

R |

T=0

.2 s

, R =

6]

SITE CLASSES AB

SITE CLASS C

SITE CLASS D

(b) T = 2.0 s

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0CR

P[C

R |

T =

2.0

s, R

= 6

]

SITE CLASSES AB

SITE CLASS C

SITE CLASS D

Figure 4.16. Effect of site conditions on empirical cumulative distributions of CR:

(a) T=0.2s; and (b) T=2.0s.

(a) T = 0.5 s

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 2.0 4.0 6.0 8.0 10.0CR

P [

CR |

T=0

.5 s

, R =

6]

MC

TK

O-O

(b) T = 2.0 s

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0CR

P [

CR

| T=2

.0 s

, R =

6]MC

TK

O-O

Figure 4.17. Effect of hysteretic behavior on empirical cumulative distribution of CR:

(a) T=0.5s; and (b) T=2.0s.

4.3.5.2 Parametric probability distribution of CR

In the previous section it was observed that the empirical distribution of CR follows a skewed

distribution with longer tails moving toward upper values. Then, parametric probability

distributions such as lognormal, Gamma, Gumbel, Weibull, or Rayleigh might be adequate to

represent the observed sample distribution (Benjamin and Cornell, 1970). In this investigation,

three parametric CDF’s were evaluated to determine if they can characterize the empirical

distribution of CR: a) Lognormal; b) Weibull; and c) Rayleigh. These probability distributions

were chosen since they have the convenience over other probability distributions that can be

fully defined from two parameters (lognormal and Weibull) or only one parameter (Rayleigh).

In particular, the lognormal distribution was chosen as primary candidate since it includes

explicitly the central tendency and the dispersion, or spread, of the sample distribution. In


125

addition, it has been employed successfully by other researchers to characterize the sample

distribution of other seismic demands. The selected CDF’s are defined as follows:

Lognormal:

−Φ=

X

XX

xxF

ln

ln)ln()(

σµ

(4.16)

where Φ is the standard normal CDF while Xlnµ and Xlnσ are the central tendency and

dispersion of the probability distribution

Weibull:

( )βα xxFX ⋅−−= exp1)( (4.17)

Rayleigh:

( )2exp1)( xxFX ⋅−−= α (4.18)

where α and β are fitted parameters that can be found from the sample data. It should be noted

that the Rayleigh probability distribution is a special case of the Weibull distribution with β =

2.

To verify whether the candidate probability distributions are adequate to characterize the

empirical probability distribution of CR, the Kolmogorov-Smirnov (K-S) goodness-of-fit test

was used in this investigation (Benjamin and Cornell, 1970).

In general, it was found that the Weibull probability distribution provided the best fit to

the empirical distribution. For example, a comparison of the empirical distribution of CR,

obtained from a short-period (T=0.5 s) and long-period (T=2.0 s) system, with respect to

Lognormal (using counted median and σln X as statistical parameters), Weibull and Rayleigh

fitted distributions is illustrated in figures 4.18 and 4.19. corresponding to three different

levels of relative lateral strength. It can be seen that the Weibull distribution provides the best

fit for both systems followed by the lognormal distribution. The Rayleigh distribution does not

provide a good fit to the empirical distribution, especially for low levels of lateral strength

ratio.


126

In spite of adequately representing the empirical distribution of CR, the Weibull

distribution has the disadvantage that finding its parameters for each period of vibration and

each lateral strength ratio requires additional statistical studies, where as for the lognormal

distribution this parameters are readily available from results presented in Chapter 2 (i.e.,

central tendency and dispersion). Therefore, it was decided to use the lognormal distribution to

characterize the empirical probability distribution of CR can be evaluated as follows:

[ ]

−Φ−=>

R

R

C

CRRR

cRTcCP

ln

ln)ln(1,|

σ

µ (4.19)

where Φ is the standard normal cumulative distribution function, RClnµ is the mean of the

natural logarithm of the inelastic displacement ratio and RClnσ is the standard deviation of the

natural logarithm of CR.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0CR

P [C

R |

R =

4.0

]

data, T = 0.5 sWeibull fitRayleigh fitLognormal fit

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0CR

P [C

R |

R =

2.0

]


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0CR

P [C

R |

R =

6.0

]


Figure 4.18. Comparison of parametric CDF with respect to empirical distribution of CR for a short-

period system (T=0.5 s) and three levels of lateral strength.


127

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0CR

P [C

R |

R =

4.0

]

data, T = 2.0 sWeibull fit

Rayleigh fitLognormal fit

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0CR

P [C

R |

R =

2.0

]


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0CR

P [C

R |

R =

6.0

]


Figure 4.19. Comparison of three parametric CDF with respect to empirical distribution of CR for a

long-period system (T=2.0 s) and three levels of lateral strength.

4.3.5.3 Selection of statistical parameters for the lognormal probability distribution

of CR

The lognormal probability distribution has the convenience that only two statistical parameters

are needed to describe the cumulative distribution of CR, one parameter that describes the

central value of the sample data and another parameter that characterize the dispersion or

uncertainty around the central tendency measure. It should be noted that several measures of

central tendency and dispersion can be obtained from the sample data. For example, the

sample mean, counted median and geometric mean (i.e., exp of mean of the natural logarithm

of the data, µlnX) are commonly used measures of central tendency and can be used to compute

the first parameter while the coefficient of variation, COV, or standard deviation of the natural

logarithm of the data, σln X, are typical descriptors of dispersion and can be used to compute

the second parameter. The definition of these statistical measures is reported in Appendix B. It

should be mentioned that if the empirical CDF of any seismic demand truly follows a


128

lognormal CDF, µlnX and σlnX are the logical statistical parameters to use. However, since the

lognormal probability distribution is an approximation of the empirical (true) distribution, it

would be necessary to find the best combination of statistical measures that lead to an

adequate fitting between the parametric and the empirical distribution. In consequence, there

is not a unique combination of statistical parameters that can be used for using the lognormal

probability distribution. For instance, a comparison of the empirical distribution of CR for a

short-period weak SDOF system (i.e., T=2.0s and R=2) with the lognormal CDF computed

using different combinations of commonly used statistical parameters is shown in figure 4.20.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0CR

P [C

R |

R =

2]

Data T = 0.2 s

Lognormal fit: geometric mean andstd. dev. Log dataLognormal fit: mean and COV

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0CR

P [C

R |

R =

2]

Data T = 0.2 s

Lognormal fit: median and std. dev.log dataLognormal fit: mean and std. dev.log data

Figure 4.20. Fitting of the empirical distribution of CR with different combination of statistical

parameters of the lognormal distribution for a short-period system (T=0.2 s).

It should be noted that some statistical measures might be more affected by the

presence of outliers than others (i.e., very large values that are significantly larger than the rest

of the sample), mainly measures of dispersion. This situation would lead to overestimation or

underestimation of the conditional probability of exceeding CR in certain range of values when

using lognormal distribution. Therefore, more robust statistical measures might be desirable.

As an alternative for estimating robust statistical measures to be used with the lognormal

CDF, Miranda and Aslani (2003) proposed to compute the ordinate of the origin, µ~ , and the

slope, ,~σ of the line given by:

( ) ( )pCR1~~lnln −Φ⋅+= σµ (4.20)


129

where p is the probability of the i-observation computed as ni /)5.0( − and 1−Φ is the inverse

of the normal cumulative distribution function. Alternative parameters µ~ and σ~ are computed

from conventional linear regression analysis using data as data points ln CR and

[ ]n/)5.01(1 −Φ − of the sorted data that lie in the inter-quartile range. For instance, the

lognormal fitting using µ~ and σ~ as statistical parameters is shown in figure 4.21. For

comparison purposes, lognormal fit with conventional statistical parameters (geometric mean

and standard deviation of the log (natural logarithm) of the data) is also showed in the same

figure. It can be seen that this approach, although slightly more complicated to compute than

other statistical measures, might lead to better approximation of the empirical CDF of CR. The

former approach was also tested with statistical results from systems with different periods of

vibration and lateral strength ratios with successful results.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0cR

P [

CR |

T =

0.2

s, R

= 2

]

Data T = 0.2 s

Lognormal fit: alternative parameters

Lognormal fit: geometric mean and std dev.Log data

Figure 4.21. Fitting of the empirical cumulative distribution of CR with different parameters of the

lognormal distribution for a short-period weak system (T=0.2 s, R=2.0).

Finally, in addition of investigating if a parametric distribution allows characterizing the

conditional probability of exceeding CR for a given period of vibration and a given level of

relative lateral strength, it is important to further investigate if the selected parametric

distribution is able to reproduce CR spectra for different percentiles. Therefore, inelastic

displacement ratios corresponding to five percentiles levels computed assuming lognormal

distribution are shown in figure 4.22 for three levels of relative lateral strength. Sample

geometric mean was used as a measure of central tendency while the dispersion was obtained


130

from the procedure suggested by Miranda and Aslani (2003). For comparison purposes,

counted percentiles, illustrated in figure 2.11 (Chapter 2), are also included in the plot. From

the figure, it can be seen that the lognormal distribution together with the statistical parameters

reproduces the counted percentiles very well, even for low and high probabilities of

exceedance (e.g., p=10% and 90%). Thus, it is believed that the lognormal CDF provides a

reasonable assumption to compute the conditional probability [ ]RTcCP RR ,|> , in equation

(4.7).

(a) R = 2

0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

p=90%p=70%p=50%p=30%p=10%Data

(b) R = 4

0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

p=90%p=70%p=50%p=30%p=10%

Data

(c) R=6

0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

p=90%p=70%p=50%p=30%p=10%Data

Figure 4.22. Comparison of counted percentiles and percentiles of CR assuming lognormal CDF (using statistical parameters from sample data) for: (a) R = 2; (b) R = 4; (c) R = 6.

4.4 Proposed Statistical Models to Estimate the Cumulative Distribution of CR

In this section, simplified models are proposed to estimate the statistical measures of central

tendency and dispersion of inelastic displacement ratios computed from earthquake ground

motions collected in firm site conditions. The selection of relevant predictors as well as the

functional form of the proposed models is based on the statistical studies described in Chapter


131

2. In Chapter 2 it was observed that both central tendency (e.g., mean or median) and

dispersion (e.g., coefficient of variation or standard deviation of the natural logarithm of data)

of CR depend on the period of vibration, T, and on the lateral strength ratio, R. Therefore, the

functional form of a nonlinear model should be a function of both T and R as follows

);,(~

θRTfCR = (4.21)

In this study, simplified nonlinear equations are proposed to estimate the central tendency

and dispersion of CR. It should be noted that the proposed equations can be used to estimate

any measure of central tendency (e.g., mean, median or geometric mean of CR) or dispersion

(e.g., COV of CR orRClnσ ) through adequate selection of parameter estimates.

4.4.1 Central Tendency Functional

The following functional form proposed in Section 2.7.2 (Chapter 2) is used to estimate

the central tendency of CR:

21

)1(1

~θθ T

RCR

⋅

−+= (4.22)

where θ1 and θ2 are parameters whose estimates are obtained through nonlinear regression

analysis. For instance, parameter estimates 1θ and 2θ and 95% confidence intervals obtained

for three measures of central tendency (i.e., sample mean, counted median and geometric

mean) were reported in Table 2.4 while parameter estimates to estimate mean CR obtained for

each firm soil site condition were reported in Table 2.5 (Section 2.7.2, Chapter 2). It should be

mentioned that equation (4.22) corresponds to a surface in the CR – R – T space as illustrated

in figure 4.23.


132

1.0

2.0

3.0

4.05.0

6.0

0.10.5

0.91.3

1.72.2

3.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0C R

R

PERIOD [s]

Figure 4.23. Mean of CR for elastoplastic SDOF systems estimated with equation (4.22).

4.4.2 Dispersion Functional

The following functional form is proposed to estimate any measure of dispersion of inelastic

displacement ratios (i.e., COV of CR, RClnσ , or σ~ obtained from equation 4.20):

αββ

σ ⋅

+⋅

+=)1.0(

11~21 TRC (4.23)

[ ]))1(exp(1 43 −−−⋅= Rββα (4.24)

where β1, β2, β3, β4, and β5 are coefficients that can be obtained through nonlinear regression

analysis using the same technique described in the previous section and employing the sample

dispersion data of interest. Since it was previously identified that the dispersion computed

from equation (4.20) leads to a good characterization of the empirical CDF when used as

statistical parameter for the lognormal CDF, fitted parameters were obtained from σ~ sample

data.

After performing non-linear regression analysis, fitted parameters for σ~

are: 0.7ˆ1 =β , 0.11ˆ

2 =β , 1.2ˆ3 =β , 7.0ˆ

4 −=β . A comparison between dispersion computed


133

following the procedure outlined in equation (4.20) and that estimated using equations (4.23)

and (4.24) is shown in figure 4.25.

(a) from sample data

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

σ

R = 6.0

R = 5.0R = 4.0

R = 3.0R = 2.0

R = 1.5

(b) from functional model

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

σ

R = 6.0

R = 5.0R = 4.0

R = 3.0R = 2.0

R = 1.5

Figure 4.24. Dispersion of CR for elastoplastic SDOF systems: (a) Dispersion computed from equation

(5.20) and σ~ sample data; (b) dispersion estimated with equations (4.23) and (4.24).

It can be seen that the proposed equation capture reasonably well the main trends,

including the saturation of dispersion as the lateral relative strength increases. Finally, it

should be mentioned that equation (4.23) also represents a surface in the RCσ~ – R – T space.

For illustration purposes, the fitted equation using is showed in figure 4.24.

1.0

2.0

3.04.0

5.06.0

0.10.5

0.91.3

1.72.2

3.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4σ

R

PERIOD [s]

Figure 4.25. Dispersion of CR for elastoplastic SDOF systems estimated with equation (4.23) and

parameter estimates obtained from σ~ obtained from equation (4.20).


134

4.4.3 Evaluation of Proposed Functional Models to Estimate Conditional

Probability of CR

Since our objective is to compute the continuous maximum inelastic displacement hazard

curve for a wide range of periods of vibration and levels of relative lateral strength, it is of

interest to verify if the proposed simplified equations introduced in the previous section to

estimate central tendency and dispersion of CR can also provide a good fit when using both a

lognormal but not with statistical parameter but with approximate parameters obtained through

regression analyses.

Figure 4.26a shows the empirical CDF of CR for a system with T = 0.5 s and R = 4.0. The

lognormal CDF computed with sample parameters (i.e., mean of ln CR as measure of central

tendency and σ~ as parameter of dispersion) is illustrated in the same figure. It can be observed

that the lognormal CDF adequately follows the empirical CDF for this period of vibration and

level of lateral strength ratio. In addition, a comparison of the empirical CDF of CR and the

lognormal CDF computed with statistical parameters estimated by the proposed equations is

illustrated in figure 4.26b. In computing the lognormal CDF given in equation (4.15),

RClnµ was obtained from the following relationship: 2ln 5.0ln σµ −= RC C

R, where RC was

estimated from equation (4.22) using parameter estimates obtained from sample mean of CR

and the dispersion σ was estimated from equations (4.23) and (4.24) employing parameter

estimates obtained from σ~ sample data. From the figure, it can be observed that the use of

proposed equations to estimate statistical parameters of the lognormal cumulative distribution

also lead to a good agreement with respect to the empirical distribution of CR. The graphic

representation of the K-S test corresponding to a 90% confidence level is also showed in both

figures. It should be mentioned that the adequacy of using equations (4.22) and (4.23) to

compute the cumulative lognormal distribution of CR was verified for other periods of

vibration and lateral strength ratios. Therefore, it is believed that the functional forms of

equations (4.22) and (4.23), with adequate parameter estimates, provide a good way to

estimate of the central tendency and dispersion of CR.


135

(b)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0cR

P[C

R |

T=0

.5s,

R =

4]

Data, T = 0.5s

Lognormal with parameters from proposedequationsK-S test, 90% confidence

(a)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0cR

P[C

R |

T=-

.5s,

R =

4]

Data, T = 0.5s

Lognormal with parameters from sample data

K-S test, 90% confidence

Figure 4.26. Lognormal CDF: (a) using parameters from sample data; (b) using parameters from

proposed equations (4.20)-(4.24).

As mentioned in Section 2.3 (Chapter 2), one way of considering the dispersion of CR

consist on computing inelastic displacement ratios corresponding to different percentiles. CR

associated to different percentiles (i.e., percentile expresses the probability of exceeding a

specified value) computed from statistical results were illustrated in figure 2.11 (Chapter 2). In

this investigation, percentiles of CR were computed by assuming a lognormal distribution with

central tendency and dispersion estimated from equations (4.22) and (4.23). A comparison

between CR spectra corresponding to 5 different percentiles computed from the statistics of CR

and those computed from the lognormal assumption are shown in figure 4.26. From the figure,

it can be seen that a very good estimation is obtained by assuming that CR is lognormally

distributed. Furthermore, it can be observed that equations (4.22) and (4.23) yield a good

estimate of the central tendency and dispersion to estimate the conditional probability of CR.


136

(a) R = 2

0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

Datap=90%p=70%p=50%p=30%p=10%

(b) R = 4

0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

Datap=90%p=70%p=50%p=30%p=10%

( c ) R = 6

0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

CR

Datap=90%p=70%p=50%p=30%p=10%

Figure 4.27. Comparison of counted percentiles and percentiles of CR assuming lognormal CDF (using statistical parameters estimated from equations 4.22-4.24) for: (a) R = 2; (b) R = 4; (c) R = 6. 4.5 Evaluation of the Proposed Approach to Compute )( iδλ

The main goal of this investigation is the probabilistic estimation of i∆ . This goal can be

reached by generating site-specific maximum inelastic displacement hazard curves, for a given

period of vibration and relative lateral strength. This curves express mean annual frequency of

maximum inelastic displacement demand, i∆ , exceeding a specified value of iδ , )( iδλ .

Assuming a Poissonian distribution of earthquake occurrence at the specified site and for the

small values that are of interest in earthquake engineering, )( iδλ is approximately equal to the

annual probability of exceeding iδ , ][ iiP δ≥∆ . This information is of valuable interest since

uniform hazard spectra of ∆i the can be developed from a family of )( iδλ curves

corresponding to different periods of vibration and different levels of lateral strength. To


137

illustrate the proposed approach, a family of )( iδλ for a site in a region of high seismicity was

generated in this study and it is described in the following sections.

4.5.1 Spectral Displacement Seismic Hazard Curves

In order to compute the seismic hazard curve of ∆i, the seismic hazard at the designated site

should be available. The site-specific seismic hazard information usually is provided from

seismologists in terms of the mean annual frequency of exceeding peak ground acceleration

(PGA) or spectral acceleration (Sa) at various periods, computed from the elastic response of

SDOF systems having 5% of damping ratio. For example, this information is readily available

from the United States Geological Survey (USGS) for any site in the U.S. (Frankel and

Leyendecker, 2001).

Then, the seismic hazard curve of Sa for the Stanford Campus corresponding to five

periods of vibration (T = 0.2 s., 0.3 s., 0.5 s., 1.0 s., and 2.0 s.) was obtained from the USGS

(Frankel and Leyendecker, 2001). However, the cited reference only provides discrete values

of mean annual frequency of exceeding for Sa, )( asλ , while integration of equation (4.1)

requires a continuous seismic hazard curve. To overcome this situation, a fourth-order

polynomial model was found adequate to approximate the seismic hazard curve in terms of the

natural logarithm of Sa and )( aSλ to obtain a continuous function. The proposed model has

the following functional form:

4104

3103

2102101010 )()()()( aaaaa SLogSLogSLogSLogSLog ⋅+⋅+⋅+⋅+= βββββλ

(4.25)

Therefore, conventional linear regression analysis was used to obtain the parameter

estimates that provide the best fit of each seismic hazard curve corresponding to each period

of vibration.

From structural dynamics theory, it is well-established that spectral displacement Sd is

related to pseudo spectral acceleration Sa through the following relationship

ad ST

S2

2

4π= (4.26)


138

Then, the mean annual frequency of exceeding a certain displacement demand, )( dsλ , was

obtained for each of the aforementioned periods of vibration.

4.5.2 Maximum Inelastic Displacement Demand Hazard Curves

A total of 25 maximum inelastic displacement demand hazard curves corresponding to five

periods of vibration (T = 0.2, 0.3, 0.5, 1.0, and 2.0s) and five yielding strength coefficients (Cy

= 0.1, 0.2, 0.4, 0.6 and 0.8) were computed as part of this investigation by performing

numerical integration of equation (4.1). The resulting maximum inelastic displacement

demand hazard curves are showed from figure 4.28 to figure 4.32. Its linear elastic spectral

displacement seismic hazard curves counterpart is also plot for reference purposes. Maximum

inelastic seismic hazard curves allow estimating the mean annual frequency of exceeding a

threshold inelastic displacement demand which also represents approximately the probability

of exceeding a certain inelastic displacement demand. For example, for a system with T=0.5s

and Cy=0.1 (e.g., weak system) the mean annual frequency of exceeding 10 cm (4 in) is 2.88

times the mean annual frequency of exceeding the same inelastic displacement of a system

with the same period of vibration but with Cy=0.8 (e.g., strong system). Both systems are

expected to behave nonlinearly for this seismic displacement demand level.

T = 0.2 s

0.00001

0.0001

0.001

0.01

0.1

1

0.01 0.1 1 10 100 1000Maximum Inelastic Displacement, ∆ i [cm]

λ (∆i)

Cy=0.1

Cy=0.2

Cy=0.4

Cy=0.6

Cy=0.8

USGS (Elastic)

Figure 4.28. Maximum inelastic displacement demand hazard curve corresponding to T=0.2 s.


139

T = 0.3 s

0.00001

0.0001

0.001

0.01

0.1

1


λ (∆i)

Cy=0.1

Cy=0.2

Cy=0.4

Cy=0.6

Cy=0.8

USGS (Elastic)

Figure 4.29. Maximum inelastic displacement demand hazard curve corresponding to T=0.3s.

T = 0.5 s

0.00001

0.0001

0.001

0.01

0.1

1

0.01 0.1 1 10 100 1000Maximum Inelastic Displacement, ∆i [cm]

λ (∆i)

Cy=0.1

Cy=0.2

Cy=0.4

Cy=0.6

Cy=0.8

USGS (Elastic)



140

T = 1.0 s

0.00001

0.0001

0.001

0.01

0.1

1

0.01 0.1 1 10 100 1000Maximum Inelastic Displacement, ∆i [cm]

λ (∆i)

Cy=0.1

Cy=0.2

Cy=0.4

Cy=0.6

Cy=0.8

USGS (Elastic)


T = 2.0 s

0.00001

0.0001

0.001

0.01

0.1

1


λ (∆i)

Cy=0.1

Cy=0.2

Cy=0.4

Cy=0.6

Cy=0.8

USGS (Elastic)



141

λ(∆i)

0.0001

0.001

0.01

0.1

1

0.01 0.1 1 10 100 1000

Maximum Inelastic Displacement, ∆i [cm]

(a) Cy = 0.1

T = 0.2s

T = 0.3s

T = 0.5s

T = 1.0s

T = 2.0s

λ(∆i)

0.0001

0.001

0.01

0.1

1

0.01 0.1 1 10 100 1000

Maximum Inelastic Displacement, ∆i [cm]

(b) Cy = 0.4

T = 0.2s

T = 0.3s

T = 0.5s

T = 1.0s

T = 2.0s

Figure 4.33. Maximum inelastic displacement demand hazard curve as a function of period of

vibration for: (a) Cy = 0.1; and (b) Cy = 0.4.

A comparison of maximum inelastic displacement demand hazard curves as a function of

period of vibration and for two levels of Cy are showed in figure 4.33. As expected, it can be

observed that )( iδλ depends on both, the lateral strength and the period of vibration of the

systems. For example, for a long-period (T = 2.0 s) weak (Cy = 0.1) system the mean annual

frequency of exceeding 10 cm (4 in) is about 2.64, 7.28, 15.89, and 28 times the mean annual

frequency of exceeding the same inelastic displacement demand of a system with the same

strength but with periods of vibration of 1.0, 0.5, 0.3, and 0.2 seconds, respectively. On the


142

other hand, a stronger system (Cy = 0.4) with a period of vibration of 2.0 s. would experience

an inelastic displacement demand of 10 cm with a exceedance probability of about 2.36, 3.71,

5.42, and 6.57 times than that of a system with identical strength but with periods of vibration

of 1.0, 0.5, 0.3, and 0.2 seconds, respectively.

4.5.3 Uniform Hazard Spectra of Maximum Inelastic Displacement Demand

Maximum inelastic displacement demand hazard curves can be used to build uniform hazard

spectra of maximum inelastic displacement, MID-UHS. MID-UHS represents the maximum

inelastic displacement demand ordinates corresponding to the same probability of exceedance,

as a function of period of vibration and the lateral strength of the system. For example, MID-

UHS spectra corresponding to five yield strength coefficients and for 10% and 2% exceedance

probability in 50 years (e.g., 475 and 2475 years return period, respectively) are shown in

figure 4.34. As can be expected, weaker structures (i.e., with low yield strength coefficient)

are more susceptible to experience larger maximum inelastic deformation demands than

stronger structures (i.e., with high yield strength coefficient). For example, for a 10% chance

of being exceeded in 50 years, a system with T=0.5s and Cy=0.1 will experience maximum

inelastic displacement demand 2.22 times larger than that experienced by the same system but

with Cy=0.8. However, for a 2% probability of exceeding in 50 years, the same system with

Cy=0.1 would experience maximum inelastic displacement demands twice as high as a system

with the same period but with Cy=0.8.

Therefore, it should be pointed out that MID-UHS are very useful for the seismic

assessment of existing structures. In addition, this type of spectra can be used to establish

performance limit-states based on maximum inelastic displacement demands.


143

(a) 10% in 50 yrs

0

5

10

15

20

25

30

35

40

0.0 0.5 1.0 1.5 2.0 2.5

PERIOD [s]

∆ i [cm]

Cy = 0.1

Cy = 0.2Cy = 0.4

Cy = 0.6Cy = 0.8

Elastic

(b) 2% in 50 yrs

0

10

20

30

40

50

60

70

80

90

0.0 0.5 1.0 1.5 2.0 2.5PERIOD [s]

∆ i [cm]

Cy = 0.1

Cy = 0.2Cy = 0.4

Cy = 0.6Cy = 0.8

Elastic

Figure 4.34. Uniform hazard spectra of maximum inelastic displacement demand corresponding to: (a)

10% in 50 years; (b) 2% in 50 years.

4.6 Summary

A simplified probabilistic approach to estimate maximum inelastic displacement demands, i∆ ,

of single-degree-of-freedom (SDOF) systems was introduced in this chapter. The suggested

approach allows generating maximum inelastic displacement demand hazard curves, )( iδλ ,

as well as uniform hazard spectra of maximum inelastic displacement demand, MID-UHS,

corresponding to different return periods which can be used in performance-based assessment

methodologies for the evaluation of existing structures that can be represented as SDOF

systems (e.g., bridge piers). For illustration purposes, the proposed approach was evaluated for

a firm soil site in a region of high-seismicity located in California. Furthermore, the suggested


144

approach can be implemented for evaluating )( iδλ and MID-UHS in sites with different soil

conditions (e.g., soft soil sites).

The proposed simplified approach makes use of constant-strength inelastic displacement

ratios, CR, in order to estimate the maximum inelastic displacement demand, i∆ , from

maximum elastic displacement demand, Sd. Based on previous studies, it is assumed that the

dependence of CR on earthquake magnitude and distance to the earthquake source is negligible

which allow computing the expected value of CR without the need of attenuation relationships.

Furthermore, the approach assumes that CR and Sd are statistically uncorrelated random

variables, which imply that the expected values of CR only depends on the period of vibration

and the level of lateral strength ratio of the SDOF system.

Therefore, in the probabilistic estimation of i∆ outlined in equation (4.1), the evaluation

of the seismic hazard at the site can be separated from the evaluation of the conditional

cumulative distribution of CR. Thus, the proposed approach takes advantage of already

available site-specific hazard curves provided by the United States Geological Survey in the

United States (Frankel and Leyendecker, 2000).

The underlying assumptions inherent in the suggested simplified approach were evaluated

and the main observations that support them are summarized as follows:

1. It was found that earthquake magnitude does not significantly affect CR ordinates for

periods of vibration longer than about 1.0 s and with lateral strength ratios smaller than 4.

Nevertheless, some dependence of CR on magnitude was observed for short-period weak

systems (i.e., weak systems relative to the ground motion intensity). Similar observations

were found for the dependence of CR on distance to the source.

2. It was found that long-duration ground motions might lead to larger CR ordinates than

short-duration records for systems with lateral strength ratios greater than 4 in the short-

and medium-period region, which mean that strong motion duration might influence the

amplitude of CR.

3. The relative error derived by neglecting the statistical correlation between CR and Sd is not

constant over the period region under consideration. In general, the relative error

decreases as the level of lateral strength ratio decreases.


145

4. Empirical probability distribution of CR conditioned on several parameters (e.g., period of

vibration, lateral strength ratio, soil conditions and hysteretic behavior) exhibited a non-

symmetrical shape with respect to the central value, including longer tails moving towards

upper values. To characterize the empirical probability distribution of CR, it was found

that parametric probability distribution functions such as lognormal and Weibull are

adequate and provide a good representation. An improved measure of dispersion

previously proposed by Miranda and Aslani (2003) was tested and it is recommended to

be used as statistical parameter for the parametric lognormal distribution of CR.

Simplified nonlinear equations to estimate statistical parameters of CR were proposed. The

proposed equations have adequate functional forms that reproduce the observed central

tendency and dispersion trend of CR with changes in the period of vibration and the lateral

strength ratio. The use of the simplified equations to estimate statistical parameters of the

lognormal probability distribution allows computing percentiles of CR, which are in good

agreement with counted percentiles in the inter-quartile range.

___________________________________________________________________________________ Chapter 5 Maximum Residual Displacement Demands of SDOF Systems

146

Chapter 5

Residual Displacement Demands of SDOF Systems

5.1 Introduction

The evaluation of residual deformation demands has received little attention in the past. A

literature survey revealed that there have been relatively few investigations that have

systematically studied residual deformation demands of structures subjected to earthquake

excitation and most of them have focused their attention in the response of single-degree-of-

freedom (SDOF) systems (Mahin and Bertero, 1981; MacRae and Kawashima, 1997;

Kawashima et al., 1998; Borzi et al., 2002; Pampanin et al., 2002). It should be noted that

previous studies evaluated residual displacement demands from the nonlinear response of

SDOF systems experiencing constant displacement ductility levels under earthquake

excitation. However, as explained in Chapter 2, the structure’s displacement ductility capacity

is not known during the evaluation of existing structures and, thus, deformation demands

based on constant lateral strength ratio are desirable for incorporation of performance-based

assessment methodologies. In addition, while previous investigations provided information

about the parameters that may influence the amplitude of residual deformation demands,

limited information about the dispersion in the estimation of central tendencies was reported.

Furthermore, information about expected residual deformation demands in short period

structure, and their associated dispersion, is still needed.

The objective of this Chapter is to present the results of a comprehensive statistical study

aimed to estimate residual deformation demands of SDOF systems. For that purpose, two

approaches are introduced: (a) a direct approach, which consists of estimating the residual

displacement through empirical information about the ratio of residual displacement demand

at the end of the excitation to the maximum elastic displacement demand for SODF systems

with known relative strength; and (b) an indirect approach, in which the residual displacement

is computed by first estimating the maximum inelastic displacement demand and then

multiplying it by the ratio of residual displacement demand to maximum inelastic


147

displacement demand for constant-relative strength SDOF systems. For both approaches, the

effects of period of vibration, level of lateral strength, local firm site conditions, type of

earthquake ground motion are investigated. In addition, the effect of post-yield stiffness and

hysteretic behavior are also addressed. Special emphasis is given to the study of the

uncertainty in the estimation of residual displacement. Finally, simplified equations to

estimate residual deformation demands are proposed to facilitate the estimation of residual

displacements in performance-based earthquake design methodologies.

5.2 Review of Previous Studies on Residual Displacement Demands of SDOF

Systems

The purpose of this section is to provide an overview of some of the main results from

previous investigations on residual deformation demands of SDOF systems. An additional

literature review of prior studies related to multi-degree-of-freedom (MDOF) systems is

offered in Chapter 7. Pioneering observations about residual deformation demands of

elastoplastic and stiffness-degrading SDOF systems were first provided by Mahin and Bertero

(1981). They reported mean residual displacements demands of elastoplastic systems

subjected to 10 records as high as 45% of mean maximum inelastic displacement demands

with high levels of dispersion (i.e., coefficients of variation close to 1.0). Years later,

motivated from large permanent deformations observed in bridge piers after the 1995 Hyogo-

Ken-Nambu earthquake, MacRae and Kawashima (1997) performed the first systematic study

aimed to evaluate residual deformation demands. The authors studied residual deformation

demands from the response of inelastic single -degree-of-freedom (SDOF) oscillators subjected

to a set of 11 earthquake ground motions recorded in Japan. They pointed out that the post-

yield stiffness ratio (i.e., ratio of post-yield stiffness to initial elastic stiffness, α) is a

parameter that can have a significant influence on the amplitude of residual deformations. In

particular, they noted that residual deformations decreases as post-yie ld stiffness ratio

increases and that negative post-yield stiffness ratio could yield residual deformation demands

close to maximum inelastic displacement demands in P-∆ sensitive SDOF systems. In

addition, the authors mentioned that the residual displacement ratio (i.e., ratio of the maximum

residual displacement demand to the maximum inelastic displacement demand) is not

significantly influenced by earthquake magnitude, distance to the source, type of soil


148

condition, or period of vibration of the system. Based on the former analytical studies,

Kawashima and his co-workers (1998) proposed a residual displacement response spectrum to

be used in the design of new bridge piers in Japan. The suggested spectrum depends on the

post-yield stiffness ratio of the system (named bilinear factor) which is computed from the

design target ductility factor and the yield displacement of the system.

Later, Borzi et al. (2001) reported the ratio of residual displacement demand to maximum

inelastic displacement demand for SDOF systems having two levels of negative post-yield

stiffness ratio (a = -1% and -2%) and experiencing specific levels of displacement ductility

when subjected to a set of 364 earthquake ground motions. The authors confirmed the

observations made by MacRae and Kawashima (1997) that the amplitude of residual

deformation demands, with respect to maximum deformation demands, depends on the target

displacement ductility and the level of negative post-yield stiffness ratio.

More recently, Pampanin et al. (2002) studied residual deformation demands normalized

with respect to maximum inelastic displacement demands of four equivalent SDOF systems

having similar properties (i.e., first-mode period of vibration, mass and lateral strength) of four

RC framed buildings when subjected to a set of 20 earthquake ground motions scaled to two

representative levels of seismic intensity. They considered three different hysteretic behaviors

(i.e., Bilinear, Takeda and Self-Centering) with positive, zero, and negative post-yield stiffness

ratios to represent the global behavior of the RC building frames. In particular, the authors

noted that the normalized residual deformation demands and their dispersion depend on the

post-yield stiffness ratio, the seismic intensity and the hysteretic rule of the nonlinear SDOF

system. Their study was motivated from the recent emergence of innovative self-centering

structural systems (e.g., unbonded precast reinforced concrete or post-tensioned steel moment

resisting connections) that provide adequate inelastic deformation capacity while significantly

controlling or eliminating residual deformations during earthquake excitation. Within this

scope, they also suggested a methodology, in a displacement-based design format similar to

the so-called displacement coefficient method of FEMA 356 (FEMA, 2000), to incorporate

residual displacement checking in the design procedure of new frame structures or the

evaluation of exiting structures.


149

5.3 Evaluation of Residual Displacement Demands

In this Chapter two approaches are proposed to compute residual deformation demands of

SDOF systems. The approaches are referred to as direct and indirect depending on the

information that is needed to estimate the residual deformation demand. Next, a detailed

explanation of the suggested approaches is presented.

5.3.1 Direct Approach

The residual displacement ratio, Cr, is defined as the absolute value of residual displacement

demand, r∆ , measured at the end of the ground motion excitation divided by the maximum

lateral elastic displacement demand, Sd, on systems with the same mass and initial stiffness

(i.e., same period of vibration) when subjected to the same earthquake ground motion. In both

cases displacements are relative to the ground. Mathematically this is expressed as:

d

rr S

C∆

= (5.1)

In equation (5.1), r∆ is computed in SDOF systems with constant yielding strength

relative to the strength required to maintain the system elastic (i.e., constant relative strength).

The relative strength ratio, R, was defined in Section 2.2 (Chapter 2). Figure 5.1 shows an

example of displacement time histor ies of elastic and inelastic SDOF systems having T=0.5 s

and subjected to the well-known north south component of the 1940 El Centro record. For this

system, the residual displacement ratio is 1.86in/2.25in=0.83. Central tendencies and

dispersion of Cr can be computed directly from statistical studies of SDOF systems subjected

to an adequate suite of earthquake ground motions.


150

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

0 5 10 15 20 25 30 35Time [s]

∆ [ in]

Elastic

ElastoplasticC r = 0.83

Sd = 2.25 in∆r = 1.86 in

El Centro NS-1940 T = 0.5 sec, ζ = 5%, Cy =0.015

Figure 5.1. Example of direct approach computation using displacement time -history computed for the

NS component of the 1940 El Centro record.

5.3.2 Indirect Approach

Another way to estimate the residual displacement demand is through the ratio of residual

displacement demand, r∆ , to the maximum (peak) displacement demand, i∆ , as follows:

i

r∆∆

=γ (5.2)

The residual ratio, γ, is a measure of the amplitude of the residual displacement demands

at the end of the excitation compared to the maximum inelastic displacements demands of a

nonlinear constant-relative strength SDOF system under an earthquake ground motion

excitation. The residual ratio is reported in absolute value hereafter.

It should be noted that multiplying and dividing equation (5.2) by the maximum elastic

displacement demand yields a relationship between the previously defined residual

displacement ratio, Cr, and the inelastic displacement ratio, CR, defined in section 2.2 (Chapter

2) in the following form:

R

r

d

d

i

rCC

SS

=⋅∆∆

=γ (5.3)

Therefore, an estimation of Cr can be done from the product of CR and γ as follows:


151

Rr CC ⋅= γ (5.4)

The above relationship takes advantage of previously published statistical studies on CR,

such as those presented in Chapter 2, and it represents and indirect way of computing residual

deformation demands from maximum inelastic displacement demands of SDOF systems. An

example of the computation of residual ratio for the same system described in the previous

section is shown in figure 5.2. For this system, residual ratio yields 0.61 (γ=1.86/3.09=0.61),

which means that the residual deformation demand is 61% of the maximum inelastic demand.

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

0 5 10 15 20 25 30 35Time [s]

∆ [ in]

Elastic

Elastoplastic

Sd = 2.25 in∆r = 1.86 in

El Centro NS-1940 T = 0.5 sec, ζ = 5%, Cy =0.015

∆i = 3.09 in

CR = 1.37 C r = 0.83 γ = 0.61

Figure 5.2.Example of indirect approach computation using displacement time -history computed for the

NS component of the 1940 El Centro record.

5.4 Hysteretic Models Considered in This Study

There are very few studies that have investigated the influence of hysteretic behavior on

residual displacement demands of SDOF systems (Mahin and Bertero, 1981; Kawashima et

al., 1998; Borzi et al., 2001; Pampanin et al., 2002). Therefore, it is of particular interest to

further examine the effect of hysteretic behavior on the estimation of residual deformation

demands. For this purpose, two parameters in the force-deformation hysteretic characteristics

were studied: (a) the effect of positive post-yield stiffness ratio, α, in bilinear systems; and (b)

the effect of the unloading stiffness in stiffness-degrading systems.


152


In this investigation, three suites of earthquake ground motions were employed to evaluate

residual deformation demands in SDOF systems. The first set contains 240 earthquake ground

motions that were recorded in firm soil sites during various seismic events in California. The

sond set contains 100 acceleration time histories recorded on stations located on very soft soil

sites in the former bed-lake of Mexico City, which are referred to as soft-soil records

hereafter. The third set includes 44 earthquake ground motions recorded close the causative

fault denominated near-fault ground motions. A complete description of each ground motion

set was presented in Chapters 3 and 4 of this dissertation. In addition, a complete list of the

ground motions contained in each set as well as key parameters is reported in Appendix A.

5.6 Statistical Results Using Direct Approach

In this section, the main results of a comprehensive statistical study aimed at evaluating the

central tendency and dispersion of residual deformation demands through residual

displacement ratios computed from elastoplastic SDOF systems are reported. In particular, the

effects of different soil conditions, relative lateral strength, positive post-yield stiffness and

hysteretic behavior on Cr are discussed in this section. The effect of hysteretic behavior in

bilinear and stiffness-degrading systems comprises a new set of results which are described in

Section 5.7.3.

5.6.1 Central Tendenc ies for Different Firm Soil Conditions

A total of 72,000 residual inelastic displacement ratios Cr were computed for elastoplastic

systems as part of this investigation (corresponding to 240 acceleration time histories, 50

periods of vibration and 6 levels of relative strength). Then sample mean residual

displacement ratios were computed by averaging results for each period of vibration, each

lateral strength ratio, and each group of local site conditions at the recording station. Figure

5.3 shows mean residual displacement ratios, Cr, corresponding to each of the firm site

conditions considered here. It can be seen that, in general, constant-strength residual

displacement ratios show the same tendency regardless of the local site conditions. A similar


153

observation was done for inelastic displacement ratios, CR, reported in Section 2.6.1.1

(Chapter 2).

SITE CLASSES AB(mean of 80 ground motions)

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

SITE CLASS C(mean of 80 ground motions)

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

SITE CLASS D(mean of 80 ground motions)

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 5.3. Mean residual displacement ratios for NEHRP site classes AB, C and D.

In general, two spectral regions can be identified. In the first spectral region, residual

displacement ratios are characterized for being larger than 1 (e.g., for T < 0.4 s), which means

that residual displacement demands are larger than maximum elastic displacement demands,

and in the second region residual deformation demands are smaller than maximum elastic

displacement demands. The limiting period that divides the region were residual displacement

demands are, on average, greater than maximum elastic displacement demands from the

spectral region where the opposite occurs, depends on the relative lateral strength ratio and on

the local site conditions. In general, this limiting period increases as the level of lateral

strength ratio decreases (e.g., weaker structures and/or stronger ground motion intensities). For

example, for structures in a site class D, residual displacement demands become larger than


154

maximum elastic displacement demands for periods smaller than about 0.25 s for systems with

a lateral strength ratio of 3 and for periods smaller than about 0.6 s for lateral strength ratios of

6. For periods of vibration smaller than, approximately, 1.0 s the spectral ordinates of Cr

strongly depends on the period of vibration and on the lateral strength ratio. For periods of

vibration longer than about 1.0 s Cr follows a fairly constant trend, which suggest that, in this

spectral region, Cr does not depend on the period of vibration. Furthermore, in this spectral

region, the ordinates of Cr tend to saturate as the lateral strength increases.

5.6.2 Central Tendencies for all Site Classes

Sample mean and counted median of constant-strength residual displacement ratios

corresponding to all 240 ground motions considered in this investigation, regardless of the soil

conditions at the recording station, are shown in figures 5.4a and 5.4b. As expected, the same

trend of Cr is observed for both mean and median values of Cr. In general, residual

displacement ratios are strongly depended on the period of vibration and on the level of lateral

strength for periods of vibration smaller than 1.0 s while residual displacement ratios are

approximately period independent for periods of vibration longer than 1.0s. In addition, it is

confirmed that the limiting period that divides residual displacement ratios larger than one

from residual displacement ratios smaller than one depends on the lateral strength ratio.

(a) mean of 240 ground motionsSITE CLASSES AB,C,D

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(b) median of 240 ground motionsSITE CLASSES AB,C,D

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 5.4. Residual displacement ratios for all 240 earthquake ground motions recorded in NEHRP site

classes AB, C and D: (a) Mean; (b) median.


155

5.6.3 Dispersion of Cr

The evaluation of the variability of seismic demands , also often referred to as record-to-record

variability, has become an important issue in the implementation of performance-based

earthquake engineering procedures. This section reports the dispersion of residual

displacement ratios introduced in the preceding section and discusses its implications in

performance-based design.

(a) SITE CLASSES AB

0.0

0.4

0.8

1.2

1.6

2.0

2.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV Cr

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(b) SITE CLASSES C

0.0

0.4

0.8

1.2

1.6

2.0

2.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV Cr

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(c) SITE CLASSES D

0.0

0.4

0.8

1.2

1.6

2.0

2.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV Cr,EP

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 5.5. Coefficient of variation of residual displacement ratios corresponding to: (a) Site classes

AB, (b) site class C, and (c) site class D.

Similarly to the evaluation of inelastic displacement ratios, dispersion was quantified by

computing coefficients of variation (COV) of residual displacement ratios. This statistical

parameter was computed for each period of vibration, each level of relative strength, and each

set of earthquake ground motions. Coefficients of variation of Cr ratios corresponding to site

classes AB, C and D are shown in figures 5.5a, 5.5b and 5.5c, respectively. From these


156

figures, it can be observed that dispersion of Cr is larger than that reported for CR (Section

2.6.2, Chapter 2). In general, regardless of the firm soil condition, a similar trend can be

identified which is characterized by two regions. In the first region, for periods of vibration

between 0.15 s and 0.5 s, dispersion increases significantly as the period of vibration

decreases. In the second region, for periods of vibration longer than 0.5 s, dispersion does not

significantly changes as the period of vibration increases. In particular, in the short spectral

region it seems that ground motions recorded on site class C lead to larger dispersion than that

computed for site classes AB and D.

Next, dispersion of mean Cr corresponding to ground motions from all site classes

considered herein are presented in figure 5.6. It can be seen that, in general, dispersion

increases as the level of lateral strength ratio increases. Dispersion is particularly high for

short periods of vibration (e.g., T < 0.25 s) regardless of the level of lateral strength ratio. For

periods longer than about 0.5 s, for a given level of lateral strength ratio, COV does not

significantly change with variation on the period of vibration. It should be noted that

dispersion of Cr is much higher than that of CR (see Section 2.6.2), which means that the

estimation of residual deformation demands involves larger variabilities than those involved in

the estimation of maximum displacement demands. However, it should be noted that for

periods of vibration longer than about 0.5 s, the variability of Cr is less dependent on changes

of lateral strength ratio than CR.

SITE CLASSES AB,C,D

0.0

0.4

0.8

1.2

1.6

2.0

2.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV Cr

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 5.6. Coefficient of variation of residual displacement ratios for all 240 ground motions recorded

in NEHRP site classes AB, C and D.


157

Similarly to the procedure described in Section 2.6.2.2 (Chapter 2), counted percentiles

representing 5 residual displacement ratio non-exceedance probabilities Cr were evaluated for

three levels of relative lateral strength and are shown in figure 5.7. The large variability in the

estimation of Cr is clearly visible in these figures. For example, a system with period of

vibration of 1.0 s and lateral strength ratio of 4 has 10% probability that Cr exceeds

approximately 1.0 (i.e., that the residual displacement demand exceeds the peak elastic

displacement demand). In addition, there is 40% of probability (i.e., 40% that Cr will be

between the curves associated to percentiles 30% and 70%) that Cr be between 0.22 and 0.6

and 80% probability that Cr ranges between 0.07 and 1.0. In addition, for systems with R= 4

and when the period of vibration is longer than 1.0s, it can be seen that even though the

median Cr (p=50%) is approximately 0.4, there is 80% probability that Cr would be

approximately between 0.07 and 1.02, which means that in this spectral region residual

displacements could be as high as the elastic spectral displacement. For a short period system,

for example T = 0.5 s, Cr would be approximately between 0.08 and 1.57 in 80% of the cases.

(a) R = 2

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

p = 90%p = 70%p = 50%p = 30%p = 10%

(b) R = 4

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

C r

p = 90%

p = 70%

p = 50%

p = 30%

p = 10%

(c) R = 6

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

C r

p = 90%p = 70%

p = 50%p = 30%

p = 10%

Figure 5.7. Residual displacement ratios corresponding to different counted percentiles for: (a) R =2; (b) R=4; and (c) R = 6.


158

An important observation is that the limiting period that divide the region where residual

displacement demands are larger than maximum elastic displacement demands, for a given

lateral strength ratio, depends on the percentile level. For example, a system with T=1.0 s and

R=4 would have 80% of probability that the limiting period be between 0.1 s and 1.05 s.

Therefore, from the above observations it is evident that dispersion on residual displacement

ratios is significant and it should be taken into account when evaluating residual displacement

demands. A comparison of counted percentiles and those computed assuming a lognormal

distribution of Cr will be shown in Section 6.3.5.2 (Chapter 6).

5.6.4 Effect of Firm Soil Conditions

It has been reported that soil conditions do not have a significant effect in residual deformation

demands (MacRae and Kawashima, 1997). However, there is no further evidence that

confirms the former observation. Thus, it is particularly interesting to explore the influence of

firm soil conditions on the residual displacement ratios. For that purpose, the same sets of

earthquake ground motions representative of site classes AB, C and D, which were described

in Section 2.4 (Chapter 2), were employed to compute Cr corresponding to each ground

motion set. Then, mean residual displacement ratios for site classes AB, site class C and site

class D normalized by mean residual displacement ratios computed considering all 240 ground

motions from the three site classes were computed and are shown in figures 5.8a, 5.8b and

5.8c, respectively.


159


0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr,D/Cr,ABCD

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5


0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr,C/Cr,ABCD

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(a) SITE CLASSES AB(mean of 240 ground motions)

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr,AB /Cr,ABCD

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 5.8. Mean residual displacement ratios of each group normalized by mean ratios from all ground

motions: (a) Site classes AB; (b) site class C; (c) site class D.

In general, it can be seen that the level of overestimation produced by neglecting the effect

of site conditions (e.g., Cr computed for a given site class is smaller than Cr computed from all

240 ground motions regardless of the soil conditions) or underestimation (e.g., Cr computed

for a given site class is larger than Cr computed from all 240 ground motions regardless of the

soil conditions) depends on the level of relative lateral strength and on the spectral region.

However, for each site class, a clear trend of underestimation or overestimation of Cr,

depending on lateral strength ratio and period of vibration, can not be identified. In general, it

can be observed that if the effect of soil conditions are neglected in the estimation of mean Cr

for structures built on rock site conditions (site classes AB) and mean Cr computed from all

240 ground motions are used instead, it would tend, in general, to overestimate residual

displacement demands for systems with period of vibration shorter than 1.5 s. For structures

placed on site class C with exception of systems with very short period of vibration, the use of

mean Cr computed from the whole earthquake ground motion set would generally produce an


160

overestimation of residual deformation demands on the order of 15%. In addition, for

structures built on site class C the use of mean Cr computed from all site classes considered

here would produce, on average, underestimations of residual deformation demands.

From the observations made in Chapter 2, it was found that local firm soil conditions have

a relatively small effect on the ordinates of CR. However, from the above observations, it is

concluded that that the effect of firm soil conditions on Cr is, in general, larger than that

observed in CR and, it is recomended that this effect be considered while estimating residual

deformation demands.


In previous sections it was noted that Cr depends on both the period of vibration and the level

of relative lateral strength. Therefore, a further look at the influence of the lateral strength ratio

on residual displacement ratios is desirable and it is presented in this section. Residual

displacement ratios plotted as a function of lateral strength ratio and corresponding to three

periods of vibration are shown in figure 5.9. For a short period system (e.g., T=0.5 s), it can be

seen that median Cr (p=50%) increases at a nonlinear rate as R increases. This observation

holds true also for low and high probabilities of exceeding Cr (p=90 and 10%, respectively). In

addition, it can be seen that for longer periods of vibration (e.g., T=1.0 and T=2.0 s), median

values of Cr do not significantly increase as the lateral strength ratio increases, suggesting that

for intermediate and long periods, and similarly to CR, the central tendency of Cr is not

strongly affected by the relative lateral strength of the system. In addition, it can also be seen

that the variability in the estimation of Cr slightly increases as R increases (see figure 5.6) .

Another observation, particularly true for T = 0.5 s, is that the 10% and 90% counted

percentiles are not symmetrically positioned with respect to the median (p=50%) suggesting

that the probability distribution of Cr is right-skewed with respect to its central tendency. This

observation can be explained since Cr values are always positive.


161

(a) T = 0.5s

0.0

1.0

2.0

3.0

4.0


Cr

p = 90%

p = 50%

p = 10%

(b) T = 1.0s

0.0

1.0

2.0

3.0

4.0

1.0 2.0 3.0 4.0 5.0 6.0 7.0


Cr

p = 90%

p = 50%p = 10%

(c) T = 2.0s

0.0

1.0

2.0

3.0

4.0

1.0 2.0 3.0 4.0 5.0 6.0 7.0


Cr

p = 90%

p = 50%

p = 10%

Figure 5.9. Effect of lateral strength ratio on residual displacement ratios: (a) T=0.5s; (b) T=1.0s; and

(c) T=2.0s.

5.6.6 Effect of the Frequency Content (Type of Ground Motion )

The effect of the frequency content on the evaluation of residual deformation demands has not

been systematically assessed in previous studies. In particular, earthquake ground motions

recorded in stations located on soft soil deposits, which are characterized by narrow-band

motions with low frequency content, or acceleration time histories recorded near the causative

fault with forward directivity effects, which have showed la rge velocity pulses, may influence

residual deformation demands. Therefore, it is important to evaluate the effect of this type of

ground motions on residual displacement ratios and, thus, in residual deformation demands. In

order to evaluate the effect of the frequency content on the estimation of residual displacement

ratios and, in particular, on residual deformation demands , four suites of earthquake ground

motions were considered in this investigation. The first two set comprises ground motions

recorded on soft soil conditions (i.e., 100 acceleration time histories collected in the former

bed-lake of Mexico City and 18 ground motions recorded at mud sites in the San Francisco


162

Bay Area). The third set consists of 40 ground shaking records obtained near the causative

fault (fault-normal near-fault records) and the final set includes all 240 ground motions

collected on rock or firm soil site conditions, which has been used in previous discussions.

The aforementioned sets of ground motions were already described in Chapters 2 and 3.

Residual displacement ratios for elastoplastic SDOF systems were computed for each

ground motion set and statistically organized as described in Section 5.6.1. Mean residual

displacement ratios corresponding to soft-soil sites (San Francisco Bay Area and Mexico

City) , near-fault sites and firm soil sites (it is repeated for comparison purposes) are illustrated

in figures 5.10a, 5.10b, 5.10c and 5.10d, respectively. It should be noted that Cr ratios for soft-

soil sites were obtained for normalized periods of vibration with respect to the predominant

period of the ground motion while Cr ratios for near-fault sites were obtained by normalizing

the periods of vibration with respect to the pulse period as discussed in Section 3.6.7 (Chapter

3).

(b) Mexico City Soft Soil Set(mean of 100 ground motions)

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

Cr

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(d) Site Classes AB,C,D(mean of 240 ground motions)

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

C r

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(a) San Francisco bay Area Set (mean of 18 ground motions)

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

Cr

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(c) Near-Fault set(mean of 40 ground motions)

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tp

C r

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 5.10. Mean residual displacement ratios computed from: (a) San Francisco Area set; (b) Mexico

City set; (c) near-fault set; and (d) firm soil set (site classes AB,C,D).


163

It can be seen that in spite of the important differences in frequency content and

bandwidth (e.g., soft soil records collected in Mexico City have narrower bandwidth and

shorter frequency content that those recorded in the bay area of San Francisco), Cr ratios for

both soft soil sites follow a similar trend. Moreover, there is also a similarity with their CR

counterparts (see figure 3.7, Chapter 3). However, Cr ratios for soft-soil conditions, which are

characterized by low-frequency narrow band signals, exhibit significantly different trend than

that observed for Cr ratios computed from earthquake ground motions recorded in firm site

conditions. Inelastic displacement ratios computed for near-fault ground motions exhibits

some similarities to the Cr for soft-soil conditions.

Despite the large differences in frequency content, Cr ratios computed with records these

four ground motion sets, two characteristic regions. In the first region, Cr tends to be larger

than 1 (e.g., residual deformation demands are larger than maximum elastic displacement

demands) while in the second region Cr tends to be smaller than 1 (e.g., residual deformation

demands are smaller than maximum elastic displacement demands). In the former region, it

can be seen that Cr significantly decreases when the period of vibration is close, or equal, to

the predominant period of the ground motion of soft soil records. A slightly similar trend can

be identified for Cr ratios computed with near-fault ground motions. Finally, it should be noted

that the limiting normalized period where residual deformation demands are larger than the

maximum elastic displacement demand depends on the level of relative lateral strength and the

type of earthquake ground motion. Therefore, it can be concluded that the type of ground

motion has significant effect on the ordinates of Cr and, thus, in the evaluation of residual


In addition to computing central tendency of Cr from different type of ground motions, the

dispersion associated in the estimation of Cr was computed and is illustrated in figure 5.11. It

can be seen that dispersion of residual displacement ratios computed from ground motions

recorded on soft-soil sites is very different than that of near-fault ground motions for

normalized periods smaller than 1.0. It can be observed that dispersion of Cr computed from

near-fault ground motions significantly increases for periods of vibration smaller than the

pulse period. In general, it can be concluded that the variability in estimating Cr is also higher

than that reported for CR when soft-soil and near-fault records are considered.


164

(d) Site Clasess AB,C,D set

0.0

0.4

0.8

1.2

1.6

2.0

2.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV Cr

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(a) San Francisco Bay Area

0.0

0.4

0.8

1.2

1.6

2.0

2.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

COV Cr

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(b) Mexico City Soft Soil Set

0.0

0.4

0.8

1.2

1.6

2.0

2.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

COV C r

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(c) Near-Fault Set

0.0

0.4

0.8

1.2

1.6

2.0

2.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tp

COV C r

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 5.11. Coefficients of variation of Cr corresponding to: (a) Soft-soil records in the San Francisco

Bay Area; (b) soft soil records in Mexico City; (c) near-fault records; and (d) firm-soil records.

5.6.6.1 Effect of pulse period on Cr

In Chapter 3, it was identified that the pulse period associated to the velocity pulse of

near-fault ground motions has significant effect on the ordinates of inelastic displacement

ratios. Then, it is also interesting to review if the pulse period also influences the ordinates of

residual displacement ratios and, more specifically on the amplitude of residual deformation

demands in SDOF systems. For that purpose, three sets containing 12 near-fault ground

motions with three different ranges of pulse periods were assembled. In this Section, median

values are reported since this statistical measure is less influenced by outliers and it provides a

better measure of central tendency for small samples. Then, a comparison of median Cr

computed from ground motions in three ranges of pulse periods for lateral strength ratios

equal to two, four and six are shown in figure 5.12. It can be seen that for R=2, pulse period

has a significant effect on Cr for short normalized periods of vibration with respect to the pulse


165

period (e.g., T/Tp < 0.5). Furthermore, the influence of pulse period increases as the system

becomes weaker relative to the intensity of the ground motions (i.e., for higher values of R). In

particular, it can be seen that Cr computed for near-fault ground motions having pulse periods

smaller than 1.0 s tend to be higher than Cr computed for ground motions with longer pulse

periods for T/Tp ratios smaller than about 1. It should be noted that the smallest difference in

Cr ordinates occurs when the period of vibration is close, or equal, to the pulse period of the

ground motions.

It is concluded that the pulse period of the velocity pulse present in near-fault type ground

motions has an important effect on Cr ordinates and, thus, in residual deformation demands.

Therefore, pulse periods should be adequately characterized while estimating residual


(a) R = 2

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

Cr

Tp < 1.0 sec. 1.0 sec. < Tp < 2.0 sec. Tp > 2.0 sec. median

(b) R = 4

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

C r

Tp < 1.0 sec. 1.0 sec. < Tp < 2.0 sec. Tp > 2.0 sec. median

(c) R = 6

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

Tp < 1.0 sec 1.0 sec < Tp < 2.0 sec.

Tp > 2.0 sec. median

Figure 5.12. Effect of the pulse period of near-fault ground motions on median Cr for different levels of

lateral strength ratio: (a) R=2; (b) R = 4; and (c) R = 6.


166


Past studies on SDOF systems have noted that the magnitude of residual deformation demands

is influenced by the hysteretic behavior (MacRae and Kawashima, 1997; Pampanin et al.,

2002). However, a clear trend of central tendencies and its variability has not been identified.

Therefore, in this section the effect of hysteretic behavior on Cr is studied in three parts. The

first part investigates the effect of positive post-yield stiffness on Cr using non-degrading

bilinear systems having three different positive post-yielding stiffness ratios (i.e., post-yield

stiffness normalized by initial stiffness). The second part explores the influence of unloading

stiffness on Cr using three types of stiffness-degrading SDOF systems while the third part

considers three types of stiffness- and-strength degrading SDOF systems. Relevant results are

introduced next.

5.6.7.1 Effect of positive post-yield stiffness ratio

Mean residual displacement ratios computed from bilinear systems having two different levels

of positive post-yield stiffness ratio (α = 1% and 3%) are shown in figures 5.13a and 5.13b., It

can clearly be seen that SDOF systems having positive post-yield stiffness ratio lead to

smaller residual displacement ratios and, in consequence, to smaller residual deformation

demands than those computed from systems with elastic -perfectly plastic hysteretic behavior

(see figure 6.4a). This observation coincides with previous investigations (e.g., MacRae and

Kawashima, 1997; Pampanin et al., 2002). In particular, it should be noted that, with

exception of very short periods of vibration, for a given lateral strength ratio , Cr ratios for

bilinear systems are less influenced by the period of vibration than their counterparts

computed for elastoplastic systems,. Furthermore, it seems that Cr ratios are not strongly

dependent on the level of lateral strength ratio in a wide range of spectral periods as compared

to Cr ratios computed for elastic -perfectly plastic systems, which particularly true for α=3%.

In addition, it should be noted that, for a given lateral strength ratio , Cr slightly decreases as

the positive post-yield stiffness ratio increases.


167

(b) α = 3%SITE CLASSES AB,C,D


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(a) α = 1%SITE CLASSES AB,C,D


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 5.13. Effect of post-yield stiffness ratio, α, on mean residual displacement ratios:

(a) α=1%; and (b) α=3%.

In addition of studying the influence of positive post-yield stiffness ratio in the estimation

of residual deformation demands it is also interesting to investigate if positive post-yield

stiffness ratio of bilinear systems leads to smaller dispersion (also referred as record-to-record

variability) than that observed in elastoplastic systems. Coefficients of variation (COV) of Cr

computed for two bilinear systems having post-yield stiffness ratio of 1% and 3% is shown in

figures 5.14a and 5.14b. Comparing figures 5.6 and 5.14 it can be seen that the dispersion in

the short spectral region (e.g., periods of vibration shorter than 0.5 s) of both bilinear systems

with positive post-yield stiffness ratio is much smaller than that of elastoplastic systems.

However, dispersion is just slightly smaller for both bilinear systems than that observed in

elastoplastic systems in the medium and period region (e.g., periods of vibration longer than

0.5 s). It should be noted that an increase of post-yield stiffness ratio in bilinear systems (e.g.,

from 1% to 3%) does not decrease the dispersion of residual deformation ratio significantly

over the period spectral region. Furthermore, systems having positive post-yield stiffness

exhibit large variability in Cr (e.g., COV around of 0.8).


168


0.0

0.4

0.8

1.2

1.6

2.0

2.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV Cr

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5


0.0

0.4

0.8

1.2

1.6

2.0

2.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV C r

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 5.14. Coefficients of variation of Cr computed from bilinear systems with: (a) α = 1%, (b) α =

3%.


The influence of unloading stiffness on residual displacement ratios was evaluated by

modifying the parameter HC in the modified three-parameter model (described in Section 2.5,

Chapter 2), while keeping the other parameters constant. In order to study the effect of

unloading stiffness of stiffness-degrading systems on residual displacement ratios, four values

of parameter HC were chosen in this study: 0.1, 1.5, 2.5 and ∞→ . A value of

HC ∞→ reproduces the modified-Clough model (Mahin and Bertero, 1976) while values of

2.5 and 1.5 simulate two types of Takeda model (Takeda et al., 1970). Both modified-Clough

and Takeda models have been extensively used to represent the hysteretic behavior of

reinforced concrete (RC) elements failing in a flexural mode under cyclic loading. A value of

HC = 0.1 reproduces an origin-oriented hysteretic behavior. In this section, only 80 ground

motions recorded in site class AB were considered.


169

(c) HC =1.5SITE CLASSES AB


0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(b) HC = 2.5SITE CLASSES AB


0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(d) HC = 0.1SITE CLASSES AB


0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(a) HC SITE CLASSES AB


0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 5.15. Mean residual displacement ratios computed from stiffness of stiffness-degrading systems

with different unloading stiffness: (a) HC; (b) HC=2.5; (c) HC=1.5; (d) HC=0.1.

Mean residual displacement ratios computed for each type of stiffness-degrading behavior

are shown in figures 5.15a, 5.15b, 5.15c and 5.15d, respectively. It can be observed that Cr for

stiffness-degrading systems holds the same trend previously observed and described for

elastoplastic systems. However, it should be noted that mean Cr ordinates for stiffness-

degrading systems are smaller than mean Cr ordinates for elastoplastic systems, which means

that residual displacement demands for stiffness-degrading systems are, on average, smaller

than their counterparts computed from elastoplastic systems. This observation can be

explained since the hysteresis loops of stiffness-degrading systems of the type considered in

this study tends to re-center to the origin during cyclic excitation while systems with

elastoplastic behavior do not return to the origin once large excitations have taken place,

therefore leading to larger residual deformation demands a the end of the earthquake ground

motion shaking. For illustration purposes, figure 5.16 shows a comparison of the nonlinear


170

response of a SDOF system subjected to the component 360 of the record collected in station

54399 during the 1994 Northridge earthquake considering three types of hysteretic behavior.

Elastoplastic

-20

-15

-10

-5

0

5

10

15

20

-2.0 -1.0 0.0 1.0 2.0Displacement [cm]

For

ce [k

g]

Origin-oriented model

-20

-15

-10

-5

0

5

10

15

20

-2.0 -1.0 0.0 1.0 2.0Displacement [cm]

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 2 4 6 8 10 12 14 16 18 20Time [s]

Dis

plac

emen

t [c

m]

Elastoplastic

Modified-Clough

Origin-oriented

Modified-Clough model

-20

-15

-10

-5

0

5

10

15

20

-2.0 -1.0 0.0 1.0 2.0Displacement [cm]

Figure 5.16. Comparison of hysteretic response and displacement time -history from a SDOF system

(T=1.0s, ζ =5%, Cy=0.015) subjected to the records collected in station 24399 comp. 360 (1994 Northridge earthquake) considering three types of hysteretic behavior.

It should be mentioned that the aforementioned observation has also been reported by

other researchers (Pampanin et al. , 2002; Luco et al., 2004). Then, it is expected that structural

members and systems exhibiting wide hysteresis loops (e.g., steel buckling restrained braces,

eccentric brace frames) during seismic excitation could experience larger residual deformation

demands than structures suffering loading and unloading stiffness degradation during strong

ground shaking.

Another observation is that, depending on the lateral strength ratio, Cr decreases as the

unloading stiffness decreases for systems with periods of vibration smaller than about 1.0 s.

However, for periods longer than 1.5 s, Cr, in general, does not significantly increases as the

lateral strength ratio increases. The former observation is valid for all levels of unloading

stiffness.


171

Besides studying the variation in central tendency of residual displacement ratios

corresponding to different hysteretic behaviors, it is important to examine the corresponding

dispersion. Coefficients of variation of residual displacement ratios for each stiffness-

degrading hysteretic behavior are shown in figures 5.17a, 5.17b, 5.17c and 5.17d.


0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV C r

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5


0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV C r

R = 6.0R = 5.0R = 4.0R = 3.0

R = 2.0R = 1.5

(c) HC = 1.5SITE CLASSES AB

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV Cr

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5


0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV C r

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 5.17. Coefficients of variation of residual displacement ratios corresponding to different stiffness-degrading systems: (a) HC ∞→ ; (b) HC=2.5; (c) HC=1.5; (d) HC=0.1.

The first general observation is that the dispersion of Cr computed for systems having

stiffness-degrading hysteretic characteristics is larger than that associated to elastoplastic

systems in the short period region (e.g., periods of vibration shorter than about 0.5 s), but it is

similar the medium- and long-period spectral region if the system does not exhibit severe

unloading stiffness (e.g., Modified-Clough- or Takeda-type hysteretic behavior). Examining

the figures, it can be seen that the variability in the estimation of Cr is larger for systems


172

having origin-oriented hysteretic properties (e.g., unloading hysteretic branches which tend to

unload towards the origin) than for systems with unloading stiffness equal to or similar to the

elastic stiffness (e.g., Modified-Clough- or Takeda-type hysteretic behaviors), which means

that dispersion tends to increase as the unloading stiffness reduces. It should also be

mentioned that the influence of lateral strength ratio on dispersion tends to increase, in

general, as the unloading stiffness decreases.

5.7 Statistical Results using the Indirect Approach

As was introduced in Section 5.2.1, an alternative way of computing residual displacement

demands is through the relationship between the residual displacement demand and the

maximum inelastic displacement demand. This ratio is called residual ratio, γ, which also

represents the ratio between Cr and CR. In this section, results of a comprehensive statistical

study aimed at evaluating central tendencies and dispersion of residual ratios are presented. In

particular, the effects of relative lateral strength, soil conditions at the recording station,

frequency content and hysteretic behavior are presented and discussed.

5.7.1 Central Tendencies for Different Soil Conditions

Mean ratios of residual deformation demand to maximum deformation demand were

computed for elastoplastic systems subjected to each set of earthquake ground motions

collected in three different firm soil sites and are shown in figures 5.18a, 5.18b and 5.18c. For

the sake of simplicity, this ratio is named residual ratio hereafter. The first observation is that,

with exception of very short periods, the residual ratio does not significantly changes with

variations of the period of vibration for a given level of lateral strength. In addition, for a

given period of vibration, the residual ratio tends to increase as the level of lateral strength

ratio increases, which means that residual deformation demands tend to be larger with respect

to maximum inelastic deformation demands as the system becomes weaker relative to the

intensity of the ground motion, regardless of the soil conditions. However, residual ratios tend

to saturate as the level of lateral strength increases over the whole spectral region. It should be

noted that weak systems (e.g., R greater than or equal than 4) with elastoplastic hysteretic

features would sustain residual deformation demands as large as 40% of maximum


173

deformation demands. This observation has very important implications in earthquake-

resistant design since recently proposed energy-dissipation devices exhibit nearly elastic -

perfectly plastic hysteretic features and, in consequence, they could lead to large permanent

deformations.

(a) SITE CLASSES AB(mean of 80 ground motions)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆ r / ∆i

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5


0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆ r / ∆ i

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5


0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆r / ∆ i

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 5.18. Mean ratios of residual displacement demand to maximu m inelastic displacement demand

corresponding to: (a) Site classes AB; (b) site class C; and (c) site class D.

5.7.2 Central Tendencies for All Site Classes

To illustrate the central tendency of residual ratio regardless of the soil condit ions, mean and

median ratios of residual displacement demand to maximum inelastic displacement demand

computed for elastoplastic systems when subjected to all 240 ground motions are shown in

figure 5.19. As noted before, it can be seen that for a given level of relative lateral strength this

ratio exhibits a fairly constant trend over the whole spectral region considered in this study.

Then, this ratio is not strongly dependent of the period of vibration and it mainly depends on

the lateral strength ratio of the system. In general, residual displacement demands become


174

larger as the lateral strength ratio increases (e.g., weaker structures would experience larger

residual displacement demands than stronger structures). For example, residual displacement

demands can be, on average, as high as 33% and 42% of maximum inelastic displacement

demands for systems with lateral strength ratios of two and four, respectively.

SITE CLASSES AB,C,D(median of 240 ground motions)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆ r / ∆ i

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

SITE CLASSES AB,C,D(mean of 240 ground motions)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆ r / ∆ i

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 5.19. Ratios of residual displacement demand to maximum inelastic displacement demand for all

240 ground motions recorded in NEHRP site classes B, C and D: (a) Mean; (b) median.

5.7.3 Dispersion of Residual Ratio

The variability in the estimation of mean residual ratios was measured by the coefficient of

variation (COV). Coefficients of variation of residual ratios corresponding to each site class

are shown in figures 5.20a, 5.20b and 5.20c. It can be seen that COV are not significantly

influenced by the period of vibration or the level of lateral strength for systems subjected to

earthquake ground motions recorded on site classes AB and C. In these cases, and ignoring the

effect of lateral strength, COV is, on average, 0.58 over the whole spectral region. However,

more scatter of COV as a function of period of vibration and lateral strength ratio can be seen

for systems subjected to acceleration time histories recorded on site class D.


175

(c) SITE CLASS D

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV ∆r/∆ i

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(b) SITE CLASS C

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV ∆r / ∆ i

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(a) SITE CLASSES AB

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV ∆r/∆ i

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 5.20. Coefficients of variation of residual ratios corresponding to: (a) Site classes AB; (b) site

class C; (c) site class D.

In addition, dispersion of residual ratios computed using all 240 earthquake ground

motions is shown in figure 5.21. It can be observed that COV of residual ratios are not

strongly dependent on the period region or the level of lateral strength. It ranges between 0.46

and 0.71. Assuming that the level of relative lateral strength is negligible and averaging COVs

over the whole spectral region yields a COV of 0.58.

From the results presented in this section and those of Section 5.5.3, it can be concluded

that the estimation of mean residual ratios involves smaller levels of dispersion than that

associated in the estimation of mean inelastic displacement ratios. This observation might

suggest the indirect approach is more adequate to estimate residual deformation demands in a

reliability-based procedure of existing structures.


176

SITE CLASSES A,B,C,D

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV ∆r/∆ i

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 5.21. Coefficients of variation of residual ratios for all 240 ground motions recorded in NEHRP

site classes AB, C and D.

(a) R = 2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆ r / ∆i

p = 90%p = 70%p = 50%p = 30%p = 10%

(b) R = 4

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆r / ∆ i

(c) R = 6

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆ r / ∆i

Figure 5.22. Residual ratios corresponding to different counted percentiles for: (a) R =2; (b) R=4 and

(c) R = 6.


177

An alternative way of quantifying dispersion used in this dissertation has been through

counted percentiles. Thus, counted percentiles for residual ratio computed using data from all

240 ground motions and corresponding to three lateral strength ratios are shown in figures

5.22a, 5.22b and 5.22c. It can bee seen that the high level of dispersion is reflected in the

ranges of values associated to different probabilities of exceeding residual ratios. For example,

for a system with T=1.0 s and R=2 there is 40% probability that the residual ratio would be

between 0.21 and 0.41 while there is 80% probability that deformation ratio ranges between

0.05 and 0.59. For a system with the same period but with R=6, residual ratio would be

between 0.30 and 0.61in 40% of the cases while this ratio would range between 0.10 and 0.80

for the same system in 80% of the cases.

5.7.4 Effect of the Frequency Content (Type of Ground Motion)

The effect of the type of ground motion on residual ratios was also investigated. Mean residual

ratios computed for the suites of ground motions corresponding to soft-soil sites (San

Francisco Bay Area and Mexico City ground motion sets) as well as near-fault and firm sites

are shown in figures 5.23a, 5.23b, 5.23c and 5.23d, respectively. In general, it can be seen that

residual ratios for soft-soils sites and near-fault ground motions are not constant over the

whole spectral region and they depend on both the normalized period of vibration and the

level of lateral strength. Similarly to residual ratios for firm-soil sites, residual ratios for these

types of earthquake ground motions tend to saturate as the level of lateral strength increases.

Particularly for soft soil sites, it can be observed that ground motions recorded at mud

sites in the San Francisco Bay Area leads to larger residual ratios than those recorded in soft-

soil deposits of Mexico City. For both soft-soil sites, residual ratio decreases as the period of

vibration becomes close or equal to the predominant period of the ground motion. Finally,

residual ratios computed from near-fault ground motions slightly follow a similar trend than

that observed for soft soil sites.


178

(a) Mexico City Soft Soil Set(mean of 100 ground motions)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

∆ r / ∆ i

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5


0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

∆ r / ∆ i

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(c) Near-Fault Set(mean of 40 ground motions)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tp

∆ r / ∆ i

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(d) Site Classes AB,C,D(mean of 240 ground motions)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆ r / ∆ i

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 5.23. Mean ratios of residual displacement demand to maximum inelastic displacement demand computed for: (a) San Francisco Bay Area set; (b) Mexico City set; (c) Near-fault set, and (d) firm-site

set.

In addition to investigating the effect of the type of ground motion on residual ratios, it is

interesting to study if dispersion of residual ratios changes for different type of ground

motions. COVs associated to mean Cr ratios for each ground motion set is showed in figure

5.24. It can be seen that, in general, the variability in the estimation of mean Cr when using

soft-soil records is larger than that when using firm site records.


179

(d) Site Classes AB,C,D

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV ∆ r/∆ i

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(b) Mexico City Soft Soil Set

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

COV ∆ r / ∆ i

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(a) San Francisco bay Area Set

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tg

COV ∆ r/∆ i

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(c) Near-Fault Set

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tp

COV ∆ r/∆ i

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 5.24. Coefficients of variation of residual ratios corresponding to: (a) San Francisco Bay Area

set; (b) Mexico City set; (c) near-fault set; and (d) firm soil site set

The pulse period associated to the velocity pulse in fault-normal near-fault ground motions

was identified as an important parameter that influences the ordinates of residual displacement

ratios. Therefore, it is interesting to evaluate if residual ratios are also influenced by the pulse

period. For that purpose, the same subsets of near-fault ground motions mentioned in Section

5.7.2.1 were employed to compute residual ratios of elastoplastic systems corresponding to

ground motions with three ranges of pulse periods. Then, a comparison of median residual

ratios computed for ground motions set corresponding to each of the pulse period are shown in

figure 5.25. In general, with exception of very short normalized periods, residual ratios

increase as the system becomes weaker with respect to the ground motion intensity (i.e., as the

level of relative strength increases) and changes, for a given lateral strength ratio, as the

normalized period of vibration changes. It can be seen that residual ratios computed from


180

ground motions with pulse periods shorter than 1.0 s. exhibits a different trend than those

computed with ground motions having pulse periods longer than 1.0 s.

(b) 1.0 sec. < Tp < 2.0 sec.(mean of 12 ground motions)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tp

∆ r / ∆ i

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(a) Tp < 1.0 sec.(mean of 12 ground motions)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tp

∆ r / ∆ i

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(c) Tp > 2.0 sec.(mean of 12 ground motions)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0T / Tp

∆ r / ∆ i

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 5.25. Effect of pulse period on median residual ratios: (a) Tp < 1.0s; (b) 1.0s < Tp < 2.0s;

and (c) Tp >2.0s.


5.7.5.1 Effect of positive post-yield stiffness ratio

In Section 2.6.4.1 (Chapter 2), it was found that maximum inelastic displacement demands of

bilinear SDOF systems become smaller as the post-yield stiffness ratio increases. Then, it is

interesting to study the reduction of residual deformation demands with increment in positive

post-yield stiffness ratio, α, of bilinear SDOF systems. For this purpose, the ratio of Cr


181

computed for bilinear systems with three different values of post-yield stiffness ratio (α = 3%,

5%, and 10%) with respect to Cr computed for elastoplastic systems was obtained using all

240 earthquake ground motions from firm site conditions (see figure 5.26). This ratio

represents the ratio of residual displacement demands for bilinear systems to residual

displacement demands for elastoplastic systems. In general, it can be seen that ∆r for bilinear

systems decreases with respect to ∆r for elastoplastic systems as α increases. For a given post-

yield stiffness ratio, ∆r for bilinear systems decreases with respect to ∆r for elastoplastic

systems as the period of vibration becomes shorter and as the lateral strength ratio increases at

a fairly constant rate for periods of vibration longer than 0.5 s, but it decreases more abruptly

for periods of vibration shorter than about 0.5 s.



0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆α

=3%

/ ∆α

=0%

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5



0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆α

=5%

/ ∆α

=0%

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(c) α = 10%SITE CLASSES AB,C,D


0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆α

=10

%/ ∆

α=0

%

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 5.26. Effect of post-yield stiffness ratio on mean ratios of residual displacement demands of bilinear to elastoplastic systems : (a) α=3%; (b) α=5%; and (c) α=10%.


182


The unloading stiffness of stiffness-degrading systems was identified as an important

parameter that influences the magnitude of residual displacement ratios. Then, additional

information the influence of unloading stiffness on residual ratios is presented in this section.

For that purpose, the ratio of residual displacement demand computed for systems with

different unlading stiffness normalized with respect to maximum inelastic displacement

demand computed for stiffness-degrading systems having modified-Clough hysteretic

behavior is resented in this section. Similar values of unloading stiffness and the same set of

80 ground motions from site classes AB were considered in this study. Figure 5.27 shows

residual ratios computed from each stiffness-degrading system.



0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆ r / ∆ i

R = 6.0

R = 5.0R = 4.0

R = 3.0R = 2.0R = 1.5

(c) HC = 1.5SITE CLASSES AB


0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆ r / ∆ i

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5



0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆ r / ∆ i

R = 6.0R = 5.0R = 4.0R = 3.0

R = 2.0R = 1.5



0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

∆ r / ∆ i

R = 6.0

R = 5.0R = 4.0R = 3.0

R = 2.0R = 1.5

Figure 5.27. Mean ratios of residual displacement demand to maximum inelastic displacement demand

computed for different stiffness-degrading systems : (a) HC ∞→ ; (b) HC=2.5; (c) HC=1.5; (d) HC=0.1.


183

From the figure, it can be observed that residual ratios decreases as the unloading

stiffness decreases, which means that residual deformation demands decrease as the unloading

stiffness decreases. For a given lateral strength ratio, residual ratios vary depending on the

type of stiffness-degrading system and, in particular, on the unloading stiffness that is adopted.

For a given stiffness-degrading system, residual ratios increase as the lateral strength ratio

increases, but they tend to saturate as the level of lateral strength ratio becomes large. In

particular, it should be noted that the residual ratios for origin-oriented stiffness-degrading

systems (e.g., HC = 0.1) tend to increase as the period of vibration increases, which means

that flexible structures experiencing this type of behavior would sustain larger residual

deformation than their rigid counterparts.

5.8 Summary

The main purpose of this chapter was to evaluate residual deformation demands of single -

degree-of-freedom (SDOF) systems subjected to earthquake excitation. For that purpose, two

approaches to characterize residual deformation demands were introduced. The first approach,

named direct approach, consists of computing residual displacement demands as a function of

maximum elastic displacement demands. This ratio that makes use of residual displacement

ratios, Cr, that permit the estimation of maximum residual displacements demands from

maximum elastic displacements. The second approach, called indirect approach, is based on

computing the residual deformation demands as a function of maximum inelastic deformation

demand. This ratio was defined as the residual ratio that allows the estimation of maximum

residual displacements demands from maximum inelastic displacement demands. Thus, a

comprehensive statistical study of nonlinear SDOF systems with degrading and non-degrading

hysteretic behaviors having different levels of relative lateral strength when subjected to a

relatively large set of earthquake ground motions recorded on different soil site conditions

(e.g., firm and soft soil sites) and close causative faults (i.e., near-fault ground motions) was

conducted to obtain statistical information (e.g., central tendencies and dispersion) for both

approaches. The following conclusions are drawn from the results of this investigation.

a) Direct approach:

1. In the short period spectral region, residual displacement demands of systems with

constant relative strength are, on average, much larger than maximum elastic


184

displacement demands. In this spectral region the ratio of residual displacement to

maximum elastic displacement demand is strongly dependent on the period of

vibration, on the lateral strength ratio and on the type of hysteretic behavior. For

periods longer than 1.0 s, residual displacement ratios are not very sensitive to

changes in the period of vibration and they mainly depend on the lateral strength ratio.

2. Coefficients of variation of residual displacement ratios tend to increase with

increasing lateral strength ratios particularly when periods of vibration are in the short

spectral region. In general, dispersion of Cr is larger than for CR, which means that the

dispersion on residual displacement ratios is very important and larger than the

dispersion on maximum inelastic displacement demands, therefore it should be

carefully taken into account when evaluating residual displacements demands.

3. It was found that the effects of local firm site conditions on constant relative strength

residual displacement ratios are larger than those of constant relative strength inelastic

displacement ratios. In addition, it was showed that residual displacement ratios

computed from earthquake ground motions recorded on soft soil conditions or near

the causative fault exhibit a different trend than those computed from records

collected on rock or firm site conditions. Then, influence of local soil site conditions

(e.g., firm or soft soils) or the type of ground motion (e.g., near-fault or far-fault)

should be addressed when estimating residual deformation demands. In particular, the

pulse period of the velocity pulse showed in near-fault ground motions was found to

be an important parameter that influences the ordinates of residual displacement

ratios.

4. Residual displacement ratios of stiffness-degrading systems are significantly affected

by the level of unloading stiffness. It was observed that Cr ratios and, consequently,

residual deformation demands decreases as the unloading stiffness decreases in

stiffness-degrading systems.

b) Indirect approach:

1. Residual ratios obtained from ground motions compiled in firm soil sites increases as

the lateral strength ratio increases, but they are not very sensitive to changes in the

period of the ground motion. In addition, residual ratios are not significantly

influenced by local firm soil site conditions.


185

2. Residual ratios computed from soft soil records and near-fault records showed

variations with changes in the normalized period of vibration and the lateral strength

ratio of the system. They showed a different trend than that observed for residual

ratios computed for firm soil sites.

3. For residual ratios computed using near-fault ground motions, it was showed that the

pulse period influences the ordinates of residual ratios.

4. It was found that the estimation of residual ratios involves smaller variability (i.e.,

smaller dispersion) than that associated in the estimation of inelastic displacement

ratios.

5. Similarly to residual displacement ratios, the level of unloading stiffness of stiffness-

degrading systems has an important effect in the estimation of residual ratios. In

general, residual ratios decrease as the unloading-stiffness decreases.

___________________________________________________________________________________ Chapter 6 Probabilistic Estimation of Residual Displacement Demands of SDOF Systems 186

Chapter 6

Probabilistic Evaluation of Residual Displacement

Demands for SDOF Systems

6.1 Introduction

The evaluation of residual displacements plays an important role for determining the technical

and economical feasibility of retrofit techniques to be used in damaged structures due to

earthquake ground motion shaking. For example, as a consequence of large permanent

deformations, on the order of 1.75% drift, more than 100 reinforced concrete piers were

demolished after the 1995 Kobe earthquake even though they did not experienced collapse

(Kawashima, 2000). Furthermore, an adequate estimate of permanent deformations is

important to assess the seismic risk from aftershocks events, in particular for estimating the

probability of collapse during an aftershock of structures that were severely damaged during

the main event (Bazzuro et al., 2004; Luco et al., 2004). Thus, an adequate estimate of the

residual displacement demands that existing structures would experience as a result of

earthquake ground shaking should be of primary importance in performance-based

assessment. For this reason, recently published FEMA 356 document (FEMA, 2000) includes

limiting residual deformation drifts corresponding to different performance levels of structural

components. However, this document does not provide quantitative procedures to evaluate

these residual drift demands on exiting structures with known dynamic properties and lateral

strength.

Several researchers have provided valuable information about the parameters that

influence the magnitude of residual deformation demands during seismic excitation and some

of them emphasized the need to incorporate earthquake-design criteria accounting for residual

deformations in the design of new structures (Mahin and Bertero, 1981; MacRae and

Kawashima, 1997; Kawashima et al., 1998; Borzi et al., 2001; Pampanin et al., 2002; Sakai


and Mahin, 2003). However, none of them provided quantitative methods to evaluate residual

deformations of existing structures (e.g., buildings, bridge piers, etc.) subjected to earthquake

excitation with different levels of intensity. Furthermore, desirable methods to estimate

residual deformation demands of existing structures requires a probabilistic framework to

explicitly incorporate the large variability observed in the estimation of residual deformation

demands. Of particular importance for the development of a performance-based methodology

accounting for residual displacement demands is the estimation of the probability of exceeding

a certain level of residual deformation for a given seismic environment.

The major objective of this chapter is to propose a simplified probabilistic approach to

evaluate the mean annual frequency of exceedance of residual deformation demands on

particular structures located at a specific site, by means of structure-specific residual

displacement demand hazard curves. Families of residual displacement demand hazard curves

allow the derivation of uniform hazard spectra of residual displacement demand, for different

return periods, which can be used for establishing adequate residual deformation demand limit

states (e.g., limiting residual displacement demand with a probability of exceedance of 10% in

50 years) for performance-based assessment of existing structures. Uniform hazard spectra of

residual displacement demand are also very valuable tools in performance-based design of

new structures. In essence, the simplified approach introduced in this chapter is similar to that

outlined in Chapter 4 to estimate maximum inelastic displacement demand. Similarly to the

probabilistic estimation of maximum inelastic displacement demands, it should be mentioned

that the study described in this chapter is limited to estimate residual displacement demands of

structures that can be represented as SDOF systems such as bridge piers.

This chapter introduces the simplified probabilistic approach to estimate residual

displacement demands in Section 6.2. Next, Section 6.3 focuses on the validation and

limitations of the simplifying assumptions inherent in the simplified probabilistic approach.

For example, the effects of earthquake magnitude, distance to the earthquake source and

duration of the ground motion on residual deformation demands, evaluated by means of

residual displacement ratios, are specially evaluated and discussed. In Section 6.4, the

empirical probability distribution of residual deformation demands is examined and adequate

parametric probability distribution functions are proposed to characterize the observed

empirical distributions. In addition, adequate functional models to represent the central


tendency and dispersion of residual displacement demands are presented in Section 6.4. The

proposed simplified probabilistic approach is implemented to compute families of residual

displacement demand hazard curves for different periods of vibration and different lateral

strength coefficients (Section 6.5) and uniform hazard spectra of residual displacement

demand at firm soil sites (Section 6.6). Finally, a quantitative comparison of maximum

inelastic and residual displacement demand hazard curves is presented in Section 6.7.

6.2 Probabilistic Approach to Estimate Residual Displacement Demands

The main goal of the proposed simplified approach consists of computing the mean annual

frequency of residual displacement demand, r∆ , exceeding a specific residual displacement

demand, rδ , ( )rδλ ,at a specific site and conditioned on the structural properties of the

system, which can be expressed mathematically as follows:

[ ] dd

dyddrrr ds

dSsd

CTsSP)(

,;|)(0

λδδλ ⋅=>∆= ∫

∞ (6.1)

In the above equation, the first term inside the integrand [ ]yddrr CTsSP ,;| =>∆ δ is the

conditional probability of residual displacement demand, r∆ , exceeding a specific residual

displacement demand rδ , conditioned on the period of vibration of the system, T, the yield

strength coefficient, Cy of the system (i.e., the lateral yield strength of the system, Fy,

normalized by its weight), as well as the maximum elastic spectral displacement demand, Sd.

In addition, )( dsλ is the mean annual frequency of exceedance of the elastic displacement

demand evaluated at a ground motion intensity equal to sd, and dλ (Sd) denotes its differential

with respect to Sd (also evaluated at sd).

Similarly to the simplified probabilistic approach to estimate maximum inelastic

displacement demands introduced in Section 5.2 (Chapter 5), a key assumption in the

proposed procedure is that r∆ depends only on the structural properties of the system (e.g., T

and Cy) and the level of ground motion intensity (measured by Sd), and it is not affected by


other parameters such as earthquake magnitude and distance to the source. This simplifying

assumption allows us to separate the probabilistic estimation of the seismic hazard at a given

site (i.e., right-hand side of the integrand) from the probabilistic estimation of r∆ (i.e., left-

hand side of the integrand), which can be devised as an extension of conventional Probabilistic

Seismic Hazard Analysis (PSHA).In particular, this simplified assumption allows using

already available seismic hazard curves from the United States Geological Survey (USGS) for

any geographical location in the United States (Frankel and Leyendecker, 2000).

Conventional PSHA assumes that earthquake events follow a Poissonian probability

distribution with small rate of exceedance. It should be noted, based on this assumption, the

mean annual frequency of exceeding a given residual displacement demand expressed in (6.1)

is also approximately equal to the probability of exceeding a given residual displacement

demand, [ ] )( rrrP δλδ ≈>∆ in any given year.

In order to evaluate the conditional probability term in equation (6.1), it is convenient to

make use of the residual displacement ratio, Cr, previously defined in Section 5.2 (Chapter 5).

The residual displacement ratio is a non-dimensional parameter defined as the residual

displacement demand at the end of the earthquake excitation, r∆ , divided by the maximum

lateral elastic displacement demand, Sd, on systems with the same mass and initial stiffness

(i.e., same period of vibration) when subjected to the same earthquake ground motion. In both

cases, displacements are relative to the ground. Mathematically this is expressed as follows:

d

rr S

C∆

= (6.2)

In equation (6.1), residual displacement demands are computed for systems with constant

yield strength coefficient. For convenience, in this study the inelastic response was computed

for SDOF systems with constant yield strength relative to the strength required to maintain the

system elastic (i.e., constant relative strength). Here, the relative lateral strength is measured

by the strength ratio, R, which is defined as:

y

a

FSm

R

= (6.3)


where m is the mass of the system and Sa is the acceleration spectral ordinate. The numerator

in equation (6.3) represents the lateral strength required to maintain the system elastic, which

sometimes is also referred to as the elastic strength demand. Then, if information of Cr and Sd

is available for a given earthquake ground motion, the residual displacement demand can be

estimated as follows:

drr SC ⋅=∆ (6.4)

Substituting equation (6.4) to estimate the conditional probability of r∆ in equation (6.1)

permits the estimation of the mean annual frequency of exceeding a residual displacement

demand rδ as follows:

[ ] dd

dyddrrr ds

dSsd

CTsScCP)(

,;|)(0

λδλ ⋅=>= ∫

∞ (6.5)

In the proposed procedure, it is assumed that the expected value of r∆ can be obtained as

the product of Cr times Sd, which implies that Cr and Sd are independent random variables and

that their correlation can be neglected. Furthermore, in computing the conditional probability

term in equation (6.5), it is assumed that is equivalent to compute [ ]yRR CTcCP ,|> , where cr

is defined as drr sc δ= . Therefore, equation (6.5) can be expressed as follows:

( ) [ ] dd

dyrrr ds

dSsd

CTcCP)(

,|0

λδλ ⋅>= ∫

∞ (6.6)

It should be mentioned that a core assumption in the above formulation is the lack of

dependence of Cr on the intensity of the event and distance form the causative fault. This

assumption is based on a prior statistical study that showed little dependence of Cr on

earthquake magnitude and distance to the source (Kawashima et al., 1998). This simplified

assumption will be further verified in Section 6.3.1 and 6.3.2. In addition, to evaluate the

conditional probability inside the integrand, an examination of the empirical cumulative

distribution function of Cr will be carry out in Section 6.3.5.1 and candidate parametric

cumulative distribution functions will be evaluated in Section 6.3.5.2 of this chapter.


It was mentioned that in order to compute ( )rδλ the seismic hazard curve at the

designated site should be available for the structure’s fundamental period of interest. The site-

specific seismic hazard information usually is provided from seismologists in terms of the

mean annual frequency of exceeding peak ground acceleration (PGA) or spectral acceleration

(Sa) at various periods of vibration, computed from the elastic response of SDOF systems

having 5% of damping ratio. For example, this information is readily available from the

United States Geological Survey (USGS) for any site in the U.S. (Frankel and Leyendecker,

2001).

It should be mentioned that recently proposed procedures for the design of new structures

or for the evaluation of existing structures use the yield displacement of the structure, ∆y, as a

primary parameter (Aschheim, 2002; Paulay, 2002). Then, it is convenient to express equation

(6.5) as a function of ∆y through the following relationship:

RSd

y =∆ (6.7)

Thus, an alternative form of equation (6.5) can be defined as follows:

( ) [ ]∫∞

⋅∆=⋅>⋅=0

)(,;| d

d

dydddrdrr ds

dSsd

TsSscSCPλ

δλ (6.8)

In equation (6.8), the dynamic properties (i.e., period of vibration) and the capacity (i.e.,

yield strength coefficient, lateral strength ratio, or yield (roof) displacement) of the existing

structure (e.g., bridge piers) are known, which means that a reasonable estimate is available.

For example, nonlinear static analysis (pushover) of the structure of interest subjected to an

adequate lateral force profile provides information of the global capacity (Seneviratna and

Krawinkler, 1997). On the other hand, ambient vibration tests or empirical formulae (e.g.,

Chopra and Goel, 2000) can be used for estimating the period of vibration.

Finally, it should be mentioned that proposed simplified approach is generic and it can be

implemented to estimate the maximum inelastic displacement demand of systems subjected to

different type of ground motions (e.g., far-field and near-fault ground motions), built on

different soil conditions (e.g., rock, firm soil or soft-soil sites), having different hysteretic

behavior and having different damping ratios. In this study, the implementation of the


proposed approach is only illustrated for structures built on firm soil sites, having elastoplastic

hysteretic behavior and with equivalent damping ratio of 5%.

6.3 Evaluation of Simplified Assumptions

The simplified probabilistic approach to estimate residual displacement demands outlined in

the previous section is similar to that introduced in Section 4.2 (Chapter 4) to estimate

maximum inelastic displacement demands. Therefore, it is subjected to the same underlying

assumptions and limitations. In summary, three main simplifying assumptions have been

made in the proposed simplified probabilistic approach to estimate residual displacement

demands:(1) Sufficiency of Cr (i.e. there is a lack of dependence of Cr on earthquake source

characteristics such as earthquake magnitude, distance to the source and duration of the

ground motion); (2) Cr and Sd are statistically uncorrelated; and (3) the empirical cumulative

distribution of Cr can be represented by a parametric cumulative distribution function (e.g.,

lognormal). In a similar way to the estimation of the probabilistic estimation of maximum

inelastic displacement demand, the validity of these simplifying assumptions should be

verified as well as the limitations of the suggested approach. Then, the following sub-sections

examine the simplified assumptions using the statistical results reported in Chapter 5 of

residual displacement ratios computed for elastoplastic SDOF systems, otherwise noted, when

subjected to ground motions recorded in firm site conditions.

6.3.1 Effect of Earthquake Magnitude on Cr

There is very little evidence about the influence of earthquake magnitude on residual

deformation demands. For example, Kawashima et al. (1998) evaluated qualitatively the effect

of earthquake magnitude on residual displacement demands of SDOF bilinear systems using

63 ground motions from seismic events with magnitudes ranging from 6.6 to 7.9, but the study

only included 3 ground motions with magnitude greater than 7.5. The authors did not found a

significant correlation between residual deformation demands and earthquake magnitude.

Therefore, a further look at the influence of earthquake magnitude on residual displacement

demands is still desirable.


(b) R = 4

0.0

0.4

0.8

1.2

1.6

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

Cr

7.0 < Ms < 7.8 (24 records)6.3 < Ms < 6.9 (24 records)5.7 < Ms < 6.2 (24 records)

(a) R = 2

0.0

0.4

0.8

1.2

1.6

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

Cr

7.0 < Ms < 7.8 (24 records)6.3 < Ms < 6.9 (24 records)5.7 < Ms < 6.2 (24 records)

(c) R = 6

0.0

0.4

0.8

1.2

1.6

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

7.0 < Ms < 7.8 (24 records)

6.3 < Ms < 6.9 (24 records)

5.7 < Ms < 6.2 (24 records)

Figure 6.1. Effect of earthquake magnitude on residual displacement ratios for:

(a) R = 2; (b) R = 4; and (c) R = 6.

In order to study if residual displacement ratios are modified by earthquake magnitude,

mean residual displacement ratios were computed from three bins of records corresponding to

three ranges of surface wave magnitude (Ms). Each bin contained 24 earthquake ground

motions recorded in the same site condition (site class D). Then, mean Cr corresponding to

three different magnitude ranges and for 3 levels of lateral strength ratio are shown in figure

6.1. It can be seen that the spectral shape and amplitude of Cr is not significantly affected by

earthquake magnitude for low levels of lateral strength (i.e., R = 2). However, earthquake

magnitude seems to have more influence on Cr as the level of lateral strength increases and, in

particular, for periods of vibration between 0.5s and 1.0s. After an examination of all results, it

was found that earthquake magnitude has smaller effect on strong systems (e.g., R < 3) than in

weak systems (e.g., R > 4) in the short- and medium-period range. It should be noted that large


magnitude events do not necessarily lead to larger residual displacement ratios, regardless of

the lateral strength ratio.

(a) M1 set (5.7 < Ms < 6.2)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr,M1/Cr,M

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(b) M2 set (6.3 < Ms < 6.9)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr,M2/Cr,M

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(c) M3 (7.0 < Ms < 7.8)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr,M3/Cr,M

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 6.2. Mean Cr computed from each set of ground motions corresponding to each magnitude range normalized with respect to mean Cr from all ground motions: (a) 5.7 < Ms < 6.2; (b) 6.3 < Ms < 6.9; and

(c) 7.0 < Ms < 7.8.

To estimate approximately the level of underestimation or overestimation produced when

using Cr neglecting the effect of earthquake magnitude instead of mean Cr computed from

ground motions belonging to a magnitude range, ratios of mean Cr computed from each

ground motions set to mean Cr computed from all 72 ground motions are presented in figure

6.2. In general, it can be seen that the level of underestimation or overestimation is not

constant for any of the magnitude ranges over the whole spectral region and it depends on the

level of relative strength ratio for a given period of vibration. For example, it can be seen that

using mean Cr from all 72 ground motions would lead to important underestimations for


systems with periods of vibration between 0.5 and 1.2 s. when subjected to ground motions

collected from Ms between 6.3 and 6.9.

(a) SITE CLASSES AB

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

ρ (Cr, Ms)

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(b) SITE CLASS C

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

ρ (Cr, Ms)

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(c) SITE CLASS D

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

ρ (Cr, Ms)

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 6.3. Coefficients of correlation between Cr and earthquake magnitude for each site condition: (a) Site classes AB; (b) site class C; (c) site class D.

Another way to further quantitatively investigate the effect of earthquake magnitude on Cr

is through the coefficient of correlation,sr MC ,ρ , which measures the statistical linear

correlation between Cr and Ms. Since firm soil conditions showed larger influence on residual

displacement ratios than on inelastic displacement ratios, the correlation coefficient between

Cr and Ms was first computed from the results from each site class.


SITE CLASSES AB,C,D

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

ρ (Cr, Ms)

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 6.4. Coefficient of correlation between Cr and earthquake magnitude for all 240 earthquake ground motions recoded in NEHRP site classes AB, C, and D.

Figure 6.3 shows sr MC ,ρ as a function of period of vibration and lateral strength ratio for

each site class. From the figure, it can be noted that sr MC ,ρ is not constant over the whole

spectral region regardless of the firm soil site condition. In general, sr MC ,ρ slightly increases

as the level of lateral strength increases (i.e.,Cr has stronger correlation with earthquake

magnitude for weak structures than for strong structures). In addition, it can be observed that

sr MC ,ρ is weakly correlated (e.g., sr MC ,ρ < 0.2) for periods of vibration longer than 0.5 s,

regardless of the site class. For periods of vibration shorter than about 0.5 s, sr MC ,ρ is larger

for systems subjected to accelerograms recorded in site class D. A graphical representation of

the variation of sr MC ,ρ computed from all 240 ground motions considered in this study as a

function of both the lateral strength ratio and the periods of vibration is presented in figure 6.4.

From the figure, it can be noted that sr MC ,ρ is not constant over the whole spectral region and

that, in general, the correlation slightly increases as the level of lateral strength increases (e.g.,

Cr has stronger correlation with earthquake magnitude for weak structures than for strong

structures). In general, sr MC ,ρ is weakly correlated (e.g.,

sr MC ,ρ < 0.15) for periods of

vibration longer than 0.5 s.


Finally, a similar approach used in Section 5.5 (chapter 5) to investigate the statistical

significance of additional explanatory variables was used to study the effect of earthquake

magnitude on the residual displacement ratio. As explained in the aforementioned section, this

statistical technique can be used to evaluate qualitatively the influence of earthquake

magnitude on Cr by: (1) plotting the Cr residuals, riri CCe −= ,ˆ (where Cr,i is the computed

residual displacement ratio for the ith ground motion and rC is the sample mean), for a given T

and R; (2) performing conventional linear regression of the sei 'ˆ on earthquake magnitude;

and (3) computing the coefficient of determination, R2. The coefficient of determination is a

statistical measure that allows quantifying the strength of the relationship among sei 'ˆ and

earthquake magnitude, which can be also used to evaluate if earthquake magnitude allows

explaining the variability (measured by the residuals) in the estimation of residual

displacement ratios. As an example of this statistical approach, fitted Cr residuals versus

earthquake magnitude are shown in figure 6.5 for periods of vibration equal to 0.2s, 0.5s, 1.0s,

and 2.0s and for lateral strength ratios of 4. Similar plots were obtained for other combinations

of period of vibration and lateral strength ratio. It can be seen that the percentage in the

estimation of Cr explained by the earthquake magnitude decreases as the period of vibration

increases (e.g., for a system with T=0.2 s, 5% of the variability in the estimation of Cr is

explained by earthquake magnitude while only 2% is explained for a systems with T=0.5 s.).

From the aforementioned discussions, it can be concluded that earthquake magnitude has

larger influence on residual displacement ratios ordinates than that observed for inelastic

displacement ratios. Also, ground motions from large magnitude events do not necessarily

lead to higher residual displacement ratios. In fact, earthquake magnitude appears statistically

weakly correlated with residual displacement ratios and, furthermore, the variability in the

estimation of mean residual displacement ratios is not reduced further by accounting for

earthquake magnitude.


(c) T = 1.0 sec

y = -0.0048x + 0.032R2 = 3E-05

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5


Residual

(b) T = 0.5 sec.

y = 0.2108x - 1.058R2 = 0.0221

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0


Residual

(d) T = 2.0 sec

y = 0.0264x - 0.1267R2 = 0.0012

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5


Residual

(a) T = 0.2 sec.

y = 1.1046x - 5.9433R2 = 0.0506

-20.0

-15.0

-10.0

-5.0

0.0

5.0

10.0

15.0

20.0


Residual

Figure 6.5. Cr fitted residuals versus earthquake magnitude for R = 4: (a) T=0.2 s; (b) T = 0.5 s; (c) T = 1.0 s; and (d) T = 2.0 s.

6.3.2 Effect of Distance to the Rupture on Cr

A similar methodology employed in the previous section was followed to investigate the

influence of distance to the rupture on Cr. Therefore, three subsets of 24 ground motions

representing three different distance ranges (D, in kms) were assembled from the ground

motions corresponding to site class D. A comparison of mean Cr ratios computed from each

subset is shown in Figure 6.6 for three different levels of lateral strength.


(b) R = 4

0.0

0.4

0.8

1.2

1.6

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

Cr

10.0 < D < 20.0 (24 records) 20.1 < D < 45.0 (24 records) 45.1 < D < 160.0 (24 records)

(a) R = 2

0.0

0.4

0.8

1.2

1.6

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

Cr

10.0 < D < 20.0 (24 records)20.1 < D < 45.0 (24 records)

45.1 < D < 160.0 (24 records)

(c) R = 6

0.0

0.4

0.8

1.2

1.6

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

10.0 < D < 20.0 (24 records)20.1 < D < 45.0 (24 records)

45.1 < D < 160.0 (24 records)

Figure.6.6. Effect of nearest distance to the horizontal projection of rupture, D [kms], on residual

displacement ratios for: (a) R = 2, (b) R = 4, and (c) R = 6.

It can be seen that, for the range of distances considered in this study, mean Cr ordinates

does not significantly changes when computed from ground motions ensembles representative

of different distance ranges for systems with low lateral strength ratio (e.g., R = 2). However,

the ordinates of mean Cr become different as the system becomes weaker with respect to the

ground motion intensity (i.e., for relative lateral strength greater than 4), which suggest that

mean Cr ordinates are influenced by the distance range. In particular, the difference of mean

Cr ordinates computed for each bin become different for periods of vibration between 0.4s and

0.8s. It should be noted that ground motions recorded close the causative fault (e.g., between

10 and 20 km) do not necessarily yield larger residual displacement ratios than ground

motions recorded at larger distances to the causative fault.


(a) D1 set (10.0 < D < 20.0 km)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr,D1/Cr,D

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(b) D2 set (20.1 < D < 45.0 km)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr,D2/Cr,D

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(a) D3 set (45.1 < D < 160.0 km)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr,D3/Cr,D

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 6.7. Mean Cr computed from each set of ground motions corresponding to each distance range [D in Km] normalized with respect to mean Cr from all ground motion: (a) 10.1 < D < 20.0; (b) 20.1 <

D < 45.0; and (c) 45.1 < D < 160.0.

In order to investigate to what extend mean Cr computed from all 72 ground motion (i.e.,

neglecting the influence of distance to the source) underestimate or overestimate mean Cr

computed from each bin, Figure 6.7 shows ratios of mean Cr computed from each distance bin

to mean Cr computed from all 72 ground motions. In general, it can be seen that the level of

underestimation or overestimation is not constant for none of the distance ranges over the

whole spectral region and it depends on the level of relative strength ratio for a given period of

vibration. For example, it can be seen that using mean Cr from all 72 ground motions instead

of mean Cr computed from ground motions collected in the range of 10 to 20 km would lead

to overestimations for systems with periods of vibration between 0.05 and 0.75 s and between

1.25 s and 2.5 s.


(a) SITE CLASSES AB

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

ρ (Cr, D)

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(b) SITE CLASS C

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

ρ (Cr, D)

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(c) SITE CLASS D

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

ρ (Cr, D)

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 6.8. Coefficient of correlation between Cr and distance to the rupture for each site class: (a) Site classes AB; (b) site class C; and (c) site class D.

Further investigation about the dependence of Cr on D was done by computing the sample

coefficient of correlation between the residual displacement ratio and distance to the source,

DCr ,ρ corresponding to each period of vibration and each level of relative lateral strength. The

variation of DCr ,ρ as a function of the lateral strength ratio and periods of vibration is shown in

Figure 6.8 for each firm soil site class. In general, it can be seen that Cr is positively or

negatively correlated with D depending on the period region and lateral strength ratio of the

system, regardless of the firm soil condition. It can be observed that, for a given period,

sample correlation is not significantly influenced by the level of lateral strength ratio, which

means that the correlation between Cr and distance range is not strongly influenced by the

relative lateral strength of the system. In addition, it can be seen that Cr is weakly correlated

with distance to the rupture (e.g., DCr ,ρ < 0.2) for systems with periods of vibration longer


than about 0.5 s. However, it should be noted that the level of correlation for a given system

depends on the firm soil condition.

The sample DCr ,ρ computed from all 240 earthquake ground motions is shown in figure

6.9. It can be seen that if one takes into account all ground motions, regardless of the soil

condition where they were recorded, the sample correlation is smaller than 0.2, with exception

of very short periods of vibration, which means that Cr and D are weakly correlated.

SITE CLASSES AB,C,D

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

ρ (Cr, D)

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 6.9. Coefficient of correlation between Cr and distance to the rupture for all 240 earthquake ground motions recoded in NEHRP site classes AB, C, and D.

Finally, fitted Cr residuals versus distance to the source projection are shown in figure 6.10

for periods of vibration equal to 0.2 s., 0.5 s., 1.0 s., and 2.0 s. and for lateral strength ratio of

4. Similar plots were obtained for other combinations of period of vibration and lateral

strength ratio. It can be seen that the percentage in the estimation of Cr explained by the

distance to the source decreases as the period of vibration increases (e.g., for a system with

T=0.2 s., 5.5% of the variability in the estimation of Cr is explained by earthquake magnitude

while only 1% is explained for a systems with T=0.5 s.).

From the prior discussions, it can be concluded that residual displacement ratios ordinates

are more influenced by distance to the source than inelastic displacement ratios. It should be

noted that ground motions compiled from short distance to the source (between 10 and 20 km)


do not necessarily lead to higher residual displacement ratios than records collected at larger

distances. In addition, distance to the source seems statistically weakly correlated with

residual displacement ratios and, furthermore, the variability in the estimation of mean

residual displacement ratios is not further explained by accounting for distance to the source.

(a) T = 0.2 sec.

y = 0.0231x + 0.5179R2 = 0.0552

-20.0

-15.0

-10.0

-5.0

0.0

5.0

10.0

15.0

20.0


Residual

(b) T = 0.5 sec.

y = 0.003x + 0.2327R2 = 0.0109

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0


Residual

(c) T = 1.0 sec.

y = 0.0012x - 0.0492R2 = 0.0055

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5


Residual

(d) T = 2.0 sec.

y = -0.0011x + 0.0956R2 = 0.0056

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5


Residual

Figure 6.10. Cr fitted residuals versus earthquake magnitude for R = 4:

(a) T=0.2 s; (b) T = 0.5 s; (c) T= 1.0 s; and (d) T = 2.0 s.

6.3.3 Effect of Duration of the Ground Motion on Cr

In order to study the effect of ground motion duration, mean residual displacement ratios were

computed using two sets of short (SD) and long (LD) duration ground motions, described in

Section 4.3.3 (Chapter 4). A comparison of mean Cr computed for the SD and LD sets for

three levels of lateral strength ratio are shown in figure 6.11. It can be seen that for a lateral

strength ratio equal to 2, ground motion duration has slight effect on Cr. However, for weaker

structures relative to the intensity of the ground motion (i.e., for higher values of R) ground

motion duration can influence Cr ordinates in a wide spectral region, mainly for systems with


periods of vibration between 0.7 and 2.0 s. In this spectral region, LD records leads to larger

Cr ordinates than that of SD records. However, for periods of vibration longer than about 2.0 s.

SD records would lead to slightly larger Cr ratios than LD records. It should be noted that the

limiting period that divides the spectral region where Cr is larger than 1 (i.e., residual

displacement demands are larger than elastic displacement demands) might be influenced by

ground motion duration when the system becomes weaker.

(b) R = 4.0

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

LD

SD

(c) R = 6.0

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

LD

SD

(a) R = 2.0

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

LD

SD

Figure 6.11. Effect of strong motion duration on residual displacement ratios for:

(a) R=2; (b) R=4; and (c) R=6.

In order to quantify the effect of ground motion, that is to evaluate the differences in Cr

for records with different strong motion durations, ratios of mean Cr of each ground motion

duration group to mean Cr computed from all 40 ground motions were computed. Figures

6.12a and 6.12b show mean Cr for SD an LD ground motions normalized by mean Cr

considering both sets. For SD records, it can be seen that, if one neglects the effects of short

ground motion duration and uses mean Cr values from all 40 ground motions, instead one


would overestimate residual displacement demands for periods of vibration shorter than about

2.0 s. For periods of vibration longer than 2.0 s, neglecting ground motion duration would

yield to underestimations of residual displacement demands. Unlike SD records, the use of

mean Cr from all ground motions considered here would produce, in general, underestimations

of residual displacements demands for T < 2.0 s and overestimations for T > 2.0 s for systems

subjected to long duration records. It should be noted that the magnitude of underestimation or

overestimation depends on the level of lateral strength ratio.

(a) Short-duration records

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr,SD/Cr,SLD

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

(b) Long-duration records

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr,LD/Cr,SLD

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 6.12. Mean Cr computed from each ground motion duration set normalized with respect to mean Cr from all 40 ground motion: (a) SD records; and (b) LD records.

From the observations made in this section, it is concluded that for some spectral regions

and levels of lateral strength the effect of ground motion duration might influence the

ordinates of residual displacement ratios and, hence, the magnitude of residual deformation

demands. In addition, it should be noted that ground motion duration has a more significant

effect than local firm soil conditions.

6.3.4 Statistical Dependence Between Cr and Sd

In the proposed simplified approach, it is assumed that Cr and Sd, are statistically uncorrelated

random variables. If the latter assumption does not hold (i.e. Cr and Sd are statistically

correlated), it can lead to some systematic error in the estimation of residual displacement

demands. Therefore, it is important to have an estimation of the relative error produced over


the whole range of periods of vibration and lateral strength ratios considered in this study

when these random variables are assumed statistically independent. The examination of this

simplified assumption is similar to that outlined in Section 4.3.4 (Chapter 4) and, therefore, it

is subjected to the same limitations.

From equation (6.4), the expected value of the residual displacement, [ ]rE ∆ , can be

computed as follows:

[ ] [ ]drr SCEE ⋅=∆ (6.9)

If the residual displacement ratio and the maximum spectral displacement are

statistically correlated, the former right-hand side expectation should be estimated as follows:

][][][][][, drSCSCdrdr SECECOVCOVSECESCE

drdr⋅⋅⋅⋅+⋅=⋅ ρ (6.10)

where [ ]rCE and [ ]dSE are the expected values of Cr and Sd, respectively, dSrC ,ρ is the

sample correlation coefficient between the residual displacement ratio and the spectral elastic

displacement, while rCCOV and

dSCOV are the coefficients of variation of Cr and Sd,

respectively. On the other hand, if Cr and Sd are indeed statistically independent, then

dSrC ,ρ is equal to zero and equation (6.10) can be simplified as follows:

][][][ drr SECEE ⋅=∆ (6.11)

Similarly to the procedure illustrated in Section 4.3.4.1 (Chapter 4), one way to evaluate

the systematic error in the estimation of r∆ by assuming independence between Cr and Sd is

through the relative error between the exact equation (6.10) and the approximate one (6.11).

Therefore, the relative error, ε, in the evaluation of the expected value of the residual

displacement produced by assuming that Cr and Sd are independent is given by:

:


drdr

drdr

SCSC

SCSC

edr

adr

COVCOVCOVCOV

SCESCE

⋅⋅+⋅⋅−

=−⋅⋅=

,

,

11

][][

ρρ

ε (6.12)

where adr SCE ][ ⋅ and edr SCE ][ ⋅ are the approximate and exact expected values of ∆r,

respectively. Information about COV of Cr was reported in Section 5.5.3 (Chapter 5) of this

dissertation. It is important to point out that COV of Sd and dr SC ,ρ should be obtained from the

information employed to generate the spectral displacement hazard curve via PSHA. This

information is not readily available and, thus, an approximation of the these statistical

parameters was obtained employing the 240 earthquake ground motions collected in firm sites

assuming that they are representative of the same seismic hazard environment. Information

about COV of Sd was reported in Section 4.3.4.2 (Chapter 4).

SITE CLASSES AB,C,D

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

ρ (Cr, Sd)

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 6.13. Coefficient of correlation between Cr and Sd computed from all 240 earthquake ground motions from NEHRP site classes AB, C, and D.

From the statistical results obtained in this investigation, sample dSrC ,ρ as a function of

the period of vibration and lateral strength ratio was computed and it is illustrated in figure

6.13. The first observation is that Cr and Sd seems negatively correlated over the whole


spectral region regardless of the level of relative lateral strength. In fact, it can be seen that

sample correlation coefficients do not significantly change with increments in the lateral

strength ratio. In addition, it can be observed that Cr and Sd are weakly correlated (i.e.,

correlation coefficient smaller than 0.2) over the period range covered in this study.

The relative error as a function of T and R is shown in figure 6.14. It was found that

assuming that Cr and Sd are statistically independent tends to produce small overestimations of

the expected value of the residual displacement demand. It can be seen that the error is not

constant over the whole spectral region and that it also depends on the relative lateral strength

of the system. For example, ε decreases for periods of vibration between 0.15 and 0.8 s and it

tends to increase for periods between 0.8 and 1.7 s. For short periods of vibration (e.g., T < 0.3

s), the relative error is not very sensitive to variation in the lateral strength ratio. In general,

this overestimation of [ ]rE ∆ is relatively small (i.e., ε<0.25) for a wide range of periods of

engineering interest (i.e., T > 0.3 s), with exception of periods of vibration smaller than 0.3s

where the underestimation is significant. Finally, it should be noted that the relative error in

the estimation of residual displacement demands is larger than the relative error of estimating

maximum inelastic displacement demands, since dispersion of Cr (measured by COV) is

larger than the dispersion of CR.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

|ε|

R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5

Figure 6.14. Relative error when computing r∆ with approximate (equation 6.10) and exact (equation 6.11) formulation.


6.3.5 Cumulative Distribution of Cr

6.3.5.1 Empirical distribution of Cr

In Section 4.3.5.2 (Chapter 4) it was shown that the conditional probability of CR can be

evaluated assuming that CR is lognormally distributed. In the proposed approach, it is assumed

that the probability distribution of Cr, conditioned on T and Cy, is also lognormal. This

assumption is verified in this section.

The empirical distribution of mean Cr was obtained by treating the computed mean Cr for

a given period of vibration and lateral strength ratio as a random sample and assuming each Cr

value as an independent outcome. Thus, all Cr observations were sorted in ascending order

assigning each i-observation a probability equal to: )1( +ni , where n corresponds to 240

observations.

The influence of period of vibration on the variation of the empirical cumulative

distribution of Cr, computed for elastoplastic systems, is shown in figure 6.15 for three levels

of relative lateral strength. In general, it can be seen that, for a given Cr ordinate, the

conditional probability of exceeding Cr increases as the period of vibration decreases and as

the level of relative lateral strength increases. For example, for R=4.0, the probability of

exceeding Cr=1 (i.e., residual displacement demand is equal to maximum elastic displacement

demand) is 46% for a system with T=0.2 s while it is about 24% and 10% for systems with

periods of vibration of 0.5 and 1.0 s. It should be noted that the empirical distribution of Cr

follows a similar trend for systems with medium and long periods of vibration (e.g., T ≥ 1.0 s)

regardless of the lateral strength ratio. A similar observation was reported for the empirical

cumulative distribution of CR.


(a) R = 2.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0cr

P [

Cr |

T=t

, R =

2]

T = 0.2 s

T = 0.5 sT = 1.0 s

T = 2.0 sT = 3.0 s

(b) R = 4.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0cr

P [C

r | T

= t,

R =

4]

T = 0.2 s

T = 0.5 s

T = 1.0 s

T = 2.0 s

T = 3.0 s

(c) R = 6.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0cr

P [C

r | T

= t,

R =

6]

T = 0.2 s

T = 0.5 s

T = 1.0 s

T = 2.0 s

T = 3.0 s

Figure 6.15. Empirical cumulative distribution of Cr as a function of period of vibration for: (a) R = 2; (b) R = 4; and (c) R = 6.

In order to further study the effect of lateral strength ratio on the empirical cumulative

distribution of Cr, the counted probability of exceeding Cr for a short- and long-period system

is shown in figure 6.16 for three levels of lateral strength ratio. It can be seen that the

empirical distribution of Cr for the short-period system is different for each level of lateral

strength. For long-period systems, the empirical distribution is similar for relative weak

system and for relatively strong systems (i.e., R =4 and 6). Finally, the effects of hysteretic

behavior on empirical probability distributions of Cr for short-period system are shown in

figure 6.17. It can be seen that none of this two variables cause a significant change in the

probability distribution.


(a) T = 0.5 s

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0cr

P [C

r | R

= r

]

R = 2.0

R = 4.0

R = 6.0

(b) T = 2.0 s

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0cr

P [C

r | R

= r

]

R = 2.0

R = 4.0

R = 6.0

Figure 6.16. Empirical cumulative distribution of Cr as a function of lateral strength ratio for:

(a) T= 0.5 s.; and (b) T = 2.0 s.

(a) R = 2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0cr

P [

Cr |

T=0

.5 s

, R =

2]

MCTK-1TK-2O-O

(b) R = 4

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0cr

P [C

r | T

=0.5

s, R

= 4

]

MCTK-1TK-2O-O

(c) R = 6

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0cr

P [

Cr |

T=0

.5 s

, R=6

]

MCTK-1TK-2O-O

Figure 6.17. Effect of hysteretic behavior on empirical cumulative distribution of Cr (T=0.5s):

a) R= 2; (b) R=4; and (c) R = 6.


6.3.5.2 Parametric Distribution of Cr

In order to implement the suggested probabilistic approach, it is convenient that the empirical

distribution of residual displacement ratios be approximated by a parametric cumulative

distribution function (CDF). From the previous section, it was noted that the empirical

distribution of Cr followed a skewed distribution with longer tails moving toward upper

values. Therefore, parametric CDF’s such as Lognormal, Gamma, Gumbel, Weibull, or

Rayleigh might be adequate to represent the observed sample cumulative distribution

(Benjamin and Cornell, 1970). Similarly to inelastic displacement ratios, three parametric

CDF’s were evaluated to determine if they can characterize the empirical distribution of

residual displacement ratios: a) Lognormal; b) Weibull; and c) Rayleigh. These probability

distributions were chosen since they have the convenience over other probability distributions

that can be fully defined by using only two parameters (lognormal and Weibull) or by only

one parameter (Rayleigh). In particular, the lognormal distribution was chosen as primary

candidate since the parameters explicitly provide information of the central tendency and the

dispersion, or spread, of the sample distribution. In addition, it was found adequate to

represent the empirical distribution of inelastic displacement ratios. The selected CDF’s were

defined in Section 4.3.5.2 (Chapter 4). To verify weather the candidates probability

distributions are adequate to characterize the empirical probability distribution of residual

deformation demands the Kolgomorov-Smirnov (K-S) goodness-of-fit test was used in this

investigation (Benjamin and Cornell, 1970).

In general, it was found that Weibull probability distribution provided the best fit to the

empirical probability distribution. For example, a comparison of the empirical distribution of

Cr, obtained from a short-period (T=0.5 s) and long-period (T=2.0 s) system, with respect to

Lognormal, Weibull and Rayleigh fitted distribution is illustrated in figures 6.18 and 6.19.

corresponding to three different levels of lateral strength ratio. It can be seen that the Weibull

CDF provides the best fit for both systems followed by the lognormal CDF. It can be also

observed that the Rayleigh CDF, which is a particular case of the Weibull distribution with

parameter b=2.0, does not provide a good fit to the empirical distribution.


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0Cr

P [C

r > δ

| R

= 4

.0]


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0Cr

P [C

r > δ

| R

= 2

.0]


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0Cr

P [C

r > δ

| R

= 6

.0]


Figure 6.18. Comparison of three parametric CDF with empirical distribution of Cr for a short-period

system (T=0.5 s).

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0Cr

P [C

r > δ

| R

= 4

.0]


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0Cr

P [C

r > δ

| R

= 2

.0]


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0Cr

P [C

r > δ

| R

= 6

.0]


Figure 6.19. Comparison of three parametric CDF with respect to empirical distribution of Cr for a

long-period system (T=2.0 s).


Even though the Weibull CDF was found to provide the best fit to the empirical

probability distribution, it has the disadvantage that the parameters are not as easy to find as

for the lognormal distribution. Furthermore, the parameters of the Weibull distribution do not

explicitly reflect the central tendency and dispersion of Cr. Therefore, it was decided to use the

lognormal distribution to characterize the empirical conditional probability of Cr which can be

evaluated as follows:

[ ]

−Φ−=>

r

r

C

Crrr

cRTcCP

ln

ln)ln(1,|

σ

µ (6.13)

In addition of investigating if a parametric distribution allows characterizing the

conditional probability of exceeding Cr for a given period of vibration and a given level of

relative lateral strength, it is important to further investigate if the selected parametric

distribution is able to reproduce Cr spectra for different percentiles.

(a) R = 2

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

p = 90%p = 70%p = 50%p = 30%p = 10%Data

(b) R = 4

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

p = 90%p = 70%p = 50%p = 30%p = 10%Data

(c) R = 6

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

p = 90%p = 70%p = 50%p = 30%p = 10%Data

Figure 6.20. Comparison of counted percentiles and percentiles of Cr computed assuming lognormal

CDF (using statistical parameters from sample data) for: (a) R = 2; (b) R = 4; (c) R = 6.


A comparison between the counted and the estimated percentiles using the proposed

functional models is shown in figure 6.20. In this figure thin lines correspond to counted

percentiles, while thick lines correspond to percentiles computed assuming a lognormal

distribution with parameters computed from statistical data. In general, good agreement can be

observed for Cr corresponding to percentiles between 30% and 70%. However, extreme

exceedance probabilities are not captured.

6.4 Statistical Models to Estimate the Conditional Distribution of Cr

In this section, functional models are proposed to estimate statistical parameters of central

tendency and dispersion of residual displacement ratios to be used in conjunction with a

lognormal distribution. The selection of relevant predictors as well as the functional form of

the proposed models is based on the observations described in Chapter 5. In summary, both

central tendency (e.g.,, mean or median) and dispersion (e.g., coefficient of variation or

standard deviation of the natural logarithm of data) of Cr showed a nonlinear variation with

respect to the period of vibration, T, and the lateral strength ratio, R .Therefore, both T and R

were chosen as predictors.

6.4.1 Central Tendency Functional

The selected functional form should describe the dependence of the predicted parameter (e.g.,

median Cr) on each of the independent variables (e.g., T and R) in an adequate way. The

following functional form is proposed to estimate the central tendency of Cr (i.e. sample mean,

counted median and geometric mean):

αθθ θ

⋅

⋅+=

321

11~T

C r (6.14)

where

( )( )[ ]654 1exp1 θθθα −⋅−−⋅= R (6.15)


The functional form of equation (6.14) captures the observed trend of Cr. For example, the

parameter α accounts for saturation of Cr as the level of lateral strength ratio increases in the

medium and long period region. Parameters estimates to provide an estimate of median Cr in

equations 6.14 and 6.15 were obtained from non-linear regression analysis using the

subroutine nlinpar.m available in the statistical toolbox of MATLAB (MathWorks, 2001).

However, after obtaining parameters estimates using results from each site condition it was

decided to further simplify equations (6.14) and (6.15) to include only three parameters

depending on the site condition (i.e., site-dependent parameters). The simplified equation is

given by:

α⋅

⋅+=

brTa

C41

11~ (6.16)

( )

−⋅−−⋅= 5.01

21

exp1 Rcα (6.17)

where parameters a, b and c are site-dependent parameters. Parameter estimates, and 95%

confidence intervals, obtained from counted median of Cr for each site class are provided in

Table 6.1.

Table 6.1. Site-dependent parameter estimates and 95% confidence intervals for equations (6.16) and (6.17) obtained from median Cr.

Soil Type a b

c c.i. (a) c.i. (b) c.i. (c)

AB 5.140 1.299 3.302 2.619,7.662 1.056,1.542 1.844,4.761

C 4.005 1.626 2.473 2.119,5.891 1.386,1.866 1.406,3.540

D 8.998 1.336 5.855 5.765, 12.249 1.169,1.502 4.085,7.623

ABCD 5.698 1.409 3.644 3.747,7.650 1.241,1.577 2.534,4.755

Finally, it should be noted that the proposed functional form actually represent a surface in

the Cr – R – T space as shown in figure 6.21.


. 1

2

3

46

0.10.4

0.81.1

1.51.8

2.33.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

C r

R

PERIOD [s]

Figure 6.21. Median residual displacement ratio computed with equations (6.16) and (6.17) and parameter estimates given in Table 6.1 for site classes ABCD.

6.4.2 Dispersion Functional

In addition, a simplified equation to describe the variability of Cr (i.e., coefficient of variation,

COV, or standard deviation of the log of Cr, rClnσ ) is proposed in this investigation. The

functional form is given by:

( ) ϕβ

σ ⋅

+⋅

+=1.021

11~1 T

(6.18)

where

( )( )[ ]5.032 1exp1 −⋅−⋅= Rββϕ (6.19)

Similarly to equation (6.17), the parameter ϕ takes into account that dispersion (e.g., COV

of Cr reported in Section 5.6.3, Chapter 5) tends to saturate as the lateral strength ratio

increases in the medium- and long-period region. Similar procedure described in the last sub-

section was followed to obtain parameter estimates. For instance, after performing non-linear


regression analysis, the parameter estimates that fit sample COV of Cr, as well as 95%

confidence intervals, are given in Table 6.2.

Table 6.2. Site-dependent parameter estimates and 95% confidence intervals for equations (6.18 and 6.19) obtained from sample COV of Cr.

Soil Type 1β 2β 3β )ˆ.(. 1βic )ˆ.(. 2βic )ˆ.(. 3βic

AB 3.891 2.986 2.211 3.331,4.452 2.619,3.352 1.979,2.444

C 4.005 3.229 1.966 3.429,4.576 2.836,3.622 1.768,2.163

D 2.632 2.084 2.904 2.149, 3.116 1.739,2.422 2.537,3.271

ABCD 3.674 2.889 2.283 3.288,4.056 2.629,3.150 2.211,2.455

It should be noted that the proposed functional form of equations (6.18) and (6.19) also

represents a surface in the σ~ – R – T space as illustrated in figure 6.22.

1

2

34

56

0.05

0.40

0.75

1.10

1.45

1.80

2.30

3.00

0.0

0.4

0.8

1.2

1.6

2.0

COV C r

R

PERIOD [s]

Figure 6.22. Coefficient of variation of residual displacement ratio computed with equations (6.18) and (6.19) and parameter estimates given in Table 6.1 for site classes ABCD.


6.4.3 Evaluation of Proposed Functional Models to Estimate the Cumulative

Distribution of Cr

Besides proposing adequate functional models to represent the central tendency and dispersion

of residual displacement ratios, it is important to investigate if the proposed models provide

good estimates to be used in conjunction to the lognormal CDF in order to adequately

characterize the empirical cumulative distribution of Cr. For instance, figure 6.23a shows the

empirical CDF of Cr for a system with T = 0.5 s and R = 4.0. In addition, the lognormal CDF

computed with statistical parameters estimated by the proposed equations is illustrated in the

same figure. In computing the lognormal CDF given in equation (6.13), RClnµ was obtained

from the following relationship: rC CR

ˆlnln =µ , where rC is the median of Cr which was

estimated from equations (6.16) and (6.17) and parameter estimates given in Table 6.1. The

dispersion measure in equation (6.13), rClnσ , was estimated from the relationship:

)1ln( 2ln += COV

rCσ , where COV of Cr was estimated from equations (6.18) and (6.19)

and parameter estimates provided in Table 6.2. It should be noted that the former relationships

are adequate when the sample data is lognormally distributed, which was verified in the

preceding section. The graphic representation of the K-S test corresponding to a 90%

confidence level is also plotted in the same figure. It can be observed that, in general, the

proposed functional models employed to compute probability distribution of Cr lead to a

reasonable fit with respect to the observed data. Their use was also verified for a wide range of

periods and levels of lateral strengths with satisfactory results (for example, figure 6.23b).


(b) T = 2.0 s

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0cr

P[C

r | R

= 6

]

Data

Lognormal fit using proposed models


(a) T = 0.5 s

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0cr

P[C

r | R

= 4

]

Data, T = 0.5s

Lognormal fit using proposed models


Figure 6.23. Comparison of empirical and lognormal cumulative distribution functions of Cr (using proposed models) for two periods of vibration: (a) T = 0.5s; and (b) T = 2.0s.

In addition, percentiles of Cr can be also computed assuming lognormal distribution and

using the proposed functional models. A comparison between the counted and the estimated

percentiles using the proposed functional models is shown in figure 6.23. In general, good

agreement can be observed for Cr corresponding to percentiles between inter-quartiles (i.e.

between 25% and 75% percentiles). However, the use of the proposed models is not as good

for computing extreme exceedance probabilities of Cr (e.g., 10% and 90% percentile) for large

levels of lateral strength ratios (e.g., R > 4).


(b) R = 4

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

p = 90%p = 70%p = 50%p = 30%p = 10%Data

(c) R = 6

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

p = 90%p = 70%p = 50%p = 30%p = 10%Data

(a) R = 2

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Cr

p = 90%p = 70%p = 50%p = 30%p = 10%Data

Figure 6.24. Comparison of counted percentiles and percentiles of Cr computed assuming lognormal CDF (using statistical parameters from proposed models) for: (a) R = 2; (b) R = 4; (c) R = 6.

6.5 Evaluation of Proposed Approach to Compute ( )rδλ

Similarly to the development of maximum inelastic displacement hazard curves, )( iδλ ,

families of residual displacement hazard curves, )( rδλ , were computed for a firm soil site

located in a region of high-seismicity in California, which coincides with the Stanford campus.

As described in Section 4.5.1 (Chapter 4), the spectral elastic displacement hazard

curve, )( dSλ , for the site of interest was obtained from the United States Geological Survey

(Frankel and Leyendeker, 2001). Therefore, a total of 25 residual displacement demand hazard

curves corresponding to five periods of vibration (T = 0.2, 0.3, 0.5, 1.0, and 2.0s) and five

yielding strength coefficients (Cy = 0.1, 0.2, 0.4, 0.6 and 0.8) for each period of vibration were

computed through numerical integration of equation (6.1) employing equations (6.16) and

(6.18) to compute statistical parameters of the conditional probability of exceeding a certain


level of residual displacement demand. The range of integration covered spectral elastic

accelerations, Sa, from 0.01g to 5.0g with increments of 0.01g.

6.5.1 Residual Displacement Demand Hazard Curves

A family of residual displacement demand hazard curves corresponding to each of the

aforementioned periods of vibration is shown from figures 6.25 to 6.30. For references

purposes, the corresponding spectral elastic displacement demand seismic hazard curve, λ(Sd),

are also plotted. In general, it can be seen that for a given mean annual frequency (e.g., λ

=0.001) residual displacement demands can be larger than elastic displacements for very

short-period systems (e.g., T = 0.2 s and Cy < 0.6). For longer systems (e.g., T = 1.0s), residual

displacement demands could still be very large in comparison with elastic displacement

demands. For example, r∆ can be on the order of 18 cm for Cy = 0.1 and λ = 0.001).

T = 0.2 s

0.0001

0.001

0.01

0.1

1

0.01 0.1 1 10 100

Residual displacement, ∆r [cm]

λ (∆r)

Cy=0.1

Cy=0.2Cy=0.4

Cy=0.6Cy=0.8

USGS (Elastic)

Figure 6.25: Residual displacement demand hazard curve corresponding to five different yield strength

coefficient and T=0.2s.


T = 0.3 s

0.0001

0.001

0.01

0.1

1

0.01 0.1 1 10 100Residual displacement, ∆r [cm]

λ (∆r)

Cy=0.1Cy=0.2Cy=0.4Cy=0.6Cy=0.8USGS (Elastic)

Figure 6.26. Residual displacement demand hazard curve corresponding to five different yield strength coefficient and T=0.3s.

T = 0.5 s

0.0001

0.001

0.01

0.1

1

0.01 0.1 1 10 100

Residual displacement, ∆r [cm]

λ (∆r)


Figure 6.27. Residual displacement demand hazard curve corresponding to different yield strength coefficient and T=0.5s.


T = 1.0 s

0.0001

0.001

0.01

0.1

1

0.01 0.1 1 10 100Residual displacement, ∆r [cm]

λ (∆r)


Figure 6.28. Residual displacement demand hazard curve corresponding to different yield strength coefficient and T=1.0s.

T = 2.0 s

0.0001

0.001

0.01

0.1

1

0.1 1 10 100

Residual Displacement, ∆r [cm]

λ (∆r)

Cy=0.1

Cy=0.2

Cy=0.4Cy=0.6

Cy=0.8

USGS (Elastic)

Figure 6.29. Residual displacement demand hazard curve corresponding to different yield strength coefficient and T=2.0 s.


(a) C y = 0.1

0.00001

0.0001

0.001

0.01

0.1

0.1 1 10 100Residual displacement, ∆r [cm]

λ (∆r)

T = 2.0 s

T = 1.0 s

T = 0.5 s

T = 0.3 s

T = 0.2 s

(b) C y = 0.2

0.00001

0.0001

0.001

0.01

0.1


λ (∆r)

T = 2.0 s

T = 1.0 s

T = 0.5 s

T = 0.3 s

T = 0.2 s

(c) C y = 0.4

0.00001

0.0001

0.001

0.01

0.1


λ (∆r)

T = 2.0 s

T = 1.0 s

T = 0.5 s

T = 0.3 s

T = 0.2 s

(d) C y = 0.6

0.00001

0.0001

0.001

0.01

0.1


λ (∆r)

T = 2.0 s

T = 1.0 s

T = 0.5 s

T = 0.3 s

T = 0.2 s

(e) C y = 0.8

0.00001

0.0001

0.001

0.01

0.1


λ (∆r)

T = 2.0 s

T = 1.0 s

T = 0.5 s

T = 0.3 s

T = 0.2 s

Figure 6.30: Residual displacement demand hazard curve corresponding to different periods of

vibration for: (a) Cy = 0.1; (b) Cy = 0.2; (c) Cy = 0.4; (d) Cy = 0.6; and (d) Cy = 0.8.

A comparison of residual displacement demand hazard curves corresponding to the same

yield strength coefficient and as a function of period of vibration are shown from figure 6.30a

to figure 6.30d. As expected, it can be observed that )( rδλ depends on both, the lateral


strength ratio and the period of vibration of the system. For example, for a long-period (T =

2.0 s.) weak (Cy = 0.1) system the mean annual frequency of exceeding 10 cm (4 in) is about

2.64, 7.28, 15.89, and 28 times the mean annual frequency of exceeding the same inelastic

displacement of a system with the same strength but with periods of vibration of 1.0, 0.5, 0.3,

and 0.2 seconds, respectively. On the other hand, a stronger system (Cy = 0.4) with a period of

vibration of 2.0 s. would experience an inelastic displacement of 10 cm with a exceedance

probability of about 2.36, 3.71, 5.42, and 6.57 times than that of a system with identical

strength but with periods of vibration of 1.0, 0.5, 0.3, and 0.2 seconds, respectively Another

example is that, for a given )( rδλ , the same system (Cy = 0.1) would experience larger

residual displacement demands as the period of vibration increases. This increment in residual

displacement demand with increasing period of vibration is not constant and it depends, for a

given Cy, on the )( rδλ of interest.

6.6 Uniform Hazard Spectra of Residual Displacement Demand

Residual displacement seismic hazard curves can be used to build uniform hazard spectra of

residual displacement demand (RD-UHS) corresponding to different return periods (i.e., for

different levels of probabilities of exceeding a given level of residual displacement). For

example, RD-UHS corresponding to five yield strength coefficients and for 2% and 10%

probability of exceeding a given level of r∆ in 50 years are shown in figure 6.31. From the

figure, it can be seen that the RDUH curves follow a nonlinear trend which depends on the

structural properties of the system (i.e., period of vibration and yield strength coefficient). As

can be expected, weaker structures (i.e., with low yielding strength coefficient) are more

susceptible to experience large residual deformation demands. For example, for a system with

T=1.0s and Cy=0.4 and an elastoplastic hysteretic behavior, located on the Stanford campus

could experience residual displacement demands corresponding to 2% probability of

exceedance in 50 years (return period of 2475 years) that are excessively large (e.g., 15.7 cm).

It is believed that the proposed procedure can be very useful to establish adequate residual

displacements levels, or residual displacement drift ratios, associated to earthquakes with

different probabilities of being exceeded and corresponding to different performance levels.


(a) 10% in 50 yrs.

0

5

10

15

20

25

30

35

40

0.0 0.5 1.0 1.5 2.0 2.5

PERIOD [s]

∆ r [cm]

Cy = 0.1Cy = 0.2Cy = 0.4Cy = 0.6Cy = 0.8Elastic

(b) 2% in 50 yrs.

0

10

20

30

40

50

60

70

80

90

0.0 0.5 1.0 1.5 2.0 2.5PERIOD [s]

∆r [cm]

Cy = 0.1Cy = 0.2Cy = 0.4Cy = 0.6Cy = 0.8Elastic

Figure 6.31. Uniform hazard spectra of residual displacement demand corresponding to two return periods: (a) 10% in 50 years; and (b) 2% in 50 years.

6.7 Comparison of Displacement Demand Hazard Curves

Previously developed maximum inelastic displacement hazard curves are compared with

residual displacement hazard curves derived in this section for selected periods of vibration

and yield strength coefficients. For example, )( iδλ and )( rδλ hazard curves corresponding to

a system with T=0.5 s and Cy=0.1 (i.e. short-period weak system relative to the intensity of the

ground motion) are illustrated in figure 6.32. For reference purposes, the spectral elastic

displacement hazard curve, )( dSλ , is also plotted in the same figure. It can be seen that for a


given mean annual frequency of exceedance (MAF), large residual displacements demands

compared to maximum inelastic displacement demands can be expected. For example, for

λ=0.001 ∆r can be about 1 cm while ∆i could reach 4.5 cm. It is interesting to note that the

relationship between residual and maximum inelastic displacement demand is not constant for

a given MAF. Another comparison for a system with the same period of vibration, but higher

yield strength coefficient (i.e., short-period strong system relative to the intensity of the

ground motion) is given in figure 6.32. In this figure, it can be seen that the relationship

between residual and maximum inelastic displacement demand increases as the MAF

decreases.

λ(∆i, ∆r)

0.00001

0.0001

0.001

0.01

0.1

0.1 1 10 100Displacement, ∆ [cm]

ElasticResidual

Maximum

(a) T = 0.5 s, Cy = 0.1

λ(∆i, ∆r)

0.00001

0.0001

0.001

0.01

0.1

0.1 1 10 100

Displacement, ∆ [cm]

Elastic

Residual

Maximum

(b) T = 0.5 s, Cy = 0.8

Figure 6.32. Comparison of maximum and residual displacement demand hazard curves for a short-

period system (T=0.5s) with two levels of yield strength: (a) Cy = 0.1; and (b) Cy = 0.8.


6.8 Summary

Post-earthquake reconnaissance from recent seismic events have highlighted that the

magnitude of residual deformations play an important role in whether to repair and upgrade

severely damaged structures or to demolish them. Therefore, an adequate seismic assessment

of existing structures requires quantitative methods to evaluate not only maximum inelastic

deformation demands but also residual deformations in the event of earthquake excitation.

However, the large variability observed in the estimation of residual deformation demands

requires its explicit incorporation by means of a probabilistic framework.

This chapter proposed a simplified probabilistic approach to compute residual

displacement demands in single-degree-of-freedom (SDOF) systems with known period of

vibration and relative lateral strength with respect to the ground motion intensity. The

proposed procedure permits the computation of residual displacement seismic hazard curves,

)( rδλ , and uniform hazard spectra of residual displacement for different return periods.

These )( rδλ curves together with maximum displacement hazard curves, )( iδλ , can be very

helpful for the evaluation of existing structures that can be represented as SDOF systems (e.g.,

bridges) as well as for preliminary design of new structures. The proposed approach was

evaluated for a site located in a firm soil sites (e.g., Stanford campus), but the suggested

approach can be implemented in sites with different soil conditions (e.g., soft soil sites). In

particular, this study proposed adequate functional models to characterize central tendency

and dispersion of inelastic displacement ratios, Cr, which allow the estimation of residual

displacement demands from maximum elastic displacements. Parameters used in this chapter

correspond to elastoplastic systems but the procedure would be the same for other hysteretic

behaviors but changing the constant shown in equations (6.16) to (6.19).

While developing the suggested approach, several observations were made:

1. It was observed that Cr and earthquake magnitude and distance to the source are weakly

correlated, so the error introduce by assuming that the residual deformation ratio and the

elastic spectral displacement demand are independent leads to relatively small

overestimations of residual displacement demands.


2. Earthquake magnitude does not significantly affect Cr ordinates for periods of vibration

longer than about 1.0 s and with lateral strength ratios smaller than 4. Nevertheless, some

dependence on magnitude was observed for systems with periods of vibration shorter than

1.0 s. In the short period region, Cr computed from ground motions recorded during

earthquakes with magnitude higher than 6.3 were found to be larger than those computed

from records obtained in earthquakes with magnitudes between 5.7 and 6.2.

3. Cr ordinates of systems with are not significantly affected by distance to the source.

4. It was found that long-duration ground motions might lead to larger Cr ordinates than

short-duration records for systems with lateral strength ratios greater than 4 in the short-

and medium-period region, which mean that strong motion duration might influence the

magnitude of residual deformation demands.

5. Empirical probability distribution of Cr conditioned on several parameters (e.g., period of

vibration, lateral strength ratio, soil conditions and hysteretic behavior) exhibited a non-

symmetrical shape with respect to the central value, including longer tails moving towards

upper values. To characterize the empirical probability distribution of Cr, it was found that

parametric probability distribution functions such as lognormal and Weibull are adequate

and provide a good representation.

6. Simplified nonlinear equations to estimate central tendency and dispersion of Cr were

proposed. The proposed equations have adequate functional forms that reproduce the

observed central tendency and dispersion trend of Cr with changes in the period of

vibration and the lateral strength ratio. The use of the simplified equations to estimate

statistical parameters of the lognormal probability distribution allows computing

percentiles of Cr, which are in good agreement with counted percentiles in the inter-

quartile range.

___________________________________________________________________________________ Chapter 7 Statistical Evaluation of Maximum and Residual Deformation Demands Systems for

231

Chapter 7

Statistical Evaluation of Maximum and Residual

Deformation Demands for MDOF Systems

7.1 Introduction

There are few analytical investigations that have focused their attention on the evaluation of

residual deformation demands, mainly residual drift demands , of multi-degree-of-freedom

(MDOF) systems, representative of multi-story buildings, subjected to an ensemble of

earthquake ground motions (Gupta and Krawinker; 1999; Pampanin et al., 2002; Sabelli et al;

2002; Uang and Keggins, 2003; Medina and Krawinkler, 2003). However, with exception of

the work done by Medina and Krawinkler (2003), prior investigations did not report

noticeable trends in the amplitude and distribution of central values and dispersion of residual

drift demands with changes in the ground motion characteristics (e.g., intensity and duration

of the ground shaking) and building features (e.g., period of vibration, number of levels, etc.).

This lack of information was mainly due to the limited number of building study cases.

Therefore, statistical studies aimed to evaluate central tendency and dispersion of residual

deformation demands in MDOF systems is needed.

The purpose of this chapter is to present the results of a statistical evaluation of

deformation demands of MDOF systems subjected to earthquake ground motions

representative of the seismic hazard in California. In particular, this chapter focuses on

obtaining quantitative information about permanent lateral displacement demands (i.e.,

residual deformation demands), and its relationship with respect to maximum displacement

demands, from a family of 12 regular generic one-bay framed building models representative

of typical office building construction in a region of high seismicity in California. Thus,

relevant statistical parameters (e.g., central tendency and dispersion) of permanent lateral

displacement demands with changes in the ground motion characteristics (i.e., ground motion

intensity and duration) and building features (i.e., fundamental period of vibration, number of

stories, failure mechanism, element hysteretic behavior, and system overstrength) are reported


232

in this Chapter. For comparison purposes, statistical information of maximum deformation

demands is also presented in this chapter.

7.2 Previous Studies on Residual Deformation Demands of MDOF Systems

Limited investigations have focused their attention in the evaluation of residual deformation

demands of MDOF systems representing framed buildings (Gupta and Krawinker; 1999;

Pampanin et al., 2002; Medina and Krawinkler, 2003) as well as representing framed

buildings including braced systems (Sabelli et al; 2002; Uang and Keggins, 2003). For

example, as part of the SAC project, Gupta and Krawinkler (1999) reported residual

deformation demands (i.e., residual drift angles) of three steel moment resisting frame

(SMRF) models having 3, 9 and 20-story height designed to be representative of typical office

building construction prior to the 1994 Northridge earthquake and located in three different

seismic areas in the United States (i.e., Los Angeles, Seattle and Boston). The building models

were subjected to three suites of 20 earthquake ground motions comprised to be

representative of three hazard levels (i.e., 2%, 10% and 50% probability of being exceeded in

50 years,,which are referred as 2/50, 10/10 and 50/50 ground motion sets) which correspond to

three return periods (i.e., 2575, 475 and 72 years return period). In particular, for the region of

high-seismicity (i.e., Los Angeles), the authors noted that residual drift demands increases as

the intensity of the ground motions increases (i.e., with increasing hazard level). Although the

authors did not report explicitly, it was observed by inspection of their results that median

residual drift demands did not increase significantly, for a given ground motion hazard, with

increments in the number of stories. In addition, residual drift demands showed large

dispersion as the intensity of the ground motions increases, but not necessarily as the story

height increases. More recently, Pampanin et al. (2002) evaluated residual deformation

demands of four framed building models having 4, 8, 12, and 20 stor ies subjected to two

suites of 20 earthquake ground motions scaled to match the design spectrum of the UBC 97,

zone 4 for soil types C or D. Each suite of ground motions intended to represent a 2/50 and

10/50 hazard level. In particular, the authors noted that mean residual drift demands are very

sensitive to the hysteretic modeling of the structural components (e.g., elements modeled with

elastic-perfectly plastic force-deformation relationship exhibited larger residual drift demands

than when Takeda-type hysteretic member behavior was employed). They also noted an

increment in mean residual drift demands with increment in ground motion intensity assoicted


233

to large dispersion in the estimation of mean results, but they did not find any noticeable trend.

Finally, Medina and Krawinkler (2003) performed a limited evaluation of the median and

dispersion of residual drift demands from a family of 12 one-bay generic framed building

models, representing rigid an flexible versions of 6 building models with different number of

stories, subjected to 40 earthquake ground motions scaled to different levels of intensity. The

authors noted that median residual drift demands mimic the distribution of maximum drift

demands along the height as the level of intensity increases. They also noted that dispersion of

residual drift demand is large and non-uniform along the height.

On the other hand, buckling-restrained braced frames (BRBF) have emerged as a

promising lateral force resisting system for steel construction. This lateral system consists of

encased steel braces that provide adequate lateral stiffness as well as energy-dissipating

characteristics while avoiding buckling during seismic load reversals. However, it should be

noted that recently published analytical studies on the seismic behavior of BRBF have pointed

out that this system could lead to large residual deformation demands when subjected to

intense earthquake ground motions (Sabelli et al., 2002; Uang and Keggins, 2003). In

particular, Sabelli et al. (2003) analyzed 3-story and 6-story one-bay steel building frame

models with a chevron configuration of BRBF subjected to the same 10/50 and 2/50 ground

motion sets previously employed by Gupta and Krawinkle r (1999). The authors found that

BRBF’s lead to large permanent drift demands, on the order of 40% of maximum drift

demands corresponding to the 10/50 hazard level, since buckling-restrained braces exhibit

wide hysteresis loops which were modeled through an elastoplastic force-deformation

relationship.

7.3 Generic Frame Building Models Used in This Study

For the purpose of evaluating lateral residual deformation demands in MDOF systems, a

family of 12 regular one-bay framed building models, with fundamental periods ranging from

0.5 to 3.3 seconds and having number of stories from 3 to 18, was designed according to

current seismic provisions for structures located in a region of high seismicity in California

(FEMA, 1997b). In order to capture the seismic behavior of existing multi-story framed

buildings, special attention was done to provide adequate height-wise stiffness distribution

similar to the one observed in actual buildings, which is reflected in the deflected elastic


234

fundamental modal shape of the structure. Each frame building model was modeled as a two-

dimensional centerline frame using the computer software RUAUMOKO (Carr, 2003). In the

framed building models, inelastic deformation is concentrated in the plastic hinges that form at

both end of beam and column elements. In the plastic hinges, the nonlinear moment-curvature

relationships considered different models to reproduce typical hysteretic behaviors of

reinforced concrete and steel components. The moment capacity in the elements was

determined from the story shear force distribution according to the lateral static force

distribution suggested in current seismic provisions (FEMA, 1997b). A detailed description of

the design process and modeling assumptions of the family of generic framed models is given

in Appendix D.

7.4 Deformation Demand Measures

In order to adequately characterize deformation demands, both maximum and residual, four

basic deformation demand measures for MDOF systems are used throughout this section:

a) Maximum inter-story drift ratio (i.e.,, story drift normalized with respect to the height

of the story), IDRmax;

b) Roof drift ratio (i.e., roof displacement normalized by the total height of the building),

θroof;

c) Maximum residual inter-story drift ratio (i.e., residua l story drift normalized with

respect the height of the story), RIDRmax; and

d) Residual roof drift ratio (i.e., roof displacement normalized by the total height of the

building), θr,roof.

7.5 Ground Motion Characterization

Characterization of earthquake ground motion intensity has become a relevant issue in

Probabilistic Seismic Demand Analysis (PSDA). As discussed in Section 1.2 (Chapter 1),

several studies have used the spectral elastic acceleration at, or near, the fundamental period of

vibration of the structure, Sa (T1), as ground motion intensity measure (IM). This scalar


235

quantity has been widely employed since available seismic hazard curves are expressed in

terms of Sa (T1). However, it has been recognized that more efficient and sufficient IM’s can be

used, but its implementation in PSDA has been constrained due to the lack of availability

(Luco and Cornell, 2004). In particular, an IM based on specific levels of maximum inelastic

displacement demand computed from the response of an elastic -perfectly plastic SDOF system

has been identified as promising substitute of traditional Sa (T1) for PSDA (Luco and Cornell,

2004; Miranda and Aslani, 2002). This investigation make use of this improved IM for

characterizing the ground motion intensity of the ensemble of acceleration time histories

employed to perform nonlinear dynamic time-history analyses.

7.5.1 Ensemble of Earthquake Ground Motions

A core part of the deformation demand results reported in this chapter were obtained from the

non-linear dynamic response of the family of generic framed buildings subjected to a set of

40 acceleration time histories recorded during 12 earthquake events occurred in California.

The set of 40 earthquake ground motions, named LMSR-N, considered in this investigation

was originally assembled by Medina and Krawinkler (2003. All earthquake ground motions

included in the LMSR-N set were recorded on stiff soil or soft rock corresponding to soil type

D according to FEMA 356 document (FEMA, 2000). This set comprises earthquake records

with moment magnitude ranging from 6.5 to 6.9 and with distance source-to-site from 13 km

to 40 km. It should be mentioned that none of the records exhibit pulse-type near-fault

characteristics and they can be considered representative of a typical seismic environment

(e.g., magnitude and distance) in California. Details of the earthquake ground motion selection

(e.g., filter cut-off, etc.) and relevant parameters (e.g., peak ground acceleration, duration of

the ground motion, etc.) can be found in Medina and Krawinkler (2003).

In addition to the aforementioned LMSR-N set of ground motions, two additional sets of

earthquake ground motions were assembled in order to study the effect of duration of the

ground motion. Each set included 20 acceleration time histories having short (SD) and long

(LD) strong motion duration. In this investigation, the definition proposed by Trifunac and

Brady (1975) was used to characterize the effective duration of the ground motion. Details of

the SD and LD ground motions set are given in Appendix A.


236

7.5.2 Definition of Relative Intensity Measure

In this study, the maximum inelastic displacement demand computed from the response of an

equivalent single-degree-of-freedom system (SDOF) having elastic-perfectly plastic behavior

and the same dynamic properties (e.g., fundamental period of vibration and mass) of the

building, )( 1Ti∆ , was used as IM throughout this investigation. The advantages of using this

inelastic IM instead of traditional elastic IM are highlighted in the next subsection.

Of particular interest is to compare the seismic response of building models with different

structural and dynamic properties. Therefore, a relative inelastic IM, η, was defined as

follows:

y

i T∆

∆=

)( 1η (7.1)

where ∆y is the yield displacement of the equivalent SDOF system. From structural analysis

theory, the yield displacement of the equivalent SDOF system can be related to the global (i.e.,

roof) yield displacement of the structure, δy, roof, as follows:

11

,

φ

δ

Γ=∆

roofyy (7.2)

where Γ1φ1 is the normalized modal participation factor corresponding to the building’s first-

mode of vibration. The roof yield displacement in the above equation can be obtained from

nonlinear static (pushover) analysis of the structure under consideration assuming an adequate

lateral load pattern (Seneviratna and Krawinker, 1997). Thus, substituting equation (7.2) into

(7.3) yie lds:

11

1)(φ

ηΓ∆

∆=

y

i T (7.3)

Therefore, in this investigation all acceleration time histories were scaled up to a factor

that yields the same maximum inelastic displacement demand of an equivalent elastoplastic


237

SDOF system with the same fundamental period of vibration of the structure and

corresponding to target relative inelastic IM (define as relative intensity, for short). It should

be noted that for relative intensities smaller or equal to 1 each building model is expected to

behave in the elastic range while for relative intensities larger than 1 each building model is

expected to experience nonlinear behavior. For a building model with defined ∆y and Γ1φ1

parameters, the building nonlinear behavior is expected to increase as the relative intensity

measure increases (i.e., as the ground motion intensity increases.

In order to estimate the roof yield displacement of each generic building model, nonlinear

static analyses using a parabolic lateral load pattern were performed using the computer

software RUAUMOKO (Carr, 2003). Figure 7.1 shows normalized base shear versus roof

drift ratio (i.e., roof displacement normalized with respect to the total height) obtained for the

family of generic rigid and flexible framed building models.

(a) Generic Rigid Frames

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.0 0.2 0.4 0.6 0.8 1.0 1.2Roof Drift [%]

Nor

mal

ized

Bas

e S

hear

[V

/W]

N = 6

N = 9

N = 12

N = 15

N = 18

(b) Generic Flexible Frames

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.0 0.2 0.4 0.6 0.8 1.0 1.2Roof Drift [%]

No

rmal

ized

Bas

e S

hea

r [V

/W]

N = 6

N = 9

N = 12

N = 15

N = 18

Figure 7.1. Pushover curves for generic frame models used in this investigation: (a) Rigid models ; (b)

flexible models .

To study the variation of deformation demand measures with changes in the ground

motion intensity (i.e., changes in the relative intensity, η), each generic framed building model

was subjected to nonlinear dynamic time history analyses using the analysis software

RUAUMOKO (Carr, 2003). Therefore, for each building, each ground motion contained in

the LMSR-N ground motion set was scaled to reach six levels of relative intensity (η = 0.5, 1,

2, 3, 4, 5 and 6). Later, all analytical results were statistically organized to study the variation

of central tendency and dispersion of deformation demand measures with changes in the

relative intensity. Appendix C describes the definition of the statistical measures used

throughout this investigation for central tendency and dispersion. It should be mentioned that


238

the range of relative intensities chosen in this study are of practical interest. For this range of

relative intensities it is expected that the actual existing structures do not exhibit severe

structural degradation (i.e., combination of stiffness degradation, strength deterioration and

pinching) or dynamic instability due to P-∆ effects that might lead the building models to

collapse. Evaluation of maximum and residual deformation demands associated to, or in the

vicinity, collapse is beyond the scope of this dissertation.

7.5.3 Advantage of An Inelastic Intensity Measure

Next, several examples will be shown to illustrate the advantage of using the maximum

inelastic displacement demand computed from an elastic -perfectly plastic SDOF system

having the same fundamental period of vibration of each building model, ∆i (T1), as IM

compared to the spectral elastic displacement at the first-mode period of vibration of the

building, Sd (T1), in PSDA. The seismic response of a long-period building model (T1=2.0s)

with 18 stories, which is expected to exhibit higher-mode effects as the building experiences

larger levels of nonlinear behavior, is employed for illustration purposes. Figure 7.2 shows

the variation of counted median roof drift ratio , θroof, as well as counted 16th and 84th percentile

bands computed from the dynamic response of the GF-18R (T1=2.0 s) building frame model

and corresponding to six levels of ground motion intensity (η = 0.5, 1, 2, 3, 4, 5 and 6) using

Sd (T1) and ∆i (T1) as IM’s to scale each ground motion contained in the LMSR-N ground

motion set. From the figure, it is evident that using ∆i (T1) to scale the earthquake ground

motions leads to smaller dispersion around the central tendency of θroof (i.e., record-to-record

variability) than using Sd (T1) even for large levels of intensity associated with high nonlinear

behavior of the structure. A similar plot corresponding to the variation of IDRmax with changes

in the ground motion intensity is shown in figure 7.3. Again, it can be seen the beneficial

effect in reducing the record-to-record variability when an inelastic IM is used.


239

(b) IM-∆ i (T

1)

18-STORY (T1=2.0s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 20 40 60 80 100 120 140∆ i (T1) [cm]

θ roof [%]

p = 84%p = 50%

p = 16%

(a) IM-Sd (T1)

18-STORY (T 1=2.0s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 20 40 60 80 100 120 140Sd (T1) [cm]

θ roof [%]

p = 84%p = 50%p = 16%

Figure 7.2. Variation of θroof with changes in IM: (a) Sd (T1); (b) ∆i (T1).

(b) IM - ∆ i (T1)18-STORY (T

1 = 2.0s)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0 20 40 60 80 100 120 140∆ i (T1) [cm]

IDRmax [%]

p = 84%p = 50%

p = 16%

(a) IM - Sd (T

1)

18-STORY (T1 = 2.0s)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0 20 40 60 80 100 120 140Sd (T1) [cm]

IDRmax [%]

p = 84%p = 50%

p = 16%

Figure 7.3. Variation of IDRmax with changes in IM: (a) Sd (T1); (b) ∆i (T1).

The reduction in the variability (measured by the standard deviation of the natural

logarithm of the roof drift ratio) by using an inelastic IM instead of the traditional elastic

counterpart is illustrated in figure 7.4 for θroof and IDRmax. It can be seen that the variation of

dispersion is more stable (i.e., it does not significantly changes) with changes in the ground

motion intensity when ∆i (T1) is employed as IM. A further benefit of using an inelastic IM is

that smaller record-to-record variability also leads to a reduction in the number of acceleration

time histories required for performing nonlinear time-history analyses during PSDA (i.e., as

the level of intensity increases the dispersion and, hence, the number of records decreases).

This enhancement in efficiency (i.e., small record-to-record variability of the seismic demand

of interest conditioned on the ground motion intensity) when ∆i (T1) is used instead of Sd (T1)

as IM has also been reported by other researchers (e.g., Luco and Cornell, 2004; Miranda et

al., 2004).


240

(b)18-STORY (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0 20 40 60 80 100 120 140Sd (T1), ∆ i (T1) [cm]

σ ln IDRmax

Series1

Series3∆ i (T

1)

Sd (T1)

(a) 18-STORY (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0 20 40 60 80 100 120 140Sd (T1), ∆i (T 1) [cm]

σ ln θroof

Series1

Sd (T1)∆ i (T1)

Sd (T

1)

Figure 7.4. Influence of IM in the variation of dispersion for: (a) Roof drift ratio; (b) maximum inter-

story drift ratio.

Prior investigations that have recorded residual deformation demands, particularly residual

inter-story drift ratio, in MDOF systems have also noted that their estimation involves large,

and in some cases very large, record-to-record variability. Therefore, it is of particular interest

in this investigation to study if using an inelastic IM also leads to a reduction in the dispersion

for different levels of relative intensity. The variation of median residual roof drift ratio, θr,roof,

with changes in the relative intensity, using both the elastic and inelastic IM, computed from

the seismic response of the aforementioned long-period building model is shown in figure 7.5.

From the figures, it can be seen that although dispersion is still high compared to the range of

dispersion observed for median θroof the use of an inelastic IM leads to a slight reduction in the

record-to-record variability. Moreover, the number of outliers (i.e., very large values

compared to the rest of the sample) is considerable reduced.

Another example from the same building model is the variation of median RIDRmax with

changes in the ground motion intensity measured for both Sd (T1) and ∆i (T1), which is shown

in figure 7.6. Again, it can be seen that an inelastic IM is more efficient than the traditional

elastic IM when evaluating RIDRmax at different ground motion intensities, which is

particularly true at high levels of relative intensity. It should be noted that the use of an

inelastic IM leads to smaller dispersion and larger central value, for a given IM, than when the

elastic IM is used. This is due to larger correlation coefficient that exists between RIDRmax and

the inelastic IM than when the elastic IM is used. Finally, figure 7.7 shows the varia tion of

dispersion with changes in both IM’s. It can be seen that variation of dispersion is also more

stable when an inelastic IM is employed for performing incremental dynamic analysis.


241

(a) IM-Sd (T1)

18-STORY (T1 = 2.0s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 20 40 60 80 100 120 140Sd (T1) [cm]

θ r,roof [%]

p = 84%

p = 50%p = 16%

(a) IM-∆i (T1)18-STORY (T1 = 2.0s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 20 40 60 80 100 120 140∆i (T1) [cm]

θr, roof [%]

p = 84%

p = 50%

p = 16%

Figure 7.5. Variation of θr,roof with changes in IM: (a) Sd (T1); (b) ∆i (T1).

(b) IM- ∆ i(T1)18-STORY (T

1=2.0 s)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0 20 40 60 80 100 120 140

∆ i (T1) [cm]

RIDRmax [%]

p = 84%p = 50%p = 16%

(a) IM- Sd (T

1)

18-STORY (T1=2.0 s)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0 20 40 60 80 100 120 140Sd (T1) [cm]

RIDRmax [%]

p = 84%p = 50%p = 16%

Figure 7.6. Variation of RIDRmax with changes in IM: (a) Sd (T1); (b) ∆i (T1).

(b)18-STORY (T1 =2.0 s)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 20 40 60 80 100 120 140Sd (T1), ∆i (T1) [cm]

σ ln RIDRmax

Series2

Series1∆ i (T1)

Sd (T

1)

(a)18-STORY (T1 = 2.0 s)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 20 40 60 80 100 120 140

Sd (T1), ∆ i (T1)

σ ln θ r, roof

Series2

Series1∆ i (T1)

Sd (T

1)

Figure 7.7. Influence of IM in the variation of dispersion: (a) residual roof drift ratio; (b) maximum

residual inter-story drift ratio.


242

7.6 Statistical Evaluation of Maximum and Residual Deformation Demands

for One-Bay Generic Framed Building Models

For identification purposes, each generic frame (GF) was designated with a short

nomenclature. For example, GF-18R corresponds to the 18-story rigid frame while GF-18F

identifies their flexible counterpart. It should be noted that following discussions of

deformation demands for generic framed models is based on considering an elastic -perfectly-

plastic moment-curvature relationship in the plastic hinge region of the frame elements. The

effect of different hysteretic features (e.g., positive strain hardening, stiffness degradation,

strength deterioration, etc.) in the member moment-curvature behavior is studied separately.

7.6.1 Effect of Number of Stories

The distribution of median IDR over the height for three rigid buildings models with 3, 9 and

18 stories is shown in figure 7.8 while the same distribution obtained from their flexible

counterparts is illustrated in figure 7.9. The results were obtained using frame elements with

bilinear moment-curvature hysteretic behavior without strain-hardening.

(a) 3-STORY (T 1 = 0.5 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0IDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b) 9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0IDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(c) 18-STORY (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0IDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.8. Effect of number of stories on the height-wise distribution of median IDR for three rigid

building models: (a) GF-3R (T1=0.5s); (b) GF-9R (T1=1.185s); (c) GF-18R (T1=2.0s).


243

(b) 9-STORY (T1 = 1.902 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0IDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(c) 18-STORY (T1 = 3.311 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0IDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0

η = 4.0η = 6.0

(a) 3-STORY (T=0.793 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0IDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.9. Effect of number of stories on the height-wise distribution of median IDR for three flexible

building models: (a) GF-3F (T1=0.79 s); (b) GF-9F (T1=1.902 s); and (c) GF-18F (T1=3.311s).

From the figures, it can be seen that the drift profile changes as the number of stories

increases, for both rigid and flexible building models. However, although the amplitude of

drift demands is different, it can be seen that drift profiles follows as very similar trend for a

building model with the same number of stories, but different period of vibration for all levels

of ground motion intensity. In particular, it can be observed that the distribution of IDR along

the height becomes non-uniform as the relative intensity increases and it tends to concentrate

at specific stories. For low levels of relative intensity (e.g., η equal to or smaller than 2), the

largest drift demands over the height occurs in the upper stories while for a ground motion

intensity beyond this level the main drift concentration moves towards the lower stories. This

tendency of drift concentration is more evident for flexible frames than for rigid frames. The

larger increment in maximum deformation demand in the lower stories of tall flexible frames

(e.g., GF-18F model) is attributed to P-∆ effects. Drift concentration has also been reported for

generic frame models designed to exhibit ideal beam-hinge mechanism (Medina and

Krawinkler, 2003). It should be also noted that a secondary drift concentration tends to occur

in the upper stories, which is more evident for tall flexible frames. This secondary drift

concentration as the frame becomes taller and flexible reflects the effect of higher modes of

vibration (mainly second-mode of vibration) on the seismic response of regular framed

buildings.

In addition of investigating the distribution along the height of maximum drift demands, it

is also interesting to study if dispersion of central values is modified by the number of stories.


244

The height-wise dispersion distributions computed for each of the rigid frame models are

shown in figure 7.10 whereas similar plots obtained from the flexible frame models are shown

in figure 7.11. It can be seen that the distribution of dispersion along the height is not constant

and, in most of the stories for a particular building model, it tends to increase as the ground

motion intensity increases (i.e., dispersion increases as the building experiences larger levels

of nonlinear behavior). It should be noted that larger level of dispersion occurs in the upper

and bottom stories than in the intermediate stories as the relative intensity increases.

(c) 18-STORY (T1 = 2.0s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6

σ ln IDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b) 9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6σ ln IDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(a) 3-STORY (T1 = 0.5 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6σ ln IDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.10. Effect of number of stories on the height-wise dispersion distribution of IDR for three rigid

building models: (a) GF-3R (T1=0.5s); (b) GF-9R (T1=1.185s); and (c) GF-18R (T1=2.0s).

(a) 3-STORY (T1 = 0.793 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6σ ln IDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b) 9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6σ ln IDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η =4.0η = 6.0

(c) 18-STORY (T1 = 3.311s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6σ ln IDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.11. Effect of number of stories on height-wise dispersion distribution of IDR for three flexible

building models: (a) GF-3F (T1=0.79 s); (b) GF-9F (T1=1.902 s); (c) GF-18F (T1=3.311s).


245

(b) 9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0RIDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(a) 3-STORY (T1 = 0.5 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0RIDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

( c ) 18-STORY (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0RIDR [%]

z / H

η = 0.5

η = 1.0η = 2.0η = 3.0

η = 4.0η = 5.0

Figure 7.12. Effect of number of stories on the height-wise distribution of median RIDR for three rigid building models: (a) GF-3R (T1=0.5s); (b) GF-9R (T1=1.185s); (c) GF-18R (T1=2.0s).

(c) 18-STORY (T1 = 3.311 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0RIDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b) 9-STORY (T1 = 1.902 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0RIDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(a) 3-STORY (T1 = 0.783 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0RIDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.13. Effect of number of stories on the height-wise distribution of median RIDR for three flexible building models: (a) GF-3F (T1=0.79 s); (b) GF-9F (T1=1.902 s); (c) GF-18F (T1=3.311s).

Of particular interest in this investigation is to know the variation of median RIDR along

the height. This variation is shown in figures 7.12 and 7.13 for the same rigid and flexible

frame models, respectively. Similarly to the variation of IDR, the median RIDR profiles

changes as the number of stories increases. The median RIDR pattern tends to mimic the same

trend observed for median IDR for both rigid and flexible building models. For instance, it can

be observed that residual drift also tends to become non-uniformly distributed along the height

as the ground motion intensity increases. Moreover, it also tends to concentrate at the same


246

relative height where transient drift concentration occurred. This observation simply means

that once structural damage is triggered by large inter-story drifts demands, and it tends to

concentrate in specific stories, the building adopts a non-recoverable configuration until the

end of the earthquake excitation.

Figures 7.14 and 7.15 show the distribution of dispersion of RIDR along the height

computed for both rigid and flexible buildings. The first observation is that the dispersion of

residual drift demands is considerably higher than that computed for maximum (transient)

drift demands for all levels of ground motion intensity considered in this investigation.

Moreover, similar levels of dispersion are observed regardless of the number of stories. In

addition, dispersion does not follow a consistent pattern with increasing intensity along the

height. Large level of dispersion of residual drift demand in MDOF building models has also

been reported by Krawinkler and his co-workers (Gupta and Krawinkler, 1999; Medina and

Krawinkler, 2003) and Pampanin et al. (2002).

(c) 18-STORY ( T

1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0σ ln RIDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b)9-STORY

( T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0σ ln RIDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(a)3-STORY (T1 = 0.5 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0σ ln RIDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.14. Effect of number of stories on the height-wise dispersion distribution of RIDR for three

rigid building models: (a) GF-3R (T1=0.5s); (b) GF-9R (T1=1.185s); (c) GF-18R (T1=2.0s).


247

(b) 9-STORY

(T1 = 1.902 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0σ ln RIDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(c) 18-STORY

(T1 = 3.311 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0σ ln RIDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(a) 3-STORY (T 1 = 0.783 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0σ ln RIDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.15. Effect of number of stories on height-wise dispersion distribution of RIDR for three

flexible building models: (a) GF-3F (T1=0.79 s); (b) GF-9F (T1=1.902 s); (c) GF-18F (T1=3.311s).

The effect of the number of stories, N, on the variation of median maximum and residual

deformation demands (i.e., roof drift ratio and inter-story drift ratio) with respect to relative

intensity for both rigid and flexible generic framed buildings considered in this investigation is

further investigated next. First, the variation of median θroof and θr,roof with respect to the

relative intensity, η, for both rigid framed buildings is shown in figure 7.16 while a similar

plot for flexible buildings is illustrated in figure 7.17.

(a) Generic Rigid FramesIM - ∆i (T1)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

θ r,roof [%]

N = 18

N = 15N = 12N = 9

N = 6

N = 3

(a) Generic Rigid Frames IM - ∆i (T1)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

θ roof [%]

N = 18N = 15

N = 12

N = 9N = 6

N = 3

Figure 7.16. Effect of number of stories on generic rigid frames: (a) median θroof; and (b) median θr,roof .


248

(b) Generic Flexible Frames IM - ∆ i (T1)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

θ roof [%]

N =18N =15N =12N =9N =6

N = 3

(b) Generic Flexible FramesIM - ∆ i (T1)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆i)

θ r,roof [%]

N = 18N = 15N = 12N = 9N = 6

N = 3

Figure 7.17. Effect of number of stories on for generic flexible frames: (a) median θroof; and (b) median

θr,roof.

First, it can be seen that, when the records are scaled to the same maximum inelastic

displacement, N does not have an important influence on median θroof for any level of the

ground motion intensity. This observation is valid for both rigid and flexible buildings and for

the levels of relative intensity considered in this research. It can be also observed that roof

drift demands of flexible frames grow at a faster rate than those of rigid frame models as the

relative intensity increases. On the other hand, it can be observed that median θr,roof does not

significantly increases as N increases for any the level of relative intensity, for both rigid and

flexible generic frames. However, it should be noted that flexible frames exhibit larger

residual roof drift demands than rigid frames even though they have the same number of

stories. For example, the frame GF-18F (T1=3.35 s.) would sustain about twice residual roof

drift than its counterpart GF-18R (T1=2.0s) for a relative intensity of 6. Similarly to roof drift

demands, median residual roof drift demands of flexible frames grow at a faster rate as the

level of relative intensity increases than that on rigid frames. Finally, it should be noted that

median θ,roof grows at a faster rate than median θr,roof as the relative intensity increases.

The influence of N on the variability of estimating θroof and θr,roof is shown in figures 7.18

for rigid building models whereas a similar plot for flexible building models is presented in

figure 7.19. The first observation is that dispersion of θroof follows a smooth nonlinear

increasing trend as the level of intensity increases and as N increases, for both rigid and

flexible building models. However, for a given level of intensity, dispersion tends to saturate

as N increases. It should also be noted that dispersion tends to stabilize as the severity of the

ground motion increase. It is worthy to mention that dispersion in the estimation of θroof is


249

smaller than that reported in previous studies that have employed elastic intensity measures

(e.g., Medina and Krawinkler, 2003). On the other hand, it was found that the variability in the

estimation of θr,roof is much larger than the dispersion of θroof . This variability is shown in

figures 7.19a for rigid frames and figure 7.20a for flexible frames. In general, the levels of

dispersion range above 0.8 regardless the number of stories or the flexibility of the building.

Unlike dispersion of θroof, it can be seen that dispersion of θr,roof does not follow a clear trend

with respect to N and η. Finally, it should be mentioned that even though using ∆i (T1) as

intensity measure allows reducing the record-to-record variability inherent in the estimation of

θroof it is itself insufficient to reduce the variability in the estimation of residual roof drift

demands. However, the use of ∆i (T1) as intensity measure leads to smaller variability in the

estimation of θr,roof than the use of traditional Sd (T1).


0.0

0.1

0.2

0.3

0.4

0.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η(∆ i)

σ ln θ roof

N = 18N = 15N = 12N = 9N = 6N = 3


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η(∆ i)

σ ln θr,roof

N = 18

N = 15

N = 12

N = 6

N = 3

Figure 7.18. Effect of number of stories on dispersion of rigid frames:

(a) Dispersion of θroof; (b) dispersion of θr,roof.

(b) Generic Flexible Frames IM - ∆i (T1)

0.0

0.1

0.2

0.3

0.4

0.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η(∆ i)

σ ln θ roof

N = 18N = 15N = 12N = 9

N = 6N = 3

(b) Generic Flexible Frames IM - ∆i (T1)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η(∆ i)

σ ln θ r,roof

N = 18N = 15N = 12N = 9

N = 6N = 3

Figure 7.19. Effect of number of stories on dispersion of flexible frames:

(a) Dispersion of θroof; (b) dispersion of θr,roof.


250

Another important measure of deformation demand which is closely related to damage in

structural elements, and some deformation-dependent non-structural components, is the

maximum inter-story drift ratio over all stories, IDRmax. The relationship between median

IDRmax and relative intensity as a function of the number of stories is shown in figure 7.20a for

generic rigid frames and in figure 7.21a for their flexible counterparts. From these figures, it

can be seen that median IDRmax is larger for flexible models than for rigid models regardless of

the level of intensity, which means that building models with similar N but different period of

vibration experience larger levels of IDRmax. For a given type of building, it can be observed

that median IDRmax grows at a different rate with respect to the ground motion intensity

depending on N, which is particularly true for flexible building models.

In addition of maximum drift demands, information about the maximum residual inter-

story drift ratios is of high importance. Then, to allow a visual comparison of the amplitude of

residual deformation demands with respect to maximum deformation demands, median

RIDRmax for both flexible and rigid models is shown in figures 7.20b and 7.21b, respectively.

It can be observed that median RIDRmax increases almost linearly as the leve l of the ground

motion increases and as N increases, for both rigid and flexible models. As can be expected,

flexible building models lead to larger residual deformation demands than rigid models even if

they have the same N. Similarly to the variation of IDRmax, RIDRmax also grows at a rate that

depends on N as the relative intensity increases.

The variability of median IDRmax and RIDRmax as a function of the number of stories and

the relative intensity for rigid frame buildings is shown in figure 7.22 whereas similar plots are

illustrated in figure 7.23 for flexible building models.


0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

RIDRmax [%]

N = 18N = 15N = 12N = 9N = 6N = 3


0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

IDRmax [%]

N = 18N = 15N = 12N = 9N = 6N = 3

Figure 7.20. Effect of number of stories and ground motion intensity of rigid frames on:

(a) Median IDRmax; (b) median RIDRmax.


251

(b) Generic Flexible FramesIM - ∆i (T1)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

RIDRmax [%]

N = 18N = 15N = 12N = 9N = 6N = 3

(b) Generic Flexible FramesIM-∆i (T1)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

IDRmax [%]

N = 18N = 15N = 12N = 9

N = 6N = 3

Figure 7.21. Effect of number of stories and ground motion intensity of flexible frames on:

(a) Median IDRmax; (b) median RIDRmax.

(b) Generic Rigid FramesIM - ∆i (T1)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

σ ln RIDRmax

N = 6

N = 9

N = 12

N = 15

N = 18


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

σ IDRmax

N = 18N = 15

N = 12N = 9N = 6

N = 3

Figure 7.22. Effect of number of stories on dispersion of rigid frames:

(a) Dispersion of IDRmax; (b) dispersion of RIDRmax.


0.0

0.2

0.4

0.6

0.81.0

1.2

1.4

1.6

1.8

2.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

σ ln RIDRmax

N = 3N = 6N = 9N = 12

N = 15N = 18

(a) Generic Flexible FramesIM - ∆i (T1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

σ IDRmax

N = 18N = 15N = 12N = 9

N = 6N = 3

Figure 7.23. Effect of number of stories on dispersion of flexible frames:

(a)Dispersion IDRmax; (b) dispersion RIDRmax.


252

An initial observation is that the variability in the estimation of IDRmax tends to increase as

the N increases, but it does not seem to strongly depend on the level of relative intensity, for

both rigid and flexible building models. In general, dispersion of IDRmax is larger than that

reported for θroof. On the other hand, two trends in the dispersion of RIDRmax can be observed

for both type of building models. Initially, dispersion decreases as the level of relative

intensity decreases up to η=2 while dispersion does not vary significantly for relative

intensities larger than two. In the latter region, dispersion ranges between 0.4 and 0.6, which is

still larger than that observed in the estimation of IDRmax. Finally, with exception of η=0.5, it

seems that the level of dispersion is not very sensitive to changes in the number of stories

regardless of the level of relative intensity.

From the results reported in this section, it is clear that dispersion (i.e., record-to-record

variability) is larger during the estimation of RIDRmax than in the estimation of IDRmax and, in

consequence, it should be taken into account if residual deformation checking would like to be

incorporated into performance-based assessment of existing structures.

7.6.2 Effect of Period of Vibration

In the last section, it was noted that frames with the same number of stories but different

period of vibration could sustain different maximum and residual drift demands. Then, it is

interesting a closer look at the effect of period of vibration on both lateral deformation

demands. The relationship between deformation demands (median θroof and θr,roof ) and

fundamental period of vibration for six levels of relative intensity is illustrated in figure 7.24

for rigid building models and in figure 7.25 for flexible building models. It can be seen that, in

general, median θroof does not significantly change with respect to the period of vibration, for a

given relative intensity, for both rigid and flexible generic frames when earthquake ground

motions are scaled to the same ∆i (T1). However, for a given period of vibration, median θroof

increases as the relative intensity increases. It should be noted that median θroof grows at a

faster rate for flexible frames than for rigid frames. On the other hand, for a given relative

intensity, it can also be seen that the variation of median θr,roof is very stable (i.e., it does not

significantly changes) with changes in the period of vibration, for both type of generic frames.

For the same period of vibration, median θr,roof increases non-proportionally as the relative


253

intensity increases, although at a faster rate for flexible frames than for their rigid

counterparts.

0.50.9 1.2

1.51.8

2.00.5

1.02.0

3.04.0

6.0

0.00.5

1.01.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0θroof [%]

Period [s]

η (∆i)

(a)

0.5 0.91.2

1.5 1.82.0

0.51.0

2.03.0

4.06.0

0.00.51.0

1.5

2.0

2.53.0

3.5

4.0

4.5

5.0θr,roof [%]

Period [s]

η (∆i)

(b)

Figure 7.24. Effect of period of vibration on generic rigid framed building models:

(a) Median θroof; (b) median θr,roof.

0.8 1.31.9

2.4 2.93.3

0.51.0

2.03.0

4.06.0

0.00.51.0

1.52.0

2.5

3.0

3.5

4.0

4.5

5.0θroof [%]

Period [s]

η (∆i)

(a)

1.31.9 2.4

2.9 3.3 0.51.0

2.03.0

4.06.0

0.0

0.51.01.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0θr,roof [%]

Period [s]

η (∆i)

(b)

Figure 7.25. Effect of period of vibration on generic flexible framed building models:

(a) Median θroof; (b) median θr,roof .

The dispersion associated to the estimation of both θroof and θr,roof for rigid frame models is

presented in figures 7.26 while a similar plot for flexible building models is illustrated in

figure 7.27. In general, it can be observed that dispersion of θroof tends to increase as the period

of vibration increases and as the level of relative intensity increases, which is particularly true

for flexible building models. For instance, tall-flexible framed buildings experience more

variability in the estimation of θroof for than short-rigid framed buildings. The latter

observation can be explained since tall-flexible buildings are expected to exhibit higher mode

effects and higher nonlinear behavior as the relative intensity increases than low-rise and rigid


254

buildings. In addition, the variability in the estimation of θr,roof can be observed in figures

7.26b and 7.27b for rigid and flexible buildings. From the figure, it is evident that the

estimation of θr,roof involves larger dispersion than that of estimating θroof over the period range

covered in this investigation. Dispersion of θr,roof is, in general, larger than 0.8 but a clear trend

with respect to the period of vibration and the relative intensity was not identified.


0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Period [s]

σ ln θ roof

η = 6.0

η = 4.0η = 3.0

η = 2.0η = 1.0η = 0.5


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Period (s)

σ ln θr,roof

η = 6.0

η = 4.0η = 3.0η = 2.0

η = 1.0η = 0.5

Figure 7.26. Effect of period of vibration on dispersion for generic rigid frames: (a) Dispersion of θroof ;

and (a) dispersion of θr,roof .


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Period (s)

σ ln θr,roof

η = 6.0η = 4.0

η = 3.0η = 2.0η = 1.0

η = 0.5


0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Period [s]

σ ln θ r,roof

η = 6.0

η = 4.0η = 3.0η = 2.0

η = 1.0η = 0.5

Figure 7.27. Effect of period of vibration on dispersion for generic flexible frames: (a) Dispersion of

θroof; and (b) Dispersion of θr,roof ..

The aforementioned observations have important implications for PSDA-based

methodologies, such as that developed as part of the PEER project (Cornell and Krawinkler,

2000), since they assume that dispersion in the estimation of deformation demand parameters

due to record-to-record variability is constant regardless of the building’s fundamental period

and the expected level of nonlinearity of the structure. It was clearly shown that neither the

dispersion of θroof nor θr,roof follow a constant rate. Therefore, the variation of dispersion with


255

changes in the relative intensity should be taken into account while evaluating deformation

demands for performance-based assessment of existing structures.

The influence of period of vibration on median IDRmax and RIDRmax with changes in the

level of relative intensity was also investigated and it is illustrated in figure 7.28 for rigid

building models and figure 7.29 for flexible building models. First, it can be seen that median

IDRmax increases as the period of vibration increases and as the relative intensity increases for

both rigid and flexible models. However, as expected, flexible models yield larger maximum

drift demands than rigid counterparts. On the other hand, it can be observed that the variation

of median RIDRmax depends on changes of both the period of vibration and η. Similarly to the

observation made for IDRmax, median RIDRmax demands grow at a faster rate in flexible

models than in rigid models.

0.5 0.9 1.2 1.5 1.8 2.00.5

1.02.0

3.04.0

6.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0IDRmax [%]

Period [s]

η (∆ i)

(a)

0.5 0.9 1.2 1.5 1.8 2.00.5

1.02.0

3.04.0

6.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0RIDRm a x [%]

Period [s]

η (∆i)

(b)

Figure 7.28. Effect of period of vibration for generic rigid frames:

(a) Median IDRmax, and (b) median RIDRmax.

0.8 1.3 1.9 2.4 2.9 3.30.5

1.02.0

3.04.0

6.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0IDRmax [%]

Period [s]

η (∆i )

(a)

0.8 1.3 1.92.4 2.9

3.30.5

1.02.0

3.04.0

6.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0RIDRmax [%]

Period [s]

η (∆i)

(b)

Figure 7.29. Effect of period of vibration for generic flexible frames:

(a) Median IDRmax, and (b) median RIDRmax.


256

The variability of both maximum and residual deformation demands with respect to the

period of vibration and the relative intensity for rigid frame models is shown in figure 7.30

while a similar plot for flexible building models is shown in figure 7.31. From the figures, it

can be seen that dispersion of IDRmax for both types of frame models is not constant over the

period spectral region. Furthermore, for a given period of vibration, it changes with changes in

the level of relative intensity. Particularly for flexible frame models, dispersion tends to

increase as the fundamental period of vibration elongates. On the other hand, from the figures

it can be observed that the variability in estimating RIDRmax is, in general, larger than that of

IDRmax. It can be seen that dispersion of RIDRmax is not significantly affected by changes in the

fundamental period of vibration. Finally, although not very clear, it seems that the dispersion

of RIDRmax tends to stabilize as the level of relative intensity increases.


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Period [s]

σ ln RIDRmax

η = 6.0

η = 4.0

η = 3.0

η = 2.0

η = 1.0


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Period [s]

σ ln IDRmax

η = 6.0η = 4.0η = 3.0η = 2.0η = 1.0η = 0.5

Figure 7.30. Effect of period of vibration on dispersion for generic rigid frames: (a) Dispersion of

IDRmax; (b) dispersion of RIDRmax.


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Period [s]

σ ln RIDRmax

η = 6.0

η = 4.0

η = 3.0η = 2.0

η = 1.0

(a) Generic Flexible FramesIM - ∆i (T1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Period [s]

σ ln IDRmax

η = 6.0η = 4.0η = 3.0η = 2.0η = 1.0η = 0.5

Figure 7.31. Effect of period of vibration on dispersion for generic flexible frames: (a) Dispersion of

IDRmax; (b) Dispersion of RIDRmax.


257

7.6.3 Effect of the Type of Mechanism

The observations made in previous sections were obtained from the generic frame models

described in Section 7.3, which are expected to develop a “strong-column weak-beam”

mechanism. In current seismic provisions, this mechanism is fomented by specifying a beam-

to-column moment ratio. For example, the ACI-2000 provisions (ACI, 2003) requires that the

sum of the flexural moment capacity of the columns, Mc, must be greater than, or equal to, 1.2

times the flexural moment capacity of the beams, Mb, that converges in any joint,

∑ ∑≥ bc MM 2.1 . However, based on experimental evidence, this approach has been

criticized of being insufficient to ensure the development of plastic hinges only in the beams

of reinforced concrete structures during seismic excitation (Paulay, 1983). In the AISC 1997

seismic provisions (1997) for the design of new steel structures it is required

that∑ ∑> **pbpc MM , where ∑ *

pcM is the sum of the moments in the column above and

below the joint at the intersection of the beam and column centerlines and ∑ *pbM is the sum

of the moments in the beams at the intersection of the beam and column centerlines. This

seismic provisions encourages the use of improved connections (e.g., reduced-beam section

connection commonly known as “dog-bone”) that force the plastic hinges to develop in the

beams. However, existing steel structures designed with traditional moment connections, or

with non-ductile moment connections, could develop plastic hinging in the columns as has

been reported from post-earthquake field reconnaissance or analytical investigations.

From the discussion above, it is interesting to compare maximum deformation demands

and, in particular, the residual deformation demands generated when the framed building

models develop different types of mechanism. In this section, two ideal mechanisms were

compared with the expected mechanism (referred herein as full-hinge, FH, mechanism) of the

generic frames: (a) Beam-hinge (BH) mechanism, which was obtained by increasing 100

times the original moment capacity of columns; and (b) column-hinge (CH) mechanism,

which was forced to develop by scaling up 100 times the original beam flexural capacity. The

aforementioned cases represent two extreme values of the beam-to-column moment ratio. It

should be mentioned that the stiffness distribution and, thus, the modal shapes were kept

identical to the original generic frame models. In addition, in order to isolate the effect of the


258

type of mechanism, all frame elements considered elastoplastic moment-curvature

relationships regardless of the expected type of mechanism.

To illustrate the influence of the type of mechanism, median IDR profiles obtained from

the response of the GF-9R model, considering three types of mechanism, when subjected to

six levels of relative intensity are shown in figure 7.32. From the figures, it can be seen that

both the ground motion intensity and the type of mechanism influence the variation of median

IDR along the height. For example, as would be expected, CH mechanism leads to

concentration of median IDR in the first story as the relative intensity increases, while BH

mechanism yields to more uniform distribution of median IDR along the height, although

large levels of transient drift can still be appreciated at the bottom stories. As a consequence of

drift concentration, the smallest levels of drift occur in the intermediate stories in buildings

developing FH and CH mechanisms. In addition, in all mechanisms, larger median IDR are

expected in the upper stories for low levels of intensity whereas maximum median IDR moves

towards the bottom stories as the ground motion intensity increases. This drift-migration has

also been observed by Medina and Krawinkler (2003) for ideal beam-hinge type generic frame

models. In addition, it is also observed that CH and FH mechanisms also exhibit a secondary

drift concentration on the upper stories for large levels of intensity, which is a consequence of

higher-mode effects, while this secondary drift concentration is not identified when ideal BH

mechanism is considered. In order to investigate if buildings with the same number of stories

but different periods of vibration are influenced by the type of frame mechanism, figure 7.33

shows the drift profile for the flexible building models counterparts. It can be seen that, in

general, the height-wise drift pattern variation is similar regardless of the flexibility of the

building model for the levels of intensity considered in this study.


259

(a) FHM9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0IDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b) BHM9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0IDR [%]

z / H

η = 0.5

η = 1.0η = 2.0

η = 3.0η = 4.0η = 6.0

(c) CHM 9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0IDR [%]

z / H

η = 0.5

η = 1.0η = 2.0η = 3.0

η = 4.0η = 6.0

Figure 7.32. Effect of the type of mechanism on the height-wise distribution of median IDR for GF-9R

(T1=1.185 s): (a) FH mechanism; (b) BH mechanism; (c) CH mechanism.

(a) FHM9-STORY (T1 = 1.902 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0IDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0

η = 6.0

(b) BHM9-STORY (T1 = 1.902 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0IDR [%]

z / H

η = 0.5

η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(c) CHM9-STORY (T1 = 1.902 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0IDR [%]

z / H

η = 0.5η = 1.0η = 2.0

η = 3.0η = 4.0η = 6.0

Figure 7.33. Effect of the type of mechanism on the height-wise distribution of median IDR for GF-9F (T1=1.902 s): (a) FH mechanism; (b) BH mechanism; (c) CH mechanism.

In addition of studying the effect of frame mechanism in drift profiles, it is also interesting

to observe the variation of dispersion along the height due to the type of frame mechanism.

Height-wise variations of dispersion corresponding to the types of mechanism considered here

are shown in figures 7.34 and 7.35. It can be seen that the type of mechanism has also

influence in the dispersion profiles. It is interesting to observe that although secondary drift

concentration in the upper stories is not visible in BH mechanism the largest levels of

dispersion tend to occur in upper stories. In addition, unlike drift profiles for buildings with

the same number of stories but different periods of vibration, dispersion profiles seems to be


260

influenced for the building flexibility. For example, for the flexible building that develops

ideal BH mechanism, the largest levels of dispersion occur at the upper and lower stories.

(a) FHM 9-STORY (T

1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6σ ln IDR

z / H

η = 0.5

η = 1.0η = 2.0η = 3.0

η = 4.0η = 6.0

(c ) CHM 9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6σ ln IDR

z / H

η = 0.5η = 1.0

η = 2.0η = 3.0η = 4.0

η = 6.0

(b) BHM9-STORY (T

1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6σ ln IDR

z / H

η = 0.5η = 1.0

η = 2.0η = 3.0η = 4.0

η = 6.0

Figure 7.34. Height-wise dispersion distribution of IDR for GF-9R (T1=1.185 s) model:

(a) FH mechanism; (b) BH mechanism; (c) CH mechanism.

(a) FHM9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6σ ln IDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η =4.0η = 6.0

(b) BHM9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6σ ln IDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(c) CHM 9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6σ ln IDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.35. Height-wise dispersion distribution of IDR for GF-9F (T1=1.902 s) model:


The variation of residual drift demand at the end of the excitation along the height for the

same frame model and the same mechanism features is shown in figure 7.36. From the figure,

it is observed that the height-wise distribution of median RIDR for all type of mechanisms

mimic the corresponding distribution of IDR. Therefore, both CH and FH mechanisms lead to

a concentration of median RIDR in the lower stories as the ground motion intensity increases,

while BH mechanism tends to constraint residual drift in the bottom stories. In addition, as


261

was described earlier, both FH and CH mechanism exhibit a secondary residual-drift

concentration at the top stories as the relative intensity increases. However, this feature is not

observed in the BH mechanism, which means that beam-hinge mechanism is less influenced

by the higher modes and, thus, it seems to have a re-centering characteristic that constraint

permanent drift demands. Additional residual drift profiles obtained from the flexible

counterpart are shown in figure 7.37. It can be seen that, in general, the flexible building

exhibits similar residual drift profiles than the rigid counterpart. It should be noted that, in

general, the largest residual drifts occur when the FH mechanism develops.

( c ) CHM9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0RIDR [%]

z / H

η = 0.5

η = 1.0

η = 2.0

η = 3.0

η = 4.0

η = 6.0

(a) FHM9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0RIDR [%]

z / H

η = 0.5η = 1.0

η = 2.0η = 3.0η = 4.0

η = 6.0

(b) BHM9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0RIDR [%]

z / H

η = 0.5

η = 1.0η = 2.0

η = 3.0η = 4.0

η = 6.0

Figure 7.36. Effect of the type of mechanism on the height-wise distribution of RIDR for

GF-9R (T1=1.185 s.): (a) FH mechanism; (b) BH mechanism; (c) CH mechanism.

(a) FHM9-STORY (T1 = 1.902 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0RIDR [%]

z / H

η = 0.5

η = 1.0

η = 2.0η = 3.0

η = 4.0

η = 6.0

(b) BHM9-STORY (T1 = 1.902 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0RIDR [%]

z / H

η = 0.5

η = 1.0η = 2.0

η = 3.0η = 4.0

η = 6.0

( c ) CHM9-STORY (T1 = 1.902 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0RIDR [%]

z / H

η = 0.5

η = 1.0

η = 2.0

η = 3.0

η = 4.0

η = 6.0

Figure 7.37. Effect of the type of mechanism on the height-wise distribution of RIDR for

GF-9F (T1=1.902 s.): (a) FH mechanism; (b) BH mechanism; (c) CH mechanism.


262

Finally, dispersion profiles of residual drift demands corresponding to each of the building

mechanisms are shown in figure 7.38, for the rigid building models while similar plots for the

flexible counterpart are illustrated in figure 7.39. It can be seen that all frame mechanisms

leads to larger levels of dispersion along the height. In all cases, dispersion profiles do not

follow a clear trend with changes in the relative intensity.

(a) FHM9-STORY

( T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5σ ln RIDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b) BHM 9-STORY

( T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5σ ln RIDR

z / H

η = 0.5

η = 1.0η = 2.0

η = 3.0η = 4.0

η = 6.0

(c) CHM9-STORY

(T 1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5σ ln RIDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.38. Height-wise dispersion distribution of RIDR for GF-9F (T1=1.902 s) model:


(a) FHM9-STORY

( T1 = 1.902 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0σ ln RIDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b) BHM9-STORY

(T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0σ ln RIDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(c) CHM 9-STORY

(T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0σ ln RIDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.39. Height-wise dispersion distribution of RIDR for GF-9F (T1=1.902 s) model:



263

In order to further study the influence of the type of mechanism, the variation of median

IDRmax and RIDRmax with respect to the relative intensity for the GF-9R building model is

illustrated in figure 7.40. As expected from the previous discussion, it can be seen that CH

mechanism leads to the largest median IDRmax since the building frame model experiences

drift concentration in the lowest story. In addition, it can be observed that BH and FH

mechanisms yield similar median IDRmax up to η=3, even though their height-wise distribution

of IDR is different. Similarly to the variation of median IDRmax, it can be observed that CH

mechanism leads to slightly larger median RIDRmax than FH mechanism as the ground motion

intensity increases and that ideal BH mechanism produces the smallest residual deformation

demands among the mechanism studies here.

(a) 9-STORY (T1 = 1.185 s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

IDRmax [%]

FHM

BHM

CHM

(b) 9-STORY (T1 = 1.185 s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

RIDRmax [%]

FHM

BHM

CHM

Figure 7.40. Effect of the type of mechanism for GF-9R (T1 = 1.185 s) model on:

(a) Median IDRmax ; (b) median RIDRmax

On the other hand, figure 7.41 shows the variation of dispersion, for both IDRmax and

RIDRmax, associated with each building mechanism with changes in the ground motion

intensity. It can be seen that dispersion of IDRmax follows an increasing trend regardless of the

type of mechanism but, for a given relative intensity, the level of dispersion is slightly affected

by the type of building mechanism. With respect to dispersion of RIDRmax, it can be observed

that the variability tends to stabilize for relative intensities greater than 2 for all three building

mechanism. For this ground motion intensity range, it seems that ideal BH mechanism leads to

larger dispersion than CH and FH mechanism.


264

(a) 9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

σ ln IDRmax

FHM

BHM

CHM

(b) 9-STORY (T1 = 1.185 s)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

σ ln RIDRmax

FHM

BHM

CHM

Figure 7.41. Effect of the type of mechanism for GF-9R (T1 = 1.185 s) model on:

(a) Dispersion of IDRmax ; (b) dispersion of RIDRmax

A similar plot for the GF-9F (T1=1.902 s) building is shown in figure 7.42. It can be seen

that unlike its rigid counterpart, the three types of building mechanism lead to similar variation

of median IDRmax as the ground motion intensity increases. The former observation might

suggest that for buildings with the same number of stories, but different period of vibration,

the building mechanism has smallest effect on median IDRmax for flexible structures than for

rigid counterparts. In the companion figure, it can be seen that CH mechanism still leads to

larger median RIDRmax than that when FH and BH mechanisms are expected to develop in the

flexible building model.

(a) 9-STORY (T1 = 1.902 s)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

IDRmax [%]

FHM

BHM

CHM

(b) 9-STORY (T1 = 1.902 s)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

RIDRmax [%]

FHM

BHM

CHM

Figure 7.42. Effect of the type of mechanism for GF-9F (T1 = 1.902 s) model on:

(a) Median IDRmax ; (b) median RIDRmax

In addition, the variation of dispersion of IDRmax measured from the flexible building

seems more stable than that computed for the rigid counterpart as the relative intensity

increases (see figure 7.43). For a given level of ground motion intensity, the level of


265

dispersion slightly depends on the type of mechanism. The variation of dispersion of RIDRmax

shows similar trend than that observed for the rigid building model, which might suggest that

the variability of RIDRmax is not significantly influenced for the period of vibration for framed

buildings with the same number of stories.

(a) 9-STORY (T1 = 1.902 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

σ ln IDRmax

FHM

BHM

CHM

(b) 9-STORY (T1 = 1.902 s)

0.0

0.4

0.8

1.2

1.6

2.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

σ ln IDRmax

FHM

BHM

CHM

Figure 7.43. Effect of the type of mechanism for GF-9F (T1 = 1.902 s) model on:

(a) Dispersion of IDRmax ; (b) dispersion of RIDRmax

From the previous observations, it was noted that ideal beam-hinge mechanism would be

beneficial not only to avoid damage on the columns or damage concentration but also to

constraint permanent deformations after earthquake excitation. This type of mechanism (i.e.,

strong-column weak-beam mechanism) should be promoted through increasing beam-to-

column moment ratio requirements. However, it should also be mentioned that this type of

mechanism leads to slightly larger levels of dispersion of IDRmax and RIDRmax as the ground

motion intensity increases.

7.6.4 Effect of Member Hysteretic Behavior

The influence of the hysteretic behavior on seismic demands has been a topic mainly

addressed through the response of SDOF systems, but fewer studies have been performed in

MDOF systems that confirm observations made from SDOF analyses. Due to limitation in the

analytical tools, most of the studies done in MDOF systems have considered stiffness-

degrading hysteretic behavior (e.g., modified-Clough model or Takeda model) , but very little

have included strength-and-stiffness degrading hysteretic models, with or without cyclic

deterioration (e.g., Foutch and Shi, 1998; Luco and Cornell; 2002; Ibarra and Krawinkler;


266

2003; Medina and Krawinkler; 2003). In addition, it should be noted that most of the previous

studies in MDOF systems have focused their attention in the evaluation of global or local

maximum deformation demands and very few of them have reported residual deformation

demands (Gupta and Krawinkler, 1999; Pampanin et al., 2002; Medina and Krawinkler, 2003).

Moreover, very little information is known about the impact of hysteretic member modeling

on the amplitude and distribution of residual deformation demands. For instance, Pampanin et

al. (2002) reported that using Takeda (TK) model to represent the member hysteretic behavior

in building framed models leads to smaller residual inter-story drifts than when elastic-

perfectly plastic (EPP) model is employed. This observation is consistent with the results

obtained from SDOF systems presented in Chapter 2 and from other researchers (Pampanin et

al., 2002; Luco et al., 2004) since stiffness-degrading systems exhibit a self-centering

capability that constraint residual deformation demands during cyclic excitation (i.e.,

earthquake-type excitation). However, the authors did not find any significant trend neither

with respect of central tendencies nor dispersion.

A further study on the effect of member hysteretic behavior on maximum and residual

deformation demands is described in this section. This study was divided in three parts. The

first part considered only the effect of positive member strain-hardening (i.e., positive post-

yield stiffness) in the bilinear moment-curvature relationship , M-φ , of frame elements. The

second part included stiffness-degrading M-φ behavior of the frame elements while the third

part considered strength-degrading M-φ hysteretic behavior. In order to assess the effect of

member hysteretic behavior, results for this stage are compared with baseline results obtained

from elastoplastic M-φ hysteretic behavior.

7.6.4.1 Effect of positive strain-hardening

In this subsection, the influence of positive member strain-hardening (i.e., post-yield stiffness

normalized with respect to the initial stiffness) in the moment-curvature relationship of frame

elements on maximum and residual deformation demands is investigated. Two levels of

positive member strain-hardening were considered: α = 2% and 5%. A strain-hardening of 2%

is considered typical in steel components that do not experience strength and stiffness

deterioration due to web or flange local buckling as well as low-cycle fatigue under cycling

loading (e.g., Shi and Foutch, 1998; Filiatrault et al., 2001). To illustrate its influence, the


267

height-wise distribution of median IDR and RIDR obtained from each story of the GF-9R

(T1=1.185 s) building model under three levels of ground motion intensity is shown in figures

7.44 and 7.46. Regarding the distribution of median IDR along the height, it can be seen that

an increment in strain-hardening tends to decrease the concentration of maximum drift in the

lower stories (e.g., z/H smaller than about two-thirds) as the ground motion intensity

increases. However, an increment in member strain-hardening does not have an important

effect for low levels of intensity. It should be noted that an increment in member strain

hardening (e.g., from 2% to 5%) does not significantly reduces median IDR in the upper

stories where a second drift concentration is observed.

(b) η (∆i) = 4.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

IDR [%]

z / H

α = 0.1 %

α = 2.0 %

α = 5.0 %

(c) η (∆ i) = 6.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

IDR [%]

z / H

α = 0.1 %

α = 2.0 %

α = 5.0 %

(a) η (∆i) = 2.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

IDR [%]

z / H

α = 0.1 %

α = 2.0 %

α = 5.0 %

Figure 7.44. Effect of positive member strain-hardening on the height-wise distribution of median IDR

for GF-9R (T1=1.195 s): (a) η =2.0; (b) η = 4.0; (c) η = 6.0.

(a) α = 0.1%9-STORY (T

1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6σ ln IDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b) α = 2%9-STORY (T1=1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6σ ln IDR

z / H

η = 0.5η = 4.0η = 3.0η = 4.0η = 5.0η = 6.0

(c) = 5%9-STORY (T1=1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6σ ln IDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.45. Effect of positive member strain-hardening on the height-wise distribution of dispersion of

IDR for GF-9R (T1=1.185 s) model: (a) α = 0.1%; (b) α = 2%; (c) α = 5%


268

On the other hand, the distribution of dispersion of IDR along the height is shown in

figure 7.45. An interesting observation is that the level of dispersion does not decrease with

increasing member strain hardening. However, the distribution of dispersion seems to more

uniform as the relative intensity increases and as the relative increases.

(c) η (∆i) = 6.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0RIDR [%]

z / H

α = 0.1%

α = 2.0 %

α = 5.0 %

(b) η (∆i) = 4.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0RIDR [%]

z / H

α = 0.1%

α = 2.0 %

α = 5.0 %

(a) η (∆i) = 2.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0RIDR [%]

z / H

α = 0.1 %

α = 2.0 %

α = 5.0 %

Figure 7.46. Effect of positive strain-hardening on the height-wise distribution of median RIDR for

GF-9R (T1=1.185 s): (a) η =2.0; (b) η = 4.0; (c) η = 6.0.

(c) α = 5%

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5σ ln RIDR

z / H

η = 0.5

η = 1.0η = 2.0η = 3.0

η = 4.0

η = 6.0

(b) α = 2%

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5σ ln RIDR

z / H

η = 0.5

η = 1.0η = 2.0

η = 3.0η = 4.0

η = 6.0

(a) α = 0.1%

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5σ ln RIDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.47. Height-wise dispersion distribution of RIDR for GF-9R (T1=1.185 s) model:

(a) α = 0.1%; (b) α = 2%; (c) α = 5%.

Residual drift profiles obtained from the same frame model are shown in figure 7.46. It

can be seen that the effect of positive strain hardening is more noticeable on constraining

residual drift demands, even for large levels of intensity. This observation is consistent with


269

prior results from SDOF analyses (e.g., Section 5.7.3.1, Chapter 5). In addition, it can be

observed that an increment in positive member strain hardening leads to a more uniform

distribution of median RIDR along the height and, furthermore, residual deformation demand

does not significantly grow as the ground motion intensity increases. In addition, the

corresponding distribution of dispersion obtained for the same frame model is shown in figure

7.47. It should be noted that increasing member strain hardening does not reduce the levels of

dispersion, which is contrary to the tendency observed in SDOF analyses.

To further study the effect of member positive strain hardening on deformation demands,

figure 7.48 illustrates the variation of both median IDRmax and median RIDRmax for different

levels of ground motion intensity obtained from the same rigid building model. As can be

expected, an increment of member strain hardening leads to a significant reduction on both

median IDRmax and median RIDRmax even for large intensity levels considered in this

investigation. Furthermore, member strain hardening tends to bound median RIDRmax for this

frame model as the ground motion intensity increases.

(a) 9-STORY (T1 = 1.185 s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆i)

IDRmax [%]

α = 0.1 %

α = 2.0 %

α = 5.0 %

(b) 9-STORY (T1 = 1.185 s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆i)

RIDRmax [%]

α = 0.1 %

α = 2.0 %

α = 5.0 %

Figure 7.48. Effect of post-yield stiffness for GF-9R on: (a) Median IDRmax; and (b) median RIDRmax.

It is interesting to observe if the presence of strain hardening in the member moment-

curvature relationship reduces dispersion of both maximum and residual drift demands. Thus,

the variation of dispersion for IDRmax and RIDRmax with changes in the ground motion

intensity is shown in figure 7.49. It can be seen that, as expected from the results obtained

from SDOF systems, an increment of positive strain-hardening leads to smaller levels of

dispersion of IDRmax. However, unlike results obtained from SDOF analyses, an increment in

strain-hardening does not necessarily yield smaller levels of dispersion for RIDRmax. In


270

general, levels of dispersion of RIDRmax when elastoplastic M-φ is assumed are comparable

than those obtained for bilinear M-φ relationships.

(a) 9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆i)

σ ln IDRmax

α = 0.1%

α = 2.0%

α = 5.0%

(b) 9-STORY (T1 = 1.185 s)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆i)

σ ln RIDRmax

α = 0.1%

α = 2.0%

α = 5.0%

Figure 7.49. Effect of member strain hardening for GF-9F (T1 = 1.902 s) model on: (a) Dispersion of IDRmax ; (b) dispersion of RIDRmax

7.6.4.2 Effect of strength deterioration

In this subsection, the effect of strength deterioration in the member bilinear moment-rotation

relationship on both maximum and residual deformation demands is presented. This study

focuses in the seismic response of a short-period rigid building model, GF-3R (T1 = 0.5s),

since prior results in SDOF systems have showed that short period systems are more

susceptible to this effect (as discussed in Chapter 2). In particular, it is of special interest to

study if member strength deterioration alone might also constraint or might trigger excessive

residual drift demands in MDOF systems. Therefore, for the beam elements of the GF-3R

building model, plastic hinges were assigned bilinear hysteretic behavior (i.e., without

stiffness degradation or strain hardening) with two levels of member strength deterioration: (a)

Moderate strength deterioration (MSD), which attempts to reasonably reproduce the hysteretic

behavior of steel reduced beam section (RBS) connections that exhibit moderate strength

degradation at large levels of displacement ductility due to local flange or web buckling when

the beams are not laterally braced (see figure D.6, Appendix D); and (b) severe strength

deterioration (SSD), that intends to account for potential brittle behavior of bolted web-welded

flange type beam-column connections, as commonly used in California prior to the 1994

Northridge earthquake. Simulation of flexural strength deterioration in plastic hinges using


271

RUAUMOKO capabilities was explained in Section D.2.5 (Appendix D). Initially, column

frame elements were modeled with elastoplastic hysteretic behavior to isolate the effect of

strength deterioration in the beam elements.

The influence of member strength deterioration in the height-wise median IDR profiles for

the GF-3R model can be observed in figure 7.50. It can be seen that both member MSD and

SSD leads to a significantly large drift concentration in the lower stories as the relative

intensity increases, which eventually leads to unacceptable drifts that might induce collapse

when member SSD is considered, while the top story experiences small level of drift demands.

When member strength deterioration is considered, levels of dispersion along the height are

larger than those observed when strength deterioration is not considered (see figure 7.51).

(a) NSD3-STORY (T1 = 0.5 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0IDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b) MSD3-STORY (T 1 = 0.5 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0IDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(c) SSD3-STORY (T1 = 0.5 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0IDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.50. Height-wise distribution of IDR for GF-3R (T1=0.5 s): (a) NSD model; (b) MSD model; (c) SSD model.


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0σ ln IDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b) MSD3-STORY (T=0.5 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0σ ln IDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(c) SSD3-STORY (T

1 = 0.5 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0σ ln IDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.51. Height-wise dispersion distribution of IDR for GF-3R (T1=0.5 s):

(a) NSD model; (b) MSD model; (c) SSD model.


272

Next, the influence of member strength deterioration in the distribution of median RIDR

along the height for the same building model can be observed from figure 7.52. It can be seen

that both moderate and severe member strength deterioration trigger large residual drift

demands even for low levels of ground motion intensity. Comparing figures 7.50 and 7.52, it

is believed that member strength deterioration has more influence in residual drift demands

than in transient drift demands. In addition, large levels of dispersion in the estimation of

residual drift demands can be observed in figure 7.53.

(a) NSD 3-STORY (T1 = 0.5 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0RIDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b) MSD3-STORY (T1 = 0.5 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0RIDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(c) SSD3-STORY (T1 = 0.5 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0RIDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.52. Height-wise distribution of RIDR for GF-3R (T1=0.5 s):



0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0σ ln RIDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b) MSD3-STORY (T1 = 0.5 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0σ ln RIDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(c) SSD3-STORY (T 1 = 0.5 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0σ ln RIDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.53. Height-wise dispersion distribution of RIDR for GF-3R (T1=0.5 s):



273

Figure 7.54 shows the variation of median IDRmax and RIDRmax with changes in the ground

motion intensity for the GF-3R building model considering two levels of member strength

deterioration. For comparison purposes, previous results obtained with the non-degrading EPP

model are also shown in the respective figures. It can be seen that member SSD hysteretic

behavior induces significantly large median drift demands for relative intensities larger than

two compared to median drift demands computed when non-degrading member hysteretic

behavior is considered and, even, when moderate member strength deterioration is assumed. It

should also be noted that the effect of member MSD behavior is significantly different than

member non-degrading behavior on the variation of median IDRmax, but it is more significant

for the variation of median RIDRmax.

(a) 3-STORY (T1 = 0.5s)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

IDRmax [%]

SSD

MSD

EPP

(b) 3-STORY (T1 = 0.5s)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η(∆ i)

RIDRmax [%]

SSD

MSD

EPP

Figure 7.54. Effect of member strength deterioration on deformation demands GF -3R: (a) Median

IDRmax; and (b) median RIDRmax.

An additional study using the same building model was carried out considering a member

strain-hardening of 2% in the moment-curvature relationship of the column members. This

situation might be considered more realistic. Then, similar plots as those previously reported

are shown in figure 7.55. It can be seen that member SSD hysteretic behavior in the beam

elements still leads to large drift demands as the ground motion intensity increases. In

addition, it is interesting to note that member MSD hysteretic behavior does not lead to

significantly larger drifts demands than when non-degrading member hysteretic behavior is

employed in both beam and column elements.


274

(a) 3-STORY (T1 = 0.5s)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

IDRmax [%]

SSD-C2

MSD-C2

EPP

(b) 3-STORY (T1 = 0.5s)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η(∆i)

RIDRmax [%]

SSD-C2

MSD-C2

EPP

Figure 7.55. Effect of member strength deterioration on deformation demands of GF-3R:

(a) Median IDRmax; and (b) median RIDRmax.

It is believed that the results presented in this section are in good agreement with recent

field reconnaissance after earthquake events since potential fracture phenomena in steel

members of short-period structures triggers not only large maximum drifts demands but also

excessive permanent deformation demands that might represent serious difficulty to

implement retrofit schemes (SAC, 1997).

7.6.4.3 Effect of stiffness degradation

Even though the family of generic building models used throughout this investigation were

designed as representative of typical steel moment-resisting frames, it is interesting to study

different member hysteretic behaviors that are not observed in steel components. Therefore,

the well-known modified-Clough (MC) and Takeda (TK) models were used to evaluate the

influence of stiffness degradation of the element moment-curvature relationship on global

deformation demands. It should be mentioned that MC and TK models, as implemented in

RUAUMOKO (Carr, 2003) , do not account for cyclic strength degradation due to cumulative

energy nor displacement ductility under repeated loading, but they are representative of the

hysteretic behavior of well-detailed RC elements failing primarily in a flexural mode. In

addition, since MC model has unloading stiffness larger than TK model, this section also

studies the effect of unloading stiffness on maximum and residual drift demands.

Figure 7.56 illustrates a comparison of the distribution of median IDR along the height

obtained from the GF-18R (T1=2.0 s) frame model considering MC and TK member hysteretic

behavior. For reference purposes, the distribution obtained with the EPP model is also plotted


275

in the same figure. It can be seen that the distribution of median IDR along the height does not

significantly change when member stiffness degradation is considered and that maximum

median IDR located in the bottom story (i.e., z/H about 0.1) is very similar regardless of the

hysteretic modeling. It is interesting to note that both member stiffness degrading behaviors

lead to smaller drift demands in the upper stories than when non-degrading model is employed

as the relative intensity increases. A similar plot corresponding to the median residual drift

distribution along the height is shown in figure 7.57. It can be seen that member stiffness-

degrading hysteretic behavior leads to significant reduction of median RIDR along the height,

which is consistent with prior results obtained from SDOF analyses.

(b) η = 4.018-STORY (T1 = 2.0s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0IDR [%]

z / H

EPP

MC

TK

(a) η = 2.018-STORY (T1 = 2.0s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0IDR [%]

z / H

EPP

MC

TK

(c) η = 6.018-STORY (T 1 = 2.0s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0IDR [%]

z / H

EPP

MC

TK

Figure 7.56. Effect of member stiffness degradation on the height-wise distribution of median IDR for

GF-18R (T1=2.0 s.): (a) η =2.0; (b) η = 4.0; and (c) η = 6.0.

(c) η = 6.018-STORY (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0RIDR [%]

z / H

EPP

MC

TK

(a) η = 2.018-STORY (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0RIDR [%]

z / H

EPP

MC

TK

(b) η = 4.018-STORY (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5RIDR [%]

z / H

EPP

MC

TK

Figure 7.57. Effect of member stiffness degradation on the height-wise distribution of median RIDR for

GF-18R (T1=2.0 s.): (a) η =2.0; (b) η = 4.0; (c) η = 6.0.


276

Next, the influence of member stiffness degradation on median IDRmax and RIDRmax for

two rigid building models having short period, T=0.5 s, and long period, T=2.0 s, can be seen

in figures 7.58 and 7.59, respectively. It can be observed that the member stiffness degradation

does not have an important influence on the variation of median IDRmax with changes in the

relative intensity, for both building models. However, the influence of member stiffness

degradation and, in particular, the effect of the unloading stiffness has more impact on

constraining median RIDRmax as the ground motion intensity increases. The former

observation is for valid for both frame models. Between the two stiffness-degrading hysteretic

models, it can be noted that TK model leads to smaller median RIDRmax for all levels of

ground motion intensity considered in this study. This observation is explained due to the self-

centering feature of stiffness-degrading hysteretic models that increases when unloading

stiffness is reduced (e.g., TK model has smaller unloading stiffness than MC model). These

results are consistent with prior observations on SDOF systems reported in Section 5.7.3

(Chapter 5) as well as those made from other researchers (Pampanin et al., 2002).

(a) 3-STORY (T1 = 0.5 s)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η(∆ i)

IDRmax [%]

EPP

MC

TK

(b) 3-STORY (T1 = 0.5 s)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η(∆ i)

RIDRmax [%]

EPP

MC

TK

Figure 7.58. Effect of member stiffness degradation on GF -3R (T1 = 0.5 s) response:


(a) 18-STORY (T1 = 2.0 s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η(∆ i)

IDRmax [%]

EPP

MC

TK

(b) 18-STORY (T1 = 2.0 s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

RIDRmax [%]

EPP

MC

TK

Figure 7.59. Effect of member stiffness degradation on GF-18R (T1 = 2.0 s) response:



277

From the above observations and those reported for SDOF systems, there is enough

evidence to argue that structural elements exhibiting stiffness-degrading hysteretic loops

would lead to smaller residual deformation demands than those showing wide non-degrading

hysteretic loops, which means that standard steel framed buildings and buildings including

energy dissipation devices (e.g., including buckling restrained braces) are more susceptible to

sustain large permanent deformations after earthquake attack.

7.6.5 Effect of Structural Overstrength

The survival of buildings designed with older seismic codes that experienced damaged, but

without collapse, when subjected to severe earthquake events has been attributed to structural

overstrength (e.g., Osterras and Krawinkler, 1990). Sources of overstrength (OVS) arises,

among others, from the actual material properties, sizing of the members to compla in

construction requirements, etc. (e.g., Fajfar and Paulay, 1997).

For instance, in order to take into account the inherent presence of overstrength in steel

structures, the AISC seismic provisions (1997) include a system (i.e., global) overstrength

factor, Ωo, which depends on the type of lateral resisting system. For instance, Ωo = 3 for

moment-frame systems, Ωο = 2.5 for eccentrically braced frames and Ωο = 2 for other lateral

resisting systems. This Ωο factor amplifies the equivalent lateral static force distribution and,

thus, the lateral static shear force distribution, to account for structural overstrength during

seismic building response. Therefore, a uniform overstrength distribution along the height is

assumed following AISC seismic provisions (1997). However, several studies have showed

that the height-wise structural overstrength distribution in actual steel and reinforced concrete

framed structures is not uniform and it tends to be larger in the upper stories than in the

bottom stories (e.g., Calderoni et al., 1997; Teran-Gilmore, 2004). In particular, Calderoni et

al. (1997) showed that considering a uniform distribution of column overstrength does not

significantly improve the probability of failure in the upper stories. The authors found that a

step-wise increasing variation along the height, similar to that estimated in actual multi-story

buildings, decreases the probability of failure in both the upper and bottom stories.

The issue of global overstrength and height-wise overstrength is taken into account in the

evaluation of deformation demands for the family of generic frames. For that purpose, global

(system) overstrength, Ωo=2 was assumed which implies that the base shear capacity of each


278

frame was increased twice. Then, two cases were considered to distribute the additional

strength along the height: (a) uniform; and (b) non-uniform. The uniform case simply consists

of multiplying the lateral equivalent force distribution, obtained from equation D.6 (Appendix

D), obtained for each frame. The non-uniform variation of overstrength along the relative

height (z/H) of the buildings was assumed to be equal to:

λ

−+=

Hz

OVSOVSHzOVS u )1()( 1 (7.4)

where OVS1 is the expected overstrength in the first story, OVSu is the expected

overstrength in the upper story, and λ is a non-dimensional parameter that controls the

variation of the OVS along the relative height of the structure. Several examples of the

increments in OVS along the height are shown in figure 7.62a. In particular, values of λ equal

to 1 and 2 correspond to triangular and parabolic variations of OVS along the relative height

of the structure, respectively. In this investigation, it was considered the height-wise OVS

distribution shown in black line (OVS1=1, OVSu=1.8, λ=2.55) of figure 7.60a. In addition,

Figure 7.60b shows the normalized shear force shear force for the three cases considered in

this investigation.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5Normalized shear force distribution

z / H

w/o-OVS

U-OVS NU-OVS

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0height-wise OVS distribution

z / H

λ = 2.55λ = 2.0λ = 1.5λ = 1.0

Figure 7.60. Height-wise strength distribution: (a) parabolic variation per NEHRP provisions; (b)

overstrength distribution along the height considered in this study.


279

To illustrate the effect of OVS in the evaluation of deformation demands, figure 7.61

shows the height-wise median IDR distribution obtained from the response of the GF-9R

(T1=1.185s) building model without and with OVS. It can be seen that the amplitude of

median IDR in the bottom stories decreases for both cases of OVS compared to the case

without OVS for all levels of ground motion intensity. As a consequence, excessive drift

concentration is restricted. However, it should be noted that when uniform OVS is considered,

as suggested in AISC seismic provisions (1997), the upper stories experiences larger drift

demands than when OVS is neglected. When non-uniform OVS distribution is considered, the

distribution of median IDR along the height becomes more stable than when uniform OVS is

assumed and, furthermore, it does not lead to excessive levels of drift in the upper stories.

(c) NU-OVS9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0IDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(a) w/o-OVS9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0IDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b) U-OVS9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0IDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.61. Height-wise distribution of median IDR for GF-9R (T1=1.185 s) building model:

(a) Without overstrength; (b) uniform overstrength; (c) non-uniform overstrength.

In addition, figure 7.62 shows the median RIDR profiles obtained from the same building

model with and without considering OVS. Similar observations can be made: (1) The

consideration of non-uniform OVS along the height leads to smaller residual drift demands in

the bottom stories, avoiding residual drift concentration; (2) the distribution of median RIDR

becomes more uniform; and (3) it does not increases residual drift demands in the upper

stories.


280

(a) w/o-OVS9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0RIDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(c) NU-OVS 9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0RIDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b) U-OVS9-STORY (T 1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0RIDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.62. Height-wise dispersion distribution of median RIDR for GF-9R (T1=1.185 s) building

model: (a) without over strength; (b) uniform over strength; (c) non-uniform over strength.

7.6.6 Effect of Ground Motion Duration

Nowadays, there is still a controversy about the influence of ground motion duration on the

seismic response of structures. This controversy begins with the selection of an adequate

definition of strong ground shaking duration of recorded acceleration time histories that

adequately represent the time interval when a significant amount of energy content from the

earthquake ground shaking is released. Motivated by this issue, Bommer and Martinez-Pereira

(1996) published a comprehensive review about the merits and shortcomings of about 30

definitions proposed in the literature. They point out that an adequate definition should have a

physical significance based on a good correlation with geophysical parameters (e.g.,

earthquake magnitude, rupture history, etc.). Among several definitions, the most widely

measure of strong ground motion duration for earthquake engineering purposes is due to

Trifunac and Brady (1975). The authors defined significant strong motion duration, tD, as the

time interva l from 5% to 95% of the Arias intensity computed from each single acceleration

time-history (Arias, 1970). The merit of this definition is that the use of Arias Intensity has

strong correlation with observed earthquake damage in short period structures as well as

structures susceptible of liquefaction potential, but a limitation is that it does not explicitly

takes into account differences in ground motion frequency content as well as the source

geophysical features.


281

With respect to the earthquake building response, there is limited information about the

influence of strong motion duration on seismic demands of MDOF systems representative of

actual regular framed multi-story buildings. Some of the previous investigations on the effect

of strong motion duration considered equivalent SDOF systems representing the dynamic

properties of MDOF systems (e.g., Bernal, 1992; 1998), but it is well accepted that this

approach is limited to capture global seismic response of buildings behaving in the

fundamental vibration mode and, thus, higher mode effects as well as lateral stiffness and

strength variation along the height, typical of real framed multi-story buildings, are neglected

using this approach.

In this section, the influence of strong motion duration on both maximum and residual

drift demands of MDOF systems is evaluated using two generic rigid frame models

representing a short-period (GF-3R, T1=0.5 s) and a long-period (GF-18R, T1=2.0s) buildings.

Both building models were subjected to the suites of long and short duration records described

In this investigation, strong motion duration, tD, was computed according to the definition

proposed by Trifunac and Brady (1975). As a remainder, the short duration set (s20-SD)

contained 20 ground motions with tD ranging between 8.8 s and 15.9 s, while the long-duration

set (s20-SD) comprised 20 records with tD ranging between 25.7 s and 51.7 s. It should be

mentioned that the 20 short-duration records are a subset of the s40-LMSR-N set. A complete

list of the earthquake ground motions considered in this investigation is given in Appendix A.

As an example of the effect of strong motion duration on the deformation demands of

generic building models, figure 7.63 shows the distribution of median IDR along the height

obtained from the GF-18R (T1=2.0s) building model when subjected to the s40-LMSR-N, s20-

SD and s20-LS suites of ground motions scaled to different relative intensities. Similar

profiles of median IDR can be observed regardless of the suite of ground motion employed,

which might suggest that strong motion duration does not significantly change the height-wise

median IDR pattern. As described in Section 7.7.2, main concentration of median IDR occurs

in the lower portion while a secondary drift concentration appears in the upper portion of the

frame model as the ground motion intensity increases. The letter effect can be attributed to the

presence of higher mode effect as the building model experiences large levels of nonlinear

behavior. However, it is important to note that long-duration records lead to larger drift

demands in the upper stories than that induced when short-duration records are used in the

analyses. Moreover, long-duration records seem to decrease drift concentration in the lower

stories as compared with the response obtained from short-duration records. Drift demands in


282

the intermediate stories seem insensitive of the ground motion duration, with exception of

large levels of relative intensity. On the other hand, it is interesting to observe if strong motion

duration also modifies the height-wise distribution of dispersion. This distribution is illustrated

in figure 7.64. It can be seen that long-duration records lead to larger dispersion than that

computed from short-duration records as the ground motion intensity increases. It should be

noted that the largest dispersion computed from long-duration records tends to move towards

the upper stories, which is particularly true for relative intensities smaller than 6.

Similarly to maximum drift demands, profiles of median RIDR along the height

corresponding to the same building model and for increasing levels of intensity are shown in

figure 7.65. It can also be observed that long-duration ground motions lead to larger median

residual drift demands in the upper stories than short-duration records. On the other hand, with

exception of relative intensity equal to 6, it seems that short duration records induces larger

median residual drift demands in the bottom stories than long-duration ground motions.

Finally, it can be seen that the dispersion of RIDR along the height does not follow a clear

trend with changes in the ground motion intensity (see figure 7.66). However, it is evident that

the levels of dispersion of RIDR are much higher than for IDR regardless of the ground motion

duration characteristics.

(a) s40-LMSR-N18-STORY (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0IDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(c) s20-LD18-STORY (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0IDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b) s20-SD18-STORY (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0IDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.63. Height-wise distribution of median IDR for GF-18R (T1=2.0 s) building model obtained

from three suites of ground motions: (a) s40-LMSR-N; (b) s20-SD; and (c) s20-LD.


283

(a) s40-LMSR-N18-STORY (T1 = 2.0s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8σ ln IDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(c) s20-LD18-STORY (T1 = 2.0s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8σ ln IDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b) s20-SD18-STORY (T1 = 2.0s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8σ ln IDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.64. Height-wise dispersion distribution of IDR for GF-18R (T1=2.0 s) building model obtained



0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0RIDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(c) s20-LD18-STORY (T 1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0RIDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b) s20-SD18-STORY (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0RIDR [%]

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.65. Height-wise distribution of median RIDR for GF-18R (T1=2.0 s) building model obtained



284

(a) s40-LMSR-N

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.4 0.8 1.2 1.6 2.0 2.4σ ln RIDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(c) s20-LD

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.4 0.8 1.2 1.6 2.0 2.4σ ln RIDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b) s20-SD

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.4 0.8 1.2 1.6 2.0 2.4σ ln RIDR

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 7.66. Height-wise dispersion distribution of RIDR for GF-18R (T1=2.0 s) building model

obtained from three suites of ground motions: (a) s40-LMSR-N; (b) s20-SD; and (c) s20-LD.

To further study the influence of ground motion duration on deformation demands, the

variation of median IDRmax and RIDRmax for a short-period building model (GF-3R, T1=0.5s),

when subjected to the short-duration and long-duration ground motion sets, with changes in

the relative ground motion intensity is shown in figure 7.67. For comparison purposes, the

same variation obtained from the baseline s40-LMSR-N set is also plotted in the same figure.

It can be seen that, in general, strong motion duration does not have a significant influence on

the variation of median IDRmax and RIDRmax for the short-period building model, which is

particularly true for low levels of ground motion intensity (i.e., η < 3.0). Moreover, it seems

that long-duration records lead to slightly smaller deformation demands than short-duration

records for relative intensities greater than two. However, as can be seen in figure 7.68, strong

motion duration has more effect on the variability of IDRmax and RIDRmax as the ground motion

increases.


285

(a) 3-STORY (T1 = 0.5 s)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

IDR [%]

s40-LMSR-N

s20-SD

s20-LD

(b) 3-STORY (T1 = 0.5 s)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

RIDR [%]

s40-LMSR-N

s20-SD

s20-LD

Figure 7.67. Effect of ground motion duration on GF-3R (T1 = 0.5 s) response:


(a) 3-STORY (T1 = 0.5 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

σ ln IDRmax

s40-LMSR-N

s20-SD

s20-LD

(b) 3-STORY (T1 = 0.5 s)

0.0

0.4

0.8

1.2

1.6

2.0

2.4

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

σ ln RIDRmax

s40-LMSR-N

s20-SD

s20-LD

Figure 7.68. Effect of the ground motion duration on dispersion of GF-3R (T1 = 0.5 s): (a) Dispersion of

IDRmax; (b) dispersion of RIDRmax.

In addition of investigating a short-period building model, the variation of median lateral

deformation demands for a long-period building model (GF-18R, T1=2.0s) is shown in figure

7.69. Again, it seems that the variation of median IDRmax and RIDRmax is not significantly

affected by the ground motion duration. However, as illustrated in figure 7.70, dispersion

derived from the use of long-duration records is considerably higher than that of using short-

duration records.

In order to investigate if building models with the same number of stories, but different

period of vibration, are susceptible to strong motion duration, the GF-18F (T1=3.31s) was

subjected to the same suites of ground motions and the results are shown in figure 7.71. From

the figure, it can be seen that strong motion duration has more effect on the variation of

median IDRmax for the flexible frame model than that on the rigid counterpart. However,

smaller influence of ground motion duration is appreciated in the variation of median RIDRmax.


286

On the other hand, as was observed for the rigid long-period building model, long-duration

records yields larger levels of dispersion of IDRmax than short-duration records for the range of

ground motion intensities covered in this investigation (see figure 7.72).

(a) 18-STORY (T1 = 2.0 s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η(∆ i)

IDRmax [%]

s40-LMSR-N

s20-SD

s20-LD

(b) 18-STORY (T1 = 2.0 s)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η(∆ i)

IDRmax [%]

s40-LMSR-N

s20-SD

s20-LD

Figure 7.69. Effect of ground motion duration on GF-18R (T1 = 2.0 s) response: (a) Median IDRmax; and (b) median RIDRmax.

(b) 18-STORY (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η(∆ i)

σ ln IDRmax

s40-LMSR-Ns20-SD

s20-LD

(a) 18-STORY (T1 = 2.0 s)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η(∆ i)

σ ln IDRmax

s40-LMSR-Ns20-SD

s20-LD

Figure 7.70. Effect of the ground motion duration on dispersion of GF-18R (T1 = 2.0 s) model: (a) Dispersion of IDRmax ; (b) dispersion of RIDRmax

(a) 18-STORY (T1 = 3.31 s)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

IDRmax [%]

s40-LMSR-N

s20-SD

s20-LD

(b) 18-STORY (T1 = 3.31 s)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆i)

RIDRmax [%]

s40-LMSR-N

s20-SD

s20-LD

Figure 7.71. Effect of ground motion duration on GF-18R (T1 = 3.31 s) response:



287

(a) 18-STORY (T1 = 3.31 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆i)

σ ln IDRmax

s40-LMSR-N

s20-SD

s20-LD

(a) 18-STORY (T1 = 3.31 s)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆i)

σ ln RIDRmax

s40-LMSR-N

s20-SD

s20-LD

Figure 7.72. Effect of the ground motion duration on dispersion of GF-18R (T1 = 3.31 s) model:

(a) Dispersion of IDRmax; (b) dispersion of RIDRmax.

In order to study if strong motion duration has an effect on the variation of median IDRmax

and RIDRmax when member MSD hysteretic behavior is considered, the GF-3R (T1=0.5 s)

frame having beam members with type of behavior was analyzed. The results are presented in

figure 7.73. It can be seen that even in the presence of moderate strength deterioration in the

beam member behavior, the effect of strong motion duration seems small for both median

IDRmax and RIDRmax. Similarly to central values, the variation of dispersion with changes in

the ground motion intensity is not significantly affected by ground motion duration (see figure

7.44).

(a) 3-STORY (T1 = 0.5s)MSD model

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

IDRmax [%]

s40-LMSR-N

s20-SD

s20-LD

(b) 3-STORY (T1 = 0.5s)MSD model

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆i)

RIDRmax [%]

s40-LMSR-N

s20-SD

s20-LD

Figure 7.73. Effect of ground motion duration on GF-3R (T1 = 0.5 s) response considering moderate

member strength deterioration: (a) Median IDRmax; and (b) median RIDRmax.


288

(a) 3-STORY (T 1 = 0.5s)MSD model

0.0

0.1

0.2

0.3

0.4

0.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

σ ln IDRmax

s40-LMSR-N

s20-SD

s20-LD

(a) 3-STORY (T1 = 0.5s)MSD model

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

σ ln RIDRmax

s40-LMSR-N

s20-SD

s20-LD

Figure 7.74. Effect of the ground motion duration on dispersion of GF-3R (T1 = 0.5 s) model

considering moderate member strength deterioration: (a) Dispersion of IDRmax ; (b) dispersion of RIDRmax.

Therefore, from the observations made in the this sections, there is evidence to believe

that strong motion duration, as defined by Trifunac and Brady (1975), does not have a

significant impact in the variation of median IDRmax and, moreover, in the variation of median

RIDRmax of MDOF systems as the ground motion intensity increases. However, long-duration

records seems to increase both maximum and residual drift demands in the upper stories for

long-period structures as well as to increase levels of dispersion.

7.7 Summary

The main goal of this chapter was to obtain statistical information of primary deformation

demands (i.e., maximum and residual) of MDOF systems that can be used in probabilistic

evaluation of existing regular framed structures. For that purpose, a family of 12 regular one-

bay framed building models, with fundamental periods ranging from 0.5 to 3.3 s and having

number of stories from 3 to 18, were designed and modeled according to current seismic

provisions for structures located in a region of high seismicity. In order to capture the seismic

behavior of existing multi-story framed buildings, special attention was done to provide

adequate height-wise stiffness distribution similar to the one observed in actual buildings,

which is reflected in the deflected elastic fundamental modal shape of the structure. The

building models concentrate inelastic deformation in plastic hinges that form at both ends of

beam and column elements. Plastic hinges were modeled with different nonlinear moment-

curvature relationships to reproduce typical hysteretic behaviors of reinforced concrete and


289

steel components. The ir moment capacity was determined from the story shear force

distribution according to lateral static force distribution suggested in current seismic

provisions. In order to assess their seismic response, the building models were primarily

subjected to a suite of 40 earthquake ground motions scaled to reach different levels of

intensity. In addition, two sets of 20 earthquake ground motions having short and long strong

motion duration were also considered. In this investigation, an inelastic scalar intensity

measure was used instead of the traditional elastic intensity measure based on scaling each

record to the spectral acceleration corresponding to the fundamental period of the structure,

Sa(T1), which has been commonly used in PSDA studies. The inelastic intensity measure

employed in this study consists of scaling each record to reach the same maximum inelastic

displacement of an elastic -perfectly plastic SDOF system having the same dynamic properties

(i.e., yield displacement and period of vibration) of the build ing model.

In particular, several parameters that might influence the amplitude of residual

deformation demands in MDOF systems were investigated: a) Number of stories; b)

fundamental period of vibration; c) member hysteretic behavior; d) type of mechanism; e)

height-wise overstrength; and d) strong motion duration. It should be mentioned that the

evaluation of dispersion (i.e., record-to-record variability) was specially addressed. In

addition, maximum lateral deformation demands were also recorded to provide a context of

the amplitude of residual deformation demands during earthquake excitation.

The first important observation made in this chapter was that the use of ∆i(T1) as IM

reduced the dispersion of residual deformation demands, in addition of reducing the dispersion

of maximum deformation demands. In general, the level of reduction in dispersion increases

as the relative intensity increases (i.e., the structure experiences larger levels of inelastic

deformation as a consequence of increasing intensity of the ground motion). However, in spite

of the improvement in efficiency, the level of dispersion is not constant and it still depends on

the level of relative intensity (i.e., dispersion increases as the relative intensity increases).

Next, a summary of the most relevant observations from the evaluation of deformation

demands made in this chapters are reported next:

1. Effect of ground motion intensity. Median maximum and residual deformation demands

(e.g., IDRmax and RIDRmax) increase as the ground motion intensity increases. However,

their rate of increment is different. Between building models with the same number of


290

stories but different period of vibration, deformation demands for flexible models grow at

a faster rate than that for rigid counterparts.

2. Effect of number of stories. Median θroof and θr,roof demands, for a given level of

intensity, are not significantly affected for the number of stories. The dispersion of θroof

seems to increase as the number of stories increases, for a given relative intensity, but at a

non-constant rate. In general, dispersion of θr,roof is much higher than dispersion of θroof,

but any significant trend with respect to the number of stories was identified. On the other

hand, IDRmax tends to increase as the number of stories and relative intensity increases.

Similar trend was observed for RIDRmax, but the influence of the number of stories is less

significant. It was noted that the dispersion of IDRmax is higher than the dispersion of θroof,

and it also appears to slightly increase at a non-uniform rate, for a given relative intensity,

as the number of stories increases. It was of interest to observe that was, in general,

smaller that the dispersion of θr,roof, although still is considerably high (i.e., around 0.4 and

0.6).

3. Effect of the period of vibration. It was found that both θroof and θr,roof do not change

significantly with variation of the period of vibration, for a given relative intensity, when

η (∆i) is used as IM. In addition, dispersion of θroof tends to increase as the period of

vibration increases and as the relative increases. However, for a given period of vibration,

dispersion tends to saturate as the relative intensity increases. On the other hand,

dispersion of θr,roof does not follow a clear trend with changes in relative intensity and

period of vibration. Unlike θroof, IDRmax increases as both the period of vibration and the

relative intensity increases. The rate of increment is higher for generic flexible buildings

than for their rigid counterparts. However, RIDRmax does not significantly grow as the

period of vibration increases, for both rigid and flexible building models. Similarly to the

dispersion of θroof, the dispersion of IDRmax follows an increasing rate as the period of

vibration increases, while dispersion of RIDRmax ranges between about 0.4 and 0.6 for the

ranges of periods covered in this study.

4. Effect of the type of mechanism. For a building with the same number of stories, but

different period of vibration, the type of mechanism has more influence on a flexible

building than on the rigid counterpart. It was also observed that encouragement of

buildings developing beam-hinge mechanism (i.e., by increasing beam-to-column ratio)


291

constraint RIDRmax in addition of avoiding damage concentration in specific stories and

damage in the columns.

5. Effect of the member hysteretic modeling. Strain-hardening in the moment-curvature,

M-φ , relationship was found to constraint not only IDRmax but also RIDRmax. In particular,

a strain-hardening of 2% produced a significant reduction of RIDRmax besides that it leads

to a more uniform height-wise distribution. In addition, it was found that an increase in

unloading stiffness in stiffness-degrading M-φ relationships leads to smaller RIDRmax as

compared to non-degrading M-φ relationships, but the effect of unloading stiffness in

IDRmax is negligible. However, both IDRmax and RIDRmax increase when member M-φ

relationship exhibits strength deterioration. The increment in both deformation demands

depends on the level of strength deterioration and the level of relative intensity.

6. Effect of overstrength. Global (system) overstrength constraints both maximum and

residual deformation demands for all levels of relative intensity considered in this study.

The inclusion of non-uniform overstrength along the height leads to a more uniform

distribution of IDR and RIDR, constraining drift concentration at the bottom stories. In

addition, uniform overstrength might lead to larger IDR and RIDR in the upper stories

than when non-uniform overstrength is considered.

7. Effect of ground motion duration. Unlike previous results obtained from SDOF

systems, ground motion duration does not have a significant effect on both IDRmax and

RIDRmax for building models exhibiting FH mechanism with non-degrading member M-φ

relationship. Moreover, even considering moderate strength deterioration in the member

M-φ relationship, ground motion duration did not affect the variation of median IDRmax

and RIDRmaxwith changes in the ground motion intensity.

_______________________________________________________________________________ Chapter 8 Statistical Evaluation of Deformation Ratios of MDOF Systems

292

Chapter 8

Statistical Evaluation of Deformation Demand Ratios

for MDOF Systems

8.1 Introduction

The evaluation of structural performance of existing multi-story framed building structures

under earthquake ground shaking usually involves the estimation of the maximum roof

displacement demand, uroof, and its corresponding peak roof drift ratio, θroof. The peak roof

displacement of a regular framed building whose nonlinear behavior is dominated by the first

mode of vibration (i.e., higher mode effects do not significantly contribute to the seismic

response) can be related to the maximum displacement demand of an equivalent SDOF system

through modification factors. For example, in the Nonlinear Static Procedure (NLS)

introduced in the FEMA 356 document (FEMA, 2000) the target roof displacement is

estimated from the so-called displacement coefficient method, which includes several

modification factors to account for nonlinear response (e.g., C0 is a modification factor that

relates spectral displacement and likely building roof displacement, C1 is a modification factor

to relate expected maximum inelastic displacements to displacements calculated from a linear

elastic analysis, C2 is a modification factor to represent the effect of hysteretic behavior on the

maximum displacement response, C3 is a modification factor to represent increments in

displacements due to P-∆ effects). However, additional information about the height-wise

distribution of deformation demands (both maximum and residual) and, in consequence, on

the distribution of expected damage along the height is needed to fully assess the seismic

performance of existing buildings and to determine future retrofit strategies. In particular,

information about eventual drift concentration in a single story is of primary importance

during the conceptual preliminary design phase to evaluate potential soft-story mechanism or

global instability during future seismic events. On the other hand, engineers might need a

rapid estimation of permanent (residual) lateral deformation demands for building


293

performance assessment or during the conceptual predesign phase when estimates about the

maximum deformation demands (e.g., maximum inter-story drift ratio, IDRmax) are only

available.

Several researchers have suggested the use of deformation ratios (i.e., normalized lateral

deformation demands) to be incorporated in simplified displacement-based design procedures

for the rapid performance assessment of existing structures or during the conceptual

preliminary design phase of new structures, when available information is scarce and rigorous

detailed analyses are not justifiable (e.g., Qi and Moehle, 1991; Teran-Gilmore, 1996;

Miranda, 1997; Seneviratna and Krawinkler; 1997, Gupta and Krawinkler, 1999, Pampanin et

al., 2002; Aschheim, 2004; Ghobarah, 2004; Teran-Gilmore, 2004). The primary deformation

ratio that has been suggested is the ratio of IDRmax normalized with respect to θroof. This ratio

has sometimes been named coefficient of distortion (c.o.d.) in the literature. Another example

of deformation ratios is the maximum residual inter-story drift normalized with respect to the

maximum inter-story drift over all stories (Pampanin et al., 2002).

Therefore, the purpose of this chapter is to present the results of a statistical evaluation of

deformation ratios of multi-degree-of-freedom (MDOF) systems. In particular, this chapter

focuses on obtaining quantitative information about maximum and residual deformation ratios

from the seismic response of a family of 12 regular generic one-bay framed building models

representative of typical office building construction in a region of high seismicity in

California. Thus, relevant statistical parameters (e.g., central tendency and dispersion) of

lateral deformation ratios with changes in the ground motion characteristics (i.e., ground

motion intensity and duration) and building features (i.e., fundamental period of vibration,

number of stories, failure mechanism, element hysteretic behavior, and system overstrength)

are reported. This investigation is aimed to provide statistical information which can be used

for approximately estimating maximum inter-story drift and residual inter-story drift from

deformations ratios during rapid evaluation of existing structures or during the conceptual

preliminary design phase of new structures.

8.2 Previous Findings on Deformation Ratios for MDOF Systems

Several researchers have reported that the ratio of IDRmax normalized with respect to θroof is a

very useful parameter that can be used for approximately estimating IDRmax from θroof (Qi and


294

Moehle, 1991; Teran-Gilmore, 1996; Gupta and Krawinkler, 1999; Medina and Krawinkler,

2003; Ghobarah, 2004). For example, Teran-Gilmore (1996) found that this ratio is not

constant and it increases as the nonlinear behavior (i.e., displacement ductility) increases and

as the fundamental period of vibration elongates in shear-type MDOF systems. More recently,

Medina and Krawinkler (2003) reported that this ratio does not vary significantly with changes

in the ground motion intensity and that it mainly depends on the structure’s fundamental

period of vibration. Based on their statistical results, the former authors proposed simple linear

equations to predict this ratio as a function of the period of vibration of flexible and rigid

frame models exhibiting beam-hinge mechanism.

8.3 Deformation Demand Measures

In order to adequately characterize deformation demands ratios, both residual and maximum,

three basic deformation demand ratios for MDOF systems are used throughout this Section:

a) Ratio of the maximum inter-story drift ratio over the height to the maximum roof drift

ratio:

roof

IDRθ

β max= (8.1)

b) Maximum residual deformation ratio, which is defined as follows:

max

maxmax IDR

RIDR=γ (8.2)

c) Roof residual deformation ratio, which is defined as follows:

roof

roofrroof θ

θγ ,= (8.3)

d) Story residual deformation ratio, which is defined as follows:


295

i

ii IDR

RIDR=γ (8.4)

Deformation ratio β intends to provide information about concentration of maximum

deformation demands in any story since they are normalized with respect to their

corresponding roof quantities. Residual deformation ratios γi, γmax and γroof are aimed to

provide information about residual deformation measures with respect to maximum

deformation measures.

8.4 Evaluation of Maximum Deformation Ratios for MDOF Systems

In this section, statistical results on the ratio of IDRmax normalized with respect to θroof obtained

from the seismic response of the family of one-bay generic frame building models, described

in Section 7.3 and Appendix D, is reported. In particular, the effects of the number of stories,

period of vibration, member hysteretic behavior and type of mechanism are discussed in this

section. Special attention is given to the dispersion in the estimation of central values. This

information is of high importance in probabilistic-based assessment procedures that might use

this ratio to estimate drift concentration in multi-story framed buildings.


The variation of median IDRmax/θroof ratio as a function of the number of stories, N, and the

level of relative intensity is shown in figures 8.1a and 8.1b for both rigid and flexible building

models. It can be seen that this ratio follows a linear trend and it tends to increase as the N

increases and as the level of relative intensity increases. The former observation is valid for

both rigid and flexible building models. In particular, median ratios are slightly larger for

flexible building models than of their rigid counterparts, which means that framed buildings

with the same number of stories but different period of vibration would exhibit slightly

different inter-story drift concentration regardless of the level of ground motion intensity (e.g.,

a flexible building would tend to experience larger drift concentration than its rigid

representation). Another observation is that this ratio tends to saturate as N increases, for both


296

rigid and flexible buildings, for all the levels of relative intensity considered in this

investigation. It should be noted that the values of median IDRmax/θroof obtained in this study

are larger than those reported by Medina and Krawinkler (2003) using beam-hinge generic

frame models. This observation suggests that median IDRmax/θroof ratio depends not only in the

number of stories but also on the type of building mechanism as has been suggested by

Miranda (1999).


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η(∆ i)

IDRmax /θroof

N = 18N = 15N = 12N = 9N = 6N = 3


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

IDRmax/θ roof

N = 18N = 15N = 12N = 9N = 6N = 3

Figure 8.1. Effect of number of stories on median IDRmax /θroof ratio:

(a) Generic rigid frames; (b) generic flexible frames.

The dispersion in the estimation of IDRmax/θroof ratio as a function of N and the relative

intensity for both rigid and flexible frames is shown in figure 8.2. From the figures, it can be

seen that dispersion does not significantly changes as N increases for a relative intensity, η,

greater than 2.0. In addition, it seems that dispersion is not very different for rigid and flexible

building models and it remains in the range of 0.2 and 0.3. It should be noted that dispersion

levels reported in this section are larger than those mentioned by Medina and Krawinkler

(2003), which also suggests that the type of building mechanism also influences the dispersion

of IDRmax/θroof ratio.


297

(a) Generic rigid frames

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

σ

N = 18N = 15N = 12N = 9N = 6N = 3

(b) Generic flexible frames

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η(∆ i)

σ

N = 18N = 15N = 12N = 9N = 6N = 3

Figure 8.2. Effect of number of stories on dispersion of IDRmax /θroof ratio: (a) Generic rigid frames; (b) generic flexible frames.

8.4.2 Effect of Period of Vibration

In addition of investigating the influence of the N and η(∆i) on median IDRmax/θroof ratio, it is

interesting to study the variation of this ratio with respect to the fundamental period of

vibration at different intensity levels (See figure 8.3). From the figure, it can be seen that the

median IDRmax/θroof ratio and the period of vibration follow an increasing linear relationship

for all levels of ground motion intensity, which means that larger drift concentration is

expected for long-period structures than for short-period ones. For a given period of vibration,

the amplitude of median IDRmax/θroof ratio tends to increase as the level of relative intensity

increases (i.e., as the level of nonlinearity that the building is expected to suffer under

earthquake excitation increases). The aforementioned observations are valid for both rigid and

flexible building models.


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Period [s]

IDRmax/θ roof

η = 6.0η = 4.0

η = 3.0η = 2.0

η = 1.0η = 0.5


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Period [s]

IDRmax/θ roof

η = 6.0η = 4.0

η = 3.0η = 2.0

η = 1.0η = 0.5

Figure 8.3. Effect of period of vibration on median ratio of IDRmax to θroof: (a) Generic rigid frames; (b)

generic flexible frames.


298

The variation of median IDRmax/θroof ratio from both rigid and flexible frames as a function

of period of vibration and level of relative intensity is shown in figure 8.4. From the figure, it

can be seen that this ratio shows a very stable trend, which means that this ratio increases as

the period of vibration increases and as the relative intensity increases regardless of the type of

building model.

0.50.9

1.31.8

2.43.3

0.51.0

2.03.0

4.06.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5IDRmax/θroof

Period [s]

η (∆i)

Figure 8.4. Variation of median IDRmax /θroof ratio for both generic rigid and flexible frame models as a

function of the period of vibration and the ground motion intensity.

The dispersion of IDRmax/θroof ratio as a function of the period of vibration for different

levels of relative intensity is shown in figure 8.5. It can be observed that dispersion tends to

increase as the period of vibration increases for levels of relative intensity smaller or equal

than two, but it remains very stable over the period region for relative intensities greater than

two. Then, when the structure is expected to behave nonlinearly, dispersion in the estimation

of IDRmax/θroof ratio would be between 0.2 and 0.3. This level of dispersion is similar than that

computed for IDRmax and θroof.


299


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Period [s]

σ

η = 6.0

η = 4.0η = 3.0

η = 2.0η = 1.0

η = 0.5


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Period [s]

σ

η = 6.0

η = 4.0

η = 3.0

η = 2.0

η = 1.0

η = 0.5

Figure 8.5. Effect of period of vibration on dispersion of IDRmax /θroof ratio: (a) Generic rigid frames; (b)

generic flexible frames.

8.4.3 Effect of the Frame Mechanism

The variation of median IDRmax/θroof ratios, as a function of the relative intensity, computed for

two building models with the same number of stories (N=9), but different period of vibration

when they develop the three types of frame mechanism described in Section 7.5.3 (Chapter 7)

is shown in figure 8.6. It can be seen that column-hinge (CH) mechanism leads to larger drift

concentration as η(∆i) increases for both building models than when the building models

develop full-hinge (FH) and beam-hinge (BH) mechanisms, which reflects that CH

mechanism exhibits larger drift concentration than the other frame mechanisms. It should be

noted that FH mechanism follows a similar trend as CH mechanism. On the other hand, BH

mechanism tends to be more stable as the relative intensity increases for the rigid building

model, but the flexible building model also shows an increasing trend as the other frame

mechanism. In addition of showing the variation of central values, the variation of dispersion

with changes in the relative intensity for the same building models is shown in figure 8.7. It

can be seen that dispersion of IDRmax/θroof ratio is not significantly affected for the type of

mechanism for the flexible building; however, CH mechanism leads to larger levels of

dispersion in the case of the rigid building model.


300

(a)9-STORY (T1 = 1.185 s)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

IDRmax/θroof

FHM

BHM

CHM

(b)9-STORY (T1 = 1.902 s)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

IDRmax/θ roof

FHM

BHM

CHM

Figure 8.6. Effect of the frame mechanism on IDRmax/θroof ratio for two building models: (a) GF-9R

(T1=1.185 s); and (b) GF-9F (T1=1.902 s).

(a) 9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

σ IDRmax/θ roof

FHM

BHM

CHM

(b) 9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

σ IDRmax/θ roof

FHM

BHM

CHM

Figure 8.7. Effect of the frame mechanism on dispersion of IDRmax/θroof ratio for two building models:

(a) GF-9R (T1=1.185 s); and (b) GF-9F (T1=1.902 s).

8.4.4 Effect of the Member Hysteretic Behavior

Next, the variation of median IDRmax/θroof ratios obtained for the GF-9R building model

considering three levels of positive member strain hardening (α=0.1%, 2% and 5%) are shown

in figure 8.8. It can be seen that the variation of median IDRmax/θroof with changes in the

relative intensity tends to stabilize when member strain hardening is considered. Furthermore,

dispersion of this ratio decreases with increment in the member strain hardening for all levels

of ground motion intensity.


301

(a)9-STORY (T1 = 1.185 s)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

IDRmax/θ roof

α = 0.1%

α = 2.0%

α = 5.0%

(a) 9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

σ IDRmax/θ roof

α = 0.1%

α = 2.0%

α = 5.0%

Figure 8.8. Effect of member strain hardening on IDRmax/θroof ratio for GF-9R model:

(a) Median; (b) dispersion.

To investigate the effect of member stiffness-degrading hysteretic behavior on median

IDRmax/θroof ratio, figure 8.9 shows the variation of median IDRmax/θroof with respect to η(∆i)

obtained for a short-period (GF-3R) and a long-period (GF-18R) building models exhibiting

FH mechanism and considering two types of member stiffness-degrading hysteretic behavior

(i.e., Modified-Clough model, MC, and Takeda model, TK) as well as non-degrading

elastoplastic (EPP) hysteretic behavior. It can be seen that the hysteretic modeling of the

members does not have a significant effect on the variation of median IDRmax/θroof ratio as the

ground motion intensity increases for both short-period and long-period building models,

which support the idea that IDRmax/θroof ratio is a very stable parameter.

(b) 18-STORY (T1 = 2.0 s)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

IDRmax / θ roof

EPP

MC

TK

(a) 3-STORY (T1 = 0.5 s)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

η(∆ i)

IDRmax/θ roof

EPP

MC

TK

Figure 8.9. Effect of member stiffness-degrading hysteretic behavior on median IDRmax/θroof ratio:

(a) GF-3R (T1=0.5s); (b) GF-18R(T1=2.0s).


302

8.4.5 Effect of the Ground Motion Duration

Finally, the influence of ground motion duration on median IDRmax/θroof ratio is shown in

figure 8.10 for the same building models. It can be seen that strong motion duration has little

effect on this ratio for the short-period and long-period building models. Moreover, it seems

that long-duration records yield slightly smaller median IDRmax/θroof ordinates than short-

duration records.

(b) 18-STORY (T1 = 2.0 s)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i) [cm]

IDRmax / θ roof

s40-LMSR-N

s20-SD

s20-LD

(a) 3-STORY (T1 = 0.5 s)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i) [cm]

IDRmax / θ roof

s40-LMSR-N

s20-SD

s20-LD

Figure 8.10. Effect of ground motion duration on median IDRmax/θroof ratio: (a) GF-3R (T1=0.5s); (b)

GF-18R (T1=2.0s).

8.5 Evaluation of Residual Deformation Ratios for One-Bay Generic Frames

An indirect measure of the amplitude of residual deformation demands is obtained by

normalizing residual (permanent) deformation demands with respect to the maximum

(transient) deformation demand for each level of relative intensity. This normalization was

named residual deformation ratio and it can be found from the ratio of IDRmax and RIDRmax

(i.e., γmax), the ratio of θroof and θr,roof (i.e., γroof), or the ratio of RIDR and IDR at each story

level (i.e., γ). For example, the variation of residual deformation ratio along the height for

three rigid frame models having three different periods of vibration (i.e., GF-3R, GF-9R and

GF-18R) is shown in figure 8.11. It can be seen that the residual deformation ratio is not

constant along the height and it tends to increase as the ground motion intensity increases,

although it tends to saturate for large levels of relative intensity considered in this

investigation. In addition, it can be observed that the location of the maximum residual


303

deformation ratio changes with variation of the relative intensity. For example, the maximum

γ ratio for a relative intensity of two occurs at z/H = 0.889 while the maximum γ ratio for

relative intensity of three occurs at z/H = 0.333 for the GF-9R building model. It should be

noted that, for a given relative intensity, the maximum residual ratio in any story along the

height, γ, might differ from the residual ratio, γmax, computed from the ratio of the maximum

deformation demands, IDRmax and RIDRmax. As an example, it was found that, for a relative

intensity of 6, the maximum median residual deformation ratio along the height, γ, for the GF-

18R frame is 0.61 at z/H = 0.333 while median γmax is 0.48, which represents a difference of

27% (from figures 7.8c and 7.12c it was observed that median IDRmax and RIDRmax are located

at z/H = 0.11).

3-STORY (T1 = 0.5 s)EPP, set40, IM-∆ i (T1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0γ

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

9-STORY (T1 = 1.185s)EPP, set40, IM-∆ i (T1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0γ

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

18-STORY (T1 = 2.0 s)EPP, set40, IM-∆ i (T1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0γ

z / H

η = 0.5η = 1.0η = 2.0η = 3.0

η = 4.0η = 6.0

Figure 8.11. Height-wise distribution of residual deformation ratio for three rigid frame models: (a) GF-

3R (T1=0.5s); (b) GF-9R (T1=1.185s); and (c) GF-18R (T1=2.0s).

In addition of gathering information about the central tendency, it is important to evaluate

the dispersion (i.e., record-to-record variability) around central values of residual deformation

ratios. This issue is of high importance in the development of a probabilistic approach to

estimate residual deformation demands. For example, the height-wise variation of the

dispersion of γ (measured by the standard deviation of the natural logarithm of the data) with

changes in the relative intensity computed for the aforementioned building models is shown in

figure 8.12. It can be seen that the estimation of γ involves very large variability, which is

much higher than that inherent to the estimation of IDRmax and RIDRmax (e.g., see figures 7.10

and 7.14, Chapter 7). Then, implementation of residual deformation ratios for the estimation


304

of residual deformation demands from maximum deformation demands should take into

consideration the large variability inherent in the estimation of central tendencies.

(a)9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5σ ln γ

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0γ

z / H

p = 75%

p = 50%

p = 25%

Figure 8.12. Height-wise dispersion distribution of residual deformation ratio for GF-9R (T1=1.185s).

Therefore, based on an the statistical results obtained in this investigation, the effect of the

number of stories, period of vibration, type of mechanism, hysteretic modeling of the frame

elements and ground motion duration on residual deformation ratios corresponding to different

levels of relative intensity is discussed in the following subsections.


In this section, the variation of median γroof and γmax with respect to N and the ground motion

intensity is investigated. Figure 8.13 illustrates this variation of γroof for both rigid and flexible

frames. From the figure, it can be seen that median γroof of both types of frame models

increases nonlinearly as the ground motion intensity increases at a nonlinear rate.

Furthermore, median γroof seems to stabilize for large levels of intensity. In general, rigid and

flexible building models would have different median γroof which mean that building with the

same N but different period of vibration would have different median γroof.


305

(a) Generic Rigid FramesIM - ∆ i (T1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

γroof

N = 3

N = 6

N = 9N = 12

N = 15

N = 18

(b) Generic Flexible FramesIM - ∆ i (T1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η(∆ i)

γroof

N = 3

N = 6

N = 9N = 12

N = 15

N = 18

Figure 8.13. Effect of number of stories on median γroof: (a) Generic rigid frames; (b) generic flexible

frames.

A similar plot for median γmax is shown in figure 8.14. For both types of frame models,

median γmax grows at a nonlinear trend up to relative intensities smaller than three, and it tends

to stabilize for larger relative intensities. It can be seen that frame models with the same N but

different period of vibration would have different γmax ratio. In general, median γmax is larger

than median γroof for both rigid and flexible frames.

(a) Generic Rigid BuildingsIM - ∆i (T1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

γmax

N = 3N = 6N = 9N = 12

N = 15N = 18

(b) Generic Flexible BuildingsIM - ∆ i (T1)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

γmax

N = 3N = 6N = 9N = 12N = 15N = 18

Figure 8.14. Effect of number of stories on median γmax: (a) Generic rigid frames; (b) generic flexible

frames.

8.5.2 Effect of the Period of Vibration

The effect of period of vibration and ground motion intensity on residual ratios, γroof and γmax,

is shown in figures 8.15 and 8.16. As noted in the previous section, residual ratios increase

rapidly when frame models are subjected to ground motion intensities larger than or equal to


306

two. In addition, for a given level of relative intensity, the influence of period of vibration on

residual ratios is not significant for flexible frame models and it is slightly significant for rigid

frame models.

0.50.9

1.21.5

1.82.0

0.00.5

1.02.0

3.04.0

6.0

0.00.10.20.30.40.50.60.70.8

0.91.0

γroof

Period [s]

η (∆ i)


0.81.3

1.92.4

2.93.3

0.00.5

1.02.0

3.04.0

6.0

0.00.10.20.30.40.50.60.70.8

0.91.0

γroof

Period [s]

η (∆ i)


Figure 8.15. Effect of period of vibration on median γroof: (a) Generic rigid frames; (b) generic flexible frames.

0.5 0.91.2 1.5

1.82.0

0.00.5

1.02.0

3.04.0

6.0

0.00.10.20.30.40.50.60.70.80.91.0

γmax

Period [s]

η (∆i)


0.8 1.31.9 2.4

2.93.3

0.00.5

1.02.0

3.04.0

6.0

0.00.10.20.30.40.50.60.70.80.91.0

γmax

Period [s]

η (∆i)


Figure 8.16. Effect of period of vibration on median γmax: (a) Generic rigid frames; (b) generic flexible

frames.

8.5.3 Effect of the Frame of Mechanism

In this section the effect of the type of mechanism in the residual ratio is investigated. For that

purpose, the height-wise distribution of residual ratio for two building with the same number

of stories but different period of vibration (i.e., GF-9R and GF-9F models) when they develop


307

the types of mechanism (i.e., FH, BH, and BH mechanism) described in Section 7.6.3

(Chapter 7), as a function of the relative intensity, is shown in figures 8.17 and 8.18.

Comparing both figures, it can be observed that, for the same expected mechanism, the

distribution of γ along the height changes from the rigid to the flexible building models even

though they have the same number of stories. It can be seen that for both building models the

distribution of γ is not uniform along the height for all three types of mechanism. However, it

is observed that residual ratio depends on the intensity of the ground motion and, furthermore,

it seems to saturate as the relative intensity increases regardless of the mechanism developed.

Unlike the observations made for IDR and RIDR profiles, FH and CH mechanism do not lead

to concentration of the residual ratio in any of the stories.

(a) FHM 9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0γ

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(c) CHM9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0γ

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b) BHM9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0γ

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 8.17. Height-wise distribution of median residual deformation ratio for three frame mechanism

models: (a) FH mechanism; (b) BH mechanism; and (c) CH mechanism.

(a) FHM9-STORY (T1 = 1.902 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0γ

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(c) CHM9-STORY (T1 = 1.902 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0γ

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b) BHM9-STORY (T1 = 1.902 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0γ

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 8.18. Height-wise distribution of median residual deformation ratio for three frame mechanism

models: FH mechanism; (b) BH mechanism; and (c) CH mechanism.


308

To further extend our understanding of this deformation parameter, the variation of

medianγmax and γroof for both building models with respect to the relative intensity and the type

of mechanism is shown in figures 8.19 and 8.20. For the rigid building model, it can be seen

that median γroof for the three types of mechanism follows a similar trend as the intensity

increases. However, median γmax tends to be larger when the FH mechanism develops in the

building than when ideal CH and BH mechanism occur. On the other hand, it can be observed

that both CH and BH lead to similar median γmax and γroof for the flexible building model, but

FH mechanism still lead to the largest residual deformation ratios.

(a) 9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

γmax

FHM

BHM

CHM

(b)9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

γ roof

FHM

BHM

CHM

Figure 8.19. Effect of the type of mechanism on GF-9R (T1 = 1.185 s) response: (a) Median γmax; (b)

median γroof.

(a) 9-STORY (T1 = 1.902 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

γmax

FHM

BHM

CHM

(b)9-STORY (T1 = 1.902 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

γ roof

FHM

BHM

CHM

Figure 8.20. Effect of the type of mechanism on GF-9F (T1 = 1.902 s) response: (a) Median γmax; (b)

median γroof.


309

8.5.4 Effect of the Positive Strain-Hardening

Bilinear member moment-curvature relationships are typically used to represent the hysteretic

behavior of steel components that do not exhibit early lateral or local buckling. Therefore, it is

interesting to study if the distribution of residual deformation ratios changes when member

strain hardening is considered in the member moment-curvature relationship. For that purpose,

the height-wise distribution of median residual deformation ratio obtained from the GF-9R (T1

= 1.185 s.) model is shown in figure 8.21 corresponding to three different values of positive

strain hardening (α = 0.1%, 2% and 5%) in the bilinear member moment-curvature

relationship.

(b) α = 2%9-STORY (T1 = 1.185s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0γ

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(c) = 5%9-STORY (T

1 = 1.185s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0γ

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(a) α = 0.1%9-STORY (T

1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0γ

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 8.21. Height-wise distribution of median residual deformation ratio along the height for GF-9R

(T1=1.185 s) considering three levels of strain hardening: (a) α=0.1%; (b) α=2%; (c) α=5%.

From the figure above, it is evident that an increment in member strain hardening (e.g.,

from 0.1% to 2%) tends to constraint median γ along the height for all levels of relative

intensity. Furthermore, it seems that the height-wise distribution of median γ tends to be more

uniform as the level of positive strain hardening increases. However, it was observed that the

presence of strain hardening did not considerably reduce the level of dispersion along the

height (not shown).

Figure 8.22 shows the effect of member strain hardening in the variation of median γmax

and γroof with changes in the relative intensity obtained from the same GF-9R (T1=1.185 s)

building model. It can be seen that the ordinates of median γmax are larger than the ordinates of


310

median γroof even when member strain hardening is present. It is interesting to note that

including member strain hardening decreases both residual deformation ratios for relative

intensities larger than two, which reflects that member strain hardening is more beneficial for

constraining residual drift demands than transient drift demands.

(a) 9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

γmax

α = 0.1%

α = 2%

α = 5%

(a) 9-STORY (T1 = 1.185 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

γroof

α = 0.1%

α = 2%

α = 5%

Figure 8.22. Effect of member strain hardening on median residual deformation ratios for GF-9F (T1 =

1.902 s): (a) Median γmax; (b) median γroof.

8.5.5 Effect of the Member Degrading Behavior

In this subsection, the effect of member stiffness degradation and, particularly, the influence

of unloading stiffness in the member moment-curvature relationship on residual deformation

ratios is investigated. To illustrate the effect of member stiffness degradation on the

distribution of median γ along the height, figure 8.23 shows the median γ profiles obtained

from a long-period rigid (GF-18R, T1=2.0s) model considering non-degrading member

behavior and member MC and TK hysteretic behavior. It can be seen that the amplitude of

median γ decreases when member stiffness degradation is considered. Furthermore, decreasing

the unloading stiffness (i.e., TK model has smaller unloading stiffness than MC model) leads

to smaller median residual deformation ratios, which is expected since decreasing unloading

stiffness decreases residual drift demands. In particular, it can be seen that the distribution of

median γ becomes more uniform along the height when member stiffness-degrading models

are considered.


311

(b) MC18-STORY (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0γ

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(a) EPP18-STORY (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0γ

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(c) TK 18-STORY (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0γ

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 8.23. Distribution of median residual deformation ratio along the height for GF-18R (T1=2.0 s)

considering three member hysteretic behaviors: (a) EPP; (b) MC; and (c) TK.

Next, the variation of median γmax with changes in the level of relative intensity computed

for the GF-3R (T1=0.5s) and GF-18R (T1=2.0s) building models, considering non-degrading

and two types of stiffness-degrading hysteretic modeling in the moment-curvature relationship

of frame elements, is shown in figures 8.23a and 8.23b, respectively. It can be seen that for

both building models median γmax increases at a nonlinear rate as the relative intensity

increases for the three types of hysteretic modeling. It is evident that considering non-

degrading hysteretic behavior leads to higher γmax ratios than those considering member

stiffness-degrading behavior for both short-period and long-period frames. It should be noted

that median γmax does not significantly changes with changes in the ground motion intensity for

relative intensities greater than two. In fact, it should also be noted that regardless of the

member hysteretic modeling the residual deformation ratio tends to saturate as the level of

relative intensity increases. In addition, it can be observed that the long-period building

experiences larger values of median γmax from stiffness-degrading behavior than the short

period building, which reflects that long-period buildings might sustain larger residual drift

demands relative to maximum drift demands. Furthermore, the frames where TK model is

considered yield to smaller γmax ratios than when MC model is employed, which means that

decreasing unloading stiffness (i.e., TK has smaller unloading stiffness than MC model) leads

to smaller residual deformation ratios. The former observations are consistent with previous

results reported by Pampanin et al. (2002).


312

(a) 3-STORY (T1 = 0.5 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 10 20 30 40∆ i (T1) [cm]

γmax

EPP

MC

TK

(b) 18-STORY (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 20 40 60 80 100 120 140

∆ i (T1) [cm]

γmax

EPP

MC

TK

Figure 8.24. Effect of member stiffness degradation on median γmax for two building models: (a) GF-3R

(T1=0.5s); (b) GF-18R(T1=2.0s).

The variation of median γmax with changes in the relative intensity obtained from the GF-

3R (T1=0.5s) when member strength deterioration is considered is shown in figure 8.25a.

Additionally, figure 8.25b compares the variation of median γmax when member strength

deterioration is considered in the beams, but the columns includes strain hardening (α=2%).

From the first figure, it can be seen that median γmax increases rapidly as the ground motion

intensity increases up to relative intensities of three and later it continues increasing at a

slower rate, which reflects that member strength deterioration might trigger excessive residual

drift demands at low levels of relative intensity. Smaller values of median γmax are obtained

when member strain hardening (α=2%) is considered in the column elements (see figure

8.25b). However, still large residual deformation ratios are expected as a consequence of

member strength deterioration.

(a) 3-STORY (T1 = 0.5s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆i)

γmax

SSD

MSD

EPP

(b) 3-STORY (T1 = 0.5s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

γmax

SSD-C2

MSD-C2

EPP

Figure 8.25. Effect of member strength deterioration on median γmax for two building models: (a)

without strain hardening in column elements; (b) with strain hardening in column elements.


313

8.5.6 Effect of Ground Motion Duration

It was found that ground motion duration does not have significant effect on the variation of

maximum and residual drift demands with changes in the ground motion intensity (Section

7.6.6, Chapter 7). However, ground motion duration has some influence in the distribution of

median IDR and RIDR along the height, especially in the upper stories, when the building is

susceptible to high-mode effects. Therefore, it is interesting to investigate if ground motion

duration has some effect in the distribution of residual deformation ratios. To illustrate this

issue, the height-wise variation of median γ obtained for a long-period building (GF-18R,

T1=2.0s) when subjected to the sets of short-duration (s20-SD) and long-duration (s20-LD)

records, described in Appendix D, is shown in figures 8.26b and 8.26c. For comparison

purposes, the results obtained from the baseline ground motion set are shown in figure 8.26a.

It can be seen that long-duration records tends to produce large values of median γ in the

upper and lower stories as the ground motion intensity increases, which might be a reflex of

the increasing participation of high-mode effects.

(c) s20-LD 18-STORY (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0γ

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η= 6.0


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0γ

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

(b) s20-SD18-STORY (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0γ

z / H

η = 0.5η = 1.0η = 2.0η = 3.0η = 4.0η = 6.0

Figure 8.26. Distribution of median residual deformation ratio along the height for GF-9R (T1=1.185 s)

considering three sets of ground motions: (a) s40-LMSR-N; (b) s20-SD; (c) s20-LD.

In order to study the effect of ground motion duration on residual deformation ratios,

building models GF-3R (T1=0.5s) and GF-18R (T1=2.0s) were subjected to the short duration

(s20-SD) and long-duration (s20-LD) ground motion sets. A comparison of the variation of


314

median γmax and γroof with changes in the ground motion intensity for the short-period building

is shown in figure 8.27 while the same plot for the long-period building is illustrated in figure

8.28. For comparison purposes, results obtained from the s40-LMSR-N set are also plotted in

the same figures. It can be seen that long-duration records leads to slightly smaller values of

γmax and γroof than short duration records for the short-period building. However, it seems that

long-duration records yield slightly larger values of median γmax for the flexible model,

especially as the ground motion intensity increases, when the building experiences larger

levels of nonlinearity.

(a) 3-STORY (T1 = 0.5 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

γmax

s40-LMSR-N

s20-SD

s20-LD

(b) 3-STORY (T1 = 0.5 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

γroof

s40-LMSR-N

s20-SD

s20-LD

Figure 8.27. Effect of ground motion duration on residual deformation ratio for GF-3R model:

(a) Median γmax; (b) median γroof .

(a) 18-STORY (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

η (∆ i)

γmax

s40-LMSR-N

s20-SD

s20-LD

(b) 18-STORY (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

η (∆ i)

γ roof

s40-LMSR-N

s20-SD

s20-LD

Figure 8.28. Effect of ground motion duration on residual deformation ratio for GF-18R:

(a) Median γmax; (b) median γroof .


315

8.6 Summary

The main goal of this chapter was to obtain statistical information of deformation ratios (i.e.,

normalized deformation demands) of MDOF systems that can be used in probabilistic

evaluation of existing regular framed structures. Four deformation ratios were considered in

this investigation: (a) Ratio of IDRmax/θroof; (b) ratio of θr,roof/θroof; (c) ratio of RIDRmax/IDRmax;

and (d) Ratio of RIDR/IDR at each story. For that purpose, the family of 12 regular one-bay

framed building models described in the previous chapter was employed to obtain statistical

information. In order to assess the seismic building response, all building models were

primarily subjected to a suite of 40 earthquake ground motions scaled to reach different levels

of intensity. In addition, two sets of 20 earthquake ground motions having short and long

strong motion duration were also considered. Similarly to the previous chapter, an inelastic

scalar intensity measure was used instead of the traditional elastic intensity measure. In

particular, several parameters that might influence the amplitude of deformation ratios in

MDOF systems were investigated: a) Number of stories; b) fundamental period of vibration;

c) member hysteretic behavior; d) type of mechanism; and e) strong motion duration. It should

be mentioned that the evaluation of dispersion (i.e., record-to-record variability) was specially

addressed.

From the results presented in this section, it is concluded that IDRmax/θroof ratio is a very

effective parameter for providing information about drift concentration and it seems strongly

dependent on the number of stories, period of vibration and intensity of the ground motion. In

addition, the amplitude of this ratio does not seem strongly affected by member stiffness-

degrading hysteretic behavior or strong motion duration. However, the variability in its

estimation is not negligible and it should be taken into account if this ratio is incorporated into

performance-based assessment of existing structures as suggested by several researchers (Qi

and Moehle, 1991; Teran-Gilmore, 1996, 2004; Miranda, 1997; Aschheim, 2004; Ghobarah,

2004). A summary of the most important observations from the evaluation of central tendency

and dispersion of the IDRmax/θroof ratio made in this chapter are reported next:

1. The IDRmax/θroof ratio is a very efficient parameter that has relatively low record-to-record

variability when the ground motions are scaled to the same inelastic intensity measure.


316

2. The variation of median IDRmax/θroof ratio shows, for a building model with a given

number of stories, an increasing linear trend as the relative intensity increases. In addition,

for a given level of ground motion intensity, it increases at an increasing smooth nonlinear

rate as the fundamental period f vibration increases.

3. Changes in median IDRmax/θroof ratio as the relative intensity increases appear to depend on

the type of building mechanism. Ideal CH mechanism yields the largest IDRmax/θroof ratio,

since this type of mechanism is associated to large story drift concentration, and ideal BH

mechanism leads to a very stable variation of IDRmax/θroof ratio, which reflects that this

building mechanism leads to a more uniform drift distribution along the height.

4. It was observed that the variation of median IDRmax/θroof ratio is not significantly affected

by member stiffness-degrading hysteretic behavior.

5. It seems that strong motion duration does not have a significant influence on median

IDRmax/θroof ratio variation, when non-degrading member hysteretic behavior is

considered.

Another deformation ratio investigated in this chapter was the ratio of the maximum

residual deformation demand normalized with respect to the maximum deformation demand.

This ratio was named residual deformation ratio, γ, and it allows an indirect way of estimating

residual deformation demands from maximum deformation demands. It should be noted that

this ratios can be computed from maximum deformation demands (i.e., RIDRmax/IDRmax), roof

drift demands (i.e., θr,roof/θroof) or deformation demands at any story (i.e., RIDR/IDR). A

summary of the most important observations from the evaluation of residual deformation

ratios made in this chapters are reported next:

1. Median maximum residual deformation ratio, γmax, does not necessarily coincide with the

maximum median residual deformation ratio along the height at any given story, γ, for a

given building under a specific level of ground motion intensity. This observation is

explained since IDRmax and RIDRmax might occur at different stories. In general, γ is

slightly larger than γmax for all levels of ground motion intensity.

2. The height-wise variation of γ is not uniform and it tends to increase at a non-constant rate

as the ground motion intensity increases. However, it should be noted that the variation of


317

γ along the height tends to saturate as the ground motion intensity increases (i.e., it does

not significantly grow as the building exhibit larger levels of nonlinearity).

3. Dispersion of γ along the height is significantly high and not uniform for all levels of

ground motion intensity considered in this investigation. In addition, it does not increase

as the level of ground motion intensity increases, which mean that the height-wise

variation of dispersion does not follow a noticeable pattern with respect of the ground

motion intensity.

4. The variation of median γroof and γmax follows a nonlinear increasing trend as the ground

motion intensity increases, regardless of the building number of stories. In general,

median γmax is larger than median γroof for a given number of stories and specific ground

motion intensity. Moreover, for building models with the same number of stories, but

different period of vibration, flexible buildings do not necessarily lead to larger residual

deformation ratios than their rigid counterparts.

5. For relative intensities larger than two, it seems that the period of vibration has little effect

on both median γroof and median γmax.

6. For the building models studied here, it seems that ideal CH and BH mechanism lead to

smaller residual deformation ratios than those obtained when FH mechanism develops.

7. It was shown that the presence of strain-hardening in the M-φ relationship leads to smaller

residual ratios. Moreover, the distribution of residual deformation ratios along the height

becomes more uniform with an increment in the member strain-hardening level (i.e., from

2% to 5%), even for large levels of relative intensity considered in this investigation.

8. It was observed that member stiffness-degrading hysteretic behavior leads to smaller and

more stable residual deformation ratios, as the ground motion intensity increases, than

when non-degrading member hysteretic behavior is assumed. Moreover, a reduction in the

level of unloading stiffness decreases residual deformation ratios.

9. It was observed that strong motion duration does not have a significant effect in the

variation of median residual deformation ratios as the ground motion intensity increases.

______________________________________________________________________________________ Chapter 9 Probabilistic Evaluation of Residual Deformation Demands of MDOF Systems 318

Chapter 9

Probabilistic Evaluation of Residual Deformation

Demands of MDOF Systems

9.1 Introduction

In modern seismic performance-based methodologies for the evaluation of existing structures it is

necessary to adequately estimate seismic demands related to the building performance (e.g.,

maximum inter-story drift ratio, peak floor acceleration, etc.) and, particularly, the probability

that seismic demands of a structure of interest exceeds pre-defined limit states (e.g., structural

deformation capacities) under different levels of intensity. This assessment approach requires a

probabilistic framework where the estimation of seismic demands and its inherent uncertainty

should be explicitly incorporated. In this context, relevant seismic demands should be estimated.

For example, it is widely accepted that the amplitude of maximum inter-story drift ratio, IDRmax,

is correlated with different damage states sustained for structural elements as well as some drift-

sensitive non-structural elements while peak floor acceleration, PFA, is related to failure of some

classes of non-structural components. Efforts in the probabilistic estimation of the previous

seismic demands have been the main core of recently proposed performance-based assessment of

existing structures. In addition of the former seismic demands, earthquake field reconnaissance

have evidenced that residual (permanent) lateral displacement demands (e.g., residual roof drift

ratio or maximum residual inter-story drift ratio) should also play an important role in the

evaluation of structural performance in addition to maximum (transient) lateral displacement

demands.

Residual displacement demands that civil engineering structures, such as buildings and

bridges, might experience after earthquake excitation might influence decision variables such as

earthquake losses and downtime. For instance, excessive permanent lateral deformations might

drive larger earthquake losses than earthquake losses associated to, for example, building collapse

due to severe structural damage. This situation might occur if a given structure would be


demolished due to excessive residual roof drifts even if the structure did not collapse or

experience severe structural damage. Unfortunately, based on statistical analyses such as those

introduced in the preceding chapter of this dissertation, it has been noted that the estimation of

residual deformation demands involves larger uncertainty than in the estimation of maximum

deformation demands. Therefore, the evaluation of residual deformation demands into

performance-based assessment of existing structures requires a probabilistic approach where

uncertainty is explicitly incorporated.

The main purpose of this chapter is to develop and illustrate a methodology aimed to evaluate

residual deformation demands into a Probabilistic Seismic Demand Analysis (PSDA) framework

(Cornell, 1996). For instance, recently proposed methodologies to evaluate seismic performance

employs PSDA a core part (Cornell and Krawinkler, 2000; Moehle and Deierlein, 2004). In the

context of PSDA, building-specific residual drift demand hazard curves can be obtained for a

given seismic environment. These curves express the mean (annual) frequency of exceeding a

threshold residual drift demand (e.g., residual roof drift). Residual drift demand hazard curves can

be compared with maximum drift demand hazard curves develop in parallel during the seismic

performance-based assessment of existing structures.

This chapter begins with the formulation of the proposed methodology to estimate residual

deformation demand hazard curves (Section 9.2). Next, an examination of the probabilistic

distribution of residual deformation demands is offered in Section 9.3. Special emphasis is given

to examine the empirical cumulative distribution of residual deformation demands and to find

appropriate parametric probability distributions that provide an adequate characterization of the

empirical cumulative distribution. In this section, adequate functional models that describe the

variation of statistical parameters of residual deformation demands with changes in the ground

motion intensity are evaluated in Section 9.4. Finally, the proposed methodology is evaluated by

computing residual deformation demand hazard curves for a short-period and a long-period

framed building model.


9.2 Formulation of Probabilistic Estimation of Residual Deformation Demands

of MDOF Systems

As mentioned in the introduction of this dissertation (Chapter 1), the proposed methodology to

estimate residual deformation demands in existing structures makes use of PSDA as a framework,

which is mathematically expressed as follows:

( ) )()(

)(|)(

0imd

imdimd

imIMedpEDPPedp IMEDP

λλ ⋅=>= ∫

∞ (9.1)

Consistent with the suggested nomenclature in current performance-based assessment

methodologies that use PSDA framework (e.g., the methodology suggested in the Pacific

Earthquake Engineering Research Center, PEER, for assessing building seismic performance),

EDP refers to an engineering demand parameter of interest (e.g., maximum inter-story drift, peak

floor acceleration, etc.) and IM denotes the ground motion intensity measure (Cornell and

Krawinkler, 2000, Moehle and Deierlein, 2004). Then, the mean (annual) frequency of exceeding,

MAF, a predefined engineering demand parameter, edp, is denoted as )(edpEDPλ while

)(imIMλ refers to the seismic hazard at the site, measured in terms of an adequate IM, evaluated

at the ground motion intensity measure level im. In addition, the term

)|( imIMedpEDPP => expresses the conditional probability of exceeding a specific edp given

that IM is equal to im. Information of )|( imIMedpEDPP => is obtained from rigorous non-

linear dynamic time history analyses performed for a specific structure subjected to an adequate

set of ground motions that represent the local seismicity (i.e., the seismic hazard threat). Finally,

the site-specific )(imIMλ should be available, which is commonly provided in terms of the

spectral elastic acceleration corresponding to a specific period of vibration, ))(( TSaSaλ . As

explained in Section 4.2 (Chapter 4), it should be noted that for the mean annual hazard rates of

interest the MAF is approximately equal to the mean annual probability.

Nowadays, modern seismic performance-based assessment methodologies for the evaluation

of standard buildings employs maximum deformation demand measures (e.g., maximum inter-

story drift ratio, IDRmax, or roof drift ratio, θroof) as the main EDP’s since they are closely related


to structural damage. Furthermore, some non-structural components are also related to maximum

deformation demands. However, it is believed that a full characterization of deformation demands

in seismic performance-based methodologies should also take into account residual deformation

demands, ∆r, which structures would sustain after earthquake excitation in addition of maximum

deformation demands. In this dissertation, the EDP’s of interest are both maximum (transient)

and residual (permanent) deformation demands measures. In this investigation, the primary

residual deformation demands are residual inter-story drift ratio in any story level, RIDR, the

maximum residual inter-story drift ratio, RIDRmax, and the residual roof drift ratio, θr, roof. Thus, in

the context of modern seismic performance-based approaches, we are interested in computing

residual deformation demand hazard curves which express the mean annual frequency (MAF) of

exceeding a certain residual deformation demand threshold for a specific building located at a

specific seismic environment. For example, we would be interested in computing the MAF of

RIDR at the ith floor of a building, iRIDRλ , or the MAF of RIDRmax along the height of the same

building maxRIDRλ . This information can be coupled with maximum deformation hazard curves

developed in parallel (e.g., the MAF of maximum inter-story drift ratio, maxIDRλ ) to evaluate the

seismic performance of a structure of interest. It should be noted that this approach actually

represents developing a vector of EDP’s, which can be incorporated into a more comprehensive

approach to estimate the MAF of exceedance of decision variables of interest (e.g., mean annual

earthquake losses).

9.2.1 Proposed Approach to Estimate r∆λ

The mean annual frequency of exceeding a given residual deformation demand can be

mathematically expressed as follows:

( ) dd

dyddr ds

dSsd

CTsSPr

)(,;|)(

0 1λ

δδλ ⋅=>∆= ∫∞

∆ (9.2)

In the above expression, ),;|( 1 yddr CTsSP =>∆ δ is the probability of ∆r exceeding a

defined residual deformation demand conditioned on the fundamental period of vibration of the


structure, T1, the yielding strength coefficient, Cy, and the elastic spectral displacement, Sd. In this

approach, Sd is the IM and the elastic spectral displacement hazard curve, )( dS sd

λ , for the

specific seismic environment should be available. In addition, it is reasonable to assume that the

probability of ∆r exceeding δ conditioned on the occurrence of a specified level of Sd is

lognormally distributed. Then, it can be computed as follows:

( )

−Φ−=∆

∆

∆

dr

dr

S

Sdr SP

|ln

|ln)ln(1|

σ

µδ (9.3)

where [ ]⋅Φ is the well-known cumulative distribution function, dr S|ln ∆µ is a central tendency

measure of ∆r conditioned on a given Sd and dr S|∆σ is a conditional measure of dispersion of ∆r

which also depends on a given level of Sd.

From the discussion offered in Section 7.5.3 (Chapter 7), there is enough evidence that the

dispersion in the estimation of EDP’s such as IDR is considerably reduced when an IM based on

the maximum inelastic displacement demand, ∆i, computed from a SDOF system having the same

fundamental period of the analyzed structure is considered as IM. Then, it is appealing to express

equation (9.2) in terms of ∆i (T1) instead of Sd (T1) as follows:

( ) ii

iyiir d

dd

CTPr

δδδλ

δδδλ)(

,;|)(0 1 ⋅=∆>∆= ∫∞

∆ (9.4)

In the above alternative expression )( iiδλ∆ is the site-specific seismic hazard curve of

maximum inelastic displacement demand of an SDOF system, which is a function of the specific

fundamental period of vibration of the system and the yield strength coefficient of the structure.

Similarly to equation (9.2), ),;|( 1 yiir CTP δδ =∆>∆ is assumed that r∆ is lognormally

distributed and it can be computed as follows:

( )

−Φ−=∆∆

∆∆

∆∆

ir

irirP

|ln

|ln)ln(1|

σ

µδ (9.5)


where ir ∆∆ |lnµ is a central tendency measure of ∆r conditioned on a given ∆i, and

ir ∆∆ |lnσ is a

conditional measure of dispersion of ∆r which depends on a given level of ∆i.

9.3 Probabilistic Distribution of Residual Deformation Demands

An explicit consideration of the uncertainty involved in the estimation of seismic demands for

structures subjected to earthquake ground shaking through a probabilistic framework requires the

knowledge of the probability distribution of the EDP’s of interest. Early attempts to characterize

the probability distribution of inelastic structural demands computed from SDOF systems were

done by McGuire (1974), Sewell and Cornell (1987) and Miranda (1993). For example, Miranda

(1993) studied the empirical distribution of inelastic strength demands obtained from statistical

results of nonlinear SDOF systems having constant displacement ductility when subjected to a set

of 124 earthquake ground motions. The author found that parametric probability distributions

such as lognormal, gamma, Gumbel type I and Weibull are adequate to represent the cumulative

probability distribution function (CDF) of inelastic strength demands of SDOF systems.

More recently, with the increasing use of PSDA methodologies, a renewed interest in the

probability distribution characterization of nonlinear seismic demands from MDOF systems has

been rise. Some studies employing PSDA (e.g., Shome and Cornell, 1999; Miranda and Aslani;

2003) suggest that the lognormal CDF is adequate to characterize the empirical distribution of

maximum deformation demands (e.g., IDRmax) conditioned on Sa(T1). In addition, this assumption

has been verified for other demand parameters such as peak floor accelerations (Miranda and

Aslani, 2003). However, the validity of this assumption to represent the empirical cumulative

distribution of residual displacement demands (e.g., RIDRmax) has not been verified and it

constitutes the discussion of this section.

9.3.1 Empirical Distribution of Residual Deformation Demands

The empirical cumulative distribution of any residual deformation demand expresses the

probability of exceeding a certain residual deformation demand conditioned on the ground motion

intensity. Thus, it is possible to obtain the empirical distribution of residual deformation demands

(e.g., θr,roof , RIDRmax, or RIDR at any story) for a specified building model and corresponding to a


given level of relative intensity (e.g., η = 6) using the non-linear response time history analyses

results reported in Chapter 8 of this dissertation. For example, for a given relative intensity and a

given generic frame building model, the set of 40 residual deformation demand results,

corresponding to each of the 40 earthquake ground motions, can be treated as a random sample.

Each outcome is sorted in ascending order and plotted with respect to its corresponding

probability (i.e., assuming each residual deformation demand as an independent outcome, i, with

a probability equal to 1/40 (0.025)). Next, the effect of several parameters (e.g., relative intensity,

period of vibration, building frame mechanism) on the empirical distribution of residual

deformation demands will be discussed using the results obtained from the family of generic

framed building models reported throughout chapter 8.

The empirical cumulative distribution of median θr,roof obtained from the results of the GF-3R

(T1=0.5s) building model, as a function of different levels of ground motion intensity, is shown in

figure 9.1a while a similar plot for median RIDRmax is illustrated in figure 9.1b . For instance,

figure 9.1a shows the probability of θr,roof exceeding a given value of residual roof drift ratio

(δr,roof). As an example, there is, approximately, 60% probability that θr,roof be smaller or equal to

δr,roof=1.0% for a relative intensity of 6 (i.e., there is a 40% probability that median θr,roof be

greater than δr,roof=1.0%). In general, for large relative intensities, it can be seen that there is a

significant difference between the maximum and minimum computed value of residual

deformation demands, which is particularly true for RIDRmax, which means that variability in the

estimation of residual deformation demands is very large. In addition, it can be observed that the

empirical cumulative distribution is not symmetric, with longer tails moving towards upper

values of residual deformation demand as the relative intensity increases. Similar plots obtained

from a long-period building (GF-18R) are shown in figure 9.2. In this case, the difference

between minimum and maximum values of residual deformation demands is similar than that

observed for the short-period building model.


(b)

GF-3R (T1 = 0.5s)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0δRIDRmax [%]

P(R

IDR

max

| η)

η = 6.0

η = 4.0

η = 3.0

η = 2.0

(a)

GF-3R (T1 = 0.5s)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5δr,roof [%]

P( θ

r,ro

of |

η)

η = 6.0

η = 4.0

η = 3.0

η = 2.0

Figure 9.1. Empirical cumulative distribution of residual deformation demands obtained from the GF-3R

(T1=0.5 s) building model for different relative intensities: (a) θr,roof; (b) RIDRmax.

(a)

GF-18R (T1 = 2.0s)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0δθr,roof [%]

P (R

IDR

max

| η)

η = 6.0

η = 4.0

η = 3.0

η = 2.0

(b)

GF-18R (T1 = 2.0s)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0δRIDRmax [%]

P (

RID

Rm

ax |

η)

η = 6.0

η = 4.0

η = 3.0

η = 2.0

Figure 9.2. Empirical distribution of residual deformation demands obtained from the GF-18R (T1=2.0 s)

building model for different relative intensities: (a) θr,roof; (b) RIDRmax.

The effect of the period of vibration on the empirical cumulative distribution of RIDRmax for a

building model with the same number of stories (N=9), but different period of vibration, is shown

in figure 9.3. It can be seen that, for a given relative intensity, the empirical probability of

exceeding some level of RIDRmax is larger for the flexible building model than for the rigid

counterpart. For example, the probability of RIDRmax exceeding a value of 2% (i.e., δRIDRmax=2%)

for a relative intensity of three is about 22% for the rigid model (GF-9R) and it increases up to

66% for the flexible model (GF-9F). Another example is that the probability of exceeding 5%

residual drift (i.e., δRIDRmax=5%), which is considered as a residual drift limit for collapse

prevention in FEMA 356 document (FEMA, 2000), is around 20% for the rigid model whereas it

increases up to about 54% for the flexible counterpart. The former observation means that


standard framed buildings with the same number of stories, but different periods of vibration,

might have larger exceedance probabilities of residual drift demands.

(a)

9-Story (T1 = 1.185 s)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0δRIDRmax [%]

P (R

IDR

max

| η)

η = 2.0

η = 3.0

η = 4.0

η = 6.0

(b)

9-Story (T1 = 1.902 s)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0δRIDRmax [%]

P (R

IDRm

ax |

)

η = 2.0

η = 3.0

η = 4.0

η = 6.0

Figure 9.3. Empirical distribution of RIDRmax for different relative intensities obtained for building models

with same number of stories but different period of vibration: (a) GF-9R; (b) GF-9F.

In addition, the influence of the type of mechanism expected to develop in the GF-9R

(T1=1.185s) building model on the empirical cumulative distribution is shown in figure 9.4.

Comparing figure 9.3a with figures 9.4a and 9.4b, it can be seen that empirical cumulative

distribution curves changes depending on the type of frame mechanism. For example, the

probability of exceeding a residual drift of 1% (i.e., δRIDRmax=1%) given a relative intensity of

three is about 68%, 34% and 74% when the building model develops a full hinge (FH), beam-

hinge (BH) and column-hinge (CH) mechanism, respectively. Therefore, it is expected that

exceedance probabilities of RIDRmax, for a given relative intensity, decrease when the building

develop an ideal beam-hinge plastic hinge mechanism.

(a) BHM

9-Story (T1 = 1.185 s)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0δRIDRmax [%]

P (

RID

Rm

ax |

)

η = 2.0

η = 3.0

η = 4.0

η = 6.0

(b) CHM

9-Story (T1 = 1.185 s)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0δRIDRmax [%]

P (

RID

Rm

ax >

|

)

η = 2.0

η = 3.0

η = 4.0

η = 6.0

Figure 9.3. Empirical cumulative distribution of RIDRmax obtained for GF-9R building model exhibiting

two types of building mechanism: (a) BH mechanism; (b) CH mechanism.


9.3.2 Parametric Probability Distribution of Residual Deformation Demands

In the last subsection it was shown that the empirical cumulative distribution of residual

deformation demands is characterized by a skew distribution with longer tails moving toward

upper values. Therefore, parametric probability distributions functions such as lognormal,

Gamma, Gumbel, Weibull, or Rayleigh might be adequate to represent the observed sample

distribution (Benjamin and Cornell, 1970). In this investigation, three parametric cumulative

distribution functions (CDF’s) were evaluated to determine if they can characterize the empirical

cumulative distribution of residual deformation demands: a) Lognormal; b) Weibull; and c)

Rayleigh. These probability distributions were chosen since they have the convenience over other

probability distributions that can be fully defined from two parameters (lognormal and Weibull)

or only one parameter (Rayleigh). In particular, the lognormal distribution was chosen as primary

candidate since it includes explicitly the central tendency and the dispersion, or spread, of the

sample distribution. In addition, it has been employed successfully for other researchers to

characterize the sample distribution of other seismic demands (e.g., maximum inter-story drift

ratio or peak floor acceleration). The definition of each parametric CDF was provided in Section

4.3.5.2 (Chapter 4).

To verify weather the candidates probability distributions are adequate to characterize the

empirical cumulative distribution of residual deformation demands the Kolmogorov-Smirnov (K-

S) goodness-of-fit test was used in this investigation (Benjamin and Cornell, 1970). For example,

a comparison of the empirical cumulative distribution of RIDRmax, corresponding to a relative

intensity of four and obtained from the GF-9R (T1=1.185s) building model, with respect to

lognormal (using counted median and σln X as statistical parameters), Weibull and Rayleigh fitted

CDF’s is illustrated in figure 9.5 It can be seen that both the lognormal and Weibull distribution

closely follows the empirical distribution. However, the Rayleigh distribution, which is a

particular case of the Weibull distribution with parameter β=2.0, does not provide a good fit to

the empirical distribution, especially for values around the median (p=50%).


η = 4.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 2.0 4.0 6.0 8.0 10.0

δRIDRmax [%]

P (

RID

Rm

ax |

)DataLognormalWeibullRayleigh

Figure 9.5. Comparison of three parametric cumulative distribution functions with the empirical

distribution of RIDRmax obtained from the GF-9R (T1=1.185 s).

In spite of adequately representing the empirical distribution of residual deformation

demands, the parameters of the Weibull distribution must be obtained for each empirical

distribution corresponding to each building model at each level of relative intensity which might

be difficult to incorporate into a continuous solution. In addition, the parameters of the Weibull

distribution do not explicitly reflect the central tendency and, very important, the dispersion of

residual deformation demands. Therefore, it was decided to use the lognormal distribution to

characterize the conditional probability of residual deformation demands in this investigation.

9.3.3 Selection of Statistical Parameters for the Lognormal Probability

Distribution of Residual Deformation Demands

It should be mentioned that several parameters of central tendency and dispersion can be obtained

from the sample data. For example, the sample mean, counted median and geometric mean (i.e.,

mean of the natural logarithm of the EDP data, µln EDP) are commonly employed parameters of

central tendency while the coefficient of variation, COV, and standard deviation of the natural

logarithm of the data, σln EDP, are typical descriptors of dispersion. If the empirical distribution of

any EDP truly follows a lognormal distribution, µln EDP and σln EDP are logical distribution


parameters. However, since the lognormal probability distribution is an approximation of the true

(empirrical) distribution, it would be necessary to find better parameters that lead an adequate

fitting between the parametric and the empirical distribution. For example, as noted by Miranda

and Aslani (2003), there might be situations were the parametric lognormal distribution

underestimate or overestimate the probability of exceeding a given edp, even though the K-S test

declares the parametric distribution as adequate, such as in the presence of outliers at large levels

of ground motion intensity. The authors suggested using alternative lognormal distribution

parameters, such as a measure of dispersion based on the inter-quartile of the data. Therefore,

there is not a unique combination of statistical parameters that can be used for the lognormal

probability distribution.

Next, several examples will be shown to demonstrate that the lognormal probability

distribution provides an adequate characterization of the empirical distribution of residual

deformation demands. An example of the lognormal fitting using the empirical cumulative

distribution of RIDRmax obtained from both rigid and flexible building model having 9 stories

when subjected to ground motion relative intensity of four is shown in figure 9.6. In both cases,

geometric mean and counted median were used as measure of central tendency while the standard

deviation of the natural logarithm of the data was used as dispersion parameters. Also illustrated

in figure 9.6 is the K-S test corresponding to a 90% confidence level. It can be seen that the

lognormal distribution follows reasonable well the sample data, even when counted median is

employed as central tendency parameter, and that all data points lies between the K-S test bands

for both building models. Another example is shown in figure 9.7 that corresponds to the

empirical distribution of RIDRmax obtained for the GF-18R building model when subjected to

relative intensities of four and six. Again, it can be seen that the lognormal probability

distribution provides a good characterization of the conditional probability of exceeding RIDRmax

for large levels of intensity.


(b) 9-Story (T1 = 1.902 s)η = 4

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0

RIDRmax [%]

P (

IDRm

ax >

|

)

DataLognormal fit: counted medianLognormal fit: geometric meanK-S Test, 90 confidence

(a) 9-Story (T1 = 1.185 s)

η = 4

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

RIDRmax [%]

P (

IDR

max

>

| )

DataLognormal fit: counted medianLognormal fit: geometric meanK-S Test, 90% confidence

Figure 9.6 Fitting of the parametric lognormal CDF of RIDRmax for: (a) GF-9R (T1=1.185s); and (b) GF-9F

(T1=1.902s).

(a) η = 418-Story (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

RIDRmax [%]

P (

RID

Rm

ax >

|

)

Data

Lognormal fit: counted median

Lognormal fit: geometric mean

K-S Test, 90% confidence

(b) η = 618-Story (T1 = 2.0 s)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

RIDRmax [%]

P (

RID

Rm

ax >

|

)

Data

Lognormal fit: counted median


K-S Test, 90% confidence

Figure 9.7 Fitting of the parametric lognormal CDF of RIDRmax for the GF-18R building model for two

relative intensities: (a) η = 4; (b) η = 6..

Similar evaluation procedure was applied to all building models and to different residual

deformation demand measures corresponding to different levels of ground motion intensity to

verify that the lognormal probability distribution is adequate. From the overall evaluation

procedure, it was concluded that the lognormal probability distribution is a reasonable assumption

to represent the empirical distribution of residual deformation demands. In addition, it was found

that geometric mean and the standard deviation of the natural logarithm of the data are adequate

descriptors of central tendency and dispersion of residual deformation demands.


9.3.4 Variation of Statistical Parameters of Residual Deformation Demands as a

Function of the Ground Motion Intensity

From the results reported in Chapter 8, it was shown that not only the central tendency (e.g.,

counted median) but also the dispersion (e.g., standard deviation of the natural logarithm of

residual drift) of residual deformation demands changes with variation of the ground motion

intensity. Therefore, this variation of statistical parameters with changes in the ground motion

intensity should be taken into account to estimate the conditional probability of exceeding a given

residual deformation demand threshold.

From the statistical results presented in the preceding chapter, it should be noted that the

variation of central tendency and dispersion of residual deformation demands with respect to the

ground motion intensity depends on the level of relative intensity, the relative location along the

building height, the period of vibration for buildings with the same number of stories, the

hysteretic modeling of the structural elements, the type of mechanism, among others. To

exemplify this observation, figure 9.8 shows the variation of median RIDR and RIDRlnσ for 5

different story levels of the GF-9R (T1=1.185s) building model with changes in the ground

motion intensity.

9-STORY (T1 = 1.185 s)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆i)

σln RIDR

Level 9Level 7Level 5Level 3Level 1

9-STORY (T1 = 1.185 s)

0.0

0.5

1.0

1.5

2.0

2.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆i)

RIDR [%]

Level 9Level 7Level 5Level 3Level 1

Figure 9.8. Variation of central tendency and dispersion of RIDR computed for five stories of the GF-9R

building model: (a) Median RIDR; (b) dispersion of RIDR ( RIDRlnσ ).

From the figures, it can be seen that median RIDR computed in the nineth and seventh story

level grows nonlinearly as the relative intensity increases, whereas in the first and third story

median RIDR grows almost linearly at a much faster rate as the relative intensity increases. Both


trends are the reflex of concentration of RIDR in the bottom stories as the ground motion intensity

increases and, in general, of the type of building failure mechanism. On the other hand, dispersion

seems to increase or decrease depending on the level of ground motion intensity and location

along the height. For example, dispersion tends to decrease for relative intensities smaller than

four, but it tends to increase for relative intensities equal to six. Similar observations were found

from an examination of central tendency and dispersion of RIDR for other building models.

In several previous studies that employed PSDA (e.g., Luco and Cornell, 2000; Medina and

Krawinkler; 2003; MacKie and Stojadinovic, 2003), a power functional form has been commonly

used to represent the variation between the central tendency of an EDP of interest and the

selected IM, which is given mathematically as follows:

bIMaEDP )(= (9.15)

where coefficients a and b are obtained from conventional linear regression analysis in the log-

log domain that, indeed, implies a linear relationship between EDP and IM. It should be noted

that coefficient a controls the slope while coefficient b reflects the level of nonlinearity. In

particular, the central tendency of the EDP is linearly proportional to the selected IM for b=1.

In addition, previous studies have assumed constant dispersion of EDP with variation of the

ground motion intensity. For example, Medina and Krawinkler (2003) assumed values of 0.24

and 0.30 for the dispersion (i.e., standard deviation of the natural logarithm of the EDP of interest

conditioned on Sa(T1)) of maximum roof drift ratio and maximum inter-story drift ratio while

computing drift demand hazard curves. However, as illustrated in the previous discussion, neither

central tendency nor dispersion of residual deformation demands showed a constant trend with

variation of ground motion intensity. Recognizing this fact while examining maximum inter-

story drift ratio and peak floor acceleration, Miranda and Aslani (2003) proposed a three-

parameter functional form to better capture the variation of central tendency and a second-order

polynomial form to represent variation of dispersion of those EDP’s with changes in the selected

IM, which are given as follows:

3)(ˆ 21αααµ IMIM⋅= (9.16)


2321 )()(ˆ IMIM βββσ ++= (9.17)

where parameters α1, α2, α3, β1, β2, and β3 are constants that can be obtained from nonlinear

regression analysis using statistical results such as those presented in Chapter 7. The authors

noted that fitted parameter estimates in equations (9.16) and (9.17) will be different depending on

the EDP of interest and location within the structure.

Therefore, the adequacy of equations (9.15), (9.16) and (9.17) to be used for characterizing

variation of residual deformation central tendency and dispersion with respect to ground motion

intensity was evaluated in this investigation. It should be noted that Miranda and Aslani (2003)

have already showed that equations (9.16) and (9.17) are adequate to estimate central tendency

and dispersion of maximum deformation demands (e.g., IDRmax) while several studies performed

by Cornell and his co-workers (e.g., Luco and Cornell, 2000) have demonstrated that equation

(9.15) leads to a good estimation of the central tendency of IDRmax. However, the adequacy of

previous equations to describe the variation of statistical measures of residual deformation

demands has not been reviewed.

Examples of the adequacy of equations (9.15) and (9.16) to represent the variation of central

tendency of θr,roof with changes in the relative intensity for a short-period (GF-3R) and a long-

period (GF-18R) building models are shown in figure 9.9. It can be seen that equation (9.16)

captures reasonable well the variation of the central tendency with changes in the relative

intensity. However, the use of equation (9.15) might lead to underestimations or overestimations,

depending on the level of intensity, to predict the central tendency of θr,roof for both building

models. It should be noted that equation (9.15) systematically leads to overestimation of median

θr,roof for low relative intensities (e.g., η(∆i) < 2.0). This overestimation is explained since residual

deformation demands only occur when the building experiences nonlinearity at some ground

motion intensity, which means that there is a ground motion intensity threshold that triggers

residual deformation demands and that it does not necessarily correspond to the origin. Similar

observations were found for flexible buildings models. Additional examples of employing

equations (9.15) and (9.16) are shown in figure 9.10 for predicting RIDRmax with changes in the

relative intensity. It can also be observed that equation (9.16) yields better fitting of the variation

of RIDRmax than equation (9.15) for the relative intensities considered in this study.


It is important to mention that in order to employ equation (9.16), at least three different

levels of ground motion intensity should be used to estimate parameters. It is also recommended

that two of these levels of ground motion intensity correspond to approximately the limits of the

range of interest (Miranda and Aslani, 2003). Although adequate to characterize the variation of

central tendency of residual deformation demands in the range of intensities covered in this

investigation, the use of equation (9.16) might lead to serious overestimations or underestimations

of central tendency values when larger relative intensities that those covered to obtain parameter

estimates through regression analysis are employed.

(a) GF-3R (T1= 0.5 s)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

θr,roof [%]

Data

Cornell

Miranda

(b) GF-18R (T1 = 2.0 s)

0.0

0.5

1.0

1.5

2.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

θr,roof [%]

Data

Cornell

Miranda

Figure 9.9. Evaluation of equations (9.15) and (9.16) to estimate median θr,roof: (a) GF-3R (T1=0.5s); (b)

GF-18R (T1=2.0s).

(a) GF-3R (T1= 0.5 s)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

RIDR [%]

Data

Cornell

Miranda

(b) GF-18R (T1 = 2.0 s)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

RIDR [%]

Data

Cornell

Miranda

Figure 9.10. Evaluation of equations (9.15) and (9.16) to estimate median RIDRmax: (a) GF-3R (T1=0.5s);

(b) GF-18R (T1=2.0s).

In addition of evaluating central tendency, equation (9.17) was evaluated to characterize

changes of dispersion with changes on the level of relative intensity. However, it was found that


although sometimes adequate to represent variations of dispersion, the use of a polynomial

functional form might lead to important errors when evaluating dispersion at higher levels of

relative intensity than those considered in this investigation. Thus, the feasibility of functional

forms given by equations (9.15) and (9.16) to predict dispersion as a function of ground motion

intensity was also tested in this investigation. The fit obtained using equations (9.15) and (9.16) to

model dispersion of θr,roof and RIDRmax computed from the building models GF-3R and GF-18R is

shown in figures 10.11 and 10.12. It can be seen that also both the power and three-parameter

functional forms provides a reasonable fit of the variation of dispersion with changes in η(∆i).

However, it should be mentioned again that the three-parameter functional form given in equation

(9.16) is reasonable adequate to represent variations of dispersion of residual deformation

demands for the range of ground motion intensities covered in this investigation, but caution

should be exercised when using this equation for relative intensities beyond the range covered in

the regression.

(a) GF-3R (T1= 0.5 s)

0.0

0.5

1.0

1.5

2.0

2.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

σ ln θr,roof

Data

Regression

(b) GF-18R (T1 = 2.0 s)

0.0

0.5

1.0

1.5

2.0

2.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

σ ln θr,roof

Data

Regression

Figure 9.11. Evaluation of equations (9.15) and (9.16) to estimate dispersion of θr,roo f: (a) GF-3R (T1=0.5s);

(b) GF-18R (T1=2.0s).


(a) GF-3R (T1= 0.5 s)

0.0

0.4

0.8

1.2

1.6

2.0

2.4

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

σ ln RIDRmax

Data

Cornell

Miranda

(b) GF-18R (T1 = 2.0 s)

0.0

0.4

0.8

1.2

1.6

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0η (∆ i)

σ ln RIDRmax

Data

Cornell

Miranda

Figure 9.12. Evaluation of equations (9.15) and (9.16) to estimate dispersion of RIDRmax: (a) GF-3R

(T1=0.5s); (b) GF-18R (T1=2.0s).

The probability of exceeding RIDRmax, assuming a lognormal distribution with probability

parameters computed using the functional form of equation (9.16), for the GF-9R and GF-9F

building models for a relative intensity of 4 are shown in figure 9.13. It can be seen that the use of

a three-parameter functional leads to adequate probability parameter estimates and, thus, to a

reasonable representation of the probability of exceeding RIDRmax.

(b) 9-Story (T1 = 1.902 s)η = 4

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0

RIDRmax [%]

P (

IDR

max

>

| )

Data

Lognormal fit: sample data

Lognormal fit: equations

(a) 9-Story (T1 = 1.185 s)η = 4

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

RIDRmax [%]

P (

IDR

max

>

| )

Data


Lognormal fit: equations

Figure 9.13. Evaluation of equation (9.16) to estimate dispersion of RIDRmax: (a) GF-3R; (b) GF-18R. 9.4 Evaluation of the Proposed Direct Approach

The proposed approach allows obtaining different residual deformation hazard curves that would

be useful for decision making process during the seismic assessment of existing structures. For

example, hazard curves of maximum residual roof drift, )( ,roofrθλ , or maximum residual inter-


story drift ratio over all stories, )( maxRIDRλ , as well as residual inter-story drift ratio at selected

story levels, )( iRIDRλ , for a designated building. Such a residual deformation demand hazard

curves can be directly compared with maximum deformation demand hazard curves counterparts

developed in parallel. Therefore, several examples of selected building models are shown to

illustrate residual and maximum deformation seismic hazard curves as well as comparisons and

suggestions when used in seismic performance-based assessment of existing structures.

In this section, the proposed direct approach outlined in section 10.2.1 is employed to

compute residual deformation demand hazard curves of a short-period structure (T1=0.5s) and a

long-period structure (T1=2.0s). For this purpose, statistical information obtained from the GF-

3R and GF-18R building models, presented in last chapter, is used to compute sample central

tendency and dispersion parameters for six levels of relative intensity. The variation of central

tendency and dispersion of residual deformation demands was computed using equation (9.15) in

order to evaluate the conditional probability of exceeding residual deformation demands at a

given level of ground motion intensity in equation (9.5). The proposed approach assumes that the

empirical distribution of residual deformation demands conditioned in any level of ground motion

intensity can be characterized by the lognormal probability distribution. Therefore, the term

),,|( 1 δδ =∆==>∆ iyyr cCtTP can be computed with equation (9.5). Finally, the mean

annual frequency (MAF) of exceeding certain levels of residual deformation is calculated with

numerical integration of equation (9.4). It should be noted that the proposed approach make uses

of alternative site-period-yield strength-specific maximum inelastic displacement hazard

curve, )( i∆λ , developed in chapter 5 of this dissertation, which represent an improvement over

other performance-based assessment procedures.

9.4.1 Elastic and Inelastic Displacement Demand Hazard Curves

In order to compute residual deformation demand hazard curves (e.g., residual drift

demand, )( ,roofrθλ ) according to approach outlined in section 9.2 of this chapter, either the

elastic displacement hazard curve, )( dSλ , or a inelastic displacement hazard curve, )( i∆λ , at the

site of interest should be available. Site-specific elastic hazard curves usually are provided from


seismologists in terms of the mean annual frequency of exceeding a given level of peak ground

acceleration, )(PGAλ , or a spectral elastic acceleration at some period of vibration, ))(( 1TSaλ .

The later hazard curve is computed from the elastic response of SDOF systems having 5% of

damping ratio, for discrete periods of vibration, T1. For example, this information can be found

from the United States Geological Survey (USGS) for any site in the U.S. (Frankel and

Leyendecker, 2001). However, it as noted in section 5.5.1 (Chapter 5) that that cited reference

only provides discrete values of )( aSλ , while integration of equation (9.1) requires a continuous

seismic hazard curve. To overcome this problem, a fourth-order polynomial model was found

adequate to approximate the elastic seismic hazard curve in terms of the logarithm of Sa and

)( aSλ to obtain a continuous function, which is reproduced as follows

4104

3103

2102101010 )()()()( aaaaa SLogSLogSLogSLogSLog ⋅+⋅+⋅+⋅+= βββββλ

(9.18)

Thus, conventional linear regression analysis was used to obtain the parameter estimates that

provide the best fit of a given elastic seismic hazard curve corresponding to each period of

vibration. Since it is it is well-established that pseudo-spectral displacement is related to pseudo

spectral acceleration, it was assumed adequate to transform )( aSλ to )( dSλ for the range of

periods of vibration and damping ratio considered in this study.

On the other hand, a procedure to obtain inelastic displacement hazard curves, )( i∆λ , for

different periods of vibration and yield strength coefficients was proposed and evaluated in

Chapter 5. In order to obtain a continuous function to be used in equation (9.4), a fourth-order

polynomial model was also found adequate to approximate )( i∆λ in terms of the maximum

inelastic displacement of a single-degree-of-freedom system. The proposed model has the

following functional form:

4104

3103

2102101010 )()()()( iiiii LogLogLogLogLog ∆⋅+∆⋅+∆⋅+∆⋅+=∆ βββββλ

(9.19)

Similarly, conventional linear regression analysis is used to obtain the parameter estimates that

provide the best fit of each seismic hazard curve corresponding to each period of vibration. Figure


9.14 compares both elastic and inelastic curves calculated with equations (9.18) and (9.19),

respectively.

T = 2.0s, Cy = 0.2

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.0 10.0 100.0 1000.0

Displacement [cm]

λ

Elastic

Inelastic

Figure 9.14. Comparison of elastic and inelastic (for a system with

T=2.0s and Cy=0.2) displacement hazard curve.

9.4.2 Residual and Maximum Deformation Demand Hazard Curves

For illustration purposes, and using the statistical data obtained in previous chapter, two types of

residual deformation hazard curves were developed for a short-period (GF-3R, T1=0.5s) and a

long-period (GF-18R, T1=2.0s) building models. Their yield strength coefficient, Cy, was

obtained from the nonlinear static analysis being approximately 0.8 for the GF-3R building model

and 0.2 for the long-period building model. Therefore, maximum inelastic displacement hazard

curves corresponding to the combination of period of vibration and yield strength coefficient

were used for computing residual drift demand, )( ,roofrθλ , and residual inter-story drift

ratio, )( maxRIDRλ , hazard curves were obtained. The resulting hazard curves are shown in figure

9.15 for the short-period building model and in figure 9.16 for the long-period model. The

residual deformation hazard curves shown in the figures allow a probabilistic assessment of

residual deformation demands (e.g., residual roof drift demand and residual inter-story drift


demand), in which the uncertainty in the ground motion hazard at a given site and in the seismic

response of a specific building are explicitly taken into account. It should be emphasized that the

suites of earthquake ground motions considered for performing the nonlinear dynamic time

history analyses of the building of interest adequately represent the seismic hazard at the site.

(b)GF-3R (T1 = 0.5 s, Cy = 0.8)

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

RIDRmax [%]

λ (RIDRmax)

(a)GF-3R (T1 = 0.5 s, Cy = 0.8)

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

θ r,roof [%]

λ (θr,roof)

Figure 9.15. Residual deformation hazard curves for GF-3R building model: (a) θr, roof; (b) RIDRmax.

(b)GF-18R (T1 = 2.0 s, Cy = 0.2)

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

RIDRmax [%]

λ (RIDRmax)

(a)GF-18R (T1 = 2.0 s, Cy = 0.2)

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

θ r, roof [%]

λ (θr, roof)

Figure 9.16. Residual deformation hazard curves for GF-18R building model: (a) θr, roof; (b) RIDRmax.

As mentioned in section 1.4.2 (Chapter 1), a simplified assumption in the closed-form

solution suggested by Cornell et al. (2002) to compute engineering demand parameter (EDP)

hazard curves is that the record-to-record variability, IMEDP|lnσ , is constant regardless of the level

of ground motion intensity. Therefore, it is interesting to examine if significant differences in the

seismic hazard curves are introduced when dispersion of residual deformation demands changes

with variation of the relative intensity are considered compared with hazard curves computed


with constant dispersion. For instance, figure 9.17 shows )( ,roofrθλ and )( maxRIDRλ curves

computed from the results of the GF-18R building model assuming both variable and constant

dispersion with changes in the ground motion intensity. From the figures, it can be seen that

larger differences occur when computing residual drift hazard than when evaluating RIDRmax drift

hazard as the MAF decreases It should be mentioned that larger dispersion in θr,roof was found

than in RIDRmax. It should be mentioned that Medina and Krawinkler (2003) also noted that

variation in the dispersion with the intensity level plays an important role in the computation of

IDRmax hazard curves. Therefore, it is believed that the variation of both central tendency and

dispersion of residual deformation demands with modifications of relative intensity should be

addressed while computing hazard curves.

(b) GF-18R (T1 = 2.0s, Cy = 0.2)

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5RIDRmax [%]

λ (RIDRmax)

Variable dispersionConstant dispersion=0.45Constant dispersion=0.60

(a)GF-18R (T1 = 2.0s, Cy = 0.2)

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

θr, roof [%]

λ (θr,roof)

variable dispersionconstant dispersion=1.0constant dispersion=0.8

Figure 9.17.Comparison of residual deformation demand hazard curves considering variable and constant variation of dispersion with changes in ground motion intensity: (a) )( ,roofrθλ ; and (b) )( maxRIDRλ .


Similarly to the development of residual deformation hazard curves, roof drift demand,

)( ,roofrθλ , and maximum inter-story drift demand, )( ,roofrθλ , hazard curves were developed in

parallel for the same short-period (GF-3R, T1=0.5s) and a long-period (GF-18R, T1=2.0s)

building models. The resulting hazard curves are shown in figure 9.18 for the short-period

building model and in figure 9.19 for the long-period model.

(b)GF-3R (T1 = 0.5 s, Cy = 0.8)

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0IDRmax [%]

λ (IDRmax)

(a)GF-3R (T1 = 0.5 s, Cy = 0.8)

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0θ roof [%]

λ (θroof)

Figure 9.18. Maximum drift hazard curves for GF-3R building model: (a) θ roof; (b) IDRmax.

(a)GF-18R (T1 = 2.0 s, Cy = 0.2)

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0θ roof [%]

λ (θroof)

(b)GF-18R (T1 = 2.0 s, Cy = 0.2)

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

IDRmax [%]

λ (IDRmax)

Figure 9.19. Maximum drift hazard curves for GF-18R building model: (a) θ roof; (b) IDRmax.


9.5 Comparison of Residual and Maximum Deformation Hazard Curves

A main part in the proposed methodology is to evaluate the magnitude of potential residual

deformation demand with respect to maximum deformation demands for decision-making

process. This evaluation can be made directly by comparing residual deformation demand hazard

curve to maximum deformation demand hazard curves. This approach also represents developing

a vector of EDPs, which can be incorporated into a more comprehensive approach to estimate

MAF of exceedance of decision variables such as the mean annual earthquake losses.

(a)GF-3R (T1 = 0.5 s, Cy = 0.8)

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5θ roof, θr,roof [%]

λ

Roof drift

Residual roof drift

(b)GF-18R (T1 = 2.0 s, Cy = 0.2)

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5θroof, θr,roof [%]

λ

Roof drift

residual roof drift

Figure 9.20.Comparison of θroof and θr,roof hazard curves obtained from two building model: (a) GF-3R (b) )

GF-18R.


For instance, figure 9.20a shows )( ,roofrθλ and )( roofθλ computed for the short-period

building (GF-3R, T1=0.5) while figure 9.20b shows a similar comparison for the long-period

building (GF-18R, T1=2.0s). For both building models, it can be seen that the difference between

roof drift demand and residual drift demand decreases as the MAF of exceedance decreases,

which mean that residual roof drift demand as large as roof drift demands are expected for rare

seismic events (e.g., 2% probability of exceedance in 50 years).

(a)GF-3R (T1 = 0.5 s, Cy = 0.8)

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

RIDRmax, IDRmax [%]

λ

Maximum

Residual

(b)GF-18R (T1 = 2.0s, Cy = 0.2)

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0RIDRmax, IDRmax [%]

λ

Maximum

Residual

Figure 9.21.Comparison of IDRmax and RIDRmax hazard curves obtained from two building model:

(a) GF-3R; and (b) GF-18R.


On the other hand, a comparison of maximum and residual inter-story drift demand hazard

curves computed for the same building models are shown in figures 10.21a and 10.21b.It can be

seen that the difference between RIDRmax and IDRmax depends on the MAF of exceedance. For the

short-period building, it can be observed that the magnitude of RIDRmax might be close to IDRmax

for MAF of exceedance on the order of 0.0001. However, for the long-period building model, it

can be seen that the difference between RIDRmax and IDRmax tends to increase as the MAF of

exceeedance increases.

9.6 Summary

An improved approach for the performance-based assessment of existing structures is introduced

in this dissertation. This approach accounts for the estimation of both global (e.g., roof residual

drift ratio) and local (e.g., maximum residual inter-story drift ratio) residual deformation demands

along with commonly accepted maximum deformation demands (e.g., peak roof drift ratio and

peak inter-story drift ratio) to assess the seismic performance of site-specific existing structures.

The suggested approach could be used for performance-based global predesign of new structures

where structural and non-structural damage control is achieved through deformation control of

both maximum and residual lateral deformation demands.

___________________________________________________________________________________ Chapter 10 Summary and Conclusions

346

Chapter 10

Summary and Conclusions

10.1 Overview

The primary objectives of this investigation were to expand our understanding on the

parameters that influence the amplitude and distribution within the structure of peak

(transient) and residual (permanent) lateral displacement demands of structures as well as to

propose simplified methods to evaluate residual displacement demands in the context of

recently introduced performance-based assessment criteria of existing structures (e.g., Cornell

and Krawinkler, 2000; Moehle and Deierlein, 2004).

In particular, this dissertation proposes a methodology to evaluate both global (e.g., roof

drift ratio, θr,roof) and local (e.g., maximum inter-story drift ratio, RIDRmax) residual

deformation demands along with commonly accepted maximum deformation demands (e.g.,

peak roof drift ratio, θroof and peak inter-story drift ratio, IDRmax) of multi-story framed

building models representative of multi-degree-of-freedom (MDOF) systems. The suggested

approach employs Probabilistic Seismic Demand Analysis (PSDA) as a framework (Cornell,

1996) and it was formulated in Chapter 9. Therefore, both maximum and residual lateral

displacement demands of multi-degree-of-freedom (MDOF) systems are expressed in terms of

the structure-specific inelastic drift demand hazard curves that express the mean annual

frequency of exceedance of a certain level of deformation demand for a given seismic

environment. This information can be related to deformation-based performance limit states

(i.e., limiting residual drift demands) linked to discrete performance capacity levels of both

individual components as well as complete systems.

Since it was observed that the estimation of residual drift demands involves large levels of

dispersion (i.e., record-to-record variability), a more efficient intensity measure, IM, was

employed to reduce the dispersion. Then, one component of the proposed methodology is the


347

use of maximum inelastic displacement demand, i∆ , of an equivalent elastoplastic single-

degree-of-freedom (SDOF) systems, corresponding to the period of vibration and yield roof

displacement of the building of interest, as IM instead of traditional elastic IM (e.g., elastic

spectral acceleration associated to the first-mode period of vibration of the structure, Sa(T1)).

However, unlike hazard curves of Sa(T1), which are customarily provided form seismologists,

maximum inelastic displacement demand hazard curves are not readily available. Thus, a

procedure to develop maximum inelastic displacement demand hazard curves, ( )iδλ , of

SDOF systems was developed and illustrated in Chapter 4. Furthermore, the procedure was

also applied to compute residual displacement seismic hazard curves, ( )rδλ , of structures that

can be modeled as SDOF systems (e.g., some kind of bridges) and it was illustrated in Chapter

6. The procedure to obtain ( )iδλ and ( )rδλ is based on information obtained from

comprehensive statistical studies of both maximum and residual displacement demands for

inelastic SDOF systems subjected to a relatively large earthquake ground motion database

containing acceleration time histories having different ground motion features and recorded on

different soil site conditions (e.g., rock, firm soil and soft soil sites). In addition, derived from

these seismic demand hazard curves, uniform hazard spectra of maximum and residual

displacement demands was developed for two return periods.

Next, the following sections summarize the main tasks performed throughout this

investigation and highlight the most important findings. In addition, a discussion about the

limitations of the proposed procedures to evaluate peak and residual displacement demands is

offered in section 10.3 while thoughts about suggested research are provided in Section 10.4.

10.2 Summary and Main Findings

10.2.1 Evaluation of Maximum Inelastic Displacement Demands of SDOF

Systems

10.2.1.1 Statistical Studies

Initially, comprehensive statistical studies to evaluate maximum inelastic displacement

demands of SDOF systems with known (constant) lateral strength ratio (i.e., lateral strength


348

required to maintain the system elastic normalized with respect to constant yield strength

capacity of the system), R, when subjected to an ensemble of 438 earthquake ground motions

representative of different ground motion characteristics (i.e., amplitude, frequency content

and strong motion duration) and collected in different soil conditions (e.g., firm soil sites and

soft soil sites) were carried out and reported in Chapters 2 and 3. For this purpose, this stage

made used of constant-relative strength inelastic displacement ratios, CR, which are

nondimensional parameters that allow the estimation of maximum lateral inelastic

displacement demand, i∆ , from maximum elastic displacement demand, Sd.

For the sake of clarity, the main findings corresponding to inelastic SDOF systems

subjected to records compiled in firm soil conditions are summarized first. Those related to the

nonlinear response to soft-soil site records and fault-normal near-fault ground motions are

reported next.

Firm soil sites (Chapter 2)

• Based on the statistical results, the equal displacement approximation (i.e., maximum

inelastic displacement demands are equal to maximum inelastic displacement

demands), which is implicitly included in most of the seismic codes worldwide, tends

to underestimate i∆ in the short spectral region (i.e., periods of vibration, T, smaller

than about 1.0s), as the period of vibration, T, decreases and as R increases. For

example, i∆ is, on average, about 1.8 times Sd for T=0.5s and R=6). Furthermore, the

equal displacement approximation tends to overestimate i∆ when median values are

used to estimate the central tendency of CR in the medium and long-period region

(e.g., i∆ is about 0.9 times Sd for T=0.5s and R=6) .

• Limiting periods dividing regions where the equal displacement rule is applicable

from those where this rule is not applicable depend primarily on the lateral strength

ratio (e.g., limiting periods are about 0.85s for R=2 and 2.5 for R=4), although they

are also influenced by local site conditions (e.g., for R=2, the equal displacement rule

is approximately correct for periods longer than about 0.45s, 0.65s and 0.80s for

structures on site classes AB, C and D, respectively), and the type of hysteretic

behavior (e.g., for R=2 and considering Modified-Clough model, Di are similar than


349

Sd for T=1.5s, but ∆i is larger than Sd for all the period range considered in this

investigation for origin-oriented models, regardless of the lateral strength ratio) . In

general, these limiting periods increase primarily as the lateral strength ratio increases

and, to a lesser degree, as the average shear wave velocity in the upper 30 m of the

site profile decreases or as the level of structural degradation increases.

• Constant-strength inelastic displacement ratios, CR, have, in general, larger dispersion

(measured by coefficient of variation, COV) than that reported for constant-ductility

inelastic displacement ratios, Cµ (Miranda, 2000) (e.g., for T=0.5s, COV of Cµ is

about 0.2 for µ=2 while COV of CR is 0.27 for R=2), primarily in the very short-

spectral region. In particular, COV of CR tends to increase as the period of vibration

shortens for systems with T shorter than about 1.0s. In addition, dispersion of CR tend

to increase with increasing R, but tend to saturate for large lateral strength ratios (i.e.,

as the system becomes weaker relative to the lateral strength required to maintain the

system elastic)..

• It was found that the effects of local site conditions on CR are slightly larger than those

reported for Cµ (Miranda, 2000). However, the effects are still relatively small,

particularly for periods longer than 1.2 s. Neglecting the effect of site conditions for

structures with periods smaller than 1.5s built on firm sites will typically result in

errors less than 20% in the estimation of mean CR, whereas for periods longer than

1.5s the errors are smaller than 10%. Differences are even smaller if the lateral

strength ratio is equal or smaller than 3.

• Maximum inelastic displacement demands of short period degrading structures are on

average larger than those on non-degrading systems. In general, the increment in

displacement produced by degradation effects increases as the strength ratio increases.

For structures with periods longer than about 0.7 s, maximum deformation of

degrading systems are on average either similar or slightly smaller than those of non-

degrading system.

• Maximum inelastic displacement demands of stiffness-degrading systems are not

significantly affected by the unloading stiffness provided that the reduction in


350

unloading stiffness is small or moderate (e.g., nonlinear systems with hysteretic

behavior represented by Modified-Clough or Takeda models). However, for systems

that unload toward the origin (i.e., origin-oriented systems) maximum inelastic

displacement demands are, on average, larger than maximum elastic displacement

demands and, therefore, the equal displacement rule should not be used for these

systems.

• A simplified equation (see equation 2.15 in Chapter 2) was derived using nonlinear

regression analyses to estimate inelastic displacement demands of structures on firm

sites exhibiting elastoplastic behavior. The functional form of the proposed equation

has been recently adopted to improve coefficient C1 in FEMA 356 (Comartin et al.,

2004).

Soft-soil sites (Chapter 3)

• Despite the important differences in the geothecnical and ground motion

characteristics of soft soil deposits in Mexico City and in the San Francisco Bay Area

(e.g., lower shear wave velocities, higher water contents, lower frequency content,

smaller bandwidth) their CR ratios show similar trends.

• For systems with T smaller than about 0.75 times the predominant period of the

ground motion, Tg (computed as the period corresponding to the peak in a 5% damped

spectral velocity spectrum), i∆ becomes larger than Sd as the T/Tg ratio decreases and

as R increases. In this spectral region, mean CR ordinates increases nonlinearly with

increasing R. For systems with T close to Tg, maximum inelastic displacements are, on

average, smaller than maximum elastic displacements (on the order of 20%, for R=2,

to 62%, for R=6, for the Mexico City set). In this spectral region, the well-known

equal displacement approximation will lead to overestimations of lateral displacement

demands of inelastic systems. This was observed for practically all ground motion

recorded on very soft soil. For systems with T that are longer more than 1.5 times Tg ,

maximum inelastic displacements are, on average, close to the maximum elastic

displacements.

• Dispersion of CR is not constant over the whole normalized period range (T/Tg),

tending to increase for short T/Tg ratios. In general, the scatter of mean CR of soft soil


351

sites is smaller than that observed for firm sites (e.g., COV of CR is about 0.34 when

computed from records collected in Mexico City and around 0.72 when computed

from firm soil records, for T=0.5s and R=6).

• For the range of earthquake magnitudes and epicentral distances considered in this

investigation, it was shown that CR ordinates are not significantly affected by

earthquake magnitude or by distance to the source.

Fault-normal near-fault records (Chapter 3)

• The most important ground motion characteristic that influences the shape and

amplitude of CR ordinates is the pulse period, Tp, of the velocity pulse (The pulse

period for each ground motion was identified by Fu and Menun (2004) using a

velocity pulse model fitted to match each of the fault-normal near-fault ground motion

components).This is particularly true for systems with T shorter than Tp. Moreover,

fault-normal near-fault ground motions with Tp shorter than 1.0s leads to a local

amplification for systems with T/Tp ratio near 0.5. Therefore, an adequate

characterization of CR spectra for this type of ground motions should explicitly take

into account Tp.

• A good correlation (i.e., correlation coefficient of 0.83 for the fault normal near-fault

ground motion set used in this investigation) between Tp and Tg was found for the suite

of fault-normal near-fault ground motions considered in this investigation. This

observation allows an alternative way of computing CR for normalized periods since

Tg can be directly computed from a 5% damped spectral velocity spectrum.

• For systems with periods of vibration smaller than about 0.85 times Tp, maximum

inelastic displacements are, on average, larger than maximum elastic displacements

(e.g., for a system with T/Tp=0.5, ∆i is on the order of 1.12 and 2.54 times Sd for R=2

and R=6, respectively).. In this spectral region mean CR ordinates increases

nonlinearly with increasing lateral strength ratio, R. For systems with periods of

vibration between 0.85 and 2.0 times Tp, maximum inelastic displacements are, on

average, smaller than maximum elastic displacements (e.g., for a system with

T/Tp=1.0, ∆i is on the order of 0.89 and 0.77 times Sd for R=2 and R=6, respectively).

In this spectral region, using the well-known equal displacement approximation will

lead to overestimations of lateral displacement demands of inelastic systems. For


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systems with periods of vibration that are more than 2.0 times Tp , maximum inelastic

displacements are, on average, close to the maximum elastic displacements.

• Dispersion of CR is not constant over the whole normalized period range (T/Tp),

tending to significantly increase for short T/Tp ratios. In general, it was found that the

scatter of mean CR for near-fault ground motions is significantly smaller when the

period of vibration is normalized with respect to Tg instead of Tp (e.g., for R=6, COV

of CR is approximately 0.53 for T/Tg=0.5 while it is about 1.77 for T/Tp=0.5).

• It was shown that CR ordinates computed for systems with T/Tg ratios larger than

about 0.5 are not significantly affected by earthquake magnitude or by distance to the

source (for the range of earthquake size and epicentral distances considered here).

Similarly, CR ordinates are not very sensitive to the peak ground velocity of the near-

fault records.

• It was found that the effect of post-yielding stiffness in limiting maximum inelastic

displacements demands is less beneficial for near-fault ground motions than for far-

field ground motions.

• An increment in the level of stiffness-and-strength degradation increases maximum

inelastic deformation demands for systems with T shorter than Tg. However, for

systems with T greater than Tg, maximum inelastic displacement demands of

degrading systems are slightly smaller than that of non-degrading systems.

10.2.1.2 Simplified Probabilistic Approach to Estimate ∆i

To account for the uncertainty inherent in the estimation of maximum inelastic displacement

demands, a robust probabilistic simplified approach was proposed in Chapter 4. The proposed

simplified approach allows computing maximum inelastic displacement demand hazard

curves, )( iδλ , that express the mean annual frequency (MAF) of exceeding a given maximum

inelastic displacement demand of elastoplastic SDOF systems having different periods of

vibration and different yield strength coefficient. Moreover, these curves allow developing

uniform hazard spectra of maximum inelastic displacement demand for structures that can be

modeled as SDOF systems.


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The proposed simplified probabilistic approach was implemented for estimating )( iδλ of

structures located in rock or firm soil sites. The proposed approach, founded in PSDA, couples

the evaluation of the conditional probability of exceeding certain inelastic displacement

demands with information of site-specific generic elastic seismic hazard curves, which are

already available in the United States (e.g., from United States Geological Survey). The

proposed simplified approach makes use of constant-strength inelastic displacement ratios, CR,

in order to estimate the maximum inelastic displacement demand, i∆ , from maximum elastic

displacement demand, Sd. The proposed simplified approach relies on three main simplified

assumptions:

• It is assumed that the dependence of CR on earthquake magnitude and distance to the

earthquake source is negligible. This assumption allows computing the expected value

of CR without the need of attenuation relationships.

• The approach assumes that CR and Sd are statistically uncorrelated random variables,

which implies that the expected value of CR only depends on the period of vibration

and the level of lateral strength ratio of the inelastic SDOF system.

• The empirical distribution of CR can be evaluated by an adequate parametric

cumulative distribution (e.g., lognormal or Weibull probability distribution).

The assumptions underlying the proposed simplified approach were carefully evaluated

and examined throughout Section 4.3 (Chapter 4). In summary, it was found some mild

dependence of CR on earthquake magnitude and distance to the source (e.g., for systems with

T< 0.5s and R>4). However, for structures in the range of engineering interest, the dependence

is small. In addition, the suggested approach makes use of CR to estimate ∆i as the product of

CR times Sd, which assumes that CR and Sd are uncorrelated random variables. This assumption

leads to relatively small errors as compared to i∆ evaluated accounting for the correlation

between CR and Sd. The size of the error depends on the local firm soil conditions. Finally,

after examining several candidates, it was concluded that the empirical distribution of CR can

be modeled by a parametric lognormal cumulative distribution. For evaluating the statistical

parameters of the lognormal distribution, the central tendency and dispersion are obtained

from simplified models that adequately capture the observed empirical trends (see equations


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4.22-4.24, Chapter 4). For illustration purposes, a family of )( iδλ and uniform hazard spectra

of maximum inelastic displacement, MID-UHS, corresponding to two return periods were

obtained for a region of high seismicity in California.

It is believed that both )( iδλ and MID-UHS are very useful for establishing performance

limit-states based on maximum inelastic displacement demands of structures that can be

represented as SDOF systems (e.g., bridge piers). Furthermore, )( iδλ allows using maximum

inelastic displacement as ground motion intensity measure, which is a key parameter in

current performance-based methodologies based on PSDA for the assessment of existing

structures and the design of new structures. However, it should be recognized that the

simplified probabilistic approach will provide more accurate estimations if an attenuation law

of CR would be available. This task was beyond the scope of this dissertation.

10.2.2 Evaluation of Residual Displacement Demands of SDOF Systems

10.2.2.1 Statistical Studies

An effort to improve our understanding on residual displacement demands of SDOF systems

was undertaken in this investigation. For this purpose, two approaches were introduced to

estimate residual displacement demands of SDOF systems. The first approach, referred to as

direct approach, consists of estimating the residual displacement demand through empirical

information about the ratio of residual displacement demand at the end of the excitation, r∆ ,

to the maximum elastic displacement demand, Sd, of SDOF systems with known relative

strength. This ratio was defined as residual displacement ratio, Cr. In the second approach,

referred to as indirect approach, the residual displacement demand is computed by first

estimating the maximum inelastic displacement demand and then multiplying it by the ratio of

residual displacement demand to maximum inelastic displacement demand (named residual

ratio, γ) for constant-relative strength SDOF systems. Therefore, a comprehensive statistical

study was conducted to evaluate Cr and γ ratios using four different ground motion ensembles


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representative of different seismic environments and main results were reported in Chapter 5.

Next, main findings obtained from this statistical study are summarized:

Direct Approach:

• It was found that mean Cr follows a similar trend than mean CR when ground motions

recorded in firm site conditions are considered. Particularly, in the short period

spectral region (e.g., T<0.5s), residual displacement demands of systems with constant

relative strength are, on average, larger than maximum elastic demands. In this

spectral region, the ratio of residual displacement to maximum elastic displacement

demand is strongly dependent on the period of vibration and the lateral strength ratio

(e.g., for R=6 and T=0.3s, r∆ is about 1.35 times Sd) as well as on the type of

hysteretic behavior. For periods longer than 1.0s, mean Cr ratios are not very sensitive

to changes in the period of vibration and they mainly depend on the lateral strength

ratio (e.g., for T=1.5s, from 0.25 for R=2 to 0.52 for R=6) and the type of hysteretic

behavior (e.g., for T=1.5s and R=2, Cr is about 0.17 when using Modified-Clough

model while it is about 0.12 when employing Takeda model).

• In general, dispersion of Cr is larger than that for CR (e.g., for T=0.5s and R=4.0, COV

of Cr is around 1.0 while COV of CR is approximately 0.67), which means that the

variability in the estimation of residual displacement demands is very important and it

should be explicitly taken into account. In particular, coefficients of variation of Cr

tend to increase with increasing R and when periods of vibration are in the short

spectral region.

• It was found that the effects of local firm site conditions on Cr are larger than those

observed on CR. In addition, it was shown that Cr computed from earthquake ground

motions recorded on soft soil conditions or near the causative fault exhibit a different

trend than those computed from records collected on rock or firm site conditions.

Therefore, influence of local soil site conditions (e.g., firm or soft soils) or the type of

ground motion (e.g., near-fault or far-fault) should be addressed when estimating

residual deformation demands.

• For near-fault ground motions, it was found that the pulse period of the velocity pulse

is an important parameter influencing the amplitude and shape of residual


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displacement ratios. Therefore, it should be explicitlt taken into account while

evaluating residual displacement demands.

• Residual displacement ratios of stiffness-degrading systems are significantly affected

by the level of unloading stiffness. It was observed that Cr ratios and, in consequently,

residual deformation demands decreases as the unloading stiffness decreases in

stiffness-degrading systems.(i.e., Takeda model leads to smaller residual displacement

demands than Modified-Clough model since Takeda model has smaller unloading

stiffness than Modified-Clough model).

Indirect Approach:

• It was found that residual ratios obtained from ground motions recorded on firm soil

sites increases as the lateral strength ratio increases, but they are not very sensitive to

changes in the period of the ground motion. In addition, residual ratios are not

significantly influenced to local firm soil site conditions.

• Residual ratios computed from soft soil records and near-fault records show variations

with changes in the normalized period of vibration and the lateral strength ratio of the

system. They showed a different trend that that observed for residual ratios computed

for firm soil sites.

• For residual ratios computed using near-fault ground motions, it was showed that the

pulse period influences the ordinates of residual ratios.

• It was found that the estimation of mean residual ratios involves smaller variability

(i.e., dispersion) than that associated in the estimation of mean inelastic displacement

ratios (e.g., COV of residual ratio is around 0.6 while COV of Cr is approximately 0.8

for a system with T=1.0s and R=6).

• Similarly to residual displacement ratios, the unloading stiffness in stiffness-degrading

systems has an important effect in the estimation of residual ratios. In general, residual

ratios decrease as the unloading-stiffness decreases (e.g., for T=1.0s and R=6, residual

ratio is around 0.24 when considering Modified-Clough model while it is

approximately 0.15 when Takeda model is employed).


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10.2.2.2 Simplified Probabilistic Approach to Estimate ∆r

From the statistical studies performed in Chapter 5, it was shown that the dispersion of Cr and

residual ratios is very important, which means that dispersion should explicitly be taken into

account when evaluating residual displacements. Therefore, a simplified probabilistic

approach that explicitly incorporates the uncertainty inherent in the estimation of residual

displacement demands was proposed in this dissertation (Chapter 6). Like the probabilistic

approach to compute )( iδλ and MID-UHS, the suggested parallel approach allows the

computation of residual displacement demand hazard curves, )( rδλ , and uniform hazard

spectra of residual deformation demand, RD-UHS. The suggested simplified approach takes

advantage of statistical information obtained from Cr and, similarly to the probabilistic

estimation of maximum inelastic displacement demands, it relies on parallel simplified

assumptions. The validity of the simplified assumptions was also investigated throughout this

chapter.

10.2.3 Evaluation of Maximum and Residual Deformation Demands of MDOF

Systems

To gain further understanding about the sensitivity of residual deformation demands in

multi-degree-of-freedom (MDOF) systems with respect to several structural (i.e., fundamental

period of vibration, number of stories, hysteretic behavior, overstrength) and ground motion

parameters (i.e., intensity and strong shaking duration), a comprehensive statistical study on

MDOF systems was carried out and the main results are reported in Chapter 7.

To perform sensitivity studies, a family of 12 multi-story framed building models, having

6 number of stories (N = 3, 6, 9, 12, 15 and 18) was carefully designed to capture the global

behavior of multi-story moment-resisting frames. For each building model with N number of

stories, a rigid and a flexible representation were designed. The rigid building models have

fundamental periods between 0.5s and 2.0s, while the flexible counterparts have fundamental

periods between 0.79s and 3.31s. The family of generic frame models was analyzed under a

set of 40 far-field earthquake ground motions scaled to reach the same level of relative


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intensity, η (defined in Section 7.5.2). In this investigation, a structure-specific relative

intensity, η, was defined based on the structure’s roof yield displacement and an inelastic

ground motion intensity measure, IM. The inelastic IM is based on the maximum inelastic

displacement demand computed from an equivalent SDOF system having the same

fundamental period of vibration and same lateral strength of the analyzed building. The

alternative inelastic IM is more efficient in reducing the record-to-record variability of

maximum deformation demands (i.e., θroof and IDRmax) as the ground motion intensity

increases (i.e., as the structure is expected to behave more nonlinearly) than traditional elastic

IM’s, even for long-period flexible building models where the presence of higher-mode effects

is expected. Furthermore, it also reduces the record-to-record variability of residual

deformation demands (e.g., θr,roof and RIDRmax) even for high levels of relative intensity

considered in this investigation. It should be recalled that maximum inelastic displacement

hazard curves were already developed in Charter 4 following a suggested simplified

probabilistic approach. Thus, its use is recommended for performing analytical studies that

make use of PSDA and, furthermore, when a vector of engineering demand parameters (e.g.,

maximum and residual deformation demands) are employed to describe building performance.

This statistical study provided valuable information about primary deformation demands

(i.e., maximum and residual deformation demands) of MDOF systems. The most relevant

observations from the evaluation of deformation demands made in Chapter 7 are reported

next:

• Effect of ground motion intensity. Median maximum and residual deformation

demands (e.g., IDRmax and RIDRmax) increase as the ground motion intensity increases.

However, their rate of increment is different. In particular, between building models

with the same number of stories but different fundamental period of vibration, median

deformation demands for flexible models grow at a faster rate than those for their

rigid counterparts.

• Effect of number of stories. For a given level of relative intensity, both median θroof

and θr,roof demands, are not significantly affected for the number of stories. In general,

for a given level of relative intensity and number of stories, dispersion of θr,roof is


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much higher than dispersion of θroof (e.g., for η=4 and a 9-story rigid building model,

dispersion of θr,roof is 4.38 times higher than that associated to θroof) but no significant

trend with respect to the number of stories was identified. In particular, dispersion of

θroof seems to increase as the number of stories increases, for a given relative intensity,

but at a non-constant rate. On the other hand, median IDRmax tends to increase as the

number of stories and relative intensity increases. A similar trend was observed for

median RIDRmax, but the influence of the number of stories is less significant. It was

noted that the dispersion of IDRmax is higher than the dispersion of θroof (e.g., for η=4

and a 9-story rigid building model, dispersion of IDRmax is 1.48 times higher than that

associated to θroof) and that, for a given relative intensity, it slightly increases at a non-

uniform rate as the number of stories increases. In addition, the dispersion of RIDRmax

was, in general, smaller that the dispersion of θr,roof, although still is considerably high

(i.e., around 0.4 and 0.6).

• Effect of the period of vibration. It was found that, for a given relative intensity,

both median θroof and θr,roof do not change significantly with changes in the period of

vibration. In addition, dispersion of θroof tends to increase as the period of vibration

increases and as the relative ground motion intensity increases. However, for a given

period of vibration, dispersion tends to saturate as the relative intensity increases. On

the other hand, dispersion of θr,roof does not follow a clear trend with changes in

relative intensity and period of vibration. Unlike median θroof, median IDRmax

increases as both the period of vibration and the relative intensity increases. The rate

of increment is higher for generic flexible buildings than for their rigid counterparts.

However, median RIDRmax does not significantly grow as the period of vibration

increases, for both rigid and flexible building models. Similarly to the dispersion of

θroof, the dispersion of IDRmax follows an increasing rate as the period of vibration

increases, while dispersion of RIDRmax ranges between about 0.4 and 0.6 for the

ranges of periods covered in this study.

• Effect of the type of mechanism. For a building with the same number of stories, but

different period of vibration, the type of mechanism has more influence on a flexible

building than on the rigid counterpart. It was also observed that encouragement of


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buildings developing beam-hinge mechanism (e.g., by increasing beam-to- column

flexural capacity ratios) constraints RIDRmax in addition of reducing damage

concentration in specific stories and delaying/reducing column damage.

• Effect of the member hysteretic modeling. Member strain-hardening in the moment-

curvature, M-φ, relationship was found to effectively control median IDRmax. In

particular, a member strain-hardening of 2% produced a significant reduction of

median RIDRmax besides that it leads to a more uniform height-wise distribution. In

addition, it was found that an increase in unloading stiffness in stiffness-degrading M-

φ relationships leads to smaller median RIDRmax as compared to non-degrading M-

φ relationships, but the effect of unloading stiffness in median IDRmax is negligible.

However, both median IDRmax and RIDRmax increase when member M-φ relationship

exhibits strength deterioration. The increment in both deformation demands depends

on the level of strength deterioration and the level of relative intensity.

• Effect of overstrength. Global (system) overstrength constraints both maximum and

residual deformation demands for all levels of relative intensity considered in this

study. The inclusion of non-uniform overstrength along the height leads to a more

uniform distribution of median story peak and residual drift ratio, which delays the

concentration of drift demands at the bottom stories. In addition, the inclusion of

height-wise uniform overstrength might lead to larger peak and residual drift demands

in the upper stories than when non-uniform overstrength is considered.

• Effect of ground motion duration. It was found that ground motion duration does

not have a significant effect on both median IDRmax and RIDRmax for building models

with non-degrading (elastoplastic) member M-φ relationship. Moreover, even

considering moderate strength deterioration in the member M-φ relationship, ground

motion duration did not affect the variation of median IDRmax and RIDRmax with

changes in the ground motion intensity.


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10.2.4 Probabilistic Estimation of Maximum and Residual Deformation

Demands of MDOF Systems

A probabilistic approach for performance-based assessment of existing structures is

introduced in this investigation. This approach accounts for the estimation of both global (e.g.,

roof residual drift ratio, θr,roof) and local (e.g., , maximum residual inter-story drift ratio,

RIDRmax) residual deformation demands along with commonly accepted maximum

deformation demands (e.g., peak roof drift ratio, θroof, and peak inter-story drift ratio, IDRmax)

to assess the seismic performance of site-specific existing structures. The suggested approach

also could be used during the preliminary design phase of new structures (Bertero and Bertero,

2002; Teran-Gilmore, 2004), where structural and non-structural damage control is achieved

through control of lateral deformation demands, but not only of maximum lateral deformation

demands but also of residual lateral deformation demands.

In the proposed approach, both maximum and residual lateral deformation demands are

expressed in terms of the mean annual frequency of exceedance of deformation demand which

can be related to deformation-based performance limit states associated to discrete

performance levels of both individual components as well as complete systems. An important

component of the proposed methodology is the use of maximum inelastic displacement

demand seismic hazard curves instead of traditional seismic hazard curves of spectral elastic

ordinates.

10.3 Limitations

10.3.1 Hysteretic Modeling

The statistical evaluation of maximum and residual displacement demands for SDOF systems

assumes that the global response of structures that can be modeled as SDOF systems exhibit

one of the hysteretic behaviors considered in this investigation (e.g., elastoplastic, stiffness-

degrading, combination of stiffness-and-strength degradition) during earthquake excitation.

However, it should be recognized that commonly used hysteretic behavior models (e.g.,

elastoplastic, Modified-Clough, Takeda, origin-oriented, etc.) represent the force-deformation


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relationships of components, or subassemblies, tested under quasi-static cyclic loading using

deformation or force control loading protocol that simulate earthquake-type loading. However,

the actual hysteretic response of structures under earthquake loading might differ from the

quasi-static cyclic loading, as has been evidenced from shake-table tests. Therefore, the results

presented in this investigation, as well as many published in the literature, should recognize

this fact.

10.3.2 Probabilistic Approach

The proposed approach to estimate maximum and residual displacement demand hazard

curves of inelastic SDOF systems relies on simplified assumptions and, therefore, has some

limitations. Similarly to other methodologies that employ PSDA as a framework, the

suggested approach assumes that inelastic displacement ratios are not strongly influenced by

the ground motions characteristics (i.e., earthquake magnitude, distance to the source and

strong motion duration) of the earthquake environment. However, some dependence was

observed for certain period regions and levels of lateral strength ratios, which might influence

the computation of the conditional probability of exceeding some level of CR or Cr. A more

accurate probabilistic approach should include attenuation relationships of maximum and

residual displacement demands.

The proposed simplified approach is generic in essence. For instance, the proposed

approach was illustrated for a firm soil site in California in California. However, different

seismic environments might be encountered in the same region (e.g., structures located in the

bay-mud area of San Francisco Bay or those near the causative fault). Therefore, site-specific

seismic hazard curves considering soft-soil sites or near-fault effects should be available

10.3.3 Multi-Story Building Models

Although promising for conducting statistical studies that involves multiple, and time

consuming, nonlinear dynamic time-history analyses, the use of one-bay generic framed

building models has some limitations that should be mentioned. First, the buildings models

intend to capture global seismic response parameters (e.g., global roof drift, maximum inter-


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story drift), but the evaluation of local measures (e.g., member ductility demand, joint seismic

demands) might be inaccurate. Some of the limitations of the building modeling that can

influence the seismic response are given as follows:

• The influence of flexural demands caused by gravity loads on formation of plastic

hinges has been neglected.

• Bending Moment-Shear force-Axial load (M-P-V) interaction is not considered.

• Soil-Foundation-Structure interaction was ignored.

• Member hysteretic behavior including severe stiffness-and-strength deterioration.

• Multi-story framed building models were modeled as two-dimensional and, thus, tri-

dimensional effects were neglected (e.g., floor torsional effects).

In addition, it should be mentioned that recent studies have shown that the contribution of

interior gravity frames in the seismic response of steel moment resisting frames leads to

smaller deformation demands even if the columns and beams in the interior gravity frames are

modeled as pinned connections (Lee and Foutch, 2002). However, in the design of the generic

frames neither an additional contribution of lateral stiffness nor strength contribution of

interior gravity frames was considered in this study.

10.3.4 Range of Ground Motion Intensity

It should be noted that the set of ground motions used in the statistical study of MDOF

systems were scaled to a range of intensities that do not drive collapse (e.g., collapse is

assumed when extreme IDRmax, on the order of 10%, occurs in any building). However, in

some cases the scaled records lead to peak inter-story drifts in excess of 10% and just in one

cases the median IDRmax demands exceeded 10%, which is the cases of the tall-flexible

building model (GF-18F, T1=3.31s) having member elastoplastic behavior and subjected to

records scaled to a relative intensity of six. Finally, it should noted that the statistics included

all results without special consideration of the individual collapse cases.


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10.4 Suggested Research

This dissertation provides an enhanced approach to explicitly incorporate the evaluation of

residual deformation demands for the performance-based assessment of existing structures or

the global preliminary design of new standard structures. However, there are many issues that

still need to be studied. Some of these issues related to estimate residual deformation demands

are described below:

• Intensity Measure. Verification of i∆ as intensity measure for characterizing ground

motion recorded on soft-soil conditions or near the causative fault is desirable.

Otherwise, alterative intensity measures that satisfy the criteria proposed by Luco and

Cornell (2004) for this seismic environment are desirable.

• Ground Motion Input. The ground motion input was considered to excite the building

models in one (horizontal) direction. However, spatial ground motion input (both

horizontal and vertical) should be considered in the analysis.

• Building Modeling. A common and key need in the field of earthquake engineering is

improved analytical tools to mimic the nonlinear behavior of structures subjected to

seismic excitation. From simulation of structural degrading member hysteretic

behavior to adequately reproduce local and global failure modes is of primary

importance to extend our understanding on the amplitude and distribution of

permanent deformation demands. In addition, it is necessary to include the

contribution of non-structural components in the analytical models. This task, and

others, involves improvement of building modeling.

• Three-Dimensional Effects. Irregularities in strength or stiffness at the floor-level

were not considered. These structural deficiencies lead to torsional effects that may

give rise to larger residual displacement demands.

• Implementation for Structures Subjected to Soft Soil Sites and Near-Fault Effects. The

approach to compute residual displacement demand seismic hazard curves and

uniform hazard spectra of residual displacement demand should be extended to

account for soft-soil sites and near-fault-effects.


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• Calibration. Perhaps, one of the most important tasks for future research is the

calibration of the proposed methodology. This task can be done by comparing

estimates of residual deformation demands with respect to recorded permanent

deformations after dynamic earthquake-type excitation, such as shaking-table tests.

For instance, Kanvinde (2003) recorded permanent deformations of small-scale one-

story steel frames tested in one-direction shaking-table at Stanford University.

Therefore, it is suggested to conduct these type of tests for a variety of ground motion

characteristics (i.e., intensity, frequency content, duration) and structural properties

(i.e., period of vibration, lateral strength) to record possible permanent deformation. In

addition, post-earthquake reconnaissance provides another way of measuring

permanent deformations for calibration purposes.

• Other Lateral Resisting Systems. It should be interesting to extend the study of

residual deformation demands to other lateral resisting systems, such as braced frames

(e.g., bucking restrained braced frames, concentrically braced frames, eccentric braced

frames) and dual frame-wall systems as well as materials (e.g., composite steel-

concrete, wood, masonry, etc.)

_____________________________________________________________________ Appendix A Earthquake Ground Motion Records

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Appendix A

Earthquake Ground Motion Records

A.1 Introduction

A total of 458 earthquake ground motions collected in different seismic environments were

used throughout this investigation. The selected earthquake records comprise 7 distinct suites

of ground motions that intend to be representative of the seismic hazard and soil conditions

encounter in sites of high seismicity. In this Appendix, a detailed description of each suite of

ground motion is given, as well as information about the main ground motion characteristics.

A.2 Suites of Earthquake Ground Motions

A.2.1 Firm Soil Sites

A total of 240 earthquake acceleration time histories recorded in the state of California in 12

different earthquakes with surface wave magnitude (Ms) ranging from 5.8 to 7.7 were used in

this study. A particularly large number of earthquake ground motions were selected in order to

assess the dispersion of the inelastic displacement ratios and in order to be able to obtain

inelastic displacement ratios corresponding to different percentiles. All the ground motions

selected have the following characteristics: (1) recorded on accelerographic stations where

enough information exists on the geological and geotechnical conditions at the site that

enables the classification of the recording site in accordance to recent code recommendations

(e.g., FEMA, 1997a; 1997b; 2000); (2) recorded on rock or firm sites with average shear wave

velocities higher than 180 m/s (600 ft/s) in the upper 30 m (100 ft) of the site profile; (3)

recorded on free field stations or in the first floor of low-rise buildings with negligible soil-

structure interaction effects; (4) recorded in earthquakes with surface wave magnitudes (Ms)


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larger than 5.7; and (5) records in which at least one of the two horizontal components had a

peak ground acceleration larger than 40 cm/s2.

The earthquake ground motions were divided into three groups according to the local site

conditions at the recording station. The first group consisted of 80 ground motions recorded

on stations located on rock with average shear wave velocities between 760 m/s (2,500 ft/s)

and 1,525 m/s (5,000 ft/s). The second group consisted of 80 records obtained on stations on

very dense soil or soft rock with average shear wave velocities between 360 m/s (1,200 ft/s)

and 760 m/s while the third group consisted of 80 ground motions recorded on stations on stiff

soil with average shear wave velocities between 180 m/s (600 ft/s) and 360 m/s. Recording

stations in the first group correspond to site class B according to recent design provisions (e.g.,

FEMA, 1997a; 1997b; 2000) while recording stations in the second and third groups

correspond to site classes C and D, respectively. Figure A.1 shows the magnitude-distance

distribution for all 240 earthquake ground motions and figure A.2 illustrates the distribution of

PGA and distance to the rupture.

A complete list of all ground motions including peak ground accelerations, earthquake

magnitude, site class at the recording station, and distance to the horizontal projection of the

fault rupture is given in Tables A.1, A.2 and A.3.

5.0

5.5

6.0

6.5

7.0

7.5

8.0

0.0 20.0 40.0 60.0 80.0 100.0 120.0

Distance to the rupture [km]

Magnitude [Ms]

NEHRP site classes B,C,D

Figure A.1. Magnitude versus distance to horizontal projection of rupture for earthquake ground

motions considered in this study.


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0

100

200

300

400

500

600

700

0.0 20.0 40.0 60.0 80.0 100.0 120.0

Distance to the rupture [km]

PGA [cm/s2]

Site Class B

Site Class C

Site Class D

Figure A.2. Peak ground acceleration (PGA) versus distance to horizontal projection of rupture for

earthquake ground motions considered in this study.

Finally, figure A.3 shows median spectral elastic acceleration, Sa, spectra scaled at T=0.5s

and 0.9s. As can be seen, the shape of the median Sa for the three groups is similar, although

the spectra computed from records collected in site class AB is slightly larger when scaled at

T=1.0s. In addition, dispersion (standard deviation of the natural logarithm of Sa, aSlnσ )

computed from ground motions collected in each site condition is illustrated in figure A.4a. It

can be seen that despite the differences in ground motion characteristics, aSlnσ follows a

similar trend regardless of local site condition (i.e., dispersion increases as the period of

vibration increases). However, it should be noted that records compiled in site classes C and

D leads to larger dispersion than those recorded in site class AB for periods of vibration in the

medium- and long-period region (i.e., for T > 0.75s). A similar plot computed from all 240

earthquake ground motions recorded in site classes AB, C and D is shown in figure A.4b. In

general, the aforementioned observations hold.


369

Table A.1. Earthquake ground motions recorded in the NEHRP Site Class AB.

Magnitude Station Comp. 1 PGA PGV Comp. 2 PGA PGV(Ms) Number [deg] [cm/s2] [cm/s] [deg] [cm/s2] [cm/s]

1971 San Fernando 6.5 Lake Hughes, Array Station 4 126 19.6 111 168.2 5.7 201 143.5 7.11971 San Fernando 6.5 Lake Hughes, Array Station 9 127 23.0 21 119.3 4.5 291 109.4 3.91979 Imperial Valley 6.8 Superstition Mountain 286 26.0 135 189.2 6.3 45 108.0 5.11984 Morgan Hill 6.1 Gilroy 1, Gavillan Coll. 47379 16.0 230 57.5 2.9 320 93.4 2.91986 Palm Springs 6.0 Silent Valley, Poppet Flat 12206 23.7 0 102.6 3.9 90 107.4 4.01986 Palm Springs 6.0 Winchester, Hidden Valley Farms 13200 49.8 90 54.6 1.3 270 56.5 1.31986 Palm Springs 6.0 Winchester, Bergman Ranch 13199 55.3 0 62.2 1.9 90 85.7 1.81986 Palm Springs 6.0 Murrieta Hot Springs, Collings Ranch 13198 61.0 0 45.9 1.8 90 49.4 1.31986 Palm Springs 6.0 Anza Fire Station 5160 46.7 225 97.0 5.8 315 65.7 4.01986 Palm Springs 6.0 Anza Red Mountain 5224 45.6 270 102.0 5.2 360 126.5 3.41986 Palm Springs 6.0 Hemet Fire Station 12331 43.3 270 141.1 4.9 360 129.4 4.91986 Palm Springs 6.0 Anza-Tule Canyon 5231 55.4 270 107.8 6.5 360 93.1 7.51987 Whittier 6.1 Mt Wilson, CIT Seismic Station 24399 22.1 0 121.3 3.3 90 171.3 4.61987 Whittier 6.1 Los Angeles, Gritfith Park Observatory 141 22.3 0 133.8 3.6 360 121.4 4.11989 Loma Prieta 7.1 Gilroy 1, Gavillan Coll. 47379 10.5 90 433.6 33.9 360 426.6 31.61989 Loma Prieta 7.1 Hollister, SAGO south cinega road surface 47189 32.4 261 70.7 10.3 351 65.3 9.31989 Loma Prieta 7.1 Monterey, City Hall 47377 42.7 90 61.1 5.8 360 68.5 3.51989 Loma Prieta 7.1 South San Francisco, Sierra Point 58539 67.6 115 57.2 7.1 205 102.7 8.81989 Loma Prieta 7.1 San Francisco, Dimond Heighs 58130 75.9 90 110.8 14.3 360 96.4 10.51989 Loma Prieta 7.1 Piedmont, Piedmont Jr. High Grounds 58338 77.2 45 81.2 8.2 315 69.6 9.11989 Loma Prieta 7.1 San Francisco, Rincon Hill 58151 78.5 90 88.5 10.4 0 78.6 6.71989 Loma Prieta 7.1 San Francisco, Pacific Heights 58131 80.5 270 60.2 12.8 360 46.3 9.21989 Loma Prieta 7.1 San Francisco, Cliff House 58132 87.4 0 73.1 11.2 90 105.7 21.01989 Loma Prieta 7.1 San Francisco, Telegraph Hill 58133 88.0 90 51.2 5.5 0 90.5 9.51989 Loma Prieta 7.1 Point Bonita 58043 88.1 297 71.4 12.9 207 69.9 11.41992 Landers 7.5 Twentinine Palms Park Maintennance Bldg 22161 41.9 0 78.7 3.7 90 59.1 4.91992 Landers 7.5 Silent Valley, Poppet Flat 12206 51.3 90 39.4 5.1 0 48.9 3.81992 Landers 7.5 Amboy 21081 68.3 0 88.3 11.0 90 143.2 20.01994 Northridge 6.8 Malibu Canyon, Griffith Observatory 5080 20.2 360 176.4 12.3 270 270.0 17.71994 Northridge 6.8 Lake Hughes, Array Station 9 24272 28.4 90 221.2 10.1 360 154.5 8.41994 Northridge 6.8 Los Angeles, Temple & Hope 24611 32.2 180 189.1 20.0 90 123.7 13.91994 Northridge 6.8 Lake Hughes, Array Station 4 24469 34.0 0 56.4 5.1 90 82.4 5.11994 Northridge 6.8 Mt Wilson, CIT Seismic Station 24399 36.9 90 130.7 5.8 360 228.5 7.41994 Northridge 6.8 Los Angeles, City Terrace 24592 37.1 90 258.0 12.8 0 310.1 14.11994 Northridge 6.8 Antelope Buttes 24310 48.6 90 99.7 4.3 0 44.9 3.61994 Northridge 6.8 San Pedro, Palos Verdes 14159 58.5 90 93.1 6.6 0 98.9 5.61994 Northridge 6.8 Leona Valley #3 24307 37.8 0 82.4 8.5 90 104.0 8.11994 Northridge 6.8 L.A. Wonderland 90017 22.7 95 109.8 8.7 185 168.6 11.81994 Northridge 6.8 Rancho Cucamonga-Deer Can 23598 80.0 90 69.6 4.2 180 49.9 5.91994 Northridge 6.8 Littlerock-Brainard Can 23595 46.9 90 70.6 6.0 180 58.8 6.0

Earthquake Name Station Name Dist. [km]


370

Table A.2. Earthquake ground motions recorded in the NEHRP Site Class C.


1952 Kern County 7.7 Santa Barbara, Courthouse 283 85.0 42 87.8 12.1 132 128.6 15.51952 Kern County 7.7 Pasadena, CIT Athenaeum 475 109.0 180 46.5 5.6 270 52.1 9.21952 Kern County 7.7 Taft Lincoln School 1095 41.0 21 152.9 15.3 111 174.5 17.51971 San Fernando 6.5 Lake Hughes, Array Station 12 128 17.0 21 346.2 17.0 291 277.9 12.71971 San Fernando 6.5 Glendale, 633 E. Broadway 122 18.0 110 265.7 10.3 200 209.1 13.41971 San Fernando 6.5 Lake Hughes #1, Fire station #78 125 25.0 21 145.5 18.0 111 108.9 11.71971 San Fernando 6.5 Castaic Old Ridge Route 110 26.0 21 309.4 15.6 291 265.4 25.91971 San Fernando 6.5 Pearblossom Pump Plant 585 36.0 0 91.5 4.7 270 120.5 5.61979 Imperial Valley 6.8 El Centro, Parachute Test Facillity 5051 14.0 225 106.9 17.8 315 200.2 16.11984 Morgan Hill 6.1 Gilroy #6, San Ysidro Microwave Site 57383 11.5 0 214.8 11.4 90 280.4 36.71984 Morgan Hill 6.1 Gilroy Gavillan college Phys Scl Bldg 47006 16.0 337 85.9 2.9 67 95.0 3.61987 Whittier 6.1 Pasadena-CIT Athenaeum 80053 15.4 180 170.5 11.5 270 98.9 6.01987 Whittier 6.1 Arleta, Nordhoff Av. Fire Station 24087 38.9 180 91.1 5.4 270 89.2 4.71987 Whittier 6.1 L.A.-116 th. St School 14403 22.5 270 288.1 17.6 360 388.1 21.01987 Whittier 6.1 Inglewood, Union Oil Yard 14196 22.5 0 246.1 18.1 90 219.3 8.91987 Whittier 6.1 Long Beach, Recreation Park 14241 29.6 90 57.2 3.5 180 53.8 5.51987 Whittier 6.1 Sylmar, Olive View Medical Center 24514 45.9 0 55.8 4.0 90 50.3 3.51987 Whittier 6.1 Riverside, Airport 13123 57.8 180 38.4 1.4 270 56.8 1.41987 Whittier 6.1 Lancaster, Medical Office Bldg FF 24526 72.2 10 59.4 2.9 100 59.6 3.01987 Whittier 6.1 Castaic, Old Ridge Route 24278 77.3 0 67.2 4.4 90 65.4 4.51987 Whittier 6.1 LA-N Westmoreland 90021 16.6 0 209.7 9.7 270 195.0 6.21987 Whittier 6.1 Panorama City-Roscoe 90007 33.0 90 102.9 9.7 180 105.9 6.21989 Loma Prieta 7.1 Gilroy, Gavillan college Phys Sch Bldg 47006 10.9 67 349.1 28.9 337 310.0 23.01989 Loma Prieta 7.1 Saratoga, Aloha Ave. 58065 12.4 90 316.2 43.5 0 494.5 41.31989 Loma Prieta 7.1 Santa Cruz, UCSC 58135 12.5 90 401.5 21.2 360 433.1 21.21989 Loma Prieta 7.1 Gilroy 6, San Ysidro Microwave site 57383 19.9 90 166.9 14.2 0 112.2 12.81989 Loma Prieta 7.1 Coyote Lake Dam, downstream 57504 21.7 285 174.7 22.6 195 154.7 13.01989 Loma Prieta 7.1 Woodside, Fire Station 58127 38.7 90 79.7 14.7 0 79.5 15.51989 Loma Prieta 7.1 Fremont, Mission San Jose 57064 42.6 90 100.5 8.8 0 117.7 11.51989 Loma Prieta 7.1 Hayward, CSUH Stadium 58219 56.7 90 82.6 6.4 0 72.5 5.61989 Loma Prieta 7.1 Berkeley, Lawrence Berkeley Lab. 58471 83.9 90 114.4 20.9 0 47.7 9.21992 Landers 7.5 Desert Hot Springs 12149 23.2 0 167.6 20.2 90 150.9 20.91994 Northridge 6.8 Castaic Old Ridge Route 24278 24.6 360 504.2 52.2 90 557.1 52.11994 Northridge 6.8 San Marino, SW Academy 24401 35.5 360 148.2 5.4 90 122.5 7.31994 Northridge 6.8 Alhambra, 900 S. Fremont 24461 37.2 360 78.3 12.7 90 99.1 9.71994 Northridge 6.8 Lake Hughes #1, Fire station #78 24271 37.7 0 84.9 9.2 90 75.2 9.41994 Northridge 6.8 Inglewood, Union Oil Yard 14196 44.7 0 89.2 7.1 90 98.9 10.31994 Northridge 6.8 L.A.-116 th. St School 14403 41.9 90 203.9 10.3 180 130.4 13.51994 Northridge 6.8 Beverly Hill-12520 Mulhol 90014 23.2 35 604.7 40.8 125 435.2 30.21994 Northridge 6.8 Rancho Palos Verdes, Hawthorne Blvd. 14404 53.8 0 71.1 5.0 90 52.7 3.3



371

Table A.3. Earthquake ground motions recorded in the NEHRP Site Class D.


1952 Kern County 7.7 Los Angeles, Hollywood Storage PE Lot 135 107.0 90 41.2 6.0 180 58.1 5.31968 Borrego Mtn 6.7 El Centro, Imperial Valley Irrigation District 117 45.0 180 127.8 26.3 270 56.3 13.21971 San Fernando 6.5 Los Angeles, Hollywood Storage Bldg. 135 23.0 90 207.0 18.9 180 167.3 14.91971 San Fernando 6.5 Vernon, Cmd Terminal 288 33.5 187 80.5 6.4 277 104.6 9.81971 San Fernando 6.5 Santa Ana, Engineering Bldg. 281 71.5 176 26.8 2.9 266 28.2 3.11979 Imperial Valley 6.8 Calexico, Fire Station 5053 10.6 225 269.6 21.2 315 196.9 16.01979 Imperial Valley 6.8 El Centro #11, McCabe Union School 5058 12.6 140 355.4 24.7 230 374.5 29.81979 Imperial Valley 6.8 El Centro #3, Pine Union School 5057 12.7 140 261.7 46.8 230 218.1 39.91979 Imperial Valley 6.8 El Centro #12 , 907 Brockman Road 931 18.0 140 138.7 11.1 230 113.4 10.71979 Imperial Valley 6.8 El Centro #13, Strobel Residence 5059 21.9 140 114.7 14.7 230 136.2 13.01979 Imperial Valley 6.8 El Centro #1, Borchard Ranch 5056 15.5 140 136.2 16.0 230 131.3 10.71979 Imperial Valley 6.8 Plaster City, Storehouse 5052 32.0 135 55.5 4.2 45 41.9 3.81979 Imperial Valley 6.8 Coachella, Canal #4 5066 49.0 45 113.6 12.5 135 125.7 15.61984 Morgan Hill 6.1 Gilroy #2, Hwy 101/Bolsa Road Motel 47380 15.1 0 153.7 5.1 90 207.8 12.61984 Morgan Hill 6.1 Gilroy #7, Mantnilli Ranch,Jamison Rd 57425 13.7 0 183.0 7.4 90 111.5 6.01984 Morgan Hill 6.1 Gilroy #3 Sewage Treatment Plant 47381 14.4 0 177.0 11.2 90 189.8 12.71984 Morgan Hill 6.1 Gilroy #4 57382 12.8 270 219.5 19.3 360 341.1 17.41987 Whittier 6.1 Bell Los Angeles Bulk Mail Center 5129 10.6 10 322.1 31.1 280 436.9 39.71987 Whittier 6.1 Vernon, Cmd Terminal 288 11.1 7 267.3 25.4 277 239.9 22.91987 Whittier 6.1 Downey, County Maintennance Bldg 14368 16.2 180 193.2 28.8 270 150.7 13.41987 Whittier 6.1 Los Angeles, Hollywood Storage Bldg. 24303 23.8 0 201.3 9.0 90 103.7 6.91987 Whittier 6.1 Century City, LA Country Club South 24390 29.6 0 57.6 3.7 90 67.2 4.21987 Whittier 6.1 Pomona 4th, and locust FF 23525 29.9 12 68.4 2.4 102 49.0 2.31987 Whittier 6.1 Long Beach, Harbor Administration Bldg 14395 32.8 0 48.2 5.5 90 68.9 3.61987 Whittier 6.1 Rancho Cucamonga, Law and Justice Center 23497 45.5 90 55.5 1.4 360 45.3 1.41987 Whittier 6.1 Rosamond, Goode Ranch 24274 89.0 0 73.8 3.5 90 50.4 3.11989 Loma Prieta 7.1 Gilroy 7, Mantelli Ranch Jamison Rd. 57425 24.2 0 221.5 16.4 90 316.6 16.61989 Loma Prieta 7.1 Gilroy 2, Hwy 101 Bolsa Road Motel 47380 12.1 90 316.3 39.1 0 394.2 32.91989 Loma Prieta 7.1 Gilroy 3, Sewage Treatment Plant 47381 14.0 90 362.0 44.7 0 531.7 35.71989 Loma Prieta 7.1 Agnews, Agnews State Hospital 57066 27.0 90 157.6 17.6 0 163.1 26.01989 Loma Prieta 7.1 Hayward, John Muir School 58393 58.9 90 136.0 11.5 0 166.5 13.71989 Loma Prieta 7.1 Oakland, 2 story 58224 76.3 290 238.3 36.1 200 187.3 19.91989 Loma Prieta 7.1 Richmond, City Hall Parking lot 58505 92.7 280 103.6 14.2 190 122.7 17.31992 Landers 7.5 Yermo, Fire Station 22074 26.3 270 240.0 51.5 360 148.6 29.71992 Landers 7.5 Palm Springs, Airport 12025 28.2 0 74.2 10.9 90 87.2 13.81992 Landers 7.5 Fort Irwin 24577 65.5 0 111.4 9.7 90 119.8 16.41992 Landers 7.5 Baker, Fire Station 32075 88.3 50 105.6 9.4 140 103.6 11.01992 Landers 7.5 Pomona, 4th and Locust FF 23525 117.6 0 65.5 12.3 90 43.2 8.51992 Landers 7.5 Hemet, Stetson Av. Fire Station 12331 69.5 0 79.4 5.6 90 95.1 5.71994 Northridge 6.8 Los Angeles, Hollywood Storage Bldg. 24303 24.8 360 381.4 22.3 90 227.0 18.1



372

(a) Scaled at T=0.5 s

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

Sa [cm/s2]

Site Classes ABCD

Soil Type AB

Soil Type C

Soil Type D

(b) Scaled at T=1.0 s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

Sa [cm/s2]

Site Classes ABCD

Soil Type AB

Soil Type C

Soil Type D

Figure A.3. Median spectral acceleration response, Sa, spectra computed for ζ = 5%:

a) Scaled at T=0.5 s; b) scaled at T = 1.0 s.

(b)ALL SITE CLASSES AB,C,D

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.5 1.0 1.5 2.0PERIOD [s]

σ ln Sa

(a)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.5 1.0 1.5 2.0PERIOD [s]

σ ln Sa

Site class AB

Site class C

Site class D

Figure A.4. Dispersion of Sa: (a) from each site class; (b) from all 240 earthquake ground motions.

A.2.2 Soft Soil Sites

A total of 118 earthquake acceleration time histories recorded on soft soil deposits were

considered in this investigation.

The first set includes 18 ground motions recorded on bay mud sites in the San Francisco

Bay Area during the 1984 Morgan Hill earthquake and the 1989 Loma Prieta earthquake. San

Francisco Bay is located in a basin about 15 km wide bounded by the active San Andreas and

Hayward fault zones. This region is characterized by a wide variety of geologic deposits from

rocks sites in the hill area to estuarine mud and clay deposits in the flatlands along the margins

of the bay. The bay mud area is comprised by unconsolidated, water-saturated, dark plastic


373

clay and silty clay with well-sorted silt and sand dunes in some areas. It may contain more

than 50% of water content and low shear-wave velocities in the range of 67 to 116 m/s

(Borchedt, 1996). Table A.4 summarizes the main characteristics of the records comprised in

this set.

The second set of earthquake ground motions consists of 100 acceleration time histories

recorded in the soft soil zone of Mexico City in five recent earthquakes. Figure A.4 shows a

map of the current seismic microzonation of Mexico City. Three zones can be observed, the

soft zone (“lake bed” zone), the transition zone and the firm zone (hill zone). Mexico City is

located partly on an old lake bed that was formed by the Texcoco, the Chalco and the

Xochimilco lakes. Relatively thick deposits of lacustrine clay form the soft zone. In the lake

bed zone the depth of these soft clay deposits varies from 10 m (33 ft) to 60 m (197 ft).

These clay deposits are very deformable and are characterized by very high water contents

that reach more than 400%, shear wave velocities as low as 40 m/s, and high plasticity

indexes. The firm soil (hill zone) represented by a dark gray in the figure is mainly formed

by volcanic tuff. The soils corresponding to this zone are in general hardly compressible and

exhibit a relatively high shearing strength. The transition zone lies between the hill and lake

bed zone and is characterized by the variability of the soil profile. Sequences of hard soils,

sands, silty sands and soft clays are typical soil stratigraphies of this area. Damage surveys

(Rosenlueth and Meli, 1986) showed that most of the structures that collapsed or suffered

major damage during the 1985 Michoacan earthquake were located within the zone bounded

by the dotted line in figure A.5. Table A.5 lists the second set of records considered in this

study.

Table A.4. Earthquake ground motions recorded in the San Francisco Bay Area used in this study.

Date Magnitude [Mw]

Station Name Station No.

Comp. 1 PGA [cm/s2]

Comp. 2 PGA [cm/s2]

10/17/89 6.9 Foster City (Redwood Shores) 58375 90 277.6 360 63.0 10/17/89 6.9 Treasure Island (Fire Station) 58117 0 112.0 90 97.9 10/17/89 6.9 Oakland (2-Story O.B.) 58224 290 238.1 200 187.3 10/17/89 6.9 Oakland (Outer Harbor Wharf) 58472 35 281.4 305 265.5 10/17/89 6.9 S.F. International Airport 58223 90 325.7 0 230.8 10/17/89 6.9 Emeryville 1662 260 255.0 350 255.0 10/17/89 6.9 Larkspur Ferry Terminal 1590 270 134.7 360 94.6 10/17/89 6.9 Redwood City 1002 43 270.0 233 222.0 04/24/84 6.2 Foster City (APPEL 1) 58375 40 45.1 310 66.7


374

Table A.5. Earthquake ground motions recorded in Mexico City used in this study

Date Magnitude [Ms]

Station name Station No.

Comp. 1 PGA [cm/s2]

Comp. 2 PGA [cm/s2]

09/19/85 8.1 SCT SC EW 167.9 NS 97.9

04/25/89 6.9 Alameda 01 EW 37.4 NS 45.5 04/25/89 6.9 C.U. Juarez 03 EW 37.4 NS 40.2 04/25/89 6.9 Xochipilli 06 EW 43.3 NS 43.3 04/25/89 6.9 Villa Gomez 09 EW 38.6 NS 38.6 04/25/89 6.9 P.C.C. Superficie 25 EW 42.5 NS 28.9 04/25/89 6.9 Villa del Mar 29 EW 46.5 NS 49.4 04/25/89 6.9 Jamaica 43 EW 31.2 NS 35.2 04/25/89 6.9 U. Colonia IMSS 44 EW 39.6 NS 52.3 04/25/89 6.9 Balderas 45 EW 51.4 NS 42.6 04/25/89 6.9 Rodolfo Menendez 48 EW 47.7 NS 27.7 04/25/89 6.9 San Simon 53 EW 30.5 NS 39.7 04/25/89 6.9 Tlatelolco 55 EW 32.8 NS 44.9 04/25/89 6.9 Liverpool 58 EW 40.0 NS 40.6 04/25/89 6.9 Candelaria 59 EW 45.2 NS 28.6 04/25/89 6.9 Roma RO EW 54.7 NS 45.4

10/24/93 6.6 Xochipilli 06 EW 9.9 NS 8.3 10/24/93 6.6 Villa del Mar 29 EW 11.4 NS 13.6 10/24/93 6.6 Jamaica 43 EW 8.4 NS 12.1 10/24/93 6.6 U.Colonia IMSS 44 EW 15.0 NS 12.2 10/24/93 6.6 Buenos Aires 49 EW 17.1 NS 14.4 10/24/93 6.6 Tlatelolco 55 EW 9.7 NS 8.3 10/24/93 6.6 Roma-B RO-B EW 8.5 NS 6.5 10/24/93 6.6 Roma-C RO-C EW 7.9 NS 10.5 10/24/93 6.6 SCT SC EW 10.5 NS 10.9

12/10/94 6.3 Xochipilli 06 EW 15.3 NS 16.4 12/10/94 6.3 Tlatelolco 08 EW 14.4 NS 14.8 12/10/94 6.3 Jamaica 43 EW 10.6 NS 12.1 12/10/94 6.3 Balderas 45 EW 13.7 NS 11.3 12/10/94 6.3 Buenos Aires 49 EW 15.7 NS 16.4 12/10/94 6.3 Tlatelolco 55 EW 12.8 NS 9.8 12/10/94 6.3 Cordova 56 EW 17.4 NS 17.2 12/10/94 6.3 Candelaria 59 EW 14.1 NS 14,1 12/10/94 6.3 Garibaldi 62 EW 15.1 NS 13.9 12/10/94 6.3 Roma RO EW 12.0 NS 14.2 12/10/94 6.3 Roma-A RO-A EW 16.5 NS 19.4 12/10/94 6.3 Roma-B RO-B EW 13.7 NS 10.3 12/10/94 6.3 SCT SC EW 15.0 NS 11.0

09/14/95 7.1 Alameda 01 EW 37.4 NS 45.5 09/14/95 7.1 C.U. Juarez 03 EW 25.9 NS 24.9 09/14/95 7.1 CUPJ 04 EW 26.8 NS 24.5 09/14/95 7.1 Tlatelolco 08 EW 28.5 NS 26.5 09/14/95 7.1 P. .Elias Calles 10 EW 30.0 NS 29.7 09/14/95 7.1 Jamaica 43 EW 24.3 NS 27.7 09/14/95 7.1 Tlatelolco 55 EW 19.4 NS 29.7 09/14/95 7.1 Cordova 56 EW 45.2 NS 44.1 09/14/95 7.1 Garibaldi 62 EW 25.8 NS 30.1 09/14/95 7.1 Roma RO EW 37.4 NS 28.6 09/14/95 7.1 Roma-B RO-B EW 25.0 NS 23.6 09/14/95 7.1 Roma-C RO-C EW 28.9 NS 31.1


375

Figure A.5. Location of ground motion accelerographic stations in Mexico City where records used in this study were obtained.

09 08

55 01

10 29

SC 53

59 48

43

62 25

06 45 RM

58

56 49 04

03 RO

5 km

05

44

-99.20 -99.15 -99.10 -99.05 -99.00 -98.95

Longitude

19.50

19.45

19.40

19.35

19.30

19.25

Latitude N

Major Damage Area

Transition Zone

Hill Zone

Lake Bed Zone


376

A.2.3 Fault-Normal Near-Fault

In this study, a suite of 40 fault-normal near-fault ground motions was assembled to evaluate

inelastic displacement ratios for this seismic environment. A detailed list of all grounds

motions including earthquake name, station name, and horizontal distance from the recording

site to the surface projection of the rupture, PGV, velocity pulse and pulse period, Tp, can be

found in Table A.6 It should be noted that the pulse period for each ground motion was

identified by Fu and Menun (2004) using a velocity pulse model fitted to match each of the

fault-normal near-fault ground motion components.

In addition, figure A.6 shows mean and median spectral elastic acceleration, Sa, spectra

and dispersion (measured in terms of the coefficient of variation) of Sa corresponding to the

fault-normal near-fault ground motion set. Finally, figure A.8 shows a pair-wise distribution

of peak ground velocity, PGV with respect to earthquake magnitude and distance to the

source.

Fault-Normal Near-Fault Set(40 ground motions)

0

200

400

600

800

1000

1200

1400

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Sa [cm/s2]

median

mean

Fault-Normal Near-Fault(40 ground motions)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

COV Sa

Figure A.6. (a) Mean and median Sa spectra; (b) Dispersion (COV). |

0

20

40

60

80

100

120

140

160

180

200

0.0 5.0 10.0 15.0 20.0Closest Distance to the Rupture, D [km]

PGV [cm/s]

0

20

40

60

80

100

120

140

160

180

200

5.5 6.0 6.5 7.0 7.5Earthquake Magnitude [Mw]

PGV [cm/s]

Figure A.7. (a) Peak ground acceleration (PGV) versus earthquake magnitude; (b) PGV versus closest distance to the rupture.


377

Table A.6. Fault-normal near-fault earthquake ground motions used in this study Date Magnitude

[Mw] Earthquake Name Station Name D

[km] PGA

[cm/s2] PGV

[cm/s2] Tp [s]

06/28/66 6.1 Parkfield Station 2 (Cholame #2) 0.1 466.8 75.1 1.88 02/09/71 6.6 San Fernando Pacoima dam 3.3 1443.9 114.3 1.38 05/02/83 6.4 Coalinga Pleasant Valley P.P.-yard 8.5 586.2 57.9 0.70 04/24/84 6.2 Morgan Hill Anderson Dam 2.6 244.6 36.5 0.49

Gilroy Array # 6 11.8 155.0 54.5 1.04 07/08/86 6.0 N. Palm Springs North Palm Springs 8.2 656.3 73.5 1.26

Desert Hot Springs 8.0 322.0 26.9 1.38 Whitewater Trout Farm 7.3 523.5 35.8 0.63

11/24/87 6.7 Superstition Hills El Centro Imp. Co. Cent 13.9 302.6 51.9 2.41 10/01/87 Whittier Narrows Bell Gardens-Jaboneria 9.8 251.1 19.1 0.71

Santa Fe Springs-E Joslin 10.8 390.8 23.7 0.70 09/17/89 6.9 Loma Prieta Gilroy Array #1 11.2 419.3 38.5 4.24

Gilroy Array #2 12.7 398.5 45.7 1.43 Gilroy Array #3 14.4 523.0 49.3 1.79 Gilroy Array #4 16.1 338.5 35.7 1.37 Gilroy-Gavilan Coll. 11.6 288.5 30.8 1.77 Gilroy-Historic Bldg. 12.7 276.3 31.9 1.54 Saratoga-Aloha Ave. 13.0 355.8 55.6 2.25 Saratoga-W Valley Coll. 13.7 395.3 71.3 2.16 Los Gatos 3.5 704.1 172.8 3.21 Lexington Dam 6.3 673.2 178.6 1.81

03/13/92 6.9 Erzican Erzican 2.0 424.1 120.2 2.31 06/28/92 7.3 Landers Lucerne Valley 1.1 699.9 136.0 5.54 01/17/94 6.7 Northridge Canoga Park-Topanga Can 15.8 368.6 53.5 2.02

Canyon County-W Lost Cany 13.0 457.1 53.5 1.89 Jensen Filter Plant 6.2 385.6 104.5 2.83 Newhall – Fire Station 7.1 709.8 120.9 0.93 Rinaldi Receiving 7.1 869.7 173.1 1.16 Sepulveda VA 8.9 708.1 65.5 2.99 Sylmar Converter 6.2 583.0 130.3 2.88 Sylmar Converter East 6.1 822.8 116.5 3.05 Sylmar Olive View 6.4 718.7 123.1 2.53 Newhall-W.Pico Canyon Rd. 7.1 417.6 87.7 2.18 Pacoima Dam Downstream 8.0 489.1 49.6 0.48 Pacoima Kagel Canyon 8.2 516.5 56.2 0.72

01/17/95 6.9 Hyogo-Ken-Nambu KJMA 0.6 838.0 95.7 0.86 Port Island 6.6 425.0 100.3 2.34 Takatori 4.3 771.5 173.8 2.11 JMA 3.4 1067.7 160.3 0.90

08/17/01 7.4 Kokaeli Gebze 17.0 239.4 50.3 6.47

A.2.3 Long- and Short-Duration Sets

Among several definitions, the most widely measure of strong ground motion duration for

earthquake engineering purposes was defined by Trifunac and Brady (1975). The authors

defined significant strong motion duration, td, as the time interval from 5% to 95% of the

Arias intensity computed from each single acceleration time-history. The merit of this

definition is that Arias Intensity (Kramer, 1996) has showed strong correlation with observed

earthquake damage in short period structures as well as structures susceptible of liquefaction

potential, but it does not explicitly takes into account geophysical features (Boomer and

Martinez-Pereira, 1999).


378

The strong motion duration of 40 selected earthquake ground motions was computed and,

subsequently, two sets of 20 ground motions having short strong motion duration, ranging

between 8.8 s and 15.9 s, and long strong motion duration, between 25.7 s and 51.7 s, were

comprised. Throughout this investigation, s20-SD designates the short-duration set while s20-

LD refers to the long-duration set. A complete list of each ground motion set is given in

Tables A.6 and A.7. In addition, the median spectral acceleration spectra for each set scaled at

two periods is shown in figure A.8

(b) Scaled at T = 1.0 s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Sa

(T)

/ Sa

(T=1

.0s)

s20-SDmedians20-LD

(a) Scaled at T = 0.5 s

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]

Sa

(T)

/ Sa

(T=0

.5s)

s20-SDmedians20-LD

Figure A.8. Median elastic acceleration spectra for s20-LD and s20-SD ground motion sets:

(a) Scaled to T=0.5s; b) scaled to T=1.0s.

It should be noted that strong motion duration is related to earthquake magnitude and

distance to the source (e.g., Eliopoulos and Wen; 1991). For instance, figure A.8 shows the

distribution of tD and moment magnitude from all short- and log-duration records.

0

10

20

30

40

50

60

6.0 6.5 7.0 7.5 8.0 8.5Earthquake Magnitude [Mw]

tD [km]

s20-SD

s20-LD

Figure A.8. Distribution of strong motion duration and earthquake magnitude

for the sets s20-SD and s20-LD used in this investigation.


379

Table A.6. Long-duration earthquake ground motions used in this study.

Date Magnitude [Mw]

Earthquake Name Station Name Component PGA [cm/s2]

tD [s]

09/19/85 8.0 Michoacan, Mexico Caleta de Campos 90 140.7 29.3 180 139.7 25.7 La Union 90 148.3 28.1 180 165.6 27.1 La Villita 90 121.5 41.8 180 125.4 43.4

03/03/85 8.0 Valparaiso, Chile Llolleo 10 698.3 29.2 100 436.9 30.4 Melipilla 0 673.0 30.7 90 518.0 27.3 San Felipe 80 425.1 30.3 170 303.5 31.3 Valparaiso 50 291.5 50.0 140 162.7 51.7

09/20/99 7.6 Chi-Chi, Taiwan CHY15 E 142.4 37.5 CHY25 E 156.2 33.7 CHY46 W 139.6 30.0 TCU120 W 220.8 32.6 TCU123 W 160.7 35.4 TCU138 E 191.1 34.1

Table A.7. Short-duration earthquake ground motions used in this study.

Date Magnitude [Mw]

Earthquake Name Station Name PGA [g]

tD [s]

10/15/79 6.5 Imperial Valley El Centro Array #1 0.139 8.9 Plaster City 0.057 10.7

09/17/89 6.9 Loma Prieta Gilroy Array #3 0.367 11.4 Gilroy Array #7 0.226 11.5 Palo Alto-SLAC Lab 0.112 12.5

01/17/94 6.7 Northridge LA. Centinela St. 0.322 12.4 Canoga Park-Topanga Can 0.420 10.4 LA. N Faring Rd. 0.273 8.8 LA. Fletcher Dr. 0.240 11.8 Glendale-Las Palmas 0.206 11.5 LA-Hollywood Stor FF 0.231 12.0 Lake Hughes #1 0.087 13.9 Leona Valley #6 0.178 10.4 Northridge-17645 Saticoy St. 0.159 11.0 La Crescenta-New York 0.368 15.7 LA.- Saturn St 0.474 11.6 LA- Vernon Ave. 0.153 15.9 LA-Hollywood Stor Lot 0.174 11.2

11/24/87 6.7 Superstition Hills Brawley 0.156 13.5 Plaster City 0.186 11.3

_____________________________________________________________________ Appendix B Ground Motion Characterization

380

Appendix B

Ground Motion Characterization

B.1 Introduction

A key aspect in performance-based earthquake engineering and, particularly, in site response

analysis is the selection of a suite of ground motions and the adequate ground motion

characterization. Some important characteristics of a ground motion are the intensity, duration

of the ground shaking, and frequency content. Thus, several parameters have been proposed in

the literature to represent the latter characteristics. In this Appendix, further discussion about

ground motion characterization is offered.

B.1 Definition of Frequency Content Parameters

B.1.1 Predominant Period of the Ground Motion

Several definitions to estimate the predominant period of the ground motion have been

suggested in the literature (e.g. Miranda, 1991; Rathje et al., 1998; Cuesta and Aschheim,

2001). In this investigation six definitions were evaluated: (a) predominant period based on

spectral acceleration spectrum, Ta; (b) predominant period based on spectral velocity spectrum

Tg; (c) smooth spectral predominant period, To; (d) characteristic period, ∗1T ; (e) central

period, Tc; and (f) spectral characteristic period, ∗cT based on spectral moments of the square

velocity spectra.

The first definition to characterize the ground motion frequency content is named

predominant period of the ground motion, Ta, and it is defined as the period at which the

maximum ordinate in a 5% damped acceleration spectrum occurs (Rathje et al; 1998). The

second definition was suggested by Miranda (1991) and it is defined as the period


381

corresponding to the maximum ordinate of the velocity spectrum. Both spectra are computed

from an elastic single-degree-of-freedom (SDOF) system having 5% damping ratio.

The smooth spectral predominant period second is defined as follows (Rathje et al., 1998)

[ ] [ ]

[ ] [ ]PGATSHTS

PGATSHTST

T

ia

n

iia

ia

n

iiai

o

2.1)()(ln

2.1)()(ln

1

1

−⋅

−⋅⋅

=

∑

∑

=

= (B.1)

where n is the number of periods in the acceleration response spectrum, Ti is the i-th period of

vibration, Sa (Ti) is the spectral acceleration corresponding to Ti, H[x] is a Heaviside function

(it equals 1 when x > 0 and 0 for x < 0), and PGA is the peak ground acceleration.

The third definition called characteristic period of the ground motion, ∗1T , has been

recently employed to characterize the ground motion frequency content on strength-reduction

factors (Cuesta and Aschheim, 2003) and it is defined as

max

max1 )(

)(2

a

v

ss

T π=∗ (B.2)

where (sv)max and (sa)max are the maximum ordinates in the pseudo-velocity and pseudo-

acceleration response spectra computed from elastic SDOF systems with damping ratio of 5%.

The characteristic period is defined as the period at the transition between the constant-

acceleration and constant-velocity segments of a 5% damped elastic spectrum and it is

equivalent to the characteristic period, TS, in current seismic provisions (FEMA, 2000)

The last definitions of predominant period of the ground motion are based on a modified

version of the spectral parameters λ0 and λ1 originally proposed by Vanmarcke (1972). The

spectral moments, λi, of any stationary process were defined as

∫∞

=0

)( ωωωλ dGii (B.3)

where G(ω) is the spectral density function and the spectral moments, ωi, are given by


382

2,1

1

0

1 =

= i

i

i λλ

ω (B.4)

Therefore, ω1 may be interpreted as the distance to the centroid of the spectral mass, G(ω),

from the frequency origin and ω2 as the radius of gyration of G(ω) about the frequency origin.

The spectral parameter ω2 has been called the central frequency. Following this approach, the

spectral parameters ∗0λ , ∗

1λ and ∗2λ were computed from the square velocity spectra for elastic

SDOF systems having damping ratio of 5% as follows

∑=

∗ ∆⋅=n

iiv TS

1

2,0λ (B.5)

∑=

∗ ∆⋅⋅=n

iivi TST

1

2,1λ (B.6)

∑=

∗ ∆⋅⋅=n

iivi TST

1

2,

22λ (B.7)

and the central period and spectral characteristic period were computed as:

∑

∑

=

=

∆⋅

∆⋅⋅

= n

iiv

n

iivi

c

TS

TST

T

1

2,

1

2,

(B.8)

∑

∑

=

=∗

∆⋅

∆⋅⋅

=n

iiv

n

iivi

c

TS

TST

T

1

2,

1

2,

2

(B.9)

where n is the number of periods in the square velocity spectra, Sv is the spectral velocity and

∆T is the period of vibration interval.


383

B.1.2 Bandwidth

A measure of the dispersion or spread of the spectral density function about its central

frequency in terms of the spectral parameters was defined by Vanmarcke (1972) as follows:

1012

1

20

21 ≤≤

⋅−= qq

λλλ

(B.10)

The later spectral parameter is referred in the literature as the shape factor and it provides a

measure of the bandwidth of any stationary random process.

Using the concept developed by Vanmarcke an analog measure of the ground motion

spread about the central period or bandwidth as a function of the spectral parameters

computed for the square velocity spectra was defined as

( )101

21

20

21 ≤Ω≤

⋅−=Ω

∗∗

∗

λλ

λ (B.11)

Therefore, the bandwidth definition allows defining whether a ground motion has narrowband

or broadband frequency content around its central frequency or central period.

_____________________________________________________________________ Appendix C Sample Statistical Measures

384

Appendix C

Sample Statistical Measures

Throughout this investigation, different sample statistical measures1 of central tendency and

dispersion were employed. Statistical measures of central tendency included: sample mean,

counted median and geometric mean. Measures of dispersion around central tendency

considered in this dissertation includes: coefficient of variation and standard deviation of the

log (natural) of the data. Next, a definition of each sample statistical measures is provided.

C.1 Measures of Central Tendency

Sample mean

The most widely used measure of central tendency is the sample mean of the data, which is

defined as follows

∑=

=n

iix

nx

1

1 (C.1)

where n is the total number of observations and xi is and individual observation.

Counted median

The counted median is obtained by sorting in ascending order the computed individual values

(e.g. inelastic displacement ratios from 240 time-history analysis of SDOF inelastic

systems).Thus, the counted is defined by 1 Statistical parameters are computed from the sample data and are an approximation of the true statistical parameters. Confidence intervals of these sample statistical parameters based on empirical distribution of the data are beyond this investigation.


385

+=

+12

,2

,21

ni

ni

xxx( (C.2)

Geometric mean

The geometric mean in a logical estimator of the true median, especially of the data follows a

lognormal distribution, which is defined as follows:

= ∑

=

n

iix

nx

1

ln1

expˆ (C.3)

C.2 Measures of Dispersion

Standard deviation

∑=

−−

=n

ii xx

n 1

2)(1

1σ (C.4)

Coefficient of variation

X

XX x

COVσ

= (C.5)

Standard deviation of the natural logarithm of the data

∑=

−−

=n

iXiX x

n 1

2lnln )(ln

11

µσ (C.6)


386

where:

∑=

=n

iiX x

n 1ln ln

1µ (C.7)

C.3 Relationship among Statistical Parameters

2lnln 2

1ln XX x σµ −= (C.8)

( )1ln 22ln += COVXσ (C.9)

_____________________________________________________________________ Appendix D Generic Frame Models

387

Appendix D

Generic Frame Models

D.1 Introduction

The use of one-bay generic frames to analyze the seismic performance of multi-bay building

frames has been employed by some researchers (e.g., Esteva and Ruiz, 1989; Chintanapakdee

and Chopra, 2003; Medina and Krawinkler, 2003; among others). This approach seems

appealing for seismic performance assessment since it represent a less computational effort for

performing repeated nonlinear dynamic time history analyses. However, previous modeling

efforts have included simplifications that might question the similitude of nonlinear behavior

of generic frames with respect to the actual nonlinear response of multi-bay building frames.

For example, generic frames have been commonly modeled to deform as shear-type

structures, by assuming that the beams are infinitely rigid (e.g., Esteva and Ruiz, 1989).

Another approach of modeling bi-dimensional generic frame buildings is that yielding occurs

simultaneously at the end of the beams when the frames reached their yield strength capacity

under static lateral loads while the columns remain elastic in order to represent strong-column

weak-beam mechanism (e.g., Chintanapakdee and Chopra, 2003; Medina and Krawinkler,

2003). However, field investigation after earthquake events have showed that damage of

regular framed buildings designed to develop strong-column weak-beam mechanism is

concentrated in the lower portion of the structure and, in consequence, not all plastic hinges

were developed along the height, even under strong earthquakes. On the other hand, to

promote strong-column weak-beam mechanism, building codes usually specify beam-to-

column moment ratio of 1.2. However, experimental research on beam-to-column steel or

concrete subassemblies has showed that this value is not enough to guarantee the formation of

plastic hinge exclusively in the beams (Paulay, 1986). Thus, larger beam-to-column moment

ratio in building codes would be needed to assure ideal beam-hinge mechanism. This

observation has been also reported for steel moment-resisting frames (Nakashima and

Sawaisumi, 2000). Additionally, in generic frames that developed ideal beam-hinge


388

mechanism, the relative stiffness of the members was tuned such that the first-mode shape is a

straight line which implies that inter-story drift is the same at all levels under elastic response.

From the previous observations, it is still believed that one-bay generic frame building

models provides an adequate tool to conduct extensive analytical studies of MDOF systems,

but several enhancements should be considered to evaluate the global response of multi-story

regular buildings. Therefore, in order to obtain statistical information about maximum and

residual drift demands, a family of two-dimensional one-bay generic frame models was

designed following an alternative modeling procedure, which is described in this Appendix.

D.2 Modeling of Generic Framed Buildings Used in This Study

D.2.1 Fundamental Period of Vibration

The family of one-bay bi-dimensional generic frame models consist of regular frames having

six different number of levels (N = 3, 6, 9, 12, 15, 18). Each frame model has constant story

height equal to 12 feet and beam span equal to 24 feet. In addition, two generic frames were

designed to represent a rigid and a flexible steel moment-resisting frame (SMRF) structural

system having equal number of stories. The fundamental periods of the rigid and flexible

model were obtained from following equations (Chopra and Goel, 2000)

8.0028.0 HTL = (D.1)

8.0045.0 HTU = (D.2)

where H is the total height, in feet, of the building model. Parameters to estimate TL and TU

were calibrated to emulate the mean-minus- and mean-plus-one-standard deviation of

measured fundamental periods of typical SMRF designed with a seismic code prior to the

1994 Northridge earthquake. A graphical representation of the fundamental periods for both

rigid and flexible generic frames as a function of the number of stories is shown in figure D.1.

It should be mentioned that similar equations have been proposed for the authors to estimate

fundamental periods of vibration of regular reinforced concrete framed and frame-wall

buildings (Chopra and Goel, 2000).


389

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 5 10 15 20 25

Number of stories, N

Fund

amen

tal p

erio

d, T

1 (s

)

Upper bound

Low er bound

Flexible GF

Rigid GF

Figure D.1. Fundamental periods of vibration of generic frames considered in this study.

D.2.2 Height-Wise Stiffness Distribution

The family of generic frames considered in this study were designed to exhibit a first-mode

elastic deflected shape similar to the one measured in real regular buildings instead of

assuming an ideal linear (straight) line first mode deflected shape. For that purpose, an

empirical relationship among the non-dimensional parameters ρ, αo, and the number of stories,

N, was used to find the appropriate relationship of member’s moments of inertia (Miranda and

Taghavi, 2004).

2

2

12 No

⋅=

αρ (D.3)

In the above equation, the parameter ρ, first introduced by Blume (1968), is a measure the

relative beam-to-column stiffness in multi-story framed buildings and it is defined as the ratio

of the stiffness of all beams at the mid-height story of the frame to the sum of the stiffness of

all the columns at the same story. The dimensionless parameter αo was introduced by Miranda

(1999) to control the degree of participation of overall flexural and overall shear deformations

in a simplified model of multi-story buildings aimed to approximately estimate maximum

inter-story drift demands. This parameter also controls the lateral deflected shape in his

simplified building model. It should be noted that a value of α0 equal to zero represents a pure


390

flexural model and a value equal to ∞ correspond to a pure shear model. An intermediate

value of α0 corresponds to multi-story buildings that combine shear and flexural deformations.

Based on calibration of detailed analytical building models, the author noted that structural

wall buildings usually have values of α0 between 0 and 2; buildings with dual structural

systems usually have values of α0 between 1.5 and 6; whereas moment-resisting frame

buildings usually have values of α0 between 5 and 20. Therefore, in this study a constant value

of α0 equal to 15 was used in the design of all generic frames which lead to six different

values of ρ corresponding to generic frames with N = 3, 6, 9, 12, 15 and 18.

In order to simulate the reduction in stiffness along the relative height, z/H, the following

relationship proposed by Miranda and Reyes (2002) was initially used

λ

δ

−−=

Hz

HzS )1(1)( (D.4)

where δ is the ratio of the lateral stiffness at the top of the structure to the lateral stiffness

at the base of the structure and λ is a non-dimensional parameter that controls the variation of

the lateral stiffness along the height of the structure. It should be noted that the parameter δ is

not constant for actual multi-story framed buildings with different number of stories. For

example, it is expected that low-rise steel SMRF buildings (e.g., N = 3) have uniform stiffness

distribution (i.e., same section along the height) and, hence, parameter d is equal to 1, while

reductions in stiffness along the height is expected as the number of stories increases and,

thus, parameter δ decreases. In this investigation, the parameter λ was set up as 2.0 which

represent a continuous parabolic reduction of stiffness along height. A schematic

representation of equation (D.4) is shown in figure D.2 for different values of parameter δ.

Parameter δ was calibrated for each building model having different number of stories

taking into account the stiffness distribution of several SMRF analytical models available in

the literature (e.g., 3-, 9-, and 20-story SAC buildings). The resulting stiffness distribution

along the height for each building model, using equation (D.4), was used to pre-select an

adequate moment of inertia of the columns and, employing equation (D.3) the corresponding

moment of inertia of the beams. However, it should be recognized that actual multi-story

buildings do not exhibit continuous height-wise stiffness variation. For instance, high-rise

steel SMRF buildings (e.g., N = 18) would have different sections every three or four stories.


391

Then, in order to simulate more realistically the stiffness distribution of SMRF buildings, it

was decided to use a decreasing stepwise stiffness distribution along the height that still

followed the final parabolic stiffness distribution. As an example, figure D.3 shows both

stepwise and parabolic stiffness distribution for N=9 and 18. Next, the target fundamental

period of vibration for each generic frame model (e.g., N=3, T1 = 0.5 s, rigid model) was

obtained through an iterative process of the member’s moment of inertia that still satisfy the

predefined constraints (i.e., parameters αo, ρ and δ). Table D.1 reports the final values of

parameters

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2S (z / H)

z / H

δ = 1.00δ = 0.90δ = 0.75δ = 0.50δ = 0.30δ = 0.10

Figure D.2. Height-wise stiffness variation using equation (D.4).

(a) N = 9, δ = 0.40

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2

S (z / H)

z / H

parabolic variation

stepwise variation

(b) N = 18, δ = 0.25

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2

S (z / H)

z / H

parabolic variation

stepwise variation

Figure D.3. Height-wise stiffness variation for two building models: (a) 9-story; (b) 18-story.


392

Table D.1. Parameters used in equations (D.3) and (D.4)

N αo ρ δ

3 15 2.0800 1.00

6 15 0.5208 0.50

9 15 0.2315 0.40

12 15 0.1302 0.35

15 15 0.0833 0.30

18 15 0.0579 0.25

D.2.3 Fundamental Mode Shape

In order to evaluate if the family of generic framed models deflects as actual multi-story

framed buildings, an analytical solution to compute approximately mode shapes of multi-story

buildings with uniform lateral stiffness proposed by Miranda and Taghavi (2004) was

employed for comparison purposes. The analytical solution to compute modal shapes depends

directly on parameter α0. The authors noted that variation of lateral stiffness along the height

does not have a significant effect on the modal shapes and that the effect is more pronounced

for buildings where shear-type deformations control the lateral behavior than those deforming

laterally like flexural beams. Therefore, normalized modal participation factor with respect to

the first mode shape, Γiφi, obtained from a short-period (N= 3, T1 =0.5s) and long-period

(N=18, T1 = 2.0s) rigid generic models is shown in figure D.4. For comparison purposes, Γiφi,

obtained from Miranda and Taghavi (2004) equation and a linear mode shape are also showed

in the same figure. It can be seen that Γiφi for the short-period building is in good agreement

with the analytical solution, but larger difference exists for the long-period building model.

Nevertheless, it was found that the computed Γiφi for the generic framed building models

provides a better representation of the actual deflected shape of multi-story buildings than

assuming a linear deflected shape.

Finally, it should be noted that when uniform stiffness was considered in all generic

building models a very good agreement was found with respect to the analytical solution,

which was derived specifically for uniform stiffness distribution.


393

(a) 3-STORYRigid, T1 = 0.5 s

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5Γiφi

z / H

Linear modal shape

Miranda and Taghavi (2004)

ρ = 2.08, δ =1.0

(b) 18-STORYRigid, T1 = 2.0 s

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5

Γi φi

z / H

Linear modal shape

Miranda and Taghavi (2004)

ρ = 0.06, δ = 0.25

Figure D.4. Fundamental mode shapes computed for two generic building models: (a) 3-story; (b) 18-

story.

D.2.4 Height-Wise Strength Distribution

Next, the strength (i.e., bending moment capacity) of the elements was proportioned from the

lateral shear distribution along the height computed from the NEHRP Seismic provisions

(NEHRP, 2000) as follows

∑=

=N

j

kjj

kii

bi

hw

hwVF

1

(D.6)

where w is the weight at the i-story, h is the height at the j-story and k is a parameter

defined as

≥<≤+

≤=

sec5.22sec5.25.02/)5,0(

sec5.01

1

11

1

TTT

Tk (D.7)


394

The base shear was obtained from the design spectra shown in figure D.5 which correspond to

a seismic zone 4, located in an region of high seismicity in California, and soil type S2.

NEHRP 2000, IBC 2000Design Spectrum

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0Period [s]

Sa [g]

Figure D.5. Design spectrum used for the design of generic frames.

Worldwide earthquake-resistant design philosophy encourages multi-story buildings to

behave following a strong-column weak-beam mechanism, which intends to promote inelastic

action in the beams rather than in the columns under earthquake excitation. In order to satisfy

this requirement, seismic codes establish minimum beam-to-column bending moment ratio.

For instance, in the AISC 1997 seismic provisions for the design of new steel structures it is

required that ∑ ∑> **pbpc MM , where ∑ *

pcM is the sum of the moments in the column

above and below the joint at the intersection of the beam and column centerlines and

∑ *pbM is the sum of the moments in the beams at the intersection of the beam and column

centerlines. Therefore, in the design of each generic building model a beam-to-column ratio

equal to 1.2 at each floor was initially considered to obtain the final height-wise strength

distribution.

It should be noted that equivalent lateral static force distribution follows a parabolic

distribution along the height and, hence, height-wise strength distribution follows a similar

trend. However, height-wise strength distribution of actual multi-story framed buildings might

be departing from this parabolic distribution due to overstrength, mainly in the upper stories.


395

This issue is taken into account in the evaluation of deformation demands for the family of

generic frames and it is discussed in section 8.6.5 of Chapter 8.

D.2.5 Modeling Assumptions

D.2.5.1 Modeling of the generic frames

The family of generic frames was modeled as two-dimensional centerline models using the

computer program RUAUMOKO (Carr, 2003). Rayleigh damping of 5% of critical was

assigned to the first mode for all building model and the second mode for N=3, third mode for

N=6, and fourth mode for N=9, 12 and 18. To account for P-∆ effects, the same lateral seismic

weight was considered as gravity load. It should be mentioned that global P-∆ effect was

considered, but element P-δ effect was neglected.

D.2.5.2 Frame elements and hysteretic behavior

The elements were modeled with frame elements which concentrate the inelastic response in

plastic hinges that could form at both ends of the frame elements (i.e., beams and columns).

Plastic hinge length at each end was considered as 10% of the member length. In the plastic

hinge region, several moment-curvature nonlinear relationships are available in RUAUMOKO

(Carr, 2003). It should be noted that frame elements did not consider axial deformation as well

as Moment-Shear-Axial Load (M-V-P)

A very important issue in this investigation was to study the influence of hysteretic

behavior on both residual and maximum deformation demands of MDOF systems. To provide

a baseline, the frame elements (i.e., beams and columns) of each generic building model were

initially assigned an elastoplastic moment-curvature hysteretic behavior. This simple model

has been commonly used to represent the hysteretic behavior of compact steel elements that

do exhibit neither stiffness nor strength deterioration due to local web, flange or lateral

torsional buckling. However, this is an ideal case that does not always represent the observed

hysteretic behavior of steel components during experimental testing. Therefore, strength

deterioration in the moment-curvature relationship was included to simulate a more realistic

case trough the degradation model implemented in RUAUMOKO (Carr, 2003). A schematic


396

representation of the model is showed in figure D.6. In this model, strength (e.g., plastic

moment capacity, Mp) begins degrading linearly at a curvature ductility µi until a residual

strength, αMp, at curvature ductility µu. After that, a second strength loss occurs up to a

residual strength of 1% of the initial strength at ductility µf.

Figure D.6. Flexural strength degradation model implemented in RUAUMOKO (Carr, 2003).

Therefore, two bilinear hysteretic behaviors (i.e., without stiffness degradation and strain

hardening) with two levels of strength deterioration were simulated: (a) moderate strength

deterioration (MSD), and (b) severe strength deterioration (SSD). The first case attempts to

reasonably reproduce the hysteretic behavior of steel reduced beam section (RBS) connections

that exhibit moderate strength degradation at large levels of displacement ductility due to

flange or web local buckling when the beams are not laterally braced (Bruneau et al.; 1998).

For example, figure D.6 shows the experimental and analytical simulation of a beam-to-

column subassembly with beam RBS connection tested by Engelhardt (1998).

RUAUMOKO simulation

-300

-200

-100

0

100

200

300

-0.06 -0.04 -0.02 0 0.02 0.04 0.06Beam Drift angle [rad]

Forc

e [k

ips]

Specimen DB4(after Engelhardt et al., 1998)

-300

-200

-100

0

100

200

300

-0.06 -0.04 -0.02 0 0.02 0.04 0.06Beam Drift angle [rad]

Forc

e [k

ips]

Figure D.6. Calibration of bilinear strength-degrading model using RUAUMOKO (Carr, 2003).

Curvature ductility ratio

αMp

Mp

µi µu

Strength

0.01Mp

µf


397

It should be noted that the analytical simulation does not include Baushinger effect upon

unloading and reloading. The second case intends to account for potential brittle behavior of

bolted web-welded flange type beam-column connections, as commonly used in California

prior to the 1994 Northridge earthquake. Parameters to simulate strength deterioration are

given in Table D.2. For the case of severe strength degradation, the parameters were suggested

by Filiatrault et al. (2001) after examining 53 full-scale tests of strong-column weak-beam

beam-to-column joints.

Table D.2 Parameters used to simulate strength deterioration.

Strength loss α φi φu

Moderate 0.60 3.7 13.5

Severe 0.35 4.3 8.5

In addition of the aforementioned hysteretic behaviors, the well-known modified-Clough

model (Mahin and Bertero, 1976) and Takeda (Takeda et al., 1971) models were considered to

investigate the effect of stiffness degradation and, particularly, the influence of unloading

stiffness in the evaluation of residual deformation demands of MDOF systems. The former

models are already available in the RAUUMOKO (Carr, 2003) library.

___________________________________________________________________________________ References 398

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