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SEISMIC RELIABILITY ASSESSMENT OF STRUCTURES

INCORPORATING MODELING UNCERTAINTY AND

IMPLICATIONS FOR SEISMIC COLLAPSE SAFETY

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF CIVIL AND

ENVIRONMENTAL ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Beliz Ugurhan Gokkaya

December 2015

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/rp841bh0053

© 2015 by Beliz Ugurhan Gokkaya. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/rp841bh0053

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Gregory Deierlein, Primary Adviser


Jack Baker


Eduardo Miranda

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Beliz Ugurhan Gokkaya

I certify that I have read this dissertation and that, in my opinion, it

is fully adequate in scope and quality as a dissertation for the degree

of Doctor of Philosophy.

(Gregory G. Deierlein) Principal Adviser




(Jack W. Baker)




(Eduardo Miranda)

Approved for the Stanford University Committee on Graduate Studies

Abstract

Advancements in nonlinear dynamic simulation, seismic hazard analysis, and performance-

based earthquake engineering are enabling more scientific assessment of structural

collapse risk and how the risk is controlled by building code design requirements.

Although current collapse assessment methods carefully account for the nonlinear

response of structures, most of these analyses do not explicitly capture model un-

certainties associated with variability in the structural properties and response char-

acteristics of components. Instead, modeling uncertainties are typically considered

through simplified assumptions and techniques.

This dissertation focuses on modeling uncertainty in seismic performance assess-

ment and implications on seismic collapse safety of structures. A statistical framework

assessing model parameter correlations from component tests is proposed for charac-

terizing the modeling parameters that define dynamic response at a component level

and the interactions of multiple uncertain components in structural systems. The

framework is illustrated using a dataset that is composed of over two-hundred tests

of reinforced concrete columns. Statistics of the model parameters are established

including the correlation structure of the inelastic model parameters, both within a

component and between different components. The analyses show that model pa-

rameters within structural components tend to be only mildly correlated, whereas

there are strong correlations between like parameters of different components within

a building. Uncertainty propagation methods, including Monte-Carlo simulation-

based, moment-based and surrogate (response surface and neural network) methods,

are assessed for probabilistic assessment of collapse risk. To reduce computational de-

mand of collapse risk assessment, a Bayesian approach is proposed and demonstrated

using a reinforced concrete archetype building. The results emphasize the sensitiv-

ity of collapse response to modeling uncertainties and the challenges of balancing of

computational efficiency and robust uncertainty characterization.

Impacts of modeling uncertainty are evaluated for fragility functions and mean an-

nual exceedance rates for drift limits and collapse for thirty-three reinforced concrete

archetype building configurations. Modeling uncertainty is shown to have consider-

able impacts on collapse risk. Inclusion of modeling uncertainty is shown to increase

the mean annual frequency of collapse by about 1.7 times, as compared to analyses

based on median model parameters, for a high-seismic site in California. Modeling

uncertainty has a smaller effect on drift demands at levels usually considered in build-

ing codes. A novel method is introduced to relate drift demands to collapse safety

through a joint distribution of deformation demand and capacity, taking into account

simulated instances of collapse and no-collapse. This method enables linking seismic

performance goals specified in building codes to drift limits and other acceptance

criteria. The distributions of drift demand at maximum considered earthquake level

and drift capacity of case study structures are compared with drift limits as specified

in the proposed seismic criteria for the next edition (2016) of ASCE 7.

Acknowledgement

First and foremost, I would like to thank my advisors, Professors Greg Deierlein and

Jack Baker. It has been an honor to work with them both. Professor Deierlein taught

me to always think about the big picture. Proffesor Baker set an excellent example

on being productive and conducting impactful research. This work would not have

been possible without their exceptional guidance and vision. I am truly grateful for

the feedback, motivation and support they provided.

I would also like to thank Professor Eduardo Miranda for serving in my reading

committee. His insightful comments and feedback greatly improved this dissertation.

Special thanks to Professors Anne Kiremidjian and George Hilley for being a part of

my oral defense committee.

I am grateful to Professors Art Owen, Brendon Bradley, Dimitrios Vamvatsikos

Matjaz Dolsek and Kishor Jaiswal for providing valuable feedback on various aspects

of this dissertation. I also thank Curt Haselton for providing the dataset on reinforced

concrete column tests. Meera Raghunandan is greatly acknowledged for her help

related to nonductile reinforced concrete frames.

I also want to thank the members of my research group and students, faculty and

staff associated with the John A. Blume Earthquake Engineering Center. Special

thanks to my office mates, Ting Lin and Lynne Burks, as well as Nenad Bijelic for his

friendship. I thank Abhineet Gupta, Andy Seifried, Camilo Gomez, Christophe Loth,

Cristian Cruz, Cynthia Lee, David Lallemant, Gemma Cremen, Henry Burton, Jason

Wu, Mahalia Miller, Maryia Markhvida, Reagan Chandramohan and Shrey Shahi for

the feedback they provided on this research.

This work used the Extreme Science and Engineering Discovery Environment,

which is supported by NSF OCI-1053575 and Sherlock Cluster of Stanford Univer-

sity. I would like to thank Stanford University and the Stanford Research Computing

Center for providing computational resources and support that have contributed to

these research results. I gratefully acknowledge financial support provided by National

Science Foundation (grant number CMMI-1031722), John A. Blume Earthquake En-

gineering Center, Fulbright Science and Technology Fellowship and the Shah Family

Fellowship. Any opinions, findings, and conclusions expressed in this material are

those of the author and do not necessarily reflect the views of the funding sources.

Lastly, I would like to thank my parents, Muzaffer and Turker Ugurhan, and my

brother, Mert Ugurhan, for their love and encouragement throughout my life. Most

of all, I would like to thank my husband, Atasay Gokkaya, for his love, enthusiasm,

endless support and guidance. I want to dedicate this dissertation to my family,

without whom this dissertation would not be possible.

Contents

Abstract

Acknowledgement

1 Introduction 1

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

1.2 Objectives and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Characterization of Modeling Uncertainty . . . . . . . . . . . 3

1.2.2 Propagation of Modeling Uncertainty . . . . . . . . . . . . . . 3

1.2.3 Collapse Response Assessment . . . . . . . . . . . . . . . . . . 4

1.2.4 Seismic Design Criteria . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Uncertainty in Seismic Performance Assessment 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Overview of Performance-Based Earthquake Engineering . . . . . . . 8

2.3 Sources of Uncertainty in Seismic Performance Assessment . . . . . . 9

2.4 Probabilistic Seismic Collapse Risk Assessment of a Bridge Column . 13

2.4.1 Structural Idealization and Calibration of Analysis Model . . . 14

2.4.2 Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.3 Analysis Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.4 Collapse Risk using a Median Model . . . . . . . . . . . . . . 18

2.4.5 Characterization and Propagation of Modeling Uncertainty . . 19

2.4.6 Collapse Risk in the Presence of Modeling Uncertainties . . . 20

2.5 Modeling Uncertainty from the Blind Prediction Contest of the Bridge

Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5.1 Information on the Blind Prediction Contest . . . . . . . . . . 21

2.5.2 Modeling Uncertainty from Blind Prediction Contest in Com-

parison with the Simulations . . . . . . . . . . . . . . . . . . . 21

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.A Modeling Uncertainty of Other Engineering Demand Parameters: Sim-

ulations versus Blind Prediction Contest Submissions . . . . . . . . . 34

2.A.1 Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.A.2 Accelerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.A.3 Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Model Parameter Correlation 51

3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Probabilistic Seismic Performance Assessment . . . . . . . . . . . . . 54

3.4 Assessment of Correlations of Model Parameters . . . . . . . . . . . . 58

3.4.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . 58

3.4.2 Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.4.3 Correlation Assessment Procedure . . . . . . . . . . . . . . . 59

3.5 Impacts of Correlations on Dynamic Structural Response . . . . . . . 69

3.5.1 Case Study Structure . . . . . . . . . . . . . . . . . . . . . . . 69

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.7 Data and Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.A Information on Test Groups Conducting Reinforced Concrete Column

Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4 Modeling Uncertainty Propagation Methods 81

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2 Structural Reliability Theory . . . . . . . . . . . . . . . . . . . . . . 82

4.3 Structural Reliability Methods . . . . . . . . . . . . . . . . . . . . . . 84

4.3.1 Monte Carlo Simulation-Based Methods . . . . . . . . . . . . 84

4.3.2 Moment-Based Methods . . . . . . . . . . . . . . . . . . . . . 87

4.3.3 Surrogate Methods . . . . . . . . . . . . . . . . . . . . . . . . 90

4.4 Sensitivity of Collapse Capacity to Perturbations in Model Parameters 93

4.5 Reliability Analysis of Collapse Response Assessment of a 4-Story Reinforced-

Concrete Moment-Frame Building . . . . . . . . . . . . . . . . . . . . 98

4.5.1 Structural Model, Seismic Hazard and Ground Motions . . . . 98

4.5.2 Seismic Performance Assessment . . . . . . . . . . . . . . . . 100

4.5.3 The Bootstrap for Quantifying the Uncertainty in Collapse Risk

Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.5.4 Impacts of Modeling Uncertainty . . . . . . . . . . . . . . . . 102

4.5.5 Incorporating Modeling Uncertainty using an IDA Approach . 104

4.5.6 Incorporating Modeling Uncertainty using a MSA Approach . 112

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5 Bayesian Collapse Risk Assessment 120

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.2 Proposed Method for Collapse Risk Assessment . . . . . . . . . . . . 122

5.2.1 Collapse Risk Assessment . . . . . . . . . . . . . . . . . . . . 122

5.2.2 Proposed method . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.2.3 Analysis Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.3 Selection of IM Levels . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.4 Sensitivity to Number of Structural Analyses . . . . . . . . . . . . . . 134

5.5 Robustness Analyses of Proposed Method . . . . . . . . . . . . . . . 135

5.6 Model Checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.6.1 Posterior Predictive Checking . . . . . . . . . . . . . . . . . . 141

5.6.2 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . 142

5.7 Discussion on Obtaining Prior Information . . . . . . . . . . . . . . . 145

5.8 Collapse Risk Assessment of a 4-Story Reinforced Concrete Moment

Frame Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

5.8.1 Building Site and Seismic Hazard Characterization . . . . . . 146

5.8.2 Structural Modeling and Analysis . . . . . . . . . . . . . . . . 146

5.8.3 Ground Motion Selection . . . . . . . . . . . . . . . . . . . . . 148

5.8.4 Modeling Parameter Uncertainty . . . . . . . . . . . . . . . . 149

5.8.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.8.6 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6 Impacts of Modeling Uncertainties 157

6.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.3 Seismic Performance Assessment . . . . . . . . . . . . . . . . . . . . . 161

6.4 Case Study Seismic Response Analysis with Modeling Uncertainty . . 164

6.4.1 Structural Models . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.4.2 Seismic Hazard Analysis and Ground Motion Selection . . . . 166

6.4.3 Modeling Uncertainty . . . . . . . . . . . . . . . . . . . . . . . 167

6.5 Impacts of Modeling Uncertainty on Collapse Response . . . . . . . . 170

6.5.1 Effect of Modeling Uncertainty on Collapse Fragility Parameters 170

6.5.2 Net Effect of Modeling Uncertainty on Collapse Rates . . . . . 172

6.5.3 Equivalent Value of Modeling Uncertainty . . . . . . . . . . . 174

6.6 Impacts of Modeling Uncertainty on Drift Demands . . . . . . . . . . 176

6.6.1 Influence of Modeling Uncertainty on Drift Exceedance Fragility

Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.6.2 Influence of Modeling Uncertainty on Drift Demands . . . . . 177

6.7 MCE Level Drift Demand and Acceptance Criteria . . . . . . . . . . 181

6.7.1 Framework for Obtaining Distributions of Building Drift Ca-

pacity and Demand at MCE . . . . . . . . . . . . . . . . . . . 183

6.7.2 Distributions of Building Drift Capacity and Demand at MCE

for Selected Archetype Structures . . . . . . . . . . . . . . . . 185

6.7.3 Impacts of Modeling Uncertainty on the Distributions of Build-

ing Drift Capacity and Demand at MCE . . . . . . . . . . . . 188

6.7.4 Evaluation of ASCE 7 Story Drift Acceptance Criteria - Build-

ing Archetype Results . . . . . . . . . . . . . . . . . . . . . . 190

6.7.5 Evaluation of ASCE 7 Story Drift Acceptance Criteria - Para-

metric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 193

6.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

7 Conclusions 199

7.1 Summary of Findings and Contributions . . . . . . . . . . . . . . . . 199

7.1.1 Methods for Incorporating Modeling Uncertainty in Probabilis-

tic Seismic Response Assessments . . . . . . . . . . . . . . . . 199

7.1.2 Impacts of Modeling Uncertainty on Structural Seismic Perfor-

mance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

7.1.3 Framework for Linking Seismic Response to Seismic Perfor-

mance Goals Specified in Building Codes . . . . . . . . . . . . 204

7.2 Limitations and Future Research . . . . . . . . . . . . . . . . . . . . 206

7.2.1 Structural Models and Analyses Strategies . . . . . . . . . . . 206

7.2.2 Types of Uncertainties . . . . . . . . . . . . . . . . . . . . . . 206

7.2.3 Reliability Analyses . . . . . . . . . . . . . . . . . . . . . . . . 207

7.2.4 Other Limitations . . . . . . . . . . . . . . . . . . . . . . . . . 208

7.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

List of Tables

2.1 Logarithmic standard deviations of random variables . . . . . . . . . 19

2.2 Means and standard deviations, of the residuals of the maximum ab-

solute drift ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Means and standard deviations, of the residuals of the minimum abso-

lute drift ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4 Means and standard deviations, of the residuals of the Max Abs Dis-

placement in terms of mm . . . . . . . . . . . . . . . . . . . . . . . . 34

2.5 Means and standard deviations, of the residuals of the Min Abs Dis-

placement in terms of mm . . . . . . . . . . . . . . . . . . . . . . . . 36

2.6 Means and standard deviations, of the residuals of the Max Abs Ac-

celeration in terms of g . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.7 Means and standard deviations, of the residuals of the Min Abs Accel-

eration in terms of g . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.8 Means and standard deviations, of the residuals of the Max Abs Bend-

ing Moment in terms of kN.m . . . . . . . . . . . . . . . . . . . . . . 42

2.9 Means and standard deviations, of the residuals of the Min Abs Bend-

ing Moment in terms of kN.m . . . . . . . . . . . . . . . . . . . . . . 44

2.10 Means and standard deviations, of the residuals of the Max Abs Shear

Force in terms of kN . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.11 Means and standard deviations, of the residuals of the Min Abs Shear

Force in terms of kN . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.1 Between and within test group standard deviations obtained from ran-

dom effects regression. . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2 Initial correlation coefficients obtained from random effects regression. 65

3.3 Final correlation coefficients are obtained after rounding to one signif-

icant figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.4 Correlation models used with Monte Carlo simulations. “0”, “P” and

“1” refers to the cases of No Correlation, Partial Correlation and Full

Correlation, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.5 Collapse risk metrics in terms of counted median and logarithmic stan-

dard deviation (σln) of collapse capacities obtained using alternative

models. Also shown are mean annual frequency of collapse (λc) ob-

tained by integrating empirical collapse fragility functions with seismic

hazard curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.6 Test groups conducting reinforced concrete column tests. For column

dimensions, f ′c, fy, axial load ratio, area ratio of longitudinal and trans-

verse reinforcement, “Y” and “-” are used when the tests within the

test group have similar and different properties, respectively. “N/A”

is used when the number of tests per test group is 1. . . . . . . . . . 79

3.6 Test groups conducting reinforced concrete column tests. For column

dimensions, f ′c, fy, axial load ratio, area ratio of longitudinal and trans-

verse reinforcement, “Y” and “-” are used when the tests within the

test group have similar and different properties, respectively. “N/A”

is used when the number of tests per test group is 1. . . . . . . . . . 80

4.1 Collapse risk estimates obtained using an IDA approach with alterna-

tive structural reliability methods . . . . . . . . . . . . . . . . . . . . 105

4.2 Collapse risk estimates obtained using MSA approach with alternative

structural reliability methods . . . . . . . . . . . . . . . . . . . . . . 113

4.3 Collapse risk estimates using simulation-based methods with 171 ran-

dom variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.1 Median and coefficient of variation in estimated values of λc with dif-

ferent ∆ values and MLE method. For initial fragility function, an

unbiased median collapse capacity θ = 1 g and β = 0.4 are assumed. . 138

5.2 Median and coefficient of variation of estimated values of λc with dif-

ferent θ values. For initial fragility function, ∆ = 0.5 and β = 0.4 are

assumed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.3 Bayesian p-values obtained using different scenarios . . . . . . . . . . 145

5.4 Bayesian p-values obtained for IM1 and IM2 . . . . . . . . . . . . . . 152

5.5 IM levels that are used for benchmarking collapse response of the 4-

Story R/C frame structure . . . . . . . . . . . . . . . . . . . . . . . . 153

5.6 Collapse risk estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.1 Logarithmic standard deviations of random variables . . . . . . . . . 168

6.2 Correlation of random variables defining backbone curve and hysteretic

behavior of a concentrated plasticity model . . . . . . . . . . . . . . . 169

6.3 Parameters of the joint distribution of demand and capacity and the

estimated P (Collapse) along with target P (Collapse) that is obtained

using IDA results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

6.4 Change in the parameters of the joint distribution of demand and ca-

pacity due to modeling uncertainty . . . . . . . . . . . . . . . . . . . 190

6.5 Probability of collapse and mean and median drift demand at MCE

obtained using generic and hazard consistent ground motions . . . . . 192

6.6 Story drift limits for global acceptance criteria corresponding to drift

capacities of 0.06 to 0.10 . . . . . . . . . . . . . . . . . . . . . . . . . 194

List of Figures

2.1 a) Test set-up of the reinforced concrete bridge column which was

tested full-scale on a shake table b) Structural idealization of the bridge

column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Backbone curve for concentrated plasticity model relating member mo-

ment (M) and rotation (θ) . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Structural analysis model is calibrated to match the experimental results 16

2.4 Response spectra of the ground motions in the FEMA-P695 far-field

set scaled to have the same spectral acceleration at the first mode of

the bridge column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Seismic hazard curve for the site at Los Angeles, California . . . . . . 17

2.6 a) 16%, 50% and 84% fractal IDA curves using a median model with 44

ground motions from FEMA-P695 far-field set b) Empirical and fitted

collapse fragility functions . . . . . . . . . . . . . . . . . . . . . . . . 18

2.7 a) 16%, 50% and 84% fractal IDA curves using with modeling uncer-

tainty b) Empirical and fitted collapse fragility functions . . . . . . . 20

2.8 Distributions of the residuals of the maximum absolute value of the

drift ratio (“Maximum Absolute Drift Ratio”) are illustrated using a

normal distribution. The cyan and red curves correspond to predic-

tions using simulations incorporating modeling uncertainty and blind

prediction submissions, respectively. Also shown is the error in the

values calculated using the median model (in blue). . . . . . . . . . 27

2.9 Distributions of the residuals of the maximum absolute value of the

drift ratio in the opposite direction (“Minimum Absolute Drift Ra-

tio”) are illustrated using a normal distribution. Shown in cyan and

red correspond to predictions using simulations incorporating model-

ing uncertainty and blind prediction submissions, respectively. Also

shown is the prediction using simulations with a median model in blue. 30

2.10 Residuals of the maximum absolute value of the drift ratio are obtained,

and distributions are obtained using a pooled estimate of variance for

different earthquakes. These distributions are illustrated using a nor-

mal distribution. The plots in cyan and red correspond to predictions

using simulations incorporating modeling uncertainty and blind pre-

diction submissions, respectively. Also shown is the prediction using

simulations with a median model in blue. a) Maximum absolute drift

ratio b) Maximum absolute drift ratio in the opposite direction . . . . 31

2.11 Distributions of the residuals of the maximum absolute value of the rel-

ative horizontal displacement in terms of mm at the top of the column

are illustrated using a normal distribution. Distributions are obtained

using a median model, modeling uncertainty propagation and blind

prediction submissions are shown in blue, cyan and red, respectively. . 35

2.12 Distributions of the residuals of the maximum absolute value of the rel-

ative horizontal displacement in terms of mm at the top of the column

in the opposite direction are illustrated using a normal distribution.

Distributions are obtained using a median model, modeling uncertainty

propagation and blind prediction submissions are shown in blue, cyan

and red, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.13 Residuals of the maximum absolute value of the relative horizontal

displacement are obtained for different earthquakes, and the pooled

distributions are illustrated using a normal distribution. Distributions

are obtained using a median model, modeling uncertainty propagation

and blind prediction submissions are shown in blue, cyan and red, re-

spectively. a) Maximum absolute displacement b) Maximum absolute

displacement in the opposite direction . . . . . . . . . . . . . . . . . . 38

2.14 Distributions of the residuals of the maximum absolute value of total

acceleration in terms of g at the top of the column are illustrated using a

normal distribution. Distributions are obtained using a median model,

modeling uncertainty propagation and blind prediction submissions are

shown in blue, cyan and red, respectively. . . . . . . . . . . . . . . . . 39

2.15 Distributions of the residuals of the maximum absolute value of total

acceleration in terms of g at the top of the column in the opposite

direction are illustrated using a normal distribution. Distributions are

obtained using a median model, modeling uncertainty propagation and

blind prediction submissions are shown in blue, cyan and red, respec-

tively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.16 Residuals of the maximum absolute value of the total acceleration are

obtained for different earthquakes, and the pooled distributions are il-

lustrated using a normal distribution. Distributions are obtained using

a median model, modeling uncertainty propagation and blind predic-

tion submissions are shown in blue, cyan and red, respectively. a)

Maximum absolute acceleration b) Maximum absolute acceleration in

the opposite direction . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.17 Distributions of the residuals of the maximum absolute value of bend-

ing moment at the base in terms of kN.m are illustrated using a nor-

mal distribution. Distributions are obtained using a median model,



2.18 Distributions of the residuals of the maximum absolute value of bend-

ing moment at the base in terms of kN.m in the opposite direction



prediction submissions are shown in blue, cyan and red, respectively. . 45

2.19 Residuals of the maximum absolute value of the bending moment at

the base are obtained for different earthquakes, and the pooled distri-

butions are illustrated using a normal distribution. Distributions are

obtained using a median model, modeling uncertainty propagation and

blind prediction submissions are shown in blue, cyan and red, respec-

tively. a) Maximum absolute bending moment b) Maximum absolute

bending moment in the opposite direction . . . . . . . . . . . . . . . 46

2.20 Distributions of the residuals of the maximum absolute value of shear

at the base in terms of kN are illustrated using a normal distribution.

Distributions are obtained using a median model, modeling uncertainty

propagation and blind prediction submissions are shown in blue, cyan

and red, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.21 Distributions of the residuals of the maximum absolute value of shear at

the base in terms of kN in the opposite direction are illustrated using a

normal distribution. Distributions are obtained using a median model,



2.22 Residuals of the maximum absolute value of the shear at the base

are obtained for different earthquakes, and the pooled distributions



prediction submissions are shown in blue, cyan and red, respectively. a)

Maximum absolute shear b) Maximum absolute shear in the opposite

direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.1 Ibarra et al. (2005) model for moment versus rotation of a plastic

hinge in a structure. The model parameters of interest are labeled. . . 56

3.2 Illustration of correlation within a component and correlation between

components in a structure. . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3 Residuals of beam-column model parameters (εk) are plotted against

the residuals of other tests within the test group to which they belong

for a) θcap,pl, b) EIstf/EIg, c) My, d) Mc/My, e) θpc and f) γ. A subset

of data is shown for illustrative purposes. . . . . . . . . . . . . . . . . 61

3.4 IDA results obtained using different correlation models a) median IDA

response for varying correlation levels, b) median IDA response for

Partial Correlation models, c) dispersion in IDA curves for varying

correlation levels, and d) dispersion in IDA curves for Partial Correla-

tion models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.5 Mean annual frequency of exceedance of maximum story drift ratio

using different correlation models . . . . . . . . . . . . . . . . . . . . 73

3.6 Empirical cumulative distribution functions obtained using different

correlation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.1 Tornado diagrams showing relative importance of model parameters for

the bridge column. Perturbations with ±1.7σln and ±1σln are shown

with solid and dashed lines, respectively. . . . . . . . . . . . . . . . . 94

4.2 Tornado diagrams showing relative importance of model parameters

for different structures a) 4 story frame ID: 1008 b) 12 story frame

ID: 1014 c) 12 story frame ID: 2067. Perturbations with ±1.7σln and

±1σln are shown with solid and dashed lines, respectively. . . . . . . 96

4.3 Nonlinear relationship between median collapse capacity and pertur-

bations of strength and post-capping deformation capacity of columns.

Collapse modes corresponding to different realizations of random vari-

ables are displayed. The displayed collapse mechanism corresponds

to the majority case among all collapse modes observed for a given

perturbation using the ground motions in the suite. . . . . . . . . . . 97

4.4 Seismic hazard curves at Los Angeles, CA and Memphis, TN. . . . . 99

4.5 Collapse risk using a median model versus (indicates as “With RTR”)

and a model incorporating modeling uncertainty (indicates as “With

RTR+MU”) for the 4-story reinforced concrete structure. . . . . . . . 103

4.6 Change in median collapse capacity with respect to perturbations in

model parameters. The perturbations include ±1.7σln, ±1σln and

±0.5σln. Fitted lines are shown in blue. For some variables, slopes

are determined using the region which results in higher slopes and

these lines are shown in green. . . . . . . . . . . . . . . . . . . . . . . 108

4.7 Collapse risk estimates are obtained using an IDA approach with al-

ternative structural reliability methods. a) Collapse fragility curves,

b) Collapse risk deaggregation curves at Los Angeles, c) Collapse risk

deaggregation curves at Memphis . . . . . . . . . . . . . . . . . . . . 111

4.8 Collapse risk estimates are obtained using a MSA approach with simulation-

based reliability methods. a) Collapse fragility curves, b) Collapse risk

deaggregation curves at Los Angeles, c) Collapse risk deaggregation

curves at Memphis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.1 Steps of the proposed method: a) Initial collapse fragility curve and

uncertainty in median capacity, b) Selected IM levels and prior dis-

tributions at these levels, c) Ground motions selected to match condi-

tional spectra, d) Data from nonlinear time history analyses at selected

IM levels, e) Posterior distributions at selected IM levels, and f) Final

estimate of the collapse fragility function . . . . . . . . . . . . . . . . 125

5.2 Contours showing the mean square error corresponding to different

pairs of IM values. ∆ is used as 0.4. The pairs of IM values resulting

in minimum MSE are indicated with red stars. IM values are computed

corresponding to probabilities of collapse on the initial fragility curve. 132

5.3 Pairs of IM values giving minimal MSE values. IM values are computed

corresponding to probabilities of collapse on the initial fragility curve.

IM values resulting in minimum MSE are shown with circles, whereas

IM values resulting in MSE values within 10% of the minimum MSE

are shown with dots. Also shown is the best fit line to the IM values

resulting in minimum MSE. . . . . . . . . . . . . . . . . . . . . . . . 133

5.4 Sensitivity of number of structural analyses for different values of ∆

for the site at LA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.5 Distribution of λc estimates obtained using Bayesian Method with dif-

ferent ∆ values and MLE method. For initial fragility function, an

unbiased median collapse capacity θ = 1 g and β = 0.4 are assumed.

Target λc is shown using a blue solid line. . . . . . . . . . . . . . . . 138

5.6 Distribution of λc estimates obtained using Bayesian Method with dif-

ferent θ values. For initial fragility function, ∆ = 0.5 and β = 0.4 are

assumed. Target λc is shown using a blue solid line. . . . . . . . . . . 139

5.7 Fragility curves and uncertainty in median estimates for Scenarios 1

and 2 are plotted with respect to target fragility curve. . . . . . . . . 143

5.8 Histograms of test statistics obtained using posterior predictive distri-

bution for Scenario 1. The mean, standard deviation and skewness of

the observed structural analysis data at IM2 are indicated with the

red solid lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144


bution for Scenario 2. The mean, standard deviation and skewness of

the observed structural analysis data at IM2 are indicated with the

red solid lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.10 16, 50 and 84% fractile IDA curves from SPO2IDA along with an

estimate of collapse fragility curve for the 4-story R/C structure . . . 148

5.11 Bayesian method illustrated on a 4-story reinforced concrete frame

structure in comparison with maximum likelihood estimation. . . . . 151


bution for a) IM1 b) IM2. The mean, standard deviation and skewness

of the observed structural analysis data at IM2 are indicated with the

red solid lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.13 Target collapse fragility curve along with the curve estimated using a

Bayesian approach for the 4-story reinforced concrete frame structure. 154

6.1 Illustration of seismic performance assessment a) Multiple stripes anal-

yses at different ground motion intensity levels b) Collapse and drift-

exceedance fragility functions c) Seismic hazard curve d) Collapse risk

deaggregation curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.2 a) Illustrative structural idealization of the models used in this study.

Details of beam-column connections for ductile and non-ductile frames

are provided. b) Backbone curve for concentrated plasticity model c)

Illustration of the shear and axial failure models . . . . . . . . . . . . 165

6.3 Change in median collapse capacity with respect to change in the dis-

persion values of collapse fragility curves a) With respect to structural

characteristics b) With respect to number of stories and structural

characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.4 The distribution of the change in λc due to modeling uncertainty is

shown for sites characterized power-law hazard curve and the site at

LA using blue and red boxplots, respectively. Also shown in green is

the change in λc due to modeling uncertainty using an approximate

closed form expression. . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6.5 a) Illustration of practical approaches to incorporate modeling uncer-

tainty using 12-story structure with ID 2067 b) Estimated λc using the

practical approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.6 a) Example drift-exceedance fragility curves for an 8 story structure

with ID 1012 b) Change in median collapse capacity with respect

to change in the dispersion values of fragility curves defining drift-

exceedance limit states c) Boxplots showing the change in λSDR>sdr as

a function of sdr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.7 a) ECDF of drift demands using median model and with modeling un-

certainty for the 8-story structure with ID 1012 b) Change in counted

median of drift demand distributions w.r.t. drifts for different struc-

tures c) Change in dispersion of the fitted lognormal distributions to

sdr values of no-collapses d) Change in dispersion of the fitted lognor-

mal distributions to sdr values up to median . . . . . . . . . . . . . . 179

6.8 An example IDA curve in comparison with MSA data 182

6.9 Two example IDA curves are provided with red and blue lines having

drift capacities above and below MCE level, respectively. . . . . . . . 182

6.10 a) IDA curves and drift demand at MCE and capacity histograms for

ID:1019 along with collapse fragility curve b) Probability distribution

of demand at MCE and capacity for ID:1019 . . . . . . . . . . . . . . 187

6.11 IDA curves and drift demand and capacity histograms in terms of drifts

for ID:1008 a) using median model b) with modeling uncertainty . . . 189

6.12 Back-calculated (left) and proposed (right) distributions of drift de-

mands and capacities for buildings having more than 4 stories a) Risk

Categories 1 and 2 b) Risk Category 3 and c) Risk Category 4. Red

and blue lines represent the distributions of demand and capacity, re-

spectively. Mean values are indicated with dashed black lines. . . . . 195

Chapter 1

Introduction

1.1 Motivation

Seismic design provisions in building codes aim to provide life safety of building

occupants during extreme earthquake events. This is achieved by controlling collapse

risk of structures to acceptable levels. Proposed seismic provisions for the next edition

(2016) of ASCE 7 (American Society of Civil Engineers Standards Committee, 2015)

requires a probability of collapse at a maximum considered earthquake (MCE) level

to be less than 10%. The design provisions propose a set of rules and acceptance

criteria to implicitly link structural response to the aforementioned collapse safety

goals. However, the empirical nature of seismic design requirements in building codes

makes the assessment of the implied collapse safety challenging.

Advancements in nonlinear dynamic simulation, seismic hazard analysis, and

performance-based earthquake engineering are enabling more scientific assessment

of structural collapse risk and how the risk is affected by building code design re-

quirements. FEMA-P695 (FEMA, 2009b) provides guidelines for performance-based

quantification of building seismic performance factors to be used in seismic design and

assessment of seismic collapse risk. FEMA-P58 (FEMA, 2012) and PEER Tall Build-

ing Initiative (PEER, 2010) are among the other initiatives formalizing performance-

based seismic assessment and design of structures. In spite of the considerable

1

progress made in formalizing performance-based assessment methods, significant chal-

lenges persist in reliably assessing highly nonlinear behavior up to collapse.

Collapse response of structures is characterized by highly nonlinear behavior re-

sponse of its components and its assessment requires analysis models that capture

large inelastic deformations with significant cyclic strength and stiffness degradation.

Current collapse analyses are often conducted with models that reflect the median

(or expected) properties of the structural components and do not account for model

uncertainties associated with variability in the properties and response characteristics

of the components. Furthermore, analysis models are limited due to various simplifi-

cations and their extent of capturing reality constitutes a major source of uncertainty.

While the impact of modeling uncertainties on seismic collapse safety is well recog-

nized, current techniques to characterize and propagate this uncertainty are highly

simplified and idealized. A major motivation behind these simplified approaches is

the lack of data to reliably characterize modeling uncertainties and the computational

challenges in propagating uncertainties in the analyses. Advancements in distributed

computing as well as the storage and speed of modern computers are mitigating the

need for these simplifications and can enable the application of stochastic simulations

with many random variables. However, challenges remain in collecting, characterizing

and processing data on modeling uncertainties.

1.2 Objectives and Scope

The aim of this dissertation is to improve the understanding of the impacts of mod-

eling uncertainty through robust and comprehensive probabilistic seismic collapse

safety assessments of structures. This aim is achieved through the following three

primary objectives.

• Provide system reliability methods for robust characterization of the influence

of modeling uncertainties and variability in structural component behavior on

overall building system seismic response.

2

CHAPTER 1. INTRODUCTION 3

• Rigorously assess the influence of modeling uncertainties, variability and corre-

lations in component properties on the collapse behavior on archetype buildings.

• Develop a framework for systematically linking the overall system response in

the presence of modeling uncertainty to seismic performance goals specified in

building codes.

The specific research scope is summarized in the following subsections.

1.2.1 Characterization of Modeling Uncertainty

Two essential ingredients of a probabilistic framework are (1) the identification of

the variables that are deemed random, and (2) quantification of statistical distribu-

tions representing their uncertainty. Modeling uncertainty is explicitly considered

in structural response simulations, which are idealized using concentrated plasticity

hinges, shear and axial springs. The random variables characterizing a plastic hinge

include a trilinear force-displacement backbone curve having a branch with a negative

stiffness as well as a parameter to model cyclic energy dissipation. Variability and

correlations of random variables defining concentrated plastic hinges are also studied,

where correlation structure characterized dependence of model parameters within a

component as well as across different components in a structure. Sensitivity analy-

sis is utilized for identification of relatively important model parameters for collapse

capacity predictions.

1.2.2 Propagation of Modeling Uncertainty

Selection of an uncertainty propagation method requires explicit consideration of var-

ious aspects of the problem. A major challenge is to balance computational efficiency

and robust characterization of the nonlinearity of the problem. This dissertation in-

vestigates several methods ranging from simulation-based methods to neural networks

for probabilistic assessments of seismic collapse response. The methods are used to

propagate modeling uncertainty and record-to-record- variability through nonlinear

dynamic analyses under a suite of earthquake ground motions. The relative advan-

tages and disadvantages of each method are discussed and investigated using a 4-story

reinforced concrete moment frame structure.

1.2.3 Collapse Response Assessment

Assessment of collapse response is challenging owing to the highly nonlinear and

complex behavior at collapse and variability in earthquake ground motions and the

uncertainty in structural model idealizations. A set of reinforced concrete moment

frame structures located in high seismicity sites is analyzed using hazard consistent

ground motions with a conditional spectra approach and Latin Hypercube sampling

to propagate modeling uncertainties. These buildings range in height from 1 to 20

stories, including a few designs that have varying amounts of strength irregularity up

the building height. The impacts of modeling uncertainty on building drift demands

and collapse are evaluated. The extent to which ductility and strength irregularities

influence the significance of modeling uncertainties is explored leading to guidelines to

account for modeling uncertainties in design and assessment. In addition to detailed

simulations, a simplified Bayesian method is proposed to estimate collapse fragilities

including model uncertainties.

1.2.4 Seismic Design Criteria

Risk-based assessments of ductile and non-ductile reinforced concrete frame struc-

tures are conducted using hazard-consistent ground motions and explicit considera-

tion of modeling uncertainty, yielding time-based estimates of collapse risk of code-

conforming and none-code conforming structures. The global acceptance criteria in

terms story drifts specified in ASCE 7 provisions American Society of Civil Engi-

neers Standards Committee (2015) is evaluated relative to the collapse safety goals.

The link between drift limits and collapse risk is developed through a probabilistic

framework. This framework is used to evaluate the current drift acceptance criteria

in ASCE 7.

4

CHAPTER 1. INTRODUCTION 5

1.3 Organization of Thesis

This dissertation addresses topics related to modeling uncertainty in seismic per-

formance assessment of structures, focusing on characterization and propagation of

modeling uncertainty, quantification of its impacts on seismic response of structures

and implications on seismic collapse safety.

Chapter 2 discusses the uncertainties involved in seismic performance assess-

ment. It begins with a discussion on the sources of uncertainty in seismic perfor-

mance assessment. An example assessment is conducted on a bridge column with

and without explicit consideration of modeling uncertainty. The uncertainty in the

engineering seismic response parameters of the bridge column is obtained using nu-

merical simulations as well as predictions from a blind-prediction experiment. The

uncertainty predictions are compared to the shake-table results, and the importance

of modeling uncertainty propagation in seismic performance assessment is discussed.

Chapter 3 presents a framework for quantifying model parameter correlations

from component test data. The framework utilizes random effects regression mod-

els to quantify the correlations of tests conducted by different research groups and

relates them to multiple components in a structure. The dependence of modeling

parameters that define dynamic response at a component level, as well as the de-

pendence of parameters across multiple components, are characterized. Uncertainties

and dependencies in structural component model parameters of reinforced concrete

elements are evaluated by applying this framework to a database of over two hundred

reinforced concrete beam-column tests (Haselton et al. , 2008). Furthermore, the

effects of model parameter correlations are investigated through response assessment

of a four-story reinforced concrete frame structure.

Chapter 4 discusses uncertainty propagation methods in the context of seismic

reliability assessment. Alternative reliability methods are used to illustrate proba-

bilistic assessment of collapse risk. Collapse response of a four-story reinforced con-

crete frame structure is probabilistically assessed using alternative reliability methods

with explicit consideration of variability and correlations of random variables through

nonlinear dynamic analyses. The relative merits of each method (i.e. computational

efficiency, accuracy) in assessing collapse response of structures are discussed. The

methods are grouped as simulation-based, moment-based and surrogate methods.

Simulation-based methods are particularly explored in terms of their effectiveness for

uncertainty propagation in high dimensional collapse safety assessment problems.

Chapter 5 proposes a Bayesian approach to collapse risk assessment. The ob-

jective of the method is to reduce computational demand for an equivalent statistical

efficiency by incorporating prior information on structural collapse capacity, which

can be informed by a variety of sources, including information on the building design

criteria, empirical fragility functions or nonlinear static (pushover) analysis. Sen-

sitivity analysis and robustness tests of the method are performed, followed by an

illustration of the method using a case study.

Chapter 6 discusses the impacts of modeling uncertainty on seismic performance

of structures. Impacts of modeling uncertainty are evaluated for fragility functions

and mean annual exceedance rates for drift limits and collapse as well as drift (de-

formation) demands and capacities of thirty-three archetype building configurations.

A novel method is introduced to relate drift demands at maximum considered earth-

quake intensities to collapse safety through a joint distribution of deformation demand

and capacity, taking into account simulated instances of collapse and no-collapse.

This framework enables linking seismic performance goals specified in building codes

to drift limits and other acceptance criteria. The distributions of drift demand at

maximum considered earthquake level and drift capacity of case study structures en-

able comparisons with drift limits as specified in the proposed seismic criteria for the

next edition (2016) of ASCE 7.

Chapter 7 presents major conclusions and contributions of this dissertation.

Limitations of this dissertation are discussed and opportunities for future research

are explored.

6

Chapter 2

Uncertainty in Probabilistic

Seismic Performance Assessment

of Structures

2.1 Introduction

In this chapter, we present the methodology for seismic performance assessment along

with basic concepts and terminology used frequently in the thesis. We then discuss

the sources of uncertainties involved in seismic performance assessment. An example

seismic performance assessment is provided for a reinforced concrete bridge column.

The exercise is performed with and without explicit consideration of modeling un-

certainty. The performance of the bridge column was previously modeled by several

research teams as a part of a blind seismic response prediction contest, and tested full

scale on a shake table. The distributions of the predicted engineering parameters by

the blind analysis contestants in comparison with the experimental results are com-

pared with the results of the analytical simulations propagating modeling uncertainty.

They are further compared with analysis neglecting modeling uncertainty and the im-

portance of modeling uncertainty propagation in seismic performance assessment is

discussed.

7

2.2 Overview of Performance-Based Earthquake En-

gineering

Performance-based earthquake engineering (PBEE) involves design and assessment

of engineering structures to satisfy performance goals under earthquake ground mo-

tions. It involves a probabilistic framework to systematically combine seismic hazard

assessment with structural and damage analyses to assess seismic performance of

structures from initiation of damage up to collapse. The framework further combines

loss analysis enabling assessment of performance goals in terms of economic losses,

downtime and casualty rates aftermath an earthquake (Deierlein, 2004; Krawinkler

& Miranda, 2004).

Assessment of earthquake-induced structural collapse and risk-based assessment of

structural response utilizes the first part of the PBEE triple integral, which integrates

seismic hazard assessment and nonlinear dynamic history analysis. This computation

yields the mean exceedance rate of an engineering demand parameter (edp) and it is

computed using the following equation:

λ(edp) =

∫ ∞0

P (EDP > edp|IM = im)

∣∣∣∣dλIM(im)

d(im)

∣∣∣∣ d(im) (2.1)

where λ(edp) is the (annual) rate of exceeding edp and λ(im) represents a seismic

hazard curve, which relates ground motion intensity (IM) to its respective rate of

occurrence. In Equation 2.1, dλIM (im)d(im)

is the absolute value of the derivative of the

hazard curve at IM = im. IM is used in this thesis as the 5% damped spectral

acceleration at the fundamental period of the structure, Sa(T1, 5%).

P (EDP > edp|IM = im) defines the conditional probability of exceeding edp

given IM = im. Common EDPs include peak story drift ratios (SDR) and peak floor

accelerations. Computation of λ(edp) requires P (EDP > edp|IM = im) at various

ims, defining a full distribution of structural response as a function of im. A full dis-

tribution of P (EDP > edp|IM = im) as a function of IM = im constitutes a fragility

function. Fragility functions are an important outcome of probabilistic seismic perfor-

mance assessment, which by definition are cumulative distribution functions relating

8

CHAPTER 2. UNCERTAINTY IN SEISMIC PERFORMANCE ASSESSMENT 9

conditional probabilities of exceeding certain damage states, including collapse, to

ground motion intensities.

Equation 2.1 also enables risk-based assessment of structural collapse. Evalua-

tion of structural collapse is particularly important since seismic design provisions in

building codes aim to provide adequate collapse safety of structures even in extreme

events. Collapse response of structures is characterized by highly nonlinear behavior

response of its components, and it requires structural analysis models that can capture

large inelastic deformations with significant cyclic strength and stiffness degradation

in the elements due to repeated cycles of loading. Nonlinear dynamic analysis is an

essential part of collapse simulation that enables systematic assessment of structural

collapse risk. In this thesis, we use structural models idealized using concentrated

plasticity approach in order to assess collapse response of structures. The details

of the structural idealization and collapse simulation procedure are provided in the

subsequent sections.

In this thesis, we are mainly interested in structural collapse response. A perfor-

mance metric of interest is λc, mean annual frequency of collapse. Also of interest

is structural performance in terms of SDR and drift-exceedance limit states. Perfor-

mance metrics include λ(edp), which provides a time-based assessment of structural

response as well as conditional probabilities of exceeding certain structural response

thresholds.

2.3 Sources of Uncertainty in Seismic Performance

Assessment

PBEE provides a probabilistic framework for systematic propagation of uncertainty

involved in each step of the methodology. The first step of the PBEE methodology is

seismic hazard analyses. Seismic hazard curves provide a probabilistic representation

of exceedance of various ground motion intensity levels given possible earthquakes in

the site. The uncertainties involved in seismic hazard analysis and ground motion

models are propagated through the probabilistic seismic hazard analysis framework

and the end product is rates of occurrence of ground motion intensities (Cornell,

1968).

The second step of the PBEE methodology is structural analysis. Structural

analysis is conducted using a suite of ground motions to represent the variability in

earthquake ground motions. This variability is propagated via dynamic history anal-

ysis with a Monte-Carlo simulation approach to represent record-to-record variability

in structural response predictions. In this approach, each ground motion in the suite

is treated as equally likely.

Selection of appropriate ground motions to represent record-to-record variability

in seismic performance assessment has been a major focus in the literature. Generic

ground motion sets are developed for this purpose. SAC (Somerville et al. , 1997),

LMSR (Krawinkler et al. , 2003) and FEMA-P695 records (FEMA, 2009b) are among

the common ground motion suites. The spectral shape of the ground motions affect

the seismic response quantities of interest (Haselton et al. , 2011) and ideally it should

reflect the seismic hazard characteristics of the design site. Generic ground motions do

not consider site-specific hazard characteristics and treat different sites with a similar

hazard. Ideally, a risk-based assessment should be conducted using ground motion

time histories that are compatible with the seismic hazard at the site of interest

(Lin, 2012). A number of researchers provide recommendations regarding hazard-

consistent selection ground motions (i.e Baker & Cornell, 2006; Bradley, 2010; Baker,

2011). Haselton et al. (2011) developed spectral shape factors to adjust collapse

capacities that are obtained using generic ground motion sets. In addition to spectral

shape of ground motions, record-to-record variability is affected by the selection of

ground motion intensity measures as well as the number of ground motions. Optimal

selection of these factors has been explored by others (i.e. Luco & Cornell, 2007;

Bradley, 2013; Eads, 2013).

Besides record-to-record variability, another prominent source of uncertainty in

seismic performance assessment is modeling uncertainty (Cornell & Krawinkler, 2000).

Especially, the studies assessing structural collapse emphasized the importance of

phenomenological model parameters. Challa & Hall (1994) and Gupta & Krawin-

kler (1999) analyzed seismic performance of 20-story steel moment-resisting frames

10

CHAPTER 2. UNCERTAINTY IN SEISMIC PERFORMANCE ASSESSMENT11

and found that assessment of large deformations require better characterization of

structural degradation parameters. Ibarra & Krawinkler (2005) emphasized the im-

portance of modeling uncertainties on collapse risk and found that plastic rotation

capacity and post-capping stiffness are among the most important parameters. Mi-

randa & Akkar (2003) and Adam & Jager (2012) proposed expressions relating col-

lapse response to model parameters. These included post-yield stiffness and hysteretic

behavior parameters as well as fundamental period.

Bradley (2013) presents a classification of modeling uncertainties and discusses

the potential problems in numerical response simulations due to such uncertainties.

In essence, modeling uncertainty arises from the following factors:

• Variability in physical quantities. These parameters are directly measurable

(e.g. mass, material yield strength).

• Variability in phenomenological modeling parameters. These parameters are

not measurable and are generally estimated using predictive equations as a

function of physical quantities (e.g. cyclic degradation).

• Uncertainty in structural idealizations and analysis strategies. These include

the variability among different structural models and the uncertainty due to

simplifications in a structural model (e.g. formulations of damping, one or two-

dimensional structural models).

Priestley (1998); Panagiotakos & Fardis (2001) studied the yielding behavior of

reinforced concrete elements and reported small coefficient of variations (0.16 and

0.32, respectively) for yield rotation. Haselton et al. (2008) and Lignos & Krawin-

kler (2008) reported a high coefficient of variation of about 0.6 on the plastic rotation

capacity for reinforced concrete columns and steel beam-columns, respectively. The

variability on the plastic rotation capacity is significantly larger than the variability

associated with other physical model parameters due to the underlying behavioral

effects (reinforcing bar buckling, local flange buckling, fracture) that control the de-

grading response. These large variations in degrading behavior at large inelastic

deformations have also been reported for other types of structural components and

behavior (e.g., studies by Fell et al. (2009) on buckling and fracture in steel braces,

or Kunnath et al. (2009) in simulating reinforcing bar buckling and fracture).

Explicit consideration of uncertainties related to physical and phenomenological

quantities have been of interest to many researchers. Recent work has illustrated

a variety of techniques for including these uncertainties in the context of seismic

performance assessment (e.g., Porter et al. , 2002a; Lee & Mosalam, 2005; Liel et al.

, 2009; Song & Ellingwood, 1999; Celik & Ellingwood, 2010) as well as in seismic

response analysis (e.g., Rathje et al. , 2010; Bazzurro & Cornell, 2004). Chapter 4

discusses the methods for propagating the modeling uncertainty due to physical and

phenomenological parameters along with the previous research to incorporate these

uncertainties in seismic response assessments.

The earthquake response from instrumented structures provide means for validat-

ing the uncertainty in structural response simulations. Bradley (2011) proposed a

framework for validating site response analysis using mixed-effects regression to treat

multiple observations from seismometer arrays and their corresponding predictions.

The response of Van Nuys Hotel under the Northridge earthquake was recorded and

enabled comparisons of numerical simulations to the recorded response. Aslani &

Miranda (2006); Krawinkler (2004); Browning et al. (2000) compared structural re-

sponse simulations using different analysis models to the recorded responses. The

variability in different structural idealizations are also explored by Andrade & Borja

(2006) and Kwok et al. (2008) in the context of seismic site response analyses.

Besides post-earthquake analysis of instrumented structures, blind prediction con-

test datasets provide a good opportunity to investigate modeling uncertainty. Blind

prediction data coupled with a shake table analysis enable quantification of added

uncertainty due to structural idealization and analysis strategies, and validation of

the techniques utilized for uncertainty propagation. Sattar et al. (2015) used blind

prediction contest submissions of a 7-story R/C structure, which was later tested at

a shake-table at UCSD, to quantify the uncertainty in story drifts for a given earth-

quake level. He then used these uncertainty distributions and generated realizations

of drift values to modify the IDA curves, thus combining modeling uncertainty and

12


record-to-record variability. In this chapter, modeling uncertainty in the seismic re-

sponse of a bridge column is also quantified using blind prediction contest data. This

is further compared with structural analysis simulations.

There are also modeling uncertainties due to round-off errors and discrete solutions

of a continuous problem, which are small compared to the aforementioned sources

of uncertainty. Human factors in design and construction of structures constitute

an important source of uncertainty in seismic performance assessment. Modeling

and incorporation of human errors requires qualitative assessment strategies, and

employed quality assurance strategies can change the impacts of human factors. In

this thesis, our focus is quantitative assessment of uncertainty in model parameters,

and incorporation of human errors is outside the scope of this thesis.

2.4 Probabilistic Seismic Collapse Risk Assessment

of a Bridge Column

PEER and NEES organized a blind-prediction contest of a full-scale reinforced con-

crete bridge column. The full-scale column is later tested on the NEES Large High-

Performance Outdoor Shake Table at UCSD. The experimental results along with

the construction drawings, material properties and ground motions are available in

PEER & NEES (2015).

The column was designed according Caltrans Seismic Design Criteria and Bridge

Design Specifications (Caltrans, 2010). The circular column has a diameter of 4 ft

(1.2 m) and height of 24 ft (7.2 m). To mobilize the column capacity, a 250 ton (2,245

kN) reinforced concrete block was cast on top of the column (Terzic et al. , 2015). A

snapshot of the test set-up is provided in Figure 2.1.a.

In this section, we are interested in assessing seismic collapse response of the

reinforced concrete bridge column. We provide information related to modeling and

analysis of the structure, followed by the collapse performance quantified with and

without explicit consideration of modeling uncertainties.

(a) (b)

Mass

Zero-length Rotational Spring

Figure 2.1: a) Test set-up of the reinforced concrete bridge column which was testedfull-scale on a shake table b) Structural idealization of the bridge column

2.4.1 Structural Idealization and Calibration of Analysis Model

Collapse simulation of structures requires numerical models that can reproduce the

nonlinear deformation demands and degradation in stiffness and strength in the el-

ements due to repeated cycles of loading. The component hinge model originally

developed by Ibarra et al. (2005) is capable of simulating nonlinear hysteretic be-

havior of reinforced concrete beam-column elements. It is based on a trilinear mono-

tonic backbone curve, relating member moment (M) and rotation (θ), along with

nonlinear hysteretic rules to simulate strength and stiffness degradation under cyclic

loading. This model provides a powerful alternative for collapse simulation since the

post-capping behavior can be idealized using negative stiffness with enables collapse.

The post-capping portion of the backbone curve simulates strain-softening behav-

ior related to concrete crushing, rebar buckling and fracture for reinforced concrete

structural elements. The accuracy of the simulation depends on realistic character-

ization of parameters of the phenomenological model. An example backbone curve

displaying parameters of the model is provided in Figure 2.2. Following (Ibarra et al.

, 2005), the parameters defining the backbone curve are the flexural strength (My),

maximum moment capacity (Mc), effective initial stiffness (EIstf ), plastic rotation

capacity (θcap,pl), and post-capping rotation capacity (θpc).

14


My

Mc

θ

M

θpc

θcap,pl

EIstf

Figure 2.2: Backbone curve for concentrated plasticity model relating member moment(M) and rotation (θ)

The bridge column column was intended to respond nonlinearly primarily with

a flexural behavior. In order to model the nonlinear flexural response and simulate

collapse behavior, the bridge column is idealized using concentrated plasticity. It is

modeled as a single degree of freedom structure with an elastic beam-column element

connected to a concentrated plasticity hinge at the base. The structural idealization

is provided in Figure 2.1.b. The structure is modeled and analyzed using OpenSees.

The first mode period of the structure (T1) is obtained as 1.11 sec.

Available test data include the displacement of the top of the column and bending

moment at the column base under six earthquake ground motions, that are applied se-

quentially on the column. This data along with the construction drawings are used to

calibrate parameters of the concentrated hinge model (Chandramohan et al. , 2013).

It is assumed that the calibrated parameters reflect median properties. Figure 2.3

shows the calibrated and measured moment-rotation behavior of the bridge column.

Although the calibrated model parameters capture yielding and strain hardening be-

havior, we observe some discrepancies especially in the unloading stiffness. We note

that the version of the OpenSees that is used for modeling had problems regarding

unloading stiffness and accelerated reloading stiffness and these effects are neglected

in the model.

−400 −200 0 200 400 600−1

−0.5

0

0.5

1x 10

4

Column Top Displacement (mm)

Bendin

g M

om

ent at th

e B

ase (

kN

.m)

Experimental Results

Model Prediction

Figure 2.3: Structural analysis model is calibrated to match the experimental results

2.4.2 Ground Motions

Generic ground motion sets for response history analysis provide an alternative when

there is no specific site of interest. In this study, we use the generic far-field ground

motion set of FEMA-P695 (FEMA, 2009b), consisting of 22 record pairs from large

earthquakes. Figure 2.4 shows the response spectra of the ground motions scaled to

have the same spectral acceleration at the first mode of the bridge column.

To assess collapse risk, we use a high-seismic site in downtown Los Angeles (LA),

California (CA) at 33.996◦N, 118.162◦W, with the site soil characteristics character-

ized by NEHRP class D. The hazard curve is given in Figure 2.5.

2.4.3 Analysis Strategy

Incremental dynamic analysis (IDA) is a fairly established technique used to predict

collapse (Vamvatsikos & Cornell, 2002). It involves the scaling of ground motions

to increasing ground motion intensity levels until the structure displays dynamic

instability. The IDA response is commonly plotted in terms of SDR versus the spectral

16


10−1

100

10−2

10−1

100

101

T (s)

Sa (

g)

Response spectra of ground motions

Median response spectrum

Figure 2.4: Response spectra of the ground motions in the FEMA-P695 far-field set scaledto have the same spectral acceleration at the first mode of the bridge column

10−2

10−1

100

101

10−6

10−5

10−4

10−3

10−2

10−1

100

Annual F

requency o

f E

xceedance

Sa (T1=1.1 s, 5%) (g)

Figure 2.5: Seismic hazard curve for the site at Los Angeles, California

acceleration intensity of the ground motion. Dynamic instability is characterized

by the point where IDA response fattens out, beyond which drift demands tend

to increase very rapidly. The same ground motions are used over the full-range

of considered intensities in IDA. The counted fractions of ground motions causing

collapse at each intensity level are used to calibrate a collapse fragility curve relating

probability of collapse to a predefined ground motion intensity measure (IM).

2.4.4 Collapse Risk using a Median Model

Figure 2.6.a shows the 16%, 50% and 84% fractal IDA curves. Collapse capacities ob-

tained using IDA are used to construct the empirical cumulative distribution function

(ECDF) of collapse, given in Figure 2.6.b. Also shown is a fitted collapse fragility

curve using a method of moments estimator (Baker, 2015).

0 0.05 0.1 0.150

0.5

1

1.5

2

2.5

3

Sa

(T

1=

1.1

s, 5

%)

(g)

SDR

16% fractile

50% fractile

84% fractile

Collapse Fragility

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

Sa (T1=1.1s, 5%)

Pcolla

pse

Fitted fragility function

Empirical cumulative distribution

Figure 2.6: a) 16%, 50% and 84% fractal IDA curves using a median model with 44 groundmotions from FEMA-P695 far-field set b) Empirical and fitted collapse fragility functions

The median collapse capacity (θ) of the structure is obtained as Sa(T1) = 1.83 g

with a dispersion (β) of 0.41. The fitted collapse fragility curve is integrated with a

seismic hazard curve at a high-seismic site in downtown Los Angeles using Equation

2.1 to obtain λc as 1.11E-4 collapses/year.

18


2.4.5 Characterization and Propagation of Modeling Uncer-

tainty

The five parameters defining the concentrated plasticity model (given in Figure 2.2)

along with a parameter for energy dissipation capacity for cyclic stiffness and strength

deterioration (γ) are assumed as random variables. In addition to these parameters,

equivalent viscous damping (ξ) is also assumed as random. The logarithmic standard

deviations of these parameters are listed in Table 2.1.

Table 2.1: Logarithmic standard deviations of random variables

Random Variables Logarithmic standard deviations

θcap,pl 0.59EIstfEIg

0.27

My 0.31Mc

My0.10

θpc 0.73

γ 0.50

ξ 0.60

The six parameters defining the backbone curve and hysteric rules are also as-

sumed to be partially correlated. The derivation of the partial correlation coefficients

along with the quantified coefficients are discussed in Chapter 3. ξ is assumed to be

uncorrelated from the rest of the parameters.

The calibrated parameters are assumed to represent the median values of the

model parameters. To propagate modeling uncertainty, a Monte Carlo (MC) simula-

tion approach is utilized. We obtain 4400 random realizations of model parameters

using the joint distribution of the model parameters parameters. The joint distri-

bution is idealized using a multivariate lognormal distribution. Each of the sampled

4400 model parameter realizations are matched randomly with the 44 ground motions

in the FEMA-P695 far-field set.

2.4.6 Collapse Risk in the Presence of Modeling Uncertain-

ties

Figure 2.7.a shows the 16%, 50% and 84% fractal IDA curves in the presence of

modeling uncertainties. Collapse capacities obtained using IDA are used to construct

the empirical cumulative distribution function (ECDF) of collapse, given in Figure

2.7.b. Also provided is a fitted collapse fragility curve using a method of moments

estimator.

0 0.05 0.1 0.150

0.5

1

1.5

2

2.5

3

Sa

(T

1=

1.1

s, 5

%)

(g)

SDR

16% fractile

50% fractile

84% fractile

Collapse Fragility

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

Sa (T1=1.1s, 5%)

Pcolla

pse

Fitted fragility function

Empirical cumulative distribution

Figure 2.7: a) 16%, 50% and 84% fractal IDA curves using with modeling uncertainty b)Empirical and fitted collapse fragility functions

The median collapse capacity of the structure is obtained as Sa(T1) = 1.74 g with

a dispersion of 0.60. The fitted collapse fragility curve is integrated with a seismic

hazard curve at the site in downtown Los Angeles to obtain mean annual frequency

of collapse (λc) equal to 2.67E-4 collapses/year.

When compared to the collapse risk obtained using median model parameters, we

observe that modeling uncertainty shifted (reduced) the median collapse capacity by

5% and increased the dispersion of the collapse fragility curve by 44%. The combined

effect of these changes resulted in a 141% increase in λc. λc displays a bigger change

in the presence of modeling uncertainty compared to median collapse capacity.

20


2.5 Modeling Uncertainty from the Blind Predic-

tion Contest of the Bridge Column

2.5.1 Information on the Blind Prediction Contest

PEER and NEES organized a blind-prediction contest of a full-scale reinforced con-

crete bridge column. The contest was conducted before the column was tested on the

shake table. The blind-prediction contestants were provided with structural drawings

of the specimen, material properties and ground motions. The contest included pre-

dicting structural response under six consecutive unidirectional earthquake ground

motions. Predicted responses were maximum displacements, accelerations, residual

displacements, bending moments, shear and axial forces, axial strains and curvatures

as well as the failure mode. Contestants were also asked to provide the computer

platform and the types of elements used to model the structure. After the testing of

the column was completed, another questionnaire was sent to all participants, com-

posed of questions related to modeling of the structures, and 25 out of 41 participants

responded to the questionnaire.

Predictions were submitted by 41 teams of researchers and professional engineers

from fourteen different countries. For numerical analysis of the structure, contestants

used 12 different software programs with the majority (36%) of the contestants us-

ing OpenSees. The winner was determined using an absolute error of measured to

predicted response quantities (Terzic et al. , 2015).

2.5.2 Modeling Uncertainty from Blind Prediction Contest

in Comparison with the Simulations

In the previous section, we present the analysis model and collapse risk assessment of

the bridge column with and without incorporating modeling uncertainty. In this sec-

tion, we use the same structural analysis model and the aforementioned characteriza-

tion and propagation of modeling uncertainty. The only difference in the simulations

is that we use the six earthquake ground motions that are employed in the shake-

table testing of the structure. These ground motions consist of three records from

the 1989 Loma Prieta earthquake (Mw 6.9) and one record from 1995 Kobe (Mw 6.9)

earthquake. Ground motions 1, 2 and 4 are small-to-medium intensity earthquakes,

whereas ground motions 3, 5 and 6 are high-intensity earthquakes. They are applied

sequentially to the structure to determine peak engineering quantities. We conduct

analysis with and without modeling uncertainty, and compare our results with the

experiment results as well as with the predictions submitted by the contestants.

The two major exercises conducted in this section are briefly summarized as fol-

lows:

1. We quantify the distributions of the error in predicting different engineering

parameters in the presence of modeling uncertainty, and compare these with

the ones obtained using a median model. Comparisons are also performed with

the experimental test results in order to identify the bias and variability imposed

using a median model versus models incorporating modeling uncertainty. The

main focus of our analysis is on drifts. Other engineering quantities are also

provided in Appendix 2.A for reference.

2. The analysis model parameters that are calibrated to match experimental re-

sults are assumed to represent median values. The variability around these me-

dian model parameters, which are obtained using beam-column element model

component calibration data set of Haselton et al. (2008), are propagated to

characterize the uncertainty in engineering parameters due to modeling uncer-

tainty. The resulting distributions of drifts are compared to the uncertainty in

the blind prediction contest submissions. The goal is to identify the extent of

modeling uncertainty captured using the simulation-based uncertainty propa-

gation approach versus the uncertainty in the blind prediction submissions.

22


Quantification of Modeling Uncertainty from Blind Prediction Contest

Submissions

The modeling uncertainty in the blind prediction contest submissions is quantified us-

ing the residuals between the submissions and the experimental test results. Residuals

are obtained using the equation given below.

θ∗ij = θijk + εijk εij ∼ (µij, σij) (2.2)

where θ∗ij represents the experimental test result of the ith engineering demand pa-

rameter (EDP) under earthquake ground motion j. These results are available from

Terzic et al. (2015). θijk represents the predicted value of EDPij by the contestant

indexed by k. We are interested in quantifying the first two statistical moments of

the residual εij, where µij and σij represent the mean and standard deviation of εij,

respectively.

Terzic et al. (2015) provides bar plots of the bias and coefficient of variation

of the submissions. In their study, bias is quantified as the difference between the

predicted and measured values of the parameter normalized with respect to measured

response multiplied by 100. We denote the mean bias values as biasij. In addition

to mean bias, coefficient of variation of each predicted quantity for each earthquake

and engineering demand parameter are denoted by cv(θij). The bar plots given in

Terzic et al. (2015) are digitized, and biasij and cv(θij) are obtained. Using these

parameters, we obtain µij and σij as:

µij =− biasij100

θ∗ij

σij =cv(θij)

100θ∗ij

(1 +

biasij100

) (2.3)

where biasij and cv(θij) are obtained from Terzic et al. (2015).

In addition to means and standard deviations of the residuals (µij and σij) that are

computed for each earthquake and engineering parameter, we also compute a mean

estimate and a pooled standard deviation for the residuals of each engineering pa-

rameter. Residuals of each engineering parameter, indexed by i, are then distributed

as εi ∼ (µi, σi).

Blind prediction contestants submitted predictions of engineering parameters for

ngm earthquakes. The pooled standard deviation of εi is obtained by treating each

ground motion as a separate population and by using a pooled estimate to combine

the variances of different populations. The pooled standard deviation and mean of εi

are obtained using the equations given below:

µi =1

ngm

ngm∑j=1

µij

σi =

√√√√ 1

ncontestants × ngm − ngm

ngm∑j=1

(ncontestants − 1)× σ2ij

(2.4)

where ncontestants represents the number of contestants.

As mentioned previously, the contestants of the blind prediction contest utilize

different structural models and the contestants are surveyed regarding the type of

models used. Such data (blind prediction of engineering demand parameters along

with the type of structural models) can be used to study the variability due to different

structural idealizations. Equation 2.2 and subsequent calculations can be further

extended to include parameters representing the type of models used in prediction

(i.e. another subscript can be added due to structural model). This enables direct

quantification of the uncertainty due to structural models. We note that this is out of

the scope of this study since data providing blind prediction submissions with their

structural models is not available at the time of the study.

Quantification of Modeling Uncertainty from Our Simulations

The residuals for each earthquake and engineering demand parameter that we deter-

mined from simulations is calculated as follows:

θ∗ij = θijz + εijz εij ∼ (µij, σij) (2.5)

where θ∗ij represents the experimental test result of the ith engineering demand param-

eter (EDP) under earthquake ground motion j. θijz is the simulated value of EDPij

24


using model realization z. We are interested in quantifying the first two statisti-

cal moments of the residual εij, where µij and σij represent the mean and standard

deviation of εij, respectively.

We also compute a mean estimate and a pooled standard deviation for the residu-

als of each engineering parameter, µi and σi. Residuals of each engineering parameter,

indexed by i, are then distributed as εi ∼ (µi, σi). The pooled standard deviation and

mean of εi are obtained using the equations given below:

µi =1

ngm × nsim

ngm∑j=1

nsim∑z=1

εijz

σi =

√√√√ 1

ngm × nsim − ngm

ngm∑j=1

nsim∑z=1

(εijz − µi)2(2.6)

where ngm and nsim represent the number of ground motions and the number of

simulations, respectively. Note that we use similar approaches to quantify modeling

uncertainty from blind predictions as well as using our simulations.

Modeling Uncertainty in Drifts

Among the predicted engineering quantities in the blind prediction contest, we used

the maximum absolute value of the relative horizontal displacement at the top of the

column in two directions to determine the drift (i.e. displacement predictions divided

by the length of the column). Absolute maximum value of drift ratio is denoted

by “Maximum Absolute Drift Ratio” and the absolute maximum in the opposite

direction is denoted by “Minimum Absolute Drift Ratio”, as defined in Terzic et al.

(2015).

Table 2.2 summarizes the first two statistical moments, the means and standard

deviations, of the residuals of the maximum absolute drift ratio. Means and standard

deviations are computed for the six ground motions using a simulation-based approach

with a median model as well as with modeling uncertainty. Also provided are the

corresponding values from blind prediction contest submissions.

Table 2.2: Means and standard deviations, of the residuals of the maximum absolute driftratio

EQj=1 EQj=2 EQj=3 EQj=4 EQj=5 EQj=6 µi

µij

Simulations with a Median Model 0.001 -0.004 -0.018 -0.002 -0.003 0.009 -0.003

Simulations Incorporating Modeling Uncertainty 0.001 -0.001 -0.006 0.003 -0.002 0.002 -0.001

Blind Prediction Contest Submissions 0.000 0.002 0.010 0.002 0.017 0.022 0.008

EQj=1 EQj=2 EQj=3 EQj=4 EQj=5 EQj=6 σi

σij

Simulations with a Median Model - - - - - - 0.009

Simulations Incorporating Modeling Uncertainty 0.004 0.004 0.018 0.012 0.017 0.018 0.014


We illustrate the results given in Table 2.2 using a normal distribution in Figure

2.8. Note that the data are plotted for an assumed normal distribution, which needs

further testing and may not be appropriate for modeling the residuals. We quan-

tify modeling uncertainty using the first two sample moments of the residuals and

use these parameters along with a normal distribution in Figure 2.8 for illustrative

purposes.

26


−0.06 −0.04 −0.02 0 0.02 0.04 0.060

5

10

15

20

25EQ1

(a)

−0.06 −0.04 −0.02 0 0.02 0.04 0.060

5

10

15

20

25EQ2

(b)

−0.06 −0.04 −0.02 0 0.02 0.04 0.060

5

10

15

20

25EQ3

Pro

ba

bili

ty D

en

sity (c)

−0.06 −0.04 −0.02 0 0.02 0.04 0.060

5

10

15

20

25EQ4

(d) Uncertainty from Simulations

Uncertainty from Contest Submissions

Simulation Median Model

−0.06 −0.04 −0.02 0 0.02 0.04 0.060

5

10

15

20

25EQ5

Error in Max Abs DR

(e)

−0.06 −0.04 −0.02 0 0.02 0.04 0.060

5

10

15

20

25EQ6

Error in Max Abs DR

(f)

Figure 2.8: Distributions of the residuals of the maximum absolute value of the driftratio (“Maximum Absolute Drift Ratio”) are illustrated using a normal distribution. Thecyan and red curves correspond to predictions using simulations incorporating modelinguncertainty and blind prediction submissions, respectively. Also shown is the error in thevalues calculated using the median model (in blue).

From Table 2.2, we see that incorporation of modeling uncertainty shifts the means

of the predictions towards the experimental results, i.e. the mean errors approach to 0

in the presence of modeling uncertainty. In other words, the mean bias in the results

are reduced by incorporating modeling uncertainty. To the extent that no single

model is perfect, propagating the effects of uncertainty in modeling on the analysis

model can be argued to result in reducing the mean bias, even though uncertainties

are measured from a larger data set of columns with different properties. We also

observe that modeling uncertainty increases the variability in the predictions. We

obtain a pooled estimate of standard deviation (σi) of 0.9% using analysis conducted

with median model parameters (only includes earthquake variability). The standard

deviation of the drift ratio increases to 1.4% when modeling uncertainty is propagated

(includes earthquake variability and modeling uncertainty).

We see that blind prediction contest submission have more bias compared to sim-

ulations incorporating modeling uncertainty. This was expected since our simulations

are calibrated to match the experimental test results. Note that the mean bias among

all earthquakes, µi, (maximum absolute value of drift ratio obtained using contest sub-

missions) is 0.8% drift. This is fairly small, compared to the maximum drift of 7.8%

observed in the shake-table tests.

Table 2.2 also shows that the variability obtained in analytical modeling uncer-

tainty propagation and blind prediction test submissions are comparable. Similar

standard deviations are obtained for individual earthquakes as well as pooled esti-

mates, with the analytical simulations slightly over estimating the variability.

The earthquake ground motions of 1, 2 and 4 are small to medium intensity

earthquakes (EQs), whereas EQs 3, 5 and 6 are high-intensity ones. We see that

the variability in the predicted and observed response quantities for high-intensity

earthquakes are higher. Also we observe comparably larger mean bias values for

high-intensity earthquakes. The observations of more bias and variability in high-

intensity earthquakes are expected because of the challenges in simulating highly

nonlinear structural response for the high-intensity earthquakes.

Table 2.3 summarizes the means and standard deviations, of the residuals of the

minimum absolute drift ratio. These results are illustrated using a normal distribution

in Figure 2.9.

28


Table 2.3: Means and standard deviations, of the residuals of the minimum absolute driftratio

EQj=1 EQj=2 EQj=3 EQj=4 EQj=5 EQj=6 µi

µij

Simulations with a Median Model -0.001 0.001 0.000 -0.003 -0.004 -0.022 -0.005

Simulations Incorporating Modeling Uncertainty 0.000 0.001 -0.004 0.002 0.007 -0.008 0.000

Blind Prediction Contest Submissions 0.000 -0.004 -0.002 0.001 0.007 0.003 0.001

EQj=1 EQj=2 EQj=3 EQj=4 EQj=5 EQj=6 σi

σij

Simulations with a Median Model - - - - - - 0.009

Simulations Incorporating Modeling Uncertainty 0.003 0.003 0.007 0.004 0.012 0.015 0.009


−0.06 −0.04 −0.02 0 0.02 0.040

10

20

30

40

50EQ1

(a)

−0.06 −0.04 −0.02 0 0.02 0.040

10

20

30

40

50EQ2

(b)

−0.06 −0.04 −0.02 0 0.02 0.040

10

20

30

40

50EQ3

Pro

ba

bili

ty D

en

sity (c)

−0.06 −0.04 −0.02 0 0.02 0.040

10

20

30

40

50EQ4




−0.06 −0.04 −0.02 0 0.02 0.040

10

20

30

40

50EQ5

Error in Min Abs DR

(e)

−0.06 −0.04 −0.02 0 0.02 0.040

10

20

30

40

50EQ6

Error in Min Abs DR

(f)

Figure 2.9: Distributions of the residuals of the maximum absolute value of the driftratio in the opposite direction (“Minimum Absolute Drift Ratio”) are illustrated using anormal distribution. Shown in cyan and red correspond to predictions using simulationsincorporating modeling uncertainty and blind prediction submissions, respectively. Alsoshown is the prediction using simulations with a median model in blue.

Similar observations are made for minimum absolute drift ratios. We see that

in this case modeling uncertainty, in general, decreases the mean error in predicting

a response quantity. Simulations have slightly more variability compared to contest

submissions for high-intensity earthquakes. Under the high-intensity earthquakes,

the structure behaves nonlinearly and compared to contest submissions, the slightly

higher variability obtained from simulations can be due to conservative characteriza-

tion of the variability of the model parameters defining nonlinear response.

30


The pooled distributions for maximum absolute values of drift ratio, both in the

direction where maximum is observed and the opposite direction, are illustrated in

Figure 2.10, assuming normal distributions.

−0.05 0 0.050

5

10

15

20

25

30

35

40

45

Error in Max Abs SDR

Pro

babili

ty D

ensity

Uncertainty from Simulations



−0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.030

10

20

30

40

50

60

Error in Min Abs SDR

Pro

babili

ty D

ensity




Figure 2.10: Residuals of the maximum absolute value of the drift ratio are obtained,and distributions are obtained using a pooled estimate of variance for different earthquakes.These distributions are illustrated using a normal distribution. The plots in cyan and redcorrespond to predictions using simulations incorporating modeling uncertainty and blindprediction submissions, respectively. Also shown is the prediction using simulations with amedian model in blue. a) Maximum absolute drift ratio b) Maximum absolute drift ratioin the opposite direction

Modeling Uncertainty in Collapse Probability

Failure of the column did not occur during the application of six earthquake ground

motions; only spalling of concrete was observed during the final earthquake record #

6. 33% of the contestants correctly predicted that column would not fail, whereas the

remaining 67% that column would fail.

Our analysis using the calibrated model parameters did not result in the failure of

the column. However, in the presence of modeling uncertainty, 44% of the simulations

had drifts exceeding 15%, which could be treated as collapse cases.

Comments

Blind prediction submissions represent a pool of predictions by different structural

idealizations, analysis softwares as well as different calibrations of structural model

parameters. Although the analysis input data in shake-table tests was highly con-

strained compared to a real-world problem, comparisons of blind prediction predic-

tions with experimental results might provide an effective real-world representation

of modeling uncertainty. These data sets embrace different sources of modeling un-

certainty and provide insights for characterization of epistemic modeling uncertainty

and validation of the tools used for uncertainty propagation.

In this section, we use blind prediction contest submissions predicting seismic

response of a bridge column. This blind prediction contest had 41 submissions for

each predicted engineering quantity and constitutes a small data set. Since the data

is collected as a part of a contest, the quality in the predictions can be arguable as

compared to expert judgement via a formal expert elicitation process. Furthermore,

an inherent assumption is that each contestant provide predictions with an equally

viable structural model. Factors such as these combined with data scarcity, might very

well affect the variability and bias observed in blind prediction contest submissions.

A mean estimate among the six earthquakes used in contest is computed for the

residuals of the maximum absolute drift ratio as 0.8% drift with a standard deviation

of 1.3%. The variability in the results emphasize the importance of modeling uncer-

tainty in predicting structural response. Bias in random variables can be an important

contributor to the accuracy in structural response estimation. In this chapter, model

parameters are assumed to be unbiased since simulations are calibrated to match the

experimental test results. For phenomenological model parameters, bias can be stud-

ied comparing observed (i.e. experimental) and predicted values of model parameters.

The bias values can then be incorporated in uncertainty propagation.

Simulations incorporating modeling uncertainty treat variability in physical quan-

tities and phenomenological modeling parameters. On the other hand, subject to

aforementioned limitations, blind prediction contest data is a representation variabil-

ity in physical quantities and phenomenological modeling parameter and uncertainty

in structural idealization and analysis strategies. Therefore, blind predictions can be

32


used to supplement simulations incorporating modeling uncertainty. Similar variabil-

ity is observed in drift predictions of blind prediction contest submissions and simula-

tions. However, variability in results between the blind prediction contest submissions

and simulations is much higher for calculated peak accelerations (See Appendix 3.A).

This might suggest that the modeling uncertainty is well represented in simulations

of drifts; however, its validation needs further studies with extensive blind prediction

of structural response.

2.6 Conclusions

In this chapter, we present seismic performance assessment of structures along with

the sources of uncertainties involved in the assessment framework. We provide an

example seismic collapse assessment of a reinforced concrete bridge column with and

without explicit consideration of modeling uncertainties. This structure’s perfor-

mance was blindly predicted by many researchers and engineering professionals as a

part of blind prediction contest organized by PEER and NEES. The bridge column

was later tested full-scale on a shake-table and experimental results are available. We

provide comparisons in terms of drift ratios predicted with and without incorporating

modeling uncertainty as well as those predicted by the blind prediction contestants.

Comparisons are made using residuals of the predictions from the experimental re-

sults. We observe that modeling uncertainty increases the variability in the predicted

engineering quantities and in general, decreases the mean bias of the errors compared

to using a median model. The increased variability in predicting drifts in the pres-

ence of modeling uncertainty agrees well with the variability in the drift predictions of

blind prediction submissions. The variability in the results emphasize the importance

of modeling uncertainty in predicting structural response.

2.A Modeling Uncertainty of Other Engineering

Demand Parameters: Simulations versus Blind

Prediction Contest Submissions

2.A.1 Displacements

Table 2.4: Means and standard deviations, of the residuals of the Max Abs Displacementin terms of mm

EQ1 EQ2 EQ3 EQ4 EQ5 EQ6 All EQs

µij

Simulation Median Model 4.2 -32.1 -133.9 -16.1 -21.7 69.3 -21.7

Simulation MU 4.4 -10.0 -44.2 22.2 -16.9 14.4 -6.3

Contest MU 3.4 11.8 70.9 17.4 123.2 160.0 55.2

σij

Simulation Median Model - - - - - - 65.9

Contest MU 30.9 29.3 132.2 86.8 124.9 132.7 100.1

Contest MU 31.2 35.1 127.2 70.4 118.0 119.5 92.7

34


−400 −200 0 200 400 6000

1

2

3

4x 10

−3 EQ1

(a)

−400 −200 0 200 400 6000

1

2

3

4x 10

−3 EQ2

(b)

−400 −200 0 200 400 6000

1

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4x 10

−3 EQ3

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−400 −200 0 200 400 6000

1

2

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4x 10

−3 EQ4




−400 −200 0 200 400 6000

1

2

3

4x 10

−3 EQ5

Error in Max Displacement (mm)

(e)

−400 −200 0 200 400 6000

1

2

3

4x 10

−3 EQ6


(f)

Figure 2.11: Distributions of the residuals of the maximum absolute value of the relativehorizontal displacement in terms of mm at the top of the column are illustrated using anormal distribution. Distributions are obtained using a median model, modeling uncertaintypropagation and blind prediction submissions are shown in blue, cyan and red, respectively.

Table 2.5: Means and standard deviations, of the residuals of the Min Abs Displacementin terms of mm


µij

Simulation Median Model -6.3 8.7 -0.5 -23.4 -26.0 -161.3 -34.8

Simulation MU 1.0 6.0 -29.7 14.6 52.6 -62.1 -0.1

Contest MU -1.4 -30.5 -13.8 4.1 52.7 24.6 5.1

σij


Simulation MU 24.3 23.0 49.7 26.2 86.2 109.7 68.1

Contest MU 24.1 27.1 72.8 26.9 55.4 71.8 50.9

36


−400 −200 0 200 4000

2

4

6x 10

−3 EQ1

(a)

−400 −200 0 200 4000

2

4

6x 10

−3 EQ2

(b)

−400 −200 0 200 4000

2

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6x 10

−3 EQ3

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−400 −200 0 200 4000

2

4

6x 10

−3 EQ4




−400 −200 0 200 4000

2

4

6x 10

−3 EQ5

Error in Min Displacement (mm)

(e)

−400 −200 0 200 4000

2

4

6x 10

−3 EQ6


(f)

Figure 2.12: Distributions of the residuals of the maximum absolute value of the relativehorizontal displacement in terms of mm at the top of the column in the opposite direction areillustrated using a normal distribution. Distributions are obtained using a median model,modeling uncertainty propagation and blind prediction submissions are shown in blue, cyanand red, respectively.

−400 −200 0 200 4000

1

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7x 10

−3


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−300 −200 −100 0 100 200 3000

1

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6

7

8x 10

−3


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Figure 2.13: Residuals of the maximum absolute value of the relative horizontal displace-ment are obtained for different earthquakes, and the pooled distributions are illustratedusing a normal distribution. Distributions are obtained using a median model, modelinguncertainty propagation and blind prediction submissions are shown in blue, cyan and red,respectively. a) Maximum absolute displacement b) Maximum absolute displacement in theopposite direction

2.A.2 Accelerations

Table 2.6: Means and standard deviations, of the residuals of the Max Abs Accelerationin terms of g


µij

Simulation Median Model 0.012 0.001 -0.018 0.056 0.075 0.026 0.025

Simulation MU -0.002 0.003 0.017 0.035 0.075 0.032 0.026

Contest MU -0.057 -0.175 -0.187 -0.221 -0.171 -0.184 -0.142

σij


Simulation MU 0.050 0.091 0.123 0.060 0.163 0.101 0.108

Contest MU 0.137 0.218 0.249 0.213 0.201 0.238 0.212

38


−1 −0.5 0 0.5 10

1

2

3

4EQ1

(a)

−1 −0.5 0 0.5 10

1

2

3

4EQ2

(b)

−1 −0.5 0 0.5 10

1

2

3

4EQ3

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−1 −0.5 0 0.5 10

1

2

3

4EQ4




−1 −0.5 0 0.5 10

1

2

3

4EQ5

Error in Max Acceleration (g)

(e)

−1 −0.5 0 0.5 10

1

2

3

4EQ6


(f)

Figure 2.14: Distributions of the residuals of the maximum absolute value of total accel-eration in terms of g at the top of the column are illustrated using a normal distribution.Distributions are obtained using a median model, modeling uncertainty propagation andblind prediction submissions are shown in blue, cyan and red, respectively.

Table 2.7: Means and standard deviations, of the residuals of the Min Abs Accelerationin terms of g


µij

Simulation Median Model -0.042 0.034 0.013 -0.046 0.048 -0.027 -0.003

Simulation MU -0.024 0.017 0.022 -0.018 0.098 -0.016 0.014

Contest MU -0.031 -0.156 -0.166 -0.160 -0.182 -0.152 -0.121

σij


Simulation MU 0.052 0.088 0.112 0.070 0.148 0.105 0.108

Contest MU 0.090 0.182 0.219 0.189 0.191 0.233 0.190

40


−1 −0.5 0 0.5 10

1

2

3

4EQ1

(a)

−1 −0.5 0 0.5 10

1

2

3

4EQ2

(b)

−1 −0.5 0 0.5 10

1

2

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4EQ3

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−1 −0.5 0 0.5 10

1

2

3

4EQ4




−1 −0.5 0 0.5 10

1

2

3

4EQ5

Error in Min Acceleration (g)

(e)

−1 −0.5 0 0.5 10

1

2

3

4EQ6


(f)

Figure 2.15: Distributions of the residuals of the maximum absolute value of total ac-celeration in terms of g at the top of the column in the opposite direction are illustratedusing a normal distribution. Distributions are obtained using a median model, modelinguncertainty propagation and blind prediction submissions are shown in blue, cyan and red,respectively.

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.60

2

4

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8

10

12


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−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.60

2

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6

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10


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Figure 2.16: Residuals of the maximum absolute value of the total acceleration are ob-tained for different earthquakes, and the pooled distributions are illustrated using a normaldistribution. Distributions are obtained using a median model, modeling uncertainty prop-agation and blind prediction submissions are shown in blue, cyan and red, respectively. a)Maximum absolute acceleration b) Maximum absolute acceleration in the opposite direction

2.A.3 Forces

Bending Moment

Table 2.8: Means and standard deviations, of the residuals of the Max Abs BendingMoment in terms of kN.m


µij

Simulation Median Model -34.4 -576.3 -421.2 401.1 1039.6 503.2 152.0

Simulation MU -271.1 -480.7 303.8 97.3 782.7 396.3 116.3

Contest MU -586.0 84.3 354.0 234.8 860.3 550.4 214.0

σij


Simulation MU 855.6 1573.0 2076.4 922.1 2832.2 1588.4 1827.1

Contest MU 1383.3 1570.9 1520.7 1250.4 1372.2 1509.2 1438.6

42


−1 −0.5 0 0.5 1

x 104

0

1

2

x 10−4 EQ1

(a)

−1 −0.5 0 0.5 1

x 104

0

1

2

x 10−4 EQ2

(b)

−1 −0.5 0 0.5 1

x 104

0

1

2

x 10−4 EQ3

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−1 −0.5 0 0.5 1

x 104

0

1

2

x 10−4 EQ4




−1 −0.5 0 0.5 1

x 104

0

1

2

3x 10

−4 EQ5

Error in Max Bending Moment (kN.m)

(e)

−1 −0.5 0 0.5 1

x 104

0

1

2

3x 10

−4 EQ6


(f)

Figure 2.17: Distributions of the residuals of the maximum absolute value of bending mo-ment at the base in terms of kN.m are illustrated using a normal distribution. Distributionsare obtained using a median model, modeling uncertainty propagation and blind predictionsubmissions are shown in blue, cyan and red, respectively.

Table 2.9: Means and standard deviations, of the residuals of the Min Abs BendingMoment in terms of kN.m


µij

Simulation Median Model -878.6 358.5 192.9 -924.9 877.8 -838.6 -202.2

Simulation MU -555.5 69.2 264.3 -500.8 1881.4 -591.6 120.2

Contest MU -232.6 -294.6 -962.9 109.6 165.4 606.0 -87.0

σij


Simulation MU 895.5 1525.7 1919.5 1376.4 2632.5 2148.6 2003.8

Contest MU 1273.2 1408.7 1728.4 1455.7 1104.8 1650.3 1452.3

44


−1 −0.5 0 0.5 1

x 104

0

1

2

x 10−4 EQ1

(a)

−1 −0.5 0 0.5 1

x 104

0

1

2

x 10−4 EQ2

(b)

−1 −0.5 0 0.5 1

x 104

0

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x 10−4 EQ3

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−1 −0.5 0 0.5 1

x 104

0

1

2

x 10−4 EQ4




−1 −0.5 0 0.5 1

x 104

0

1

2

3x 10

−4 EQ5

Error in Min Bending Moment (g)

(e)

−1 −0.5 0 0.5 1

x 104

0

1

2

3x 10

−4 EQ6


(f)

Figure 2.18: Distributions of the residuals of the maximum absolute value of bendingmoment at the base in terms of kN.m in the opposite direction are illustrated using anormal distribution. Distributions are obtained using a median model, modeling uncertaintypropagation and blind prediction submissions are shown in blue, cyan and red, respectively.

−6000 −4000 −2000 0 2000 4000 60000

1

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8x 10

−4


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−6000 −4000 −2000 0 2000 4000 6000 80000

1

2

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6x 10

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Figure 2.19: Residuals of the maximum absolute value of the bending moment at thebase are obtained for different earthquakes, and the pooled distributions are illustratedusing a normal distribution. Distributions are obtained using a median model, modelinguncertainty propagation and blind prediction submissions are shown in blue, cyan andred, respectively. a) Maximum absolute bending moment b) Maximum absolute bendingmoment in the opposite direction

Shear Force

Table 2.10: Means and standard deviations, of the residuals of the Max Abs Shear Forcein terms of kN


µij

Simulation Median Model 9.4 -7.7 -116.8 110.1 140.1 8.3 23.9

Simulation MU -24.2 -6.2 -39.4 58.0 135.6 25.6 23.3

Contest MU -34.3 -81.7 -421.4 -20.1 -26.2 5.2 -82.7

σij


Simulation MU 119.2 213.0 285.5 138.9 377.3 234.8 250.9

Contest MU 163.4 227.8 329.1 187.4 178.5 203.5 221.9

46


−1000 −500 0 500 10000

0.5

1

1.5

2x 10

−3 EQ1

(a)

−1000 −500 0 500 10000

0.5

1

1.5

2x 10

−3 EQ2

(b)

−1000 −500 0 500 10000

0.5

1

1.5

2x 10

−3 EQ3

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−1000 −500 0 500 10000

0.5

1

1.5

2x 10

−3 EQ4




−1000 −500 0 500 10000

0.5

1

1.5

2x 10

−3 EQ5

Error in Max Shear Force (kN)

(e)

−1000 −500 0 500 10000

0.5

1

1.5

2x 10

−3 EQ6


(f)

Figure 2.20: Distributions of the residuals of the maximum absolute value of shear at thebase in terms of kN are illustrated using a normal distribution. Distributions are obtainedusing a median model, modeling uncertainty propagation and blind prediction submissionsare shown in blue, cyan and red, respectively.

Table 2.11: Means and standard deviations, of the residuals of the Min Abs Shear Forcein terms of kN


µij

Simulation Median Model -119.5 60.5 -36.5 -114.7 70.5 -119.5 -43.2

Simulation MU -78.5 17.3 -19.3 -53.5 184.6 -96.9 -4.3

Contest MU -5.1 -102.1 -6.4 -80.3 -101.4 58.5 -33.8

σij


Simulation MU 123.7 206.5 259.3 162.8 345.5 242.6 250.8

Contest MU 163.7 214.4 212.1 222.1 178.9 236.2 206.1

48


−1000 −500 0 500 10000

0.5

1

1.5

2x 10

−3 EQ1

(a)

−1000 −500 0 500 10000

0.5

1

1.5

2x 10

−3 EQ2

(b)

−1000 −500 0 500 10000

0.5

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1.5

2x 10

−3 EQ3

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−1000 −500 0 500 10000

0.5

1

1.5

2x 10

−3 EQ4




−1000 −500 0 500 10000

0.5

1

1.5

2x 10

−3 EQ5

Error in Shear Force (kN)

(e)

−1000 −500 0 500 10000

0.5

1

1.5

2x 10

−3 EQ6


(f)

Figure 2.21: Distributions of the residuals of the maximum absolute value of shear at thebase in terms of kN in the opposite direction are illustrated using a normal distribution.Distributions are obtained using a median model, modeling uncertainty propagation andblind prediction submissions are shown in blue, cyan and red, respectively.

−1000 −500 0 500 10000

0.5

1

1.5

2

2.5

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Figure 2.22: Residuals of the maximum absolute value of the shear at the base are ob-tained for different earthquakes, and the pooled distributions are illustrated using a normaldistribution. Distributions are obtained using a median model, modeling uncertainty prop-agation and blind prediction submissions are shown in blue, cyan and red, respectively. a)Maximum absolute shear b) Maximum absolute shear in the opposite direction

50

Chapter 3

Estimation and Impacts of Model

Parameter Correlation for Seismic

Performance Assessment of

Reinforced Concrete Structures

3.1 Abstract

Advances in performance-based earthquake engineering highlight the importance of

explicit consideration of uncertainties to make robust estimations of seismic risk of

structures. Uncertainty propagation methods require the characterization of stochas-

tic dependence among the random variables used in the modeling and analysis. In

this study, a framework is presented to characterize the dependence of modeling pa-

rameters that define the nonlinear response at a component level and the interactions

of multiple components associated with a system’s response. The framework utilizes

random effects regression models on component test databases, which are composed

of multiple tests conducted by different research groups. Tests that are conducted in

similar conditions and are investigating the impacts of particular properties of com-

ponents that can effectively represent different locations in a structure are suitable

for this framework. Correlation coefficients from these regression models, reflecting

51

statistical dependency among properties of components tested by individual research

groups, are assumed to reflect correlations associated with multiple components in a

structure. Uncertainties and dependence in structural component model parameters

of reinforced concrete elements are evaluated by applying this framework to a database

of reinforced concrete beam-column tests. The analyses show that model parameters

within structural components tend to be only mildly correlated, whereas there are

strong correlations between like parameters defining different structural components

within a building. We also assess the effects of model parameter correlations on dy-

namic response of a four-story reinforced concrete frame structure. Increased levels

of correlation results in increased dispersion in the quantified dynamic response and

thus higher values of probabilities of collapse. In the case study structure, similar

lower tail response is observed in terms of collapse fragilities for uncorrelated and

partially correlated models, whereas the collapse fragilities differ at the upper tails.

Similar lower tail behavior results in similar mean annual frequency of collapse es-

timations. This work provides guidance for characterization of correlation structure

when propagating uncertainty in seismic response assessment of structures.

3.2 Introduction

Performance-based earthquake engineering enables quantification and propagation of

uncertainties in a probabilistic framework to make robust estimations of seismic risk

and loss of structures. One important source of uncertainty is the intensity and other

characteristics of ground motions. Quantification and propagation of uncertainties re-

lated to ground motion variability and seismic hazard through to engineering demand

parameters have been a major focus of this framework. Another important source is

structural modeling and analysis. Recent developments in performance-based earth-

quake engineering emphasize the importance of considering modeling uncertainties

(e.g. Bradley, 2013). The uncertainty of representing a real behavior with an ideal-

ized model and analysis method, as well as the calibration of the model parameters,

translate into modeling uncertainty. Robust characterization of seismic performance

52

CHAPTER 3. MODEL PARAMETER CORRELATION 53

of structures requires propagation of uncertainties related to structural modeling. Ex-

plicit quantification of uncertainties and characterization of dependence among the

random model parameters are essential for this purpose.

Dependence among random model parameters can be characterized using correla-

tion coefficients. Where the random variables have a multivariate normal distribution,

this provides a complete description of the dependence. However, if the joint distribu-

tion of model parameters is not multivariate normal or not known, correlations among

random variables can be used along with marginal distributions of the variables to

conveniently approximate the dependence.

Quantification of correlation of the modeling parameters that define the structural

components require data related to the parameters among different components in a

structure. Due to the scarcity of such data, current practice in system reliability anal-

ysis is to use expert judgment in quantifying correlation structure of analysis model

parameters. Haselton (2006) used a First-Order Second-Moment approach (Melchers,

1999) to propagate uncertainties in modeling variables with judgment-based correla-

tion coefficients for assessing collapse response of reinforced concrete structures and

showed that the variability in collapse capacity was highly influenced by the correla-

tion assumptions. Liel et al. (2009) also used judgmental coefficients to define the

correlation structure of meta variables defining strength and ductility of the struc-

tural components for assessing collapse response. Celarec & Dolsek (2013); Celik &

Ellingwood (2010); Pinto & Franchin (2014) propagated modeling uncertainty for seis-

mic performance assessment of reinforced concrete structures, where the correlation

structure among modeling parameters are defined using assumed coefficients.

Although the effects of correlations among random variables on system reliability

are well known, few researchers have used observational data to quantify dependence.

Idota et al. (2009) assessed the correlation of strength parameters for steel moment

resisting frames using steel coupon tests from production lots. Vamvatsikos (2014)

used the correlation coefficients by Idota et al. (2009) to study the effects of spatial

correlation of components at different locations in a building on its dynamic response.

In this study, we aim to reliably characterize correlation structure of modeling pa-

rameters that define the nonlinear cyclic response at a component level and the inter-

action of different components on the overall dynamic system level response. Random

effects regression is applied to a database of reinforced concrete column tests to infer

correlation structure of model parameters defining concentrated plasticity models.

The database is composed of tests of reinforced concrete columns tested by multiple

research groups. Correlation coefficients representing statistical dependency of the

tests conducted by a research group are assumed to reflect the correlation of different

components in a structural system. Although the reported correlation results are for

nonlinear analysis of reinforced concrete elements, the presented framework can be

applied for other types of materials or models. We also assess the effects of differ-

ent levels of correlations on dynamic response of a 4-story reinforced concrete frame

building. This assessment provides guidance for efficient and accurate characteriza-

tion of the correlation structure of modeling parameters for uncertainty propagation

studies.

To begin, an overview of probabilistic seismic performance assessment of structures

is provided along with the structural modeling and analysis techniques and the seismic

risk metrics used in this study. Following this discussion, the random variables, data

set and the framework used to assess correlation of model parameters are presented.

Finally, the quantified correlation structure of model parameters is used to study the

dynamic response of a 4-story reinforced concrete frame structure and the effects of

different levels of correlations on structural response are examined.

3.3 Probabilistic Seismic Performance Assessment

Seismic performance of structures is assessed using the probabilistic performance-

based earthquake engineering methodology (e.g. Krawinkler & Miranda, 2004; Deier-

lein, 2004). Nonlinear structural analyses are run using a suite of ground motions to

propagate uncertainties related to ground motion variability and seismic hazard. The

engineering demand parameters obtained from structural analyses are then related to

the risk of collapse and other damage states of interest.

54


The collapse risk is characterized by significant nonlinear deformations including

stiffness and strength deterioration in structural elements. Therefore, analyses of

collapse response requires models that are capable of simulating deterioration mecha-

nisms and strain softening of components. Phenomenological concentrated plasticity

models are suited for modeling collapse response of structures (Deierlein et al. , 2010).

However, model parameters that define concentrated plasticity models are generally

related to physical engineering parameters by empirical relationships. Modeling un-

certainty becomes more pronounced for collapse response simulations than for elastic

or mildly nonlinear simulations, due to both the relatively limited knowledge of model

parameters and the highly nonlinear behavior associated with collapse.

The concentrated plasticity model by Ibarra et al. (2005), which has been fre-

quently used to simulate sidesway collapse in frame structures (e.g. Zareian & Krawin-

kler, 2007; Eads et al. , 2013), is used in this study to model component response.

Specific attention is given to the correlation of model parameters used to define plastic

hinges in seismic resisting moment frames.

The elements defined by the concentrated plasticity model are assigned a trilinear

backbone curve as shown in Figure 3.1. The backbone curve has five parameters:

capping plastic rotation (θcap,pl), effective stiffness corresponding to the secant value to

40% of the yield force of the component (EIstf ), yield moment (My), capping moment

(Mc), and post-capping rotation (θpc). An additional sixth parameter, γ, is used

to define the normalized energy dissipation capacity, which controls the amount of

basic strength deterioration, post-capping strength deterioration, unloading stiffness

deterioration, and accelerated reloading stiffness deterioration.

My

Mc

Chord Rotation

Moment

θpc

θcap,pl

EIstf

Figure 3.1: Ibarra et al. (2005) model for moment versus rotation of a plastic hinge in astructure. The model parameters of interest are labeled.

In this study, we aim to reliably characterize correlation of model parameters

in a structure. Two types of correlations are considered, namely, (1) correlation

within a component, and (2) correlations between components. The former one refers

to the correlation structure of modeling parameters that define the inelastic cyclic

response at a component level, whereas the latter one refers to the interaction of

different components on overall system level response. Among the factors that might

affect these correlations are the similarities in the origin and properties of structural

materials, member geometries and details.

Illustrations of the correlations are provided in Figure 3.2. Correlation within a

component is illustrated through the interaction of θcap,pl and My of a second-story

column. Correlation between components is illustrated through the interaction of

θcap,pl and My of a first-story and second-story column, respectively. Note that these

model parameters are selected for illustrative purposes and any random variable would

have been appropriate.

56


My

Mc

Chord Rotation

Moment

θpcθcap,pl

EIstf

My

Mc

Chord Rotation

Moment

θpcθcap,pl

EIstf

within a component

between components

Figure 3.2: Illustration of correlation within a component and correlation between com-ponents in a structure.

Incremental dynamic analyses (Vamvatsikos & Cornell, 2002) involves performing

nonlinear dynamic analysis using multiple ground motions scaled to particular ground

motion intensity measure (IM) levels. Seismic capacity is inferred as the IM value

causing dynamic instability in the structure. The distribution of seismic capacities

obtained under different ground motions can be summarized by a collapse fragility

function: a cumulative distribution function defining the probability of collapse (C) at

a given IM level (P (C|IM)). The mean annual frequency of collapse (λc) is obtained

by integrating a collapse fragility curve with a site specific seismic hazard curve (Ibarra

& Krawinkler, 2005), as given in Equation 3.1.

λc =

∫ ∞0

P (C|IM = im)


d(im)

∣∣∣∣ d(im) (3.1)

where λIM(im) is the mean annual rate of exceeding the ground motion im and

dλIM (im)d(im)

defines the slope of the hazard curve at im.

Fragility functions corresponding to alternative limit states, such as exceeding

a particular value of story drift ratio, sdr, can be also obtained from incremental

dynamic analysis results. At distinct IM levels, the number of cases in which drift

response is greater than a particular value of sdr are recorded. A lognormal fragility

function is fitted to this count data using maximum likelihood estimation, where

likelihood is defined using a binomial distribution. The resulting fragility function

characterizes the probabilities of exceeding sdr limit state. These functions can simi-

larly be integrated with the seismic hazard curve in a similar fashion to Equation 3.1

to estimate the mean annual frequency (λSDR≥sdr) of exceeding a given limit state.

3.4 Assessment of Correlations of Model Parame-

ters

3.4.1 Random Variables

The concentrated hinge model of each beam-column element is defined by the six

parameters, described previously, that are considered as random variables: θcap,pl,

EIstf/EIg, My, Mc/My, θpc and γ. Haselton et al. (2008) and Panagiotakos &

Fardis (2001) provide empirical predictive equations for the model parameters defin-

ing concrete beam-columns. These equations relate column design details to the

model parameters using equations that are based on patterns in the observed data,

and judgment on expected behavior. Haselton et al. (2008) provide a full and a sim-

plified equation for some of the model parameters. We use full equations for model

parameters of interest wherever applicable in this study. Note that evaluation or ad-

vancement of the predictive equations is beyond the scope of this paper and interested

reader is referred to Haselton et al. (2008) for further information on the predictive

models.

58


3.4.2 Data Sources

Haselton et al. (2008) previously calibrated the modeling parameters to experimental

data for cyclic and lateral-load tests of reinforced concrete columns from the Pacific

Earthquake Engineering Research Center Structural Performance Database (Berry

et al. , 2004). This database provides experimental test results of reinforced concrete

columns, in terms of force-displacement histories along with information related to

reinforcement, column geometry, test configuration, axial load, and failure type.

Haselton et al. (2008) compiled a component calibration data set, consisting of 255

rectangular column tests whose failure modes are either flexure or combined flexure

and shear. The model parameters are calibrated to a cantilever column configuration,

which is modeled by an elastic element and concentrated plastic hinge at the base of

the column. The model is calibrated to match the corresponding experimental force-

displacement data. For detailed information related to the calibration procedure,

readers are referred to Haselton et al. (2008). The data set is pre-processed to

remove outliers. Furthermore, data for some parameters are missing, especially for

the parameters characterizing post-peak cyclic deterioration response. Due to the

pre-processing and these limitations, the total number of data points are 232, 197,

255, 233, 65 and 223 for θcap,pl, EIstf/EIg, My, Mc/My, θpc and γ, respectively.

The 255 column tests used for the calibration were conducted by 42 different

research laboratories, referred to as “test groups”. In Appendix 3.A, the test groups

are listed along with information related to the number of test conducted per test

group and similarities and differences between tests within a test group for column

dimensions, concrete strength (f ′c), longitudinal yield strength (fy), axial load ratio,

area of longitudinal and transverse reinforcement.

3.4.3 Correlation Assessment Procedure

We estimate correlations using residuals of the observed and predicted model param-

eter values, computed by the following equation.

ln(ykij)

= ln(ykij)

+ εkij (3.2)

In Equation 3.2, subscripts i and j represent the test group and test number,

respectively, and the superscript k indicates the random variable of interest. The

observed and predicted values of a random variable for a given test are represented

by ykij and ykij, respectively. Predicted values, ykij, are obtained in this study from the

empirical equations of Haselton et al. (2008) and Panagiotakos & Fardis (2001). εkij

is the residual obtained for random variable k from the test specified by i and j.

For each model parameter, the residual of one test is plotted with respect to the

residuals of other tests in the same test group. A subset of the data is shown for

illustrative purposes in Figure 3.3. Each test group is denoted by a specific symbol

and color combination. From Figure 3.3 the presence of correlations of each test with

respect to the group conducting the test can be inferred. The correlation pattern

is most evident for My in Figure 3.3.b. For instance, it is observed that tests from

group 1 (TG1), represented with black dots, have negative residuals; implying that the

My values of the tests conducted in that test group are consistently overestimated.

Conversely, tests from group 3 (TG3), represented with cyan dots, have positive

residuals. This pattern suggests that model parameters from a group of tests tend to

have similar residuals relative to model predictions, such that the model parameters

within a test group tend to be systemically larger or smaller than the average. This is

not surprising, considering the following observations from Table 3.6: 1) The majority

of the column tests within each group have similar geometries and dimensions. 2)

Yield strength and area ratio of longitudinal reinforcement is constant in about three-

quarters of the test groups. 3) The major difference in the tests within each group is

the level of axial load and transverse reinforcement. These features within each test

group are considered to represent the similarities between columns located in different

story levels in a structure. In addition, these similarities could further be considered

to represent various beam elements within a structure. While the validity of these

assumptions need further testing, they represent reasonable assumptions to measure

correlations among different elements within a structure.

60


−1 0 1

−1.5

−1

−0.5

0

0.5

1

1.5

εk

εk

θcap,pl

(a)

−0.5 0 0.5−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

εk

εk

EIstf

/EIg

(b)

−1 0 1

−1

−0.5

0

0.5

1

εk

εk

My

(c)

−0.4 −0.2 0 0.2−0.4

−0.2

0

0.2

0.4

εk

εk

Mc/M

y

(d)

−2 0 2

−2

−1

0

1

2

εk

εk

θpc

(e)

−1 0 1

−1.5

−1

−0.5

0

0.5

1

1.5

εk

εk

γ

(f)

Figure 3.3: Residuals of beam-column model parameters (εk) are plotted against theresiduals of other tests within the test group to which they belong for a) θcap,pl, b) EIstf/EIg,c) My, d) Mc/My, e) θpc and f) γ. A subset of data is shown for illustrative purposes.

The clustering of residuals within a test group motivated the use of random ef-

fects regression to study the correlations among model parameters. Random effects

regression, combined with the aforementioned assumptions, enables the treatment of

(1) correlations within a component, which refers to interaction of model parameters

for a given component, and (2) correlations between components, which refers to the

interaction of model parameters at different components in a structure. As mentioned

previously, Figure 3.2 illustrates the two types of correlations.

Random Effects Regression

Regression models provide estimated functional relationships between response vari-

ables and predictor variables. Random effects regression models are used when at

least one of the response variables is a categorical variable. The discrete levels for the

categorical variable are termed the “effects” in the model, and the qualifier random

implies (1) that the observed levels of a categorical predictor variable represent a

random sample from a population, and (2) that the observed categories of the predic-

tor variable do not contain all possible levels (Searle et al. , 2009; Pinheiro & Bates,

2010).

In this study, a one-way random effects model is applied to residuals from Equation

3.2 to assess the correlation structure of the model parameters. For statistical analysis,

the R software package is used (Team, 2014). The test groups are treated as a random

effect, leading to the following equation:

ln(ykij)− ln

(ykij)

= εkij

= µk + αki + εkij(3.3)

where the logarithmic residuals of each random variable, εkij, are used without any

transformation, and are fitted using the random effects model, where µk is an intercept

term corresponding to mean of the data, and αk and εk represent between- and within-

test-group variability, respectively. The αk and εk terms are independent random

variables that have zero mean and variances σ2k and τ 2k , respectively. These variances

are obtained from the regression results. From Equation 3.3 and the above definitions,

it follows that the variance of a model parameter, k, is σ2k + τ 2k .

From Equation 3.3, the covariance of the logarithms of the model parameters k

and k′ within a component j is given by:

cov(ln(ykij), ln(yk′

ij

)) = cov(εkij, ε

k′

ij )

= cov(µk + αki + εkij, µk′ + αk

′

i + εk′

ij )

= corr(αki , αk′

i )σkσk′ + corr(εkij, εk′

ij )τkτk′

(3.4)

62


where cov(·, ·) and corr(·, ·) refer to covariance and correlation of two variables. It

is implied in Equation 3.4 that cov(αki , αk′j ) = corr(αki , α

k′i )σkσk′ and cov(εki , ε

k′j ) =

corr(εki , εk′i )τkτk′ by definition. Correlation of model parameters within a component

is then:

corr(ln(ykij), ln(yk′

ij

)) =

cov(ln(ykij), ln(yk′ij

))

cov(εkij)cov(εk′ij )


ij

)) =

corr(αki , α

k′i

)σkσk′ + corr

(εkij, ε

k′ij

)τkτk′√

σ2k + τ 2k

√σ2k′ + τ 2k′

(3.5)

The covariance of the logarithms of the model parameters k and k′ between com-

ponents j and j′ is given by:

cov(ln(ykij), ln(yk′

ij′

)) = cov(µk +αki +εkij, µ

k′+αk′

i +εk′

ij′) = corr(αki , αk′

i )σkσk′ (3.6)

where cov(εkij, εk′

ij′) = 0 since varepsilonkij, εk′

ij′ are independent.

Using Equation 3.6, correlation of model parameters between components can be

shown to equal


ij′

)) =

corr(αki , α

k′i

)σkσk′√

σ2k + τ 2k

√σ2k′ + τ 2k′

(3.7)

The above equation simplifies to Equation 3.8 for assessing the correlation of the

same model parameter between components, since corr(αki , αki ) = 1 in that case.

corr(ln(ykij), ln(ykij′)) =

σ2k

σ2k + τ 2k

(3.8)

Correlation Results

Standard deviations of between-component (σk) and within-component (τk) terms

are given in Table 3.1 for the concrete column hinges. Those are obtained using

random effects regression described above. Table 3.2 shows the correlation coefficients

obtained using Equations 3.4 to 3.8 and with the values given in Table 3.1.

Table 3.1: Between and within test group standard deviations obtained from randomeffects regression.

σ τ√σ2k + τ 2k

θcap,pl 0.41 0.44 0.59EIstfEIg

0.20 0.20 0.28

My 0.26 0.10 0.30Mc

My0.07 0.08 0.10

θpc 0.24 0.69 0.73

γ 0.20 0.46 0.51

64


Table

3.2:

Init

ial

corr

elat

ion

coeffi

cien

tsob

tain

edfr

omra

nd

omeff

ects

regr

essi

on.

Com

pon

ent

iC

om

pon

ent

j

θ cap,pl i

( EI stf

EIg

) iM

yi

( Mc

My

) iθ p

ci

γi

θ cap,pl j

( EI stf

EIg

) jM

yj

( Mc

My

) jθ p

cj

γj

Componenti

θ cap,pl i

1.00

00-0

.018

30.

0578

0.2

538

0.2

083

-0.0

260

0.6

839

0.0

106

0.0

277

0.0

975

0.0

533

-0.0

202

( EI stf

EIg

) i1.

0000

0.13

54

-0.1

018

0.0

375

0.0

799

0.6

853

0.0

612

-0.0

950

-0.0

305

0.0

379

Myi

1.00

00

0.2

838

0.1

067

0.0

722

0.9

263

0.2

482

0.0

951

0.0

549

( Mc

My

) j1.0

000

0.0

077

0.1

681

0.6

728

0.0

415

0.0

192

θ pci

(sym

.)1.0

000

0.2

195

(sym

.)0.3

466

0.0

357

γi

1.0

00

0.4

102

The correlation coefficients given in Table 3.2 reflect dependence of model pa-

rameters within a component i and across different components, i and j. Table 3.3

shows the same correlation coefficients obtained after rounding the coefficients to one

significant figure. This rounding to one significant digit reflects the many simpli-

fying assumptions made in the assessment procedure of the limited data set. The

correlation structure shown in Table 3.3 is used in the rest of the paper.

We note that rounding of the correlation coefficients, and estimation of correlations

from data with missing values (for the data set used in this study, the parameters re-

lated to descending branch of the backbone curve are not recovered from many tests),

can result in a correlation matrix without the required positive semi-definiteness prop-

erty. Although our final models do not exhibit this issue, in our initial attempts on

developing the regression models, violation of positive semi-definiteness was observed.

In such cases, minor changes are needed in order to transform the resulting correlation

matrix into a positive definite one andJackel (2001) provides methods to transform

an invalid correlation matrix to a positive semi-definite one.

66


Table

3.3:

Fin

alco

rrel

atio

nco

effici

ents

are

obta

ined

afte

rro

un

din

gto

one

sign

ifica

nt

figu

re.

Com

pon

ent

iC

omp

onen

tj

θ cap,pl i

( EI stf

EI g

) iM

yi

( M c My

) iθ pc i

γi

θ cap,pl j

( EI stf

EI g

) jM

yj

( M c My

) jθ pc j

γj

Componenti

θ cap,pl i

1.0

0.0

0.1

0.3

0.2

0.0

0.7

0.0

0.0

0.1

0.1

0.0

( EI stf

EI g

) i1.

00.

1-0

.10.

00.

10.

70.

1-0

.10.

00.

0

Myi

1.0

0.3

0.1

0.1

0.9

0.2

0.1

0.1

( M c My

) j1.

00.

00.

20.

70.

00.

0

θ pc i

(sym

.)1.

00.

2(s

ym

.)0.

30.

0

γi

1.0

0.4

From Table 3.3, we observe that within a component (i.e. the left half of the

table), correlations of model parameters are rather small; the largest coefficient being

0.3, which corresponds to the correlations between Mc/My and My, and Mc/My and

θcap,pl. These values suggest moderate interactions between strength parameters and

hardening behavior. Small interactions are observed between the parameters defining

post-capping cyclic behavior within a component. For example, γ has a 0.2 correlation

with Mc/My and θpc.

Between components, like parameters have large correlation coefficients — in the

range of 0.3 to 0.9 (See the diagonal terms in the right half of the table). We see

that My followed by θcap,pl, EIstf/EIg and Mc/My have strong correlations between

components. Note that Haselton & Deierlein (2007) and Liel et al. (2009) assumed

a perfect correlation to define the interactions between like-parameters across com-

ponents. The large observed correlation coefficients for like parameters across com-

ponents is expected due to the following reason. For a parameter of interest, if the

predicted value of a test in a test group is larger than average, due to the grouping

effect present in the data set, one would expect it will be larger for the other tests

within that test group. In the setting of a structure, this translates into following:

if one finds that the predicted value of a parameter in one component is large, that

parameter is likely to be large for other components in the structure. We also observe

that correlations of different model parameters between components are small (See

the off-diagonal terms in the right half of the table); the largest being 0.2, which

corresponds to the correlation between Mc/My and My.

It is further observed that there is a small negative correlation between EIstf/EIg

and Mc/My. This is likely to be a numerical artifact, as there is no clear physical

reason why such a correlation would be negative. This correlation is small enough to

be neglected. In addition, the apparent estimation error at this level further justifies

the decision to keep only one significant figure in the correlations.

68


3.5 Impacts of Correlations on Dynamic Structural

Response

To demonstrate the impact of the correlations estimated above, and to evaluate po-

tential model simplifications for structures with many uncertain parameters, this sec-

tion presents a study of a structure’s performance while considering correlated model

parameters.

3.5.1 Case Study Structure

A reinforced concrete special moment frame structure is considered here. The building

was designed by Haselton (2006) for a high seismicity site in California in accordance

with 2003 IBC and ASCE 7-02 provisions (IBC, 2003; American Society of Civil

Engineers, 2002). The building has three bays and four stories, with a total of 12 beam

and 16 column elements. The structural system is modeled using the concentrated

plasticity approach described above, in which elements of the frame are modeled

using elastic elements with rotational springs at the ends. A Rayleigh damping of

3% is defined at the first and third mode periods of the structure, and P-∆ effects

are modeled using a leaning column. The fundamental period of the structure is

T1 = 0.94 s. The Open System for Earthquake Engineering Simulation platform is

used to analyze the structure (OpenSEES, 2015). The FEMA-P695 far-field ground

motion set of 44 ground motion components is used for structural response simulations

(FEMA, 2009b).

Structural response up to the collapse state is assessed. Monte Carlo simulation

(Kalos & Whitlock, 2009) is used for propagating uncertainties related to modeling

and ground motion variability. As mentioned previously, six parameters are identified

as random variables defining a plastic hinge. The variability in the modeling param-

eters is represented using logarithmic standard deviations (σln) for θpc, γ, θcap,pl, My,

EIstf/EIg and Mc/My of 0.73, 0.51, 0.59, 0.3, 0.28 and 0.1, respectively (Haselton

et al. , 2008). Same normalized energy-dissipation capacity (γ) is used for differ-

ent modes of cyclic energy dissipation. Furthermore, parameters defining equivalent

viscous damping and column footing rotational stiffness are assumed to be random,

with σln values of 0.6 and 0.3, respectively (Haselton, 2006; Hart & Vasudevan, 1975;

Porter et al. , 2002a). These two parameters are assumed to be independent of the

rest of the parameters. A joint distribution of model parameters is used to obtain

realizations of model parameters. A multivariate normal distribution is assumed for

the logarithms of all parameters except Mc/My. Since, by definition Mc/My is always

greater than 1, we use a one-sided truncated normal distribution for this parameter.

We simulate 4400 realizations of model parameters from this joint distribution, each

of which are randomly matched with one ground motion in the FEMA-P695 far-field

ground motion set. Analysis is then conducted by linearly scaling ground motions

up to higher intensities until structural collapse is observed. Maximum story drift

ratio (SDR) of 10% (or greater) is assumed to be an indicator of structural collapse.

Ground motion IM values are defined as 5%-damped first-mode spectral acceleration,

Sa(T1 = 0.94s, 5%).

Table 3.4 lists the correlation models used with Monte Carlo simulations. As the

name implies, the “No Correlation” model has no correlation among any component

model parameters. With six parameters identified as random variables defining the

plastic hinges in 28 elements, the No Correlation model results in 170 random vari-

ables (168 component parameters plus equivalent viscous damping and column footing

rotational stiffness). Partial Correlation A has partial correlation among component

model parameters. The correlation coefficients from Table 3.3 are used both within

and between components, for both column or a beam elements. Partial Correlation

A model is also characterized by 170 random variables. In Partial Correlation B,

correlation coefficients given in Table 3.3 define correlations within a component and

correlations of beam-to-column components. Column-to-column and beam-to-beam

parameter uncertainties are assumed to be fully correlated. Although the total num-

ber of model parameters for Partial Correlation B is also 170, the assumption of full

correlation between all beams and between all columns reduces the effective number

of random variables to 14 (six beam parameters, six column parameters, damping and

foundation stiffness). A perturbation of a random variable within a beam element

is applied identically to all beam elements. In the Full Correlation model all of the

70


elements are assumed to have perfect correlation, such that there are effectively three

random variables (one component parameter, damping and foundation stiffness).

The case study structure is a space frame and the analysis model is created in

two-dimensions for a typical three-bay frame. By using only one frame for analysis,

it is implicitly assumed that the frames in a given direction are fully correlated.

Table 3.4: Correlation models used with Monte Carlo simulations. “0”, “P” and “1” refersto the cases of No Correlation, Partial Correlation and Full Correlation, respectively.

Correlation Correlation between components Effective

Model Name within a

compo-

nent

Column-

to-

Column

Beam-to-

Beam

Beam-to-

Column

# of

R.V.s

No Correlation 0 0 0 0 170

Partial Correlation A P P P P 170

Partial Correlation B P 1 1 P 16

Full Correlation 1 1 1 1 3

Figures 3.4.a and 3.4.b show the median IDA curves among 4400 simulations using

the four correlation models. IDA yields discrete points relating ground motion inten-

sity and peak story drift ratios. The continuous IDA curves are obtained using cubic

spline interpolation as outlined in ??. Figures 3.4.c and 3.4.d show the dispersions in

the IDA curves. Dispersion in IDA curves is computed as the logarithmic standard

deviation of IM values of IDA curves corresponding to discrete SDR values, including

both collapse and no-collapse response. These IM values are obtained using the cubic

splines. The IDA response of the model with median model parameters is also given

in Figure 3.4.a and 3.4.c.

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

SDR

Me

dia

n S

a (

T1=

0.9

4 s

,5%

) (g

)

(a)(a)(a)(a)

No Correlation

Partial Correlation A

Full Correlation

Median Model

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

SDR

Me

dia

n S

a (

T1=

0.9

4 s

,5%

) (g

)

(b)(b)(b)


Partial Correlation B

0 0.02 0.04 0.06 0.08 0.10

0.1

0.2

0.3

0.4

0.5

0.6

SDR

Dis

pe

rsio

n

(c)(c)

No Correlation


Full Correlation

Median Model

0 0.02 0.04 0.06 0.08 0.10

0.1

0.2

0.3

0.4

0.5

0.6

SDR

Dis

pe

rsio

n

(d)



Figure 3.4: IDA results obtained using different correlation models a) median IDA re-sponse for varying correlation levels, b) median IDA response for Partial Correlation mod-els, c) dispersion in IDA curves for varying correlation levels, and d) dispersion in IDAcurves for Partial Correlation models.

As observed in Figure 3.4.a and 3.4.b, there is remarkably small difference between

the calculated median IDA curves for different correlation models. A small difference

between the No Correlation and Full Correlation cases begins to be observed at drift

ratios of 0.03, growing to a difference in Sa(T1, 5%) of 7% at a drift ratio of 0.1.

Median IDA response is similar for Partial Correlation A and B models.

Referring to Figures 3.4.c and 3.4.d, the dispersion in IDA curves is significantly

72


different for the assumed correlations. The differences tend to increase for SDR’s

greater than 0.03. Partial Correlation A and No Correlation models yield similar vari-

ability until SDR=0.03, and at SDR=0.1, the difference in dispersion values for these

two cases is 12%. At SDR=0.1, the difference in dispersion between No Correlation

and Full Correlation models is 47%. The Median model consistently underestimates

dispersion, where for SDR=0.1, a difference of 17% is observed between dispersion

values of Median and Partial Correlation models. Partial Correlation A and B mod-

els give similar response both in terms median IDA response and dispersion in IDA

curves.

The fragility functions and corresponding and λSDR≥sdr values are obtained for

alternative values of sdr. Figure 3.5 shows λSDR≥sdr with respect to sdr using the

assumed correlation models, where the λSDR≥sdr begin to differ for SDR values greater

than approximately 0.03. The plots show, for example, that the previously observed

differences in IDA response have an insignificant effect on λSDR≥sdr. On the other

hand, the λSDR≥sdr for the Full Correlation case is 30% to 110% less than the No

Correlation model for drift values of 0.05 and 0.1, respectively.

0 0.02 0.04 0.06 0.08 0.110

−4

10−3

10−2

sdr

λS

DR

≥ s

dr

(a)(a) No Correlation


Full Correlation

Median Model

0 0.02 0.04 0.06 0.08 0.110

−4

10−3

10−2

sdr

λS

DR

≥ s

dr

(b) Partial Correlation A


Figure 3.5: Mean annual frequency of exceedance of maximum story drift ratio usingdifferent correlation models

Figure 3.6 shows empirical collapse cumulative distribution functions for the struc-

ture obtained using the correlation models considered in this study. At smaller IM

(Sa(T1, 5%)) levels, as the correlations among parameters increase, the structure has

a higher probability of collapse. As expected, the median model provides smaller

probabilities of collapse, especially for smaller ground motion intensities. This leads

to unconservative estimates of collapse risk as explained in the following discussions.

Note that since median collapse capacity is higher for median model, it would result

in slightly larger collapse margin ratio (i.e. ratio of the median collapse intensity to

Maximum Considered Event (MCE) intensity) (FEMA, 2009b), which can be mis-

leading from a collapse safety point of view. Uncorrelated and Partially Correlated

models have similar lower tail behavior and only differ at higher IM levels.

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

Sa (T1=0.94 s,5%) (g)

Pco

llap

se

(a)(a)

No Correlation

Partial Correlation Case A

Full Correlation

Median Model

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

Sa (T1=0.94 s,5%) (g)

Pco

llap

se

(b)



Figure 3.6: Empirical cumulative distribution functions obtained using different correla-tion models

Table 3.5 provides summary statistics of the collapse fragility curves, through

counted median and logarithmic standard deviation (σln) of collapse capacities ob-

tained using alternative models. The mean annual frequency of collapse, λc, is ob-

tained by integrating the empirical collapse fragility curves with the seismic hazard

curve of the site at LA using equation 3.1.

The differences observed at the lower tail of the fragility curve due to increasing

74


Table 3.5: Collapse risk metrics in terms of counted median and logarithmic standarddeviation (σln) of collapse capacities obtained using alternative models. Also shown aremean annual frequency of collapse (λc) obtained by integrating empirical collapse fragilityfunctions with seismic hazard curve.

Model Name Median σln λc(10−4)

Median Model (i.e. no modeling uncertainty) 1.69 0.39 1.81

No Correlation 1.57 0.42 2.65

Partial Correlation A 1.67 0.48 2.75

Partial Correlation B 1.64 0.49 3.06

Full Correlation 1.69 0.67 6.39

levels of correlation translate into pronounced differences in λc estimates. For exam-

ple, there is a factor of 2.4 difference between λc estimates obtained using uncorrelated

and perfectly correlated models. The collapse fragilities for uncorrelated and partially

correlated models are similar at the lower tail and they are observed to differ at the

upper tails. Since lower tail of the collapse fragility functions contribute the most to

λc as shown in Eads et al. (2013), the similarity in lower tail results in similar λc

estimates using these models. By assuming perfect correlation for beam-to-beam and

column-to-column interactions (Partial Correlation B), a collapse response similar to

that observed with all members are partially correlated (Partial Correlation A) are

obtained. This assumption results in significant reduction in the number of random

variables needed for modeling uncertainty propagation studies.

The effects of correlations are investigated for a 4-story moment frame structure,

which was designed to have a regular strength/stiffness distribution over the height,

and similar collapse response is obtained using No Correlation and Partial Correlation

models. For the case study structure, partial correlations were not pronounced for

trigger different modes of failure compared to uncorrelated case. Although the extent

which ductility and strength irregularities influence the significance of correlations

has not been investigated, different results are likely to be obtained for buildings with

strength irregularities, since presence of correlations, even partially, might trigger the

weaker modes of failure.

Both no and perfect correlations are rare in nature and partial correlations pro-

vide a more realistic representation. However, perfect correlation has the advantage

of reducing the effective number of random variables. In essence, perfect correla-

tions can be used if conservative results are preferred. Uncorrelated models are also

hard to justify, and if the assumption is far from reality, unconservative estimates are

obtained. Although previous studies were conducted with correlations among param-

eters closer to a perfect correlation case, for the case study structure, No Correlation

model yielded comparable results to the more realistic case of Partial Correlation.

We note that for other types of components or structures, this might not hold and it

is hard to understand the impacts of correlation without any formal analysis. This

paper outlines a correlation assessment framework and provides an evaluation of the

respective impacts of correlations.

3.6 Conclusions

In this study, we consider model parameter uncertainty in seismic performance as-

sessment of structures and characterize the dependence of modeling parameters that

define cyclic inelastic response at a component level and the interactions of multiple

components associated with a system’s response. To characterize the dependence of

modeling parameters, we present a framework that enables quantification of model

parameter correlation from component tests. The framework uses random effects

regression to capture the similarities and differences among properties of structural

components tested by individual research groups. The relationships of uncertain-

ties within and between research groups are incorporated as random effects into the

regression model and statistical dependency between the tests conducted by a test

group are assessed.

Uncertainties and dependence in structural component model parameters are eval-

uated by applying this framework to a database of reinforced concrete beam-column

tests. Correlation coefficients from these regression models, reflecting statistical de-

pendency among properties of components tested by individual research groups, are

assumed to reflect correlations among multiple components within a given structure.

76


The random treatment of research groups, combined with the aforementioned obser-

vations in the data set (i.e. similarity of column dimensions, and differences in axial

load and transverse reinforcement in the tests), provided an effective justification of

this assumption. We find that correlations between differing parameters (both within

and between components) have low to moderate correlation, with correlation coeffi-

cients ranging from -0.1 to 0.3. Like parameters across components have moderate

to high correlations, ranging from 0.3 to 0.9.

Although the estimated correlation structure is developed for reinforced concrete

columns, the presented framework can be utilized in other contexts as well. This

framework can be applied to component test databases that are composed of multiple

tests conducted by different research groups. Tests that are conducted in similar

conditions and are investigating the impacts of particular properties of components

that can effectively represent different locations in a structure are suitable for this

framework.

The impacts of parameter correlations on dynamic response of a four story rein-

forced concrete frame structure are assessed. The different correlation assumptions

do not seem to bias median IDA response, even for large drifts. Variability in IDA re-

sponse, however, is observed to be significantly different for various correlation levels.

Increased levels of correlation results in increased dispersion in the quantified dy-

namic response and thus higher values of probabilities of collapse. In the case study

structure, similar lower tail response is observed in terms of collapse fragilities for

uncorrelated and partially correlated models, whereas the collapse fragilities differ at

the upper tails. Similar lower tail behavior results in similar mean annual frequency

of collapse estimations. Assumptions regarding correlation structure of the model

parameters are observed to change mean annual frequency of collapse estimates up

to a factor of 2.4 (using uncorrelated and perfectly correlated models).

No significant difference in collapse response is observed between a model having

all partially correlated parameters versus a model having full correlation among beam-

to-beam and column-to-column parameters (and partially correlated beam-to-column

parameters). The latter assumption results in significant reduction in the number of

random variables, and so is a promising approach for considering modeling uncertainty

while also managing computational expense.

3.7 Data and Resources

The data for the reinforced concrete column tests are obtained from PEER Structural

Performance Database (http://nisee.berkeley.edu/spd/) and Curt Haselton’s

website (http://www.csuchico.edu/structural/researchdatabases/reinforced_

concrete_element_calibration_database.shtml).

78

http://nisee.berkeley.edu/spd/

http://www.csuchico.edu/structural/researchdatabases/reinforced_concrete_element_calibration_database.shtml

http://www.csuchico.edu/structural/researchdatabases/reinforced_concrete_element_calibration_database.shtml


3.A Information on Test Groups Conducting Re-

inforced Concrete Column Tests

Table 3.6: Test groups conducting reinforced concrete column tests. For column dimen-sions, f ′c, fy, axial load ratio, area ratio of longitudinal and transverse reinforcement, “Y”and “-” are used when the tests within the test group have similar and different properties,respectively. “N/A” is used when the number of tests per test group is 1.

Are the properties similar among tests?

Test

Group

Reference No.

of

Tests

Column

Dimen-

sions

f ′c fy Axial

Load

Ratio

Area

Ratio of

Longi-

tudinal

Reinf.

Area

Ratio of

Trans-

verse

Reinf.

1 Galeota et al. (1996) 24 Y Y Y - - -

2 Bayrak & Sheikh (2001) 16 - - - - - -

3 Pujol (2002) 14 Y - Y - Y -

4 Wight & Sozen (1973) 13 Y - Y - Y -

5 Matamoros (1999) 12 Y - - - - -

6 Thomson & Wallace (1994) 11 Y - - - Y -

7 Atalay & Penzien (1975) 10 Y - - - Y -

8 Saatcioglu & Grira (1999) 10 Y Y - - - -

9 Mo & Wang (2000) 9 Y - Y - Y -

10 Bayrak & Sheikh (1996) 8 Y - Y - Y -

11 Muguruma et al. (1989) 8 Y - Y - Y Y

12 Tanaka (1990) 8 - - - - - -

13 Sakai (1990) 7 Y Y - Y - -

14 Kanda et al. (1988) 6 Y - Y - Y Y

15 Legeron & Paultre (2000) 6 Y - - - Y -

16 Paultre et al. (2001) 6 Y - Y - Y -

17 Saatcioglu & Ozcebe (1989) 6 Y - - - Y -

18 Takemura & Kawashima

(1997)

6 Y - Y Y Y Y

19 Xiao & Yun (2002) 6 Y - Y - Y -

20 Xiao & Martirossyan (1998) 6 Y - Y - - -

21 Zhou et al. (1987) 6 Y - Y - Y -

22 Bechtoula (1985) 5 - - - - - -

23 Sugano (1996) 5 Y Y Y - Y -

24 Watson (1989) 5 Y - Y - Y -

25 Esaki (1996) 4 Y - Y - Y -

26 Gill (1979) 4 Y - Y - Y -

27 Soesianawati (1986) 4 Y - Y - Y -

28 Wehbe et al. (1999) 4 Y - Y - Y -

29 Ohno & Nishioka (1984) 3 Y Y Y Y Y Y

30 Sezen & Moehle (2002) 3 Y - Y - Y Y

Table 3.6: Test groups conducting reinforced concrete column tests. For column dimen-sions, f ′c, fy, axial load ratio, area ratio of longitudinal and transverse reinforcement, “Y”and “-” are used when the tests within the test group have similar and different properties,respectively. “N/A” is used when the number of tests per test group is 1.

Are the properties similar among tests?

Test

Group

Reference No.

of

Tests

Column

Dimen-

sions

f ′c fy Axial

Load

Ratio

Area

Ratio of

Longi-

tudinal

Reinf.

Area

Ratio of

Trans-

verse

Reinf.

31 Ang (1981) 2 Y - Y - Y -

32 Azizinamini et al. (1988) 2 Y - Y - Y -

33 Lynn et al. (1996) 2 Y - Y - - -

34 Lynn (2001) 2 Y - Y - Y Y

35 Ohue et al. (1985) 2 Y - - - - Y

36 Ono et al. (1989) 2 Y Y Y - Y Y

37 Zahn (1985) 2 Y - Y - Y -

38 Zhou et al. (1985) 2 Y - Y - Y Y

39 Amitsu et al. (1991) 1 N/A N/A N/A N/A N/A N/A

40 Arakawa et al. (1982) 1 N/A N/A N/A N/A N/A N/A

41 Nagasaka (1982) 1 N/A N/A N/A N/A N/A N/A

42 Park & Paulay (1990) 1 N/A N/A N/A N/A N/A N/A

80

Chapter 4

Modeling Uncertainty Propagation

Methods for Seismic Collapse

Assessment of Structures

4.1 Introduction

Robust characterization of seismic performance requires the quantification and prop-

agation of various sources of uncertainties throughout the analysis and assessment.

Previous research highlighted the importance of modeling uncertainties in seismic per-

formance predictions (i.e. Haselton & Deierlein, 2007; Liel et al. , 2009; Dolsek, 2009).

Researchers utilized different methods to characterize and propagate uncertainty with

the aim of balancing computational efficiency and robust uncertainty characterization.

Because of the computational demand involved in uncertainty propagation methods

and difficulties in modeling, judgment based factors to adjust the uncertainty in col-

lapse fragility functions are proposed in FEMA-P58 (FEMA, 2012) and FEMA-P695

(FEMA, 2009b).

This chapter provides an assessment of uncertainty propagation methods for in-

corporating modeling uncertainty in collapse response prediction. The organization

of the chapter is as follows: Structural reliability theory is presented, followed by a

discussion on structural reliability methods for seismic collapse response prediction.

81

Then, sensitivity analysis for seismic collapse assessment is discussed, and reliabil-

ity analyses for seismic collapse response prediction are illustrated using a 4-story

reinforced concrete moment frame building.

The reliability methods that are presented in this chapter are grouped as simulation-

based, moment-based and surrogate methods. We discuss relative merits of each

reliability method in assessing collapse response of structures. These include compu-

tational efficiency (which is measured in terms of number of structural analysis re-

quired), accuracy in predicting collapse risk metrics, characterization of uncertainty,

and behavior under high dimensional problems. The challenges in balancing com-

putational efficiency and robust uncertainty characterization for collapse response

predictions are discussed.

4.2 Structural Reliability Theory

Challenges in representing the complex mechanics of the system and its computations

combined with the impacts of the uncertainty in modeling and analysis makes assess-

ment of system level response involved. Robust estimation of system response requires

not only accurate idealization of the system but also propagation of the uncertainty

in modeling and analysis.

Y = g(x) (4.1)

In Equation 4.1, x = [x1, x2, ...xnx ] represents random analysis input parameters,

where nx is the total number of random variables. Y is the response of the system

and g(·) maps the analysis input to the system response. Depending on the context,

g(·) can be a computer model or an explicit function idealizing system behavior.

Expected value (E[Y ]) and variance of system response, Y , (V ar[Y ]) can then be

computed as:

E[Y ] =

∫g(x)fX(x)dx (4.2)

82

CHAPTER 4. MODELING UNCERTAINTY PROPAGATION METHODS 83

V ar[Y ] =

∫(E[Y ]− g(x))2fX(x)dx (4.3)

where fX(x) represents the probability density function of the random variables.

In a sensitivity analysis, one is interested on the relative effects of random vari-

ables on the response. Using an analysis of variance perspective, this is similar to

determining the contribution of each random variable (as well as the interactions

between random variables) to V ar[Y ].

In the context of structural reliability, one is generally interested in the failure

probability of a system. The failure probability of a system, pf , is computed using

a limit state function of random variables x. Assume g(x) represents a limit state

function where g(x) ≤ 0 defines the failure state of the system (Melchers, 1999; Pinto

et al. , 2004). pf is then computed as:

pf =

∫g(x)≤0

fX(x)dx (4.4)

where fX(x) is the joint probability density function of the random variables and the

integration is performed for all values of X leading to g(x) ≤ 0.

If fX(x) has a normal distribution and the limit state function g(x) is linear, then

pf can be computed as:

pf = Φ(g(x) ≤ 0)) = Φ

(− E[Y ]√

V ar[Y ])

)= Φ(−β) (4.5)

where Φ(·) represents the standard normal distribution function. β is called the

reliability index and is equal to E[Y ]√V ar[Y ]

(Cornell, 1969).

The points that should be considered in selecting a reliability method to evaluate

pf are listed as follows:

• The choice of random variables, x. Uncertainty propagation should include

the variability in the parameters that are important in predicting the response

variable. Also, some reliability methods can get computationally expensive

with the inclusion of many random variables and simplifications can be made

considering the relative importance of the random variables in predicting failure

response.

• The statistical distribution of the random variables. Uncertainty propagation

methods require information related to statistical distribution of variables. This

includes the level of uncertainty and the correlations among the random vari-

ables.

• Probabilities of interest, pf . Depending on pf , number of analyses required can

be estimated.

• The level of nonlinearity of the limit state function. Linear approximations

are commonly used in structural reliability; however, the applicability of these

methods depends on the problem of interest.

• Computational cost. Reliability methods have different computational demands

and depending on the computational power, appropriate methods should be

selected.

• Ease of implementation

4.3 Structural Reliability Methods

Probabilistic collapse response assessment does not have an analytical solution for de-

termining the probability of failure. Structural reliability methods, ranging from ap-

proximate methods such as First-Order Second-Moment (FOSM) to Monte Carlo sim-

ulation methods, enable uncertainty propagation for collapse response assessments.

4.3.1 Monte Carlo Simulation-Based Methods

Simulation-based methods are simple, intuitive and easy to use. They do not make

any assumptions related to limit state functions. The number of simulations are not

affected by the number of random variables, thus these methods scale well for high-

dimensional problems. The drawback of these methods is the high computational

84


demand especially for constraining low probabilities of failure, where the required

number of simulations is very high. However, the advancements in parallel computing,

speed and storage performance of computers combined with the widespread use of

supercomputers mitigate the relevance of these effects these days.

Simulation-based methods require characterization of the full distribution of ran-

dom variables and ways of probabilistic sampling to obtain realizations of model input

parameters.

Random Sampling

Random-sampling based Monte Carlo simulations are among the most common reli-

ability technique. They involve drawing pseudo-random realizations of the variables

from the joint probability density function of the random variables, fX(x). For each

realization, limit state function is evaluated to check if failure of the system is ob-

served.

Using an indicator function I(g(x) ≤ 0), which takes the value of 1 if failure is

observed and 0 otherwise, pf can be estimated as the expected value of this indicator

function.

pf = E[I(g(x) ≤ 0)] =1

nsim.

nsim.∑i=1

I(g(xi) ≤ 0) =nfail.nsim.

(4.6)

where nsim. and nfail. are the number of simulations and the observed number of

failures, respectively (Kalos & Whitlock, 2009).

It can be shown that pf estimates by random-sampling based Monte Carlo simu-

lations are unbiased (Pinto et al. , 2004) provided that fX(x) is correct.

Note that using random sampling, the low probability regions of interest can

be missed if the number of simulations are not enough. Therefore, random sampling

based simulations are computationally demanding to use for low-probability-of-failure

problems. As a rule of thumb, the number of simulations needed are approximated

as:

nsim. =1− pfδ2pf

(4.7)

where δ is the desired coefficient of variation of the pf estimator (Pinto et al. , 2004).

Monte Carlo simulation-based methods are employed by several researchers to

investigate structural response (e.g. Esteva & Ruiz, 1989; Zhang & Ellingwood, 1995;

Au & Beck, 2003; Schueller, 2007; Kim et al. , 2011). The uncertainty in seismic

site response analyses is also investigated using Monte-Carlo simulations (e.g. Rathje

et al. , 2010). In the context of seismic collapse assessment, Monte Carlo simulations

can be used to compute the probabilities of failure corresponding to discrete ground

motion intensity levels, IM. Realizations of model parameters can be drawn, and

structural analysis can be performed using ground motions having intensities of IM.

This approach will result in the probability of collapse at the given IM level using

Equation 4.6. This can repeated for multiple IM levels and the number of collapses

and no-collapses at each IM level can be used with a maximum likelihood approach

to resolve a full collapse fragility curve.

Monte-Carlo simulations can also be used with an IDA approach. Collapse ca-

pacities, in terms of ground motion intensity, can be obtained corresponding to each

sampled model realization and ground motion. These capacities are then combined

in an average sense to compute a collapse fragility curve incorporating modeling un-

certainty and ground motion variability. These approaches are illustrated in Section

4.5.

Latin Hypercube Sampling

Latin Hypercube sampling provides an efficient method for probabilistic sampling

of model input parameters. It involves dividing the multivariate sample space into

predetermined number of equiprobable, disjoint regions, and realizations of model pa-

rameters are sampled from each region (Iman & Shortencarier, 1984; Helton & Davis,

2003). This provides a significant reduction in the number of required simulations.

A realization of a random variable is obtained by randomly sampling from its

marginal distributions, Fi(·), corresponding to a specified probability. Assume j

equiprobable regions are created in the sample space. Then, sampling can be per-

formed in each region with probability j−0.5Nsim

, using a Hazen probability plotting po-

sition, as given below:

86


xj,i = F−1i

(j − 0.5

Nsim

)(4.8)

where xj,i is the jth realization of the input variable i and different probability plotting

positions can be used. The sampled values of variables are matched with each other

randomly without replacement to obtain multivariate realizations of model param-

eters. However, this approach does not preserve the desired correlation of random

variables. Therefore, a post-processing is done to retrieve the desired correlation

structure. Simulation annealing is a stochastic optimization technique that can be

used for this purpose. It involves permutation of the sampled model parameters.

Permutations are performed until the error between the desired and the generated

correlation matrices is minimized (Dolsek, 2009). There are other methods proposed

for controlling correlations in Latin Hypercube sampling (e.g. Iman & Conover, 1982;

Owen, 1994).

The results of the analysis that are conducted with the Latin Hypercube samples

can be used with Equation 4.6 to compute pf . Seismic collapse assessments can be

performed in a similar fashion to random-sampling based Monte Carlo simulations,

except this time realizations of model parameters are obtained using Latin Hypercube

sampling. Previous research has shown that the sampling variability of Latin Hyper-

cube samples are significantly less than the ones of random-sampling (i.e McKay et al.

, 1979; Helton & Davis, 2003).

4.3.2 Moment-Based Methods

Moment-based methods determine the reliability of a system through the statistical

moments of the random variables. Using a Taylor series expansion, g(x) is written

as:

g(x) = g(x0)+nx∑i=1

∂g(x0)

∂xi(xi−xi0)+

1

2

nx∑i=1

nx∑j=i

∂2g(x0)

∂xi∂xj(xi−xi0)(xj−xj0)+h.o.t (4.9)

where h.o.t. stands for higher order terms. ∂g(x0)∂xi

and ∂2g(x0)∂xi∂xj

represent first and second

partial derivatives of g(x), respectively.

Moment-based reliability methods use Taylor series expansion along with first or

second-order approximations. Partial derivatives of the random variables are com-

puted and used with Equation 4.9 to determine the corresponding pf .

First-Order Second-Moment Method

First-order second-moment method (FOSM) (Melchers, 1999; Baker & Cornell, 2003,

2008), also known as differential analysis, uses a first order approximation to the

Taylor series expansion. Using the approximation given in Equation 4.9 and omitting

h.o.t, the expected value and variance of Y = g(x) are obtained as:

E[g(x)] =g(x0) +nx∑i=1

∂g(x0)

∂xiE[xi − xi0] = g(x0)

V ar[g(x)] =nx∑i=1

(∂g(x0)

∂xi

)2

V ar[xi] + 2nx∑i=1

nx∑j=i+1

∂g(x0)

∂xi

∂g(x0)

∂xjCov[xi, xj]

(4.10)

In mean-value FOSM, x0, is used as the mean value of x. The first two moments

of Y , obtained using Equation 4.10 are then used with Equation 4.5 to compute the

reliability index β and pf .

In the context of seismic collapse assessment, Y is used as the logarithmic mean

collapse capacity considering record-to-record (RTR) variability (Haselton & Deier-

lein, 2007; Liel, 2008).

Y = g(x) + Xgm (4.11)

where Xgm is a random variable representing ground motion variability having mean

0 and variance β. Xgm is further assumed to be independent of g(x). This assumption

is arguable since ground motion characteristics interact with the analysis model to

trigger the weakest collapse mechanism in the structure. Using Equation 4.10, a

first-order Taylor approximation yields E[Y ] and V ar[Y ] as:

88


E[Y ] =g(x0)

V ar[Y ] =V ar[g(x)] + β2(4.12)

FOSM results in an unchanged logarithmic mean collapse capacity in the presence

of modeling uncertainty. Logarithmic mean capacity obtained using mean values of

the model parameters equals to the logarithmic mean capacity in the presence of

modeling uncertainty (See Equation 4.12).

Seismic response and its associated limit state function is mildly to strongly non-

linear (Vamvatsikos & Fragiadakis, 2010). This makes the first-order approximation

to the limit state function defining collapse response less accurate. Also, as the num-

ber of random variables grows, the computational demand of FOSM grows linearly.

Uncertainty characterization is achieved through the first two moments of the ran-

dom variables. For distributions that are fully specified using the first two moments,

this approach can provide a full characterization of uncertainty. However, for other

distributions, this approach introduces additional approximation.

Other Methods

As their names imply, First-Order and Second-Order Reliability Methods, abbreviated

as FORM and SORM, use a first order and second-order approximation to Taylor

series expansion, respectively. These methods aim to find the design point, which

corresponds to the point having highest likelihood of failure on the limit state function

(Ditlevsen & Madsen, 1996). They are known to be very effective for high-dimensional

problems and low probabilities of failure. However, for characterizing collapse fragility

functions, they would require solving the optimization problem for two or more ground

motion intensity levels in order to resolve the full curve (Liel et al. , 2009). Thus they

are more computationally demanding than FOSM and FOSM is generally preferred for

this reason. Ching et al. (2009) proposed a moment-matching method, which includes

selections of points to effectively represent the first two moments of a statistical

distribution. They applied the method to assess the repair costs of building structures.

In this thesis, we do not investigate the use of FORM, SORM and moment-

matching methods with collapse response assessments. They are briefly included for

a comprehensive list of structural reliability methods.

4.3.3 Surrogate Methods

The essence of surrogate methods is to define a limit state function that approximately

defines the relationship between the response Y and input variables X. After the

mapping of input variables on the response variable is constructed, this function is

used with a Monte-Carlo simulation approach to estimate the uncertainty in y.

y = f(X) + ε (4.13)

In Equation 4.13, the surrogate relationship is represented as f(X). f(X) can be

a linear function or a higher order polynomial as well as a nonlinear function. In the

following sections, we present different functional forms f(X) for estimating collapse

capacity.

Response Surface

Response surface (Myers et al. , 2009) establishes a relationship using a polynomial

form to characterize the association between input parameters and the collapse ca-

pacity. The feasibility of response surface methods heavily depends on the design of

the input variables. Experimental design techniques provide efficient alternatives for

this purpose (Montgomery, 2008). The model parameters obtained using experimen-

tal design are then mapped to response variables using the pre-specified polynomial

relationship.

Equations 4.14 and 4.15 define first-order (FO) and second-order (SO) polynomial

expressions, respectively.

y = β0 +k∑i=1

βixi + ε (4.14)

90


y = β0 +k∑i=1

βi0xi +k∑i=1

βiix2i +

k∑i=1

k∑j=i+1

βijxixj + ε (4.15)

where β·,·’s represent the regression coefficients, ε is the error term and k is the number

of random variables.

A popular experimental design technique for calibrating second order models is

central composite design (CCD). Central composite designs consists of the following

points: A center point, which corresponds to each factor chosen at their median

(or mean) values and Axial points, which corresponds to each factor chosen at their

median (or mean) values except one factor taking values below and above its median

(or mean) value and A factorial design with the factors having two levels.

The premise of the response surface is to provide an approximation to a small

region of the input variables rather than estimating the whole space of these vari-

ables. Rajashekhar & Ellingwood (1993) proposed using polynomial approximations

to limit states for reliability analysis and investigated the use of response surface with

experimental design techniques to represent the interesting regions of the limit state

function. In the context of seismic collapse reliability, the response parameter can be

used as the median collapse capacity as well as other quantiles of collapse capacity.

Liel et al. (2009) used a quadratic response surface to idealize the relationship be-

tween median collapse response capacity and the input parameters. Computational

demand depends on the number of random variables and do not scale well to high

dimensional problems. For a quadratic response surface, the number of simulations

needed increases quadratically with the number of random variables using a central

composite design.

Neural Networks

Neural networks (NNs) are commonly used for function approximations and are pop-

ular due to their flexibility and adaptability. Response parameters are predicted by

establishing nonlinear functional relationships of input parameters. Their mathemat-

ical form is given in Equation 4.16.

y = f ′

(nlayer∑j=0

wjf(nx∑i=0

wj,ixi)

)(4.16)

where nx and nlayer represent the number of random variables and number of layers

in the network, respectively. wj,i represents the weights that are applied to the input

parameters and wj is the weight that is applied to the derived feature in the hidden

layer. f(·) and f ′(·) are nonlinear functions. Sigmoid or gaussian basis functions

are commonly used for f(·). f ′(·) is the final transformation to obtain the output

variables. For classification problems, it is common to use a logistic function for f ′(·),whereas an identity function is used for regression problems.

NNs are calibrated by optimizing the weights to minimize errors between predicted

(by the NN) and observed responses. Back-propagation, which is gradient descent

method, is commonly used for this purpose. It is common practice to scale the input

parameters beforehand. To avoid overfitting, a regularization parameter can be added

to the error function. Details of the NNs and their calibration can be found in Hastie

et al. (2009).

Papadrakakis et al. (1996) and Deng et al. (2005) used NNs to approximate

limit state functions in structural reliability problems. In this study, we use NNs to

predict quantiles of collapse capacities. NNs can have multiple response quantities.

This facilitates predicting different points on the collapse fragility functions. NNs

are calibrated using data obtained from IDA to predict IM levels corresponding to

multiple quantiles of collapse capacity.

NNs also provide powerful means for classification and can be used with MSA to

predict probabilities of collapse at alternative ground motion intensities. This thesis

does not explore the use of NNs with a MSA approach but this approach can be

explored in the future.

92


4.4 Sensitivity of Collapse Capacity to Perturba-

tions in Model Parameters

Sensitivity analysis is used to infer the relative importance of individual random

variables, and this information can be further used for reducing the dimensionality of

the uncertainty propagation problems by omitting relatively less important variables.

Several researchers have conducted sensitivity analyses for understanding importance

of model parameters in the context of seismic response assessment (i.e. Porter et al.

, 2002b; Aslani & Miranda, 2006; Ibarra & Krawinkler, 2005; Celik & Ellingwood,

2010).

In this section, we conduct sensitivity analyses for determining the relative im-

portance of the random variables for the bridge column that was previously analyzed

in Chapter 2 as well as reinforced concrete moment frame buildings that have been

used in previous studies of collapse (Haselton, 2006). The reinforced concrete build-

ings are designed according to the seismic provisions of IBC (2003), and analyzed

using OpenSees. Frame elements are modeled as elastic members having nonlinear

rotational springs at their ends with a concentrated plasticity hinge model. P-Delta

effects are taken into account by using a leaning column carrying gravity loads. De-

tails of the building design and analysis model can be found in (Haselton, 2006).

The fundamental periods of the structures with ID’s 1008, 1014 and 2067 are 0.94,

2.23 and 1.97 sec, whereas bridge column has a fundamental period of 1.11 sec. We

use these structures along with the generic far-field ground motion set of FEMA-

P695, consisting of 22 record pairs from extreme events to represent ground motion

variability.

The six parameters defining the backbone curve and hysteretic behavior of a com-

ponent are treated as random variables. The variability in the modeling parameters

is represented by the following: logarithmic standard deviations, σln, for θpc, γ, θcap,pl,

EIstf,40/EIg, My and Mc/My are equal to 0.73, 0.50, 0.59, 0.27, 0.31 and 0.10, re-

spectively. Equivalent viscous damping, ξ, elastic footing rotational stiffness, Kf ,

and joint shear strength Vj, are also treated as random having σln of 0.6, 0.3 and 0.1,

respectively (Haselton, 2006; Hart & Vasudevan, 1975; Porter et al. , 2002a). It is

assumed that the calibrated values of model parameters reflect median properties.

Individual perturbations of each random variable are performed at ±1.7σln and

±1σln by holding the other variables constant at their median values. These pertur-

bations are typically used in central composite designs. We note that sensitivities

depend not only on σ but also the mean (or median) values of the model parameters.

The perturbed model parameters are analyzed using the ground motion suite with an

incremental dynamic analysis (IDA) approach. Median collapse capacity correspond-

ing to each perturbation is determined and normalized with respect to the median

collapse capacity that is obtained using unperturbed model parameters. The results

are summarized using tornado diagrams. Tornado diagram of the bridge column for

the seven random variables is provided in Figure 4.1 showing the normalized median

collapse capacity obtained under each perturbation. Note that Kf and Vj are not

used for this exercise since these parameters are not used in modeling of the bridge

column.

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

ξ

EIstf

/EIg

Mc/M

y

γ

θpc

My

θcap,pl

Normalized Median Collapse Capacity

Figure 4.1: Tornado diagrams showing relative importance of model parameters for thebridge column. Perturbations with ±1.7σln and ±1σln are shown with solid and dashedlines, respectively.

Figure 4.1 suggests that plastic rotation capacity (θcap,pl) has the most effect on the

collapse capacity followed by flexural strength (My), post-capping rotation capacity

(θpc) and energy dissipation capacity for cyclic stiffness and strength deterioration

(γ). These observations are consistent with Ibarra & Krawinkler (2005).

94


For the reinforced concrete buildings, the six random variables are used to define

a concentrated plastic hinge. Two meta hinges are utilized to define beams and

columns, resulting in 12 random variables. This list is also enhanced with ξ, Kf and

Vj. Tornado diagram of the reinforced concrete buildings are provided in Figure 4.2

showing the normalized median collapse capacity obtained under each perturbation.

From Figure 4.2, we observe that My of the beams and the columns are the most

important parameters for the frames. Also, the impacts of both of these parameters

are asymmetric; i.e. their increase and decrease does not result in the same absolute

change in the median collapse capacity. Furthermore, perturbation of My,beam in any

direction results in reduction of median collapse capacity. This is due to violation of

strong-column weak-beam design provisions, which aims to uniformly distribute drifts

over building height and thus prevent localized damage. A more pronounced effect

of My is observed for the 12 story frame with ID 2067. This is a weak story design

having 65% of strength in the first story compared to other stories. The uncertainty

in My results in a more pronounced impact on this frame. We also observe that

(Mc/My)column is the third most important parameter for this frame, whereas this

parameter is not as important in the other frames. Besides the strength parameter,

we see that θcap,pl of columns and γ of beams are other important parameters, which

is consistent with the single degree of freedom structure.

Variables having less impact on the response can be omitted in uncertainty prop-

agation to reduce computational demand in some reliability methods. Sensitivity

analysis provides an important tool not only for checking and understanding analysis

models but also for dimension reduction.

Liel et al. (2009) studied the variation in collapse mechanism with respect to

change in column ductility. Although individual perturbations of parameters provide

insight in terms of significance in behavior, they are not able to capture interactions

between model parameters. Figure 4.3 shows an intermediate result from the sensi-

tivity study on the 4 story reinforced concrete structure. Median collapse capacity is

plotted with respect to two representative random variables for illustrative purposes.

Perturbations of individual parameters at ±1.7σln are displayed together with joint

perturbations of two parameters at ±1σln to capture interaction effects. A quadratic

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Kf

Vj

(EIstf

/EIg)col

(θpc

)beam

(Mc/M

y)beam

(γ)col

(θcap,pl

)beam

(Mc/M

y)col

(EIstf

/EIg)beam

ξ

(θpc

)col

(γ)beam

(θcap,pl

)col

(My)beam

(My)col


(a)

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Kf

(EIstf

/EIg)col

(γ)col

Vj

(θpc

)beam

(θcap,pl

)beam

(Mc/M

y)beam

(θpc

)col

(Mc/M

y)col

ξ

(θcap,pl

)col

(EIstf

/EIg)beam

(γ)beam

(My)beam

(My)col


(b)

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Vj

(Mc/M

y)beam

(EIstf

/EIg)col

Kf

(θpc

)beam

(θcap,pl

)beam

(γ)col

(θpc

)col

(EIstf

/EIg)beam

ξ

(γ)beam

(θcap,pl

)col

(Mc/M

y)col

(My)beam

(My)col


(c)

Figure 4.2: Tornado diagrams showing relative importance of model parameters for dif-ferent structures a) 4 story frame ID: 1008 b) 12 story frame ID: 1014 c) 12 story frameID: 2067. Perturbations with ±1.7σln and ±1σln are shown with solid and dashed lines,respectively. 96


surface fitted through these points is also shown. Collapse modes corresponding to

different realizations of random variables are displayed. The displayed collapse mech-

anism corresponds to the majority case among all collapse modes observed for a

given perturbation using the ground motions in the suite. Different collapse modes

are obtained for different values of random variables. This emphasizes the nonlinear

relationship between random variable values and collapse capacities, and points to

the need for realistic characterization of correlations between random variables.

log ( collog ( pc col

Figure 4.3: Nonlinear relationship between median collapse capacity and perturbations ofstrength and post-capping deformation capacity of columns. Collapse modes correspondingto different realizations of random variables are displayed. The displayed collapse mech-anism corresponds to the majority case among all collapse modes observed for a givenperturbation using the ground motions in the suite.

4.5 Reliability Analysis of Collapse Response As-

sessment of a 4-Story Reinforced-Concrete Moment-

Frame Building

In this section, the techniques for propagating modeling uncertainty are demonstrated

through a case study of a 4-story reinforced concrete moment frame building that

is previously used for sensitivity analysis. We illustrate the applications of these

methods for collapse response assessment and investigate the relative merits of each

method.

4.5.1 Structural Model, Seismic Hazard and Ground Mo-

tions

The case study structure is a 4-story reinforced concrete frame that was designed

by Haselton & Deierlein (2007) according to the seismic provisions of IBC (2003)

and American Society of Civil Engineers (2010). Frame elements are modeled as

elastic members having nonlinear rotational springs at their ends with a concentrated

plasticity hinge model. The first-mode period of the structure is 0.94 s.

The variability of the random variables are discussed in Section 4.4. Details of

the statistical analyses conducted for quantification of model parameter correlations

can be found in Chapter 3, and the resulting correlation coefficients are summarized

in Table 6.2. Equivalent viscous damping, elastic footing rotational stiffness and

joint shear strength are assumed to be uncorrelated from the model parameters that

define component backbone curve and hysteretic behavior. The parameters defining

component hinges are assumed to be perfectly correlated within beam components as

well as within column components (meaning that six parameters define the variations

for all beams and six define for all columns) resulting in 15 total random variables.

We use a generic ground motion set for response history analysis since there is

no specific site of interest as well as for the generalization of the results for sites

having different seismic hazard characteristics. FEMA-P695 far-field ground motion

set, which is composed of 22 ground motion record pairs from extreme events are used

98


for this purpose. IM is used in this thesis as the 5% damped spectral acceleration at

the fundamental period of the structure, Sa(T1, 5%).

Two sites are used for seismic risk calculations, which are located in Los Angeles,

CA (33.996◦N , 118.162◦W ) and Memphis, TN (35.150◦N , 90.049◦W ). Seismic hazard

curves for these sites are given in Figure 4.4. These two sites are selected because

of the different hazard characteristics. The site at Los Angeles (LA) is located at a

high seismicity region and the hazard curve is steeper compared to the hazard curve

of Memphis.

0 0.5 1 1.5 2 2.5 310

−6

10−5

10−4

10−3

10−2

10−1

100

Sa(T1=0.94 s, 5%) (g)

An

nu

al F

req

ue

ncy o

f E

xce

ed

an

ce

Los Angeles, CA

Memphis, TN

Figure 4.4: Seismic hazard curves at Los Angeles, CA and Memphis, TN.

The hazard curves at all sites are obtained from the USGS Java Ground Motion

Parameter Calculator (USGS, 2013) and modified with a site amplification factor of

1.5 to represent the NEHRP site class D.

4.5.2 Seismic Performance Assessment

Seismic collapse capacity is usually assessed using incremental dynamic analysis (IDA),

using an approach formalized by (Vamvatsikos & Cornell, 2002). IDA methods in-

volve the scaling of multiple ground motions through a range of increasing ground

motion intensity levels until the structure displays dynamic instability. This enables

quantification of ground motion capacities, which can then be related to probability

of collapse at any ground motion intensity level.

A single-stripe analysis involves conducting structural dynamic analysis using a

suite of records all of which are scaled to the same ground motion intensity. As

the name suggests, multiple stripes analysis involves conducting multiple stripes at

multiple values of ground motion intensity. Stripes are then studied in terms of their

statistical properties (Jalayer, 2003).

As with IDA procedures, Multiple Stripes Analysis (MSA) procedures are per-

formed with multiple ground motions in order to obtain statistical values of collapse

capacity. However, IDA approaches usually involve scaling a single set of ground

motions to larger and larger values, whereas MSA procedures ideally use different

ground motions at each intensity, which ideally are selected and scaled to match the

target conditional spectra that varies with ground motion intensity. In essence, these

methods result in the same collapse response if used with the same ground motion

sets. In this study, we use both of the methods, if applicable, using the same set of

ground motions. The aim is to discuss the applicability of the reliability methods

using these two collapse assessment procedures and their respective computational

demand.

Seismic risk metics of interest include parameters defining a logarithmic collapse

fragility function (median collapse capacity, θ, and dispersion, β), mean annual fre-

quency of collapse (λc) and probability of collapse in 50 years (P (C)50yr) using a

Poisson assumption.

Since the goal in this study is to assess the performance of collapse risk obtained

using various methods for incorporating modeling uncertainty, we use a fixed ground

motion suite and we do not apply any spectral shape correction to account for site-

specific spectral shape characteristics, in order to make direct comparisons among

100


methods. The collapse risk estimates and their uncertainty obtained in this chapter

are bounded by the size of ground motion suite (i.e. 44 ground motions) and the

selection of IM.

4.5.3 The Bootstrap for Quantifying the Uncertainty in Col-

lapse Risk Estimates

The bootstrap is a resampling technique for assessing the statistical accuracy of the

estimators (Efron, 1979; Efron & Tibshirani, 1994). It involves generating samples

from the original data set with replacement, and the replicated samples enable making

inference about the underlying population.

Bootstrap operates as follows: Assume the original data set is of size n. A sample is

drawn with replacement of size n from the original data set. This is called a bootstrap

sample. It is used to reconstruct the model and a realization of the estimator is

obtained. This process is repeated B=1000 times, and B realizations of the estimator

is obtained. Then, the statistical properties of the estimator is studied using its B

realizations.

In this section, we use the bootstrap to assess the statistical variability of the

collapse risk predictions (Eads et al. , 2013). Statistical variability is measured using

coefficient of variation (cov). We use the bootstrap with simulation-based methods,

but it can be used with other reliability methods as well. As mentioned previously,

Monte Carlo simulations involve drawing realizations of model parameters. These

realizations are analyzed using nonlinear dynamic analysis to obtain structural re-

sponse. Using an IDA technique, structural response corresponds to collapse capacity

in terms of ground motion intensity. We use these capacities and repeatedly obtain

samples using the bootstrap approach.

Using a MSA technique, structural response corresponds to the binomial data of

collapse, no-collapse at various ground motion intensity levels. The binomial data are

then repeatedly sampled with replacement using the bootstrap approach. Using each

bootstrap sample, corresponding collapse fragility functions are obtained. We then

study the variability in the replicated values of the parameters defining the collapse

fragility function as well as collapse risk.

4.5.4 Impacts of Modeling Uncertainty

Collapse fragility curves are obtained using a median model and a model incorporating

modeling uncertainty. These curves are integrated with the hazard curves given

in Figure 4.4 to obtain λc. Deaggregation of λc helps identify the ground motion

intensities that contribute most in estimating λc (Eads et al. , 2013). Deaggregation

curves plot the terms of the product of terms inside the integral in Equation 2.1.

Collapse fragility functions and risk deaggregation curves for the case study struc-

ture are given in Figure 4.5. If modeling uncertainty is neglected, median collapse

capacity is overestimated by 5% and dispersion is underestimated by 20% for the

case study structure. Modeling uncertainty also shifts the dominant ground motion

intensities for computing λc towards smaller intensities. These differences translate

as underestimation of λc using only median parameters in the model (i.e. neglecting

modeling uncertainty) compared to incorporating modeling uncertainty by 50% and

30% at LA and Memphis, respectively.

102


0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

Sa(T1=0.94 s, 5%)

Pco

llap

se

With MU+RTR

With RTR

(a)

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2x 10

−6

Sa(T1=0.94 s, 5%)

Deaggra

gation o

f C

olla

pse R

isk a

t LA

With MU+RTR

With RTR

(b)

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6x 10

−7

Sa(T1=0.94 s, 5%)

Deaggra

gation o

f C

olla

pse R

isk a

t M

em

phis

With MU+RTR

With RTR

(c)

Figure 4.5: Collapse risk using a median model versus (indicates as “With RTR”) and amodel incorporating modeling uncertainty (indicates as “With RTR+MU”) for the 4-storyreinforced concrete structure.

Ground motion intensities up until 2 g are dominant contributors for λc,LA (See

Figure 4.5.b). We see that these intensities shift to higher values (up to 3 g) for

computing λc,Memphis (See Figure 4.5.c). This is because of the relative flatness of the

hazard curve at Memphis, which results in the need to cover a wide range of ground

motion intensity values.

The mode of the collapse risk deaggregation curve, which represents the ground

motion intensity that is most dominant for computing λc shifted from 1.1 g to 0.82

g for the LA site (See Figure 4.5.b). 1.1 g and 0.82 g correspond to probabilities of

collapse of 12% and 8% on their respective fragility curves. The mode of the collapse

risk deaggregation curve changed from 1.4 g to 1.13 g for Memphis. This change

corresponds to a a shift in terms of probabilities of collapse from 29% to 22%. In the

presence of modeling uncertainty, we see that lower probabilities of collapse becomes

more important. However, the important IMs are a combined effect of the collapse

fragility function and the slope of the hazard curve at the site of interest.

4.5.5 Incorporating Modeling Uncertainty using an IDA Ap-

proach

Table 4.1 lists the collapse risk results obtained using alternative reliability methods

with an IDA approach. It lists collapse risk parameters obtained using different

approaches along with their coefficient of variations. Also shown are the number of

incremental dynamic analyses conducted, which is defined as one instance of a model

realization and a ground motion analyzed until collapse. In the following sections, we

present the application and calibration of reliability methods to assess the collapse

risk of the 4-story structure. We then discuss the results obtained using each method

emphasizing their relative advantages and disadvantages.

Monte-Carlo Simulations using Random Sampling

For uncertainty propagation, simulation-based methods are easy to implement and

converge to the exact solution as the sample size increases. In this study, Monte-Carlo

Simulations (MCS) are conducted to provide a benchmark collapse capacity estimate

for the case study structure. 4400 random realizations of the model parameters are

obtained using the joint probability distribution of the random variables. They are

randomly matched with the 44 ground motions in the FEMA-P695 ground motion

set and IDA is conducted for all model realization and ground motion combinations.

This results in 4400 estimates of collapse capacity.

From Table 4.1, we see that the benchmarked collapse fragility parameters and λc

104


Table 4.1: Collapse risk estimates obtained using an IDA approach with alternative struc-tural reliability methods

Coefficient of Variation using Bootstrap

Method No

IDA’s

Notes θ β λcLA×

10−4P (C)LA,50yr λcMemphis

×10−4

P (C)Memphis,50yrθ β λcLAλcMemphis

MCS 4400 benchmark 1.66 0.49 2.13 0.011 0.83 0.004 0.01 0.01 0.03 0.02

Median 44 No MU 1.73 0.39 1.21 0.006 0.64 0.003 0.06 0.10 0.23 0.13

LHS

44 - 1.55 0.55 3.43 0.017 1.06 0.005 0.08 0.09 0.29 0.17

88 - 1.72 0.47 1.71 0.009 0.74 0.004 0.05 0.08 0.22 0.12

176 - 1.67 0.50 2.21 0.011 0.84 0.004 0.04 0.06 0.17 0.09

220 - 1.68 0.49 2.02 0.010 0.81 0.004 0.03 0.05 0.15 0.09

440 - 1.71 0.50 2.03 0.010 0.80 0.004 0.02 0.03 0.11 0.06

880 - 1.66 0.50 2.23 0.011 0.85 0.004 0.02 0.02 0.07 0.04

1100 - 1.67 0.49 2.10 0.010 0.82 0.004 0.01 0.02 0.06 0.03

FOSM1364 - 1.73 0.68 4.28 0.021 1.08 0.005

4004 - 1.73 0.67 4.23 0.021 1.07 0.005

RS

1364 10%, FO 2.04 0.55 1.49 0.007 0.60 0.003

1364 25%, FO 1.77 0.55 2.26 0.011 0.81 0.004

1364 50%, FO 1.66 0.55 2.70 0.013 0.91 0.005

1452 10%, SO 1.87 0.56 2.01 0.010 0.74 0.004

1452 25%, SO 1.70 0.56 2.62 0.013 0.89 0.004

1452 50%, SO 1.57 0.55 3.27 0.016 1.04 0.005

19844 10%, SO 1.94 0.64 2.60 0.013 0.80 0.004

19844 25%, SO 1.64 0.62 3.83 0.019 1.07 0.005

19844 50%, SO 1.59 0.60 3.83 0.019 1.09 0.005

NN

1320 10%,50% 1.65 0.44 1.73 0.009 0.77 0.004

1320 10%,25% 1.65 0.47 1.97 0.010 0.81 0.004

1320 25%,50% 1.65 0.46 1.91 0.010 0.80 0.004

2200 10%,50% 1.66 0.46 1.83 0.009 0.78 0.004

2200 10%,25% 1.56 0.44 2.04 0.010 0.86 0.004

2200 25%,50% 1.62 0.49 2.25 0.011 0.87 0.004

estimates are well-constrained using MCS with cov values ranging from 1% to 3%.

Monte-Carlo Simulations using Latin-Hypercube Sampling

Latin Hypercube Sampling (LHS) is used to draw realizations of model parameters

and different sample sizes (ranging between 44 to 1100) are used to study the ef-

fectiveness of LHS. Stochastic optimization using simulated annealing is applied to

preserve the correlation structure among the random variables. Sampled model pa-

rameters are randomly matched with the 44 ground motions ground motions in the

FEMA-P695 ground motion set and respective collapse capacities are obtained. This

results in as many collapse capacities as the number of Latin Hypercube samples.

Collapse risk estimates that are obtained using LHS are given in Table 4.1. Using

440 IDA’s with LHS, λc,LA estimates have a cov of 0.11, which is 0.08 higher compared

to the benchmark result (i.e. MCS). Reducing the computational demand by a factor

of 5 (to 88 IDAs) increases the cov to 0.22 with the difference from the benchmark

result increasing to 20%. Whereas a 10 fold reduction in computational demand

results in 61% difference with a cov of 0.29 for λc,LA.

FOSM

One factor at a time experiments, in which each variable is perturbed by holding

others constant, is used to determine the partial derivatives for FOSM computations.

Each model realization obtained using this approach is analyzed with all ground

motions in the suite. Figure 4.6 shows the change in median collapse capacity with

respect to logarithmic perturbations of each model parameter. The perturbations

include ±1.7σln, ±1σln and ±0.5σln. Partial derivatives are obtained using the slope

of the lines fitted through these points (shown in blue lines). From Figure 4.6, it

is observed that My of both beams and columns, Mc/My, θcap,pl and θpc of columns

and γ and EIstf/EIg of beams have higher absolute values of partial derivatives. We

see nonlinear patterns only due to perturbations in My of both columns and beams.

Reduction in the My of columns is observed to significantly decrease collapse capacity;

however, an increase in My of columns only slightly increases collapse capacity. With

106


increasing My of beams, an increasing trend in median collapse capacity is observed.

However, after the peak point is reached, increasing My of beams is observed to

reduce collapse capacity. At this point, beams are stronger than columns, which

causes localized story collapse mechanisms. For My of columns and beams, due

to their nonlinear relationship with median collapse capacity, slopes are determined

using the region which results in higher slopes and these lines are shown in green.

These partial derivatives are used with Equation 4.10 to compute a collapse

fragility function. The resulting collapse risk estimates are shown in Table 4.1. Also

shown in Table 4.1 are the estimates using FOSM with partial derivatives computed

using a single perturbation at ±1.7σln. This approach reduced the computational

demand by three folds (4004 versus 1364 IDA’s) compared to the former alternative

and the collapse risk estimates obtained using these two approaches are comparably

similar.

Response Surface

First-order and second-order response surfaces (RS) are calibrated using one factor at

a time experiments and CCD, respectively. Using 15 random variables, one factor at

a time experiments and CCD resulted in 31 and 451 realizations of model parameters.

CCD is also used with a reduced set of random variables. The top 4 random variables

from the sensitivity analysis (See Figure 4.2.a), namely Mycolumn, θcap,plcolumn

, γbeam and

θpccolumnare selected resulting in 33 model parameter realizations. These realizations

are analyzed using all the ground motions in the FEMA-P695 ground motion set.

The predicted response quantities include 10%, 25% and 50% (median) quantiles of

collapse capacity.

Calibrated RS’s are used with randomly sampled model parameter realizations

(having a size of 10000) to predict the corresponding response parameter. The pre-

dicted point by RS along with a β defines a full lognormal collapse fragility function.

We use a β of 0.39, which is the RTR variability that is obtained using median model

parameters. At discrete ground motion intensity levels, probability of collapse values

are obtained using these fragility curves. Their average is used to find the probability

of collapse at each IM level and a final collapse fragility curve is fit through these

−2 0 21

1.5

2

δξ

(a)

−1 0 11

1.5

2

δ(EI

stf/EI

g)col

(b)

−1 0 11

1.5

2

δ(M

y)col

(c)

−0.2 0 0.21

1.5

2

δ(M

c/M

y)col

(d)

−2 0 21

1.5

2

δ(θ

cap,pl)col

(e)

−2 0 21

1.5

2

δ(θ

pc)col

(f)

−2 0 21

1.5

2

δ(γ)

col

Me

dia

n C

olla

pse

Ca

pa

city

(g)

−1 0 11

1.5

2

δ(EI

stf/EI

g)beam

(h)

−1 0 11

1.5

2

δ(M

y)beam

(i)

−0.2 0 0.21

1.5

2

δ(M

c/M

y)beam

(j)

−2 0 21

1.5

2

δ(θ

cap,pl)beam

(k)

−2 0 21

1.5

2

δ(θ

pc)beam

(l)

−2 0 21

1.5

2

δ(γ)

beam

(m)

−1 0 11

1.5

2

δK

f

(n)

−0.2 0 0.21

1.5

2

δV

j

(o)

Figure 4.6: Change in median collapse capacity with respect to perturbations in modelparameters. The perturbations include ±1.7σln, ±1σln and ±0.5σln. Fitted lines are shownin blue. For some variables, slopes are determined using the region which results in higherslopes and these lines are shown in green.

108


points.

Good fits are obtained using CCD with 33 and 451 realizations with R2 values

greater than or equal to 0.95. However, the fit is poor for FO models calibrated using

one factor at a time experiments; R2 values are obtained ranging 0.51 to 0.67.

Table 4.1 shows the results obtained using the aforementioned strategies. The

following observations are made: The FO RS does not provide an acceptable fit,

however, good estimates of λc are obtained. Calibrating RS to 10% quantile collapse

capacity resulted in smaller differences from the benchmark result (error) for λc,LA.

We obtain a 22% and 5% error λc,LA with 451 and 33 realizations, respectively;

whereas the corresponding errors increase to 80% and 23%-54% when RS is calibrated

to 25% or 50%. Calibrating RS to either 10% or 25% quantiles resulted in minimal

error at λc,Memphis. Comparable errors are obtained using 451 and 33 realizations, with

33 realizations providing slightly smaller errors. This highlights that the number of

random variables should be selected carefully to prohibit overfitting. We also observe

that RS consistently overestimates the overall dispersion due to the assumption of a

constant RTR dispersion.

Neural Networks

In this study, multi-layer radial-basis NNs are trained using back propagation with

regularization. Input data is obtained using LHS and pre-processing is conducted to

scale and omit features. The input data is randomly divided into training, validation

and test sets. Cross-validation (Hastie et al. , 2009) is used to calibrate the network

weights as well as to determine the number of hidden units. Two output layers are

used since two points are capable of defining a lognormal fragility curve. Representing

fragility curves by two points is motivated by the work of Eads et al. (2013), who

propose selecting two IMs that have significant contributions to the collapse risk. The

two output layers are defined to predict pairs of ground motion intensities correspond-

ing to 10%, 25% and 50% (median) probabilities of collapse levels. We use ground

motion intensities at 0.9, 1.2 and 1.7 g, which in fact have probabilities of collapse of

11%, 26% and 52% but used as proxies for 10%, 25% and 50%, respectively.

Table 4.1 shows the collapse risk predictions by NNs using different calibration

strategies and sample sizes. The pair of IM values corresponding to 10% and 25%

probabilities of collapse produces slightly smaller errors in λc estimates. With this

pair of IM’s, λc,LA have an error of 7% and 4% with 30 IDAs and 50 IDAs, respectively.

We obtain minimal errors in λc values with an NN approach, in general. Having an IM

corresponding to median level results in better estimates of median collapse capacity.

Results

Figure 4.7 shows the collapse response obtained using four different methods (that are

calibrated with approximately 1300 analyses) in comparison with the benchmarked

response. Shown are collapse fragility curves and collapse risk deaggregations. These

methods provide 3-4 fold reduction in computational demand compared to MCS,

which is shown with red. Collapse risk estimates by FOSM with 1364 analyses and

LHS with 1100 analyses are shown with blue and black, respectively. Estimates of

a second-order RS, which is calibrated with 1452 analyses to 10% quantile collapse

capacity, is shown in cyan. Collapse risk estimate by NN, which is calibrated with

1320 analyses to 10% and 25% quantiles collapse capacity, is shown in magenta. All

of the methods compare comparable results to MCS given the small number (44) of

GMs.

110


0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Sa(T1=0.94 s, 5%)

Pco

llap

se

Benchmark using MCS

FOSM

LHS

RS

NN

(a)

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5x 10

−6

Sa(T1=0.94 s, 5%)

Deaggra

gation o

f C

olla

pse R

isk a

t LA

Benchmark using MCS

FOSM

LHS

RS

NN

(b)

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6

7

8x 10

−7

Sa(T1=0.94 s, 5%)

Deaggra

gation o

f C

olla

pse R

isk a

t M

em

phis

Benchmark using MCS

FOSM

LHS

RS

NN

(c)

Figure 4.7: Collapse risk estimates are obtained using an IDA approach with alternativestructural reliability methods. a) Collapse fragility curves, b) Collapse risk deaggregationcurves at Los Angeles, c) Collapse risk deaggregation curves at Memphis

LHS with 1100 analyses provides a four-fold reduction in the number of struc-

tural analyses compared to MCS and results in good estimates of the overall collapse

fragility curve and λc. λc estimates differ from the benchmarked responses by 1% with

a cov of 3-6%. A similar reduction in computational demand can also be obtained

using FOSM with λc estimates differing by 102% and 29% for LA and Memphis sites,

respectively.

Among the uncertainty propagation methods considered, close estimation of the

MCS collapse fragility curve is provided by LHS and ANN predictions. FOSM and RS

in general have considerable discrepancy from MCS. Estimation by RS is comparable

to MCS at the important lower tail of the curve. It gives good estimations of collapse

probability until 20%.

The site at LA has a steeper hazard curve and thus the collapse probabilities

at each IM are multiplied with larger slopes. This results in higher collapse risk

deaggregation values and higher λc in comparison to the site at Memphis. It is also

observed that the mode of the MCS based curve occurs at 0.82 g and 1.13 g for LA and

Memphis sites, respectively. We see that for LA site, close estimations of the MCS

based collapse deaggregation curve are provided by LHS, followed by RS and NN. As

mentioned before, the latter two methods yield good estimations at the lower tail of

the collapse fragility curve. The largest contribution to the collapse risk at Memphis

occurs at around 1.13 g. At this intensity level, collapse fragility curve estimations

are similar for NN, LHS and MCS. Therefore, they yield comparable estimations for

collapse risk at this site.

LHS is a prominent method for propagating uncertainty in collapse response pre-

dictions. This is due to their computational efficiency compared to alternative meth-

ods, statistical efficiency in collapse risk estimates and ease of implementation. Al-

though NNs are also observed to provide a good alternative for probabilistic collapse

risk assessment, issues in pre-processing and training in NNs make their implemen-

tation challenging.

4.5.6 Incorporating Modeling Uncertainty using a MSA Ap-

proach

In this section, we obtain estimates of collapse risk for the 4-story R/C moment frame

structure using simulation-based methods (random sampling and LHS) in combina-

tion with a MSA approach. We use the FEMA-P695 far-field set and randomly match

each ground motion in this set with the sampled model parameter realizations. MSA

112


Table 4.2: Collapse risk estimates obtained using MSA approach with alternative struc-tural reliability methods

Coefficient of Variation using Bootstrap

Method No of

Analysis

P ′cols θ β λcLA×10−4 P (C)LA,50yr λcMemphis×10−4P (C)Memphis,50yrθ β λcLAλcMemphis

MCS 44000 Benchmark 1.66 0.49 2.13 0.011 0.83 0.004 0.01 0.01 0.03 0.02

MCS

8800 10%,25% 1.65 0.49 2.15 0.011 0.8.4 0.004 0.02 0.04 0.04 0.02

8800 10%,50% 1.60 0.47 2.13 0.011 0.86 0.004 0.01 0.02 0.04 0.02

8800 25%,50% 1.61 0.45 1.96 0.010 0.83 0.004 0.01 0.02 0.05 0.02

13200 10%, 25%, 50% 1.61 0.46 2.08 0.010 0.85 0.004 0.01 0.02 0.04 0.02

LHS

88 10%,25% 1.46 0.36 1.90 0.009 0.89 0.004 0.48 0.48 2.85 0.32

88 10%,50% 1.80 0.52 1.87 0.009 0.74 0.004 0.12 0.27 0.65 0.26

88 25%,50% 1.87 0.82 6.64 0.033 1.24 0.006 0.31 0.54 3.30 1.57

132 10%, 25%, 50% 1.75 0.55 2.38 0.012 0.83 0.004 0.13 0.22 0.59 0.24

176 10%,25% 1.61 0.44 1.84 0.009 0.80 0.004 0.17 0.29 0.43 0.14

176 10%,50% 1.60 0.43 1.84 0.009 0.81 0.004 0.05 0.14 0.27 0.13

176 25%,50% 1.60 0.43 1.81 0.009 0.80 0.004 0.05 0.17 0.35 0.15

264 10%, 25%, 50% 1.60 0.43 1.83 0.009 0.81 0.004 0.05 0.13 0.25 0.12

352 10%,25% 1.56 0.41 1.85 0.009 0.83 0.004 0.08 0.19 0.20 0.09

352 10%,50% 1.61 0.43 1.84 0.009 0.81 0.004 0.04 0.10 0.19 0.10

352 25%,50% 1.60 0.45 2.01 0.010 0.84 0.004 0.04 0.13 0.27 0.12

528 10%, 25%, 50% 1.60 0.44 1.89 0.009 0.82 0.004 0.04 0.09 0.18 0.09

880 10%,25% 1.65 0.48 2.08 0.010 0.83 0.004 0.06 0.12 0.15 0.06

880 10%,50% 1.62 0.47 2.07 0.010 0.84 0.004 0.03 0.06 0.13 0.06

880 25%,50% 1.62 0.46 1.96 0.010 0.82 0.004 0.03 0.08 0.16 0.07

1320 10%, 25%, 50% 1.62 0.47 2.04 0.010 0.83 0.004 0.03 0.06 0.12 0.06

1320 10%,25% 1.61 0.46 2.09 0.010 0.85 0.004 0.04 0.09 0.11 0.05

1320 10%,50% 1.62 0.47 2.10 0.010 0.84 0.004 0.02 0.05 0.10 0.05

1320 25%,50% 1.62 0.48 2.17 0.011 0.85 0.004 0.02 0.07 0.14 0.06

1980 10%, 25%, 50% 1.62 0.47 2.12 0.011 0.85 0.004 0.02 0.05 0.10 0.05

2200 10%,25% 1.59 0.43 1.91 0.010 0.83 0.004 0.03 0.07 0.07 0.03

2200 10%,50% 1.65 0.47 1.92 0.010 0.80 0.004 0.02 0.04 0.08 0.04

2200 25%,50% 1.65 0.50 2.20 0.011 0.85 0.004 0.02 0.05 0.11 0.05

3300 10%, 25%, 50% 1.64 0.47 1.99 0.010 0.81 0.004 0.02 0.04 0.07 0.04

is conducted with 2 or 3 IM levels (Eads et al. , 2013). Combinations of ground mo-

tion intensities are used corresponding to 10%, 25% and 50% (median) probabilities

of collapse levels. We use ground motion intensities at 0.9, 1.2 and 1.7 g, which in fact

have probabilities of collapse of 11%, 26% and 52% but used as proxies for 10%, 25%

and 50%, respectively. The estimates that are obtained using MSA are compared

with the benchmarked collapse risk, that was previously obtained from an extensive

analysis with MCS. The results are listed in Table 4.2.

A total number of 1320 analyses are conducted using LHS by performing 660

analyses at 2 IM levels and 440 analyses at 3 IM levels. Comparing these approaches,

for the same number of analyses, smaller values of cov are obtained for θ, β, and

λc’s if used with 2 IM levels at 10% and 50% probabilities of collapse yielding similar

errors. A similar observation is made if one compares a total number of 132 analyses

conducted at 3 IM levels versus 176 analyses conducted at 2 IM’s. Although for the

latter case, the number of analyses is slightly larger, we see similar errors with less

variability for the 2 IM case. Thus, constraining 2 points on the fragility curve pro-

duces more stable results as compared to distributing the analyses efforts to multiple

IM levels. This is consistent with the results by Eads et al. (2013), who also recom-

mends to conduct analysis at 2 IM levels with more ground motion records, compare

to using 3 IM levels, in order to incorporate ground motion variability.

Among the pairs of IM levels, in general, selecting the ones corresponding to 10%

and 50% yields more stable estimates λc using LHS, especially with smaller sample

sizes. This pair also minimizes the cov of β. Obviously, having an IM corresponding

to 50% improves median collapse capacity estimates.

A cov of 10% is obtained for λc,LA with a total number of 1320 analyses with an

error of 1%. A reduction of a factor of 3.75 in computational demand increases the

cov to 20% and error to 13%.

In Figure 4.8, we show the collapse fragility functions and collapse risk deaggre-

gation curves that are obtained using MCS and LHS with 2 IM’s at 10% and 50%

probabilities of collapse. Also provided is the benchmarked collapse response. With a

factor of 10 reduction in number of structural analyses, LHS provides good estimates

of the overall collapse fragility function and collapse risk deaggregations.

114


0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

Sa(T1=0.94 s, 5%)

Pco

llap

se

Benchmark using MCS (~44000)

MCS with 2 points (8800)

LHS with 2 points (880)

(a)

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2x 10

−6

Sa(T1=0.94 s, 5%)

Deaggra

gation o

f C

olla

pse R

isk a

t LA




(b)

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6x 10

−7

Sa(T1=0.94 s, 5%)

Deaggra

gation o

f C

olla

pse R

isk a

t M

em

phis




(c)

Figure 4.8: Collapse risk estimates are obtained using a MSA approach with simulation-based reliability methods. a) Collapse fragility curves, b) Collapse risk deaggregation curvesat Los Angeles, c) Collapse risk deaggregation curves at Memphis

For assessing λc, we see a significant reduction in the number of analyses with

MSA compared to IDA. For instance, to achieve a 10% cov in λc,LA 1320 analyses

are conducted using LHS. This increases to 4400 analyses (440 IDA’s) with an IDA

approach. Thus, a factor of 3.3 reduction in total number of analyses is achieved

using MSA to obtain similar variability of 10% cov in λc. Similar observations are

also made by Eads et al. (2013) for incorporating ground motion variability.

Extension to High Dimensional Problems

Simulation-based methods enable practical treatment of uncertainties due to record-

to-record variability and modeling uncertainty for collapse safety assessment. One

significant advantage of these methods is that their computational demand does not

increase significantly as the number of random variables increases. This section ex-

plores the computational demand and statistical efficiency of simulation-based meth-

ods for high-dimensional collapse risk problems.

Component hinge model parameters for 16 columns and 12 beams for the 4-

story reinforced concrete moment frame building are now treated as random and

non-perfectly-correlated, in addition to equivalent viscous damping, elastic footing

rotational stiffness and joint shear strength. Table 6.2 is used to define the correlation

of the parameters defining the 28 components. This results in 171 random variables.

The top portion of Table 4.3 shows the collapse risk estimates obtained using

MCS with 171 random variables. Collapse risk estimates that are obtained with the

assumption of perfect correlation among beam and column hinges (corresponding of

15 effective random variables) are also shown in the top portion of Table 4.3. By

treating all of the beam and column hinges as random variables, in which correlation

among components are defined using partial correlation coefficients, the collapse risk

and parameters defining a collapse fragility curve are slightly changed; dispersion of

collapse fragility curve is reduced by a small amount and median collapse capacity

is increased by 2%. This resulted in λc,LA and λc,Memphis changing by 11% and 6%,

respectively, which are very small.

116


Table 4.3: Collapse risk estimates using simulation-based methods with 171 random vari-ables

#

RVs

# of IDA

(using

MCS)

θ β λc,LA λc,Memphis

15 4400 1.66 0.49 2.13E-04 8.32E-05

171 4400 1.70 0.48 1.90E-04 7.78E-05# of IDA

(using

LHS)

Mean Estimate among 100 estimates

for

Coefficient of variation among

100 estimates inθLHS

θMCS

βLHS

βMCS

λc,LALHS

λc,LAMCS

λc,MemphisLHS

λc,MemphisMCS

θ β λc,LA λc,Memphis

44 0.99 1.02 1.14 1.07 0.02 0.08 0.25 0.13

88 0.99 1.02 1.12 1.06 0.02 0.06 0.19 0.09

176 0.99 1.00 1.07 1.04 0.01 0.04 0.13 0.07

Among simulation-based methods, LHS provides an efficient alternative to MCS.

Given the joint probability distribution of uncertain variables, LHS ensures that the

samples are distributed evenly in accordance with the given variability of the ran-

dom variables. For treating record-to-record variability and modeling uncertainty,

every record in the ground motion suite is matched with a single simulated struc-

tural model realization that is obtained using LHS. To study the variability in the

collapse response simulations, we repeat the matching of ground motion records and

structural model realizations 100 times. The bottom-left half of Table 4.3 shows the

mean ratios among 100 estimates of median collapse capacity, dispersion, λc,LA and

λc,Memphis that are obtained using LHS to the ones of MCS. The bottom-right half

of Table 4.3 shows the coefficients of variation in the LHS estimates among 100 es-

timates. Note that cov values obtained using 100 estimates are very similar to the

ones obtained using bootstrap (See Table 4.1).

Shown in Table 4.3 are the LHS results obtained using sample sizes of 44, 88 and

176. These correspond to matching the records in the ground motion suite with 1,

2 and 4 model realizations, which correspond to, in total, 44, 88 and 176 collapse

response analyses, respectively. It is observed from Table 4.3 that with 44 IDAs, on

average, one gets close estimations of collapse fragility curve. The results are observed

to be stable with coefficients of variation for median collapse capacity and dispersion

with coefficient of variations less than 10%. For λc, however, higher coefficients of

variations are obtained. λc,LA has a coefficient of variation of 0.25 if used with with

a sample size of 44. Increasing number of analyses to 176 IDAs, the variation in the

λc,LA decreases to 0.13.

4.6 Summary

In this chapter, we present structural reliability methods for uncertainty propagation

in seismic collapse response assessment. Alternative reliability methods are used

to illustrate probabilistic assessment of collapse risk and the relative merits of each

method are discussed. The methods are grouped as simulation-based, moment-based

and surrogate methods. Simulation-based methods are particularly explored in terms

of their effectiveness for uncertainty propagation in high dimensional collapse safety

assessment problems. The results emphasize the sensitivity of collapse response to

modeling uncertainties and the challenges of balancing of computational efficiency

and robust uncertainty characterization.

We have presented probabilistic collapse assessment of a 4-story concrete frame

structure incorporating uncertainties related to ground motion and structural mod-

eling. Correlations of component model parameters are quantified and incorporated

into uncertainty propagation. MCS, LHS, FOSM, NN and RS reliability methods

are explored to incorporate uncertainty in collapse predictions. For a similar number

of analyses, FOSM is observed to have considerable error in estimating collapse risk

compared to the other methods. Although RS yields a less accurate estimate of the

overall fragility curve, when calibrated to predict the lower tail, it closely estimates

collapse risk. LHS and NN approaches are observed to provide superior estimates

of the collapse fragility curves for the case study structure compared to the other

methods. Although NNs are observed to provide a good alternative for probabilis-

tic collapse risk assessment, issues in pre-processing and training in NNs make their

implementation challenging.

For uncertainty propagation studies, treatment of collapse analyses with a large

number of random variables ( 20 or more) is challenging and requires dimension re-

duction techniques in order to use approximate methods such as FOSM, RS and NN.

118


In contrast, simulation-based methods (MCS, LHS) are robust and scalable to high

dimensionality. While simulation-based methods have high computational demands

due to repeating simulations, smart sampling techniques can be employed to reduce

the computational demands while handling cases with high dimensionality. The ex-

ample presented in this chapter illustrates the use of LHS for this purpose. LHS is

observed to be a prominent method for probabilistic seismic collapse assessment pro-

viding computational efficiency compared to other alternatives, statistical efficiency

in estimators and ease of implementation. Using LHS with 440 incremental dynamic

analyses, collapse risk is estimated with a coefficient of variation of 0.1. Similar vari-

ability can be obtained if LHS is used in combination with a multiple stripes analyses

at 2 ground motion intensity levels reducing the computational demand by a factor

of 3.3 for the structures analyzed.

Chapter 5

An Efficient Bayesian Approach for

Seismic Collapse Risk Assessment

5.1 Introduction

Building codes aim to avoid structural collapse of buildings during a major earth-

quake. This performance goal is achieved by ensuring the risk due to earthquake-

induced collapse is low. Collapse risk estimates are obtained by aggregating seismic

hazard analysis and structural collapse response simulations. Structural analysis con-

ducted with hazard-consistent ground motions combine these two components of col-

lapse risk estimation. Hazard curves provide a probabilistic representation of ground

motion intensities and by selecting a ground motion suite that is consistent with the

hazard at the site of interest, record-to-record variability is propagated in structural

response simulations.

An important aspect of collapse risk estimation is simulation of structural collapse

response. Simulation of structural collapse response requires structural models and

nonlinear analysis tools that are capable of representing large inelastic deformations

and cyclic and in-cycle deterioration in the structural elements. The accuracy of

structural collapse performance predictions is bounded by the modeling uncertain-

ties present in the simulation capabilities and our understanding of how buildings

perform under ground shaking. There have been major efforts in recent years to

120

CHAPTER 5. BAYESIAN COLLAPSE RISK ASSESSMENT 121

propagate modeling uncertainties for robust collapse performance predictions. Hasel-

ton (2006) used a first-order second-moment (FOSM) approach for this purpose and

modeled the relationship between model parameters and collapse capacity using a lin-

ear approximation. Liel et al. (2009) used a response surface method to establish a

quadratic relationship between model parameters and collapse capacity. Vamvatsikos

(2014) and Dolsek (2009), Celarec & Dolsek (2013) used a Latin-Hypercube sampling

approach to obtain model realizations to be used in a Monte-Carlo framework for

uncertainty propagation.

The aforementioned uncertainty propagation methods are shown to be applicable

to the problem of collapse risk assessment of buildings. However, balancing compu-

tational demands and accuracy remains as a major challenge. The computational

demand of simulation-based uncertainty propagation methods grow as the probabili-

ties of interest become smaller. On the other hand, the number of random variables

is a major factor determining the computational demand in surrogate methods, i.e.

response surface and FOSM.

Efficient methods have been proposed to decrease the computational demand for

collapse risk predictions. Azarbakht & Dolsek (2007), Azarbakht & Dolsek (2011)

propose ways to estimate incremental dynamic analysis response with a reduced set

of ground motion recordings, whereas Eads et al. (2013) reduces computational

demand by focusing nonlinear response history analyses to two or three ground motion

intensity levels.

Bayesian statistics facilitate the incorporation of any prior knowledge to inform

statistical inference. Singhal & Kiremidjian (1998) proposed using Bayesian statis-

tics to update fragility functions with observational building damage data. Gardoni

et al. (2002) used Bayesian updating to estimate parameters of fragility functions of

structural components using laboratory test results of reinforced concrete columns.

Jalayer et al. (2010) used a Bayesian framework for assessing the effects of structural

modeling uncertainty. Jaiswal et al. (2011) also used a Bayesian approach for com-

puting empirical collapse fragility functions combining expert opinion and field data

for global building types. They incorporated test and inspection results of structures

in order to update the prior information on the modeling uncertainties.

In this study, we present an efficient method to propagate modeling and ground

motion uncertainties. The method increases computational efficiency by incorporat-

ing prior information within a Bayesian framework. First, we present the analysis

rule with discussions on development and calibration of the method. This is followed

by sensitivity analyses of the method. Following, we provide a robustness analyses

of the proposed method. Later, we discuss ways of checking the model predictions.

Finally, we present an illustration of the method for collapse risk assessment of a

4-story reinforced-concrete moment-frame building.

5.2 Proposed Method for Collapse Risk Assess-

ment

5.2.1 Collapse Risk Assessment

Mean annual frequency of collapse (λc) is a common metric for representing collapse

risk and it is defined as:

λc =

∫ ∞0

P (C|IM = im)


d(im)

∣∣∣∣ d(im) (5.1)

where λIM is the mean annual frequency of exceedance of IM , which is obtained from

seismic hazard curves and dλIM (im)d(im)

represents the absolute value of the derivative

of λIM with respect to IM . P (C|IM = im) is the probability of collapse given

ground motion intensity measure IM = im. Collapse fragility curves are cumulative

distribution functions defining P (C|IM = im) at various IM values. It is common to

represent collapse fragility curves with a lognormal distribution (Porter et al. , 2007;

Bradley & Dhakal, 2008).

P (C|IM = im) = Φ

(ln(im)− ln(θ)

β

)(5.2)

In Equation 5.2, θ and β represent the median and logarithmic standard deviation

(dispersion) of collapse capacity, and Φ() is the cumulative distribution function of

122


a standard normal distribution. For estimating collapse fragility curves, methods

ranging from incremental dynamic analysis to multiple stripes analysis have been

proposed (Vamvatsikos & Cornell, 2002; Jalayer, 2003).

In this study, we aim to provide an efficient method for estimating collapse fragility

functions to give reliable estimates of collapse risk and we quantify collapse risk using

λc. Using a multiple stripes analysis approach, conducting structural analyses at few

ground motion intensity levels is a computationally efficient strategy (Eads et al. ,

2013; Baker, 2015). Furthermore, two points are enough to represent the full collapse

fragility function using a lognormal assumption. To increase computational efficiency

in this method, we focus our analysis efforts on constraining two points on the fragility

function.

Development of collapse risk assessment models requires characterization and

quantification of various sources of uncertainties. Each component of collapse risk

estimation, including modeling of seismic hazard and structural response and relat-

ing this response to the collapse damage state, have inherent randomness as well as

uncertainty that is caused by natural variability and lack of knowledge. These two

types of uncertainties are named aleatory and epistemic, respectively (Kiureghian &

Ditlevsen, 2009). For reliable seismic risk assessment, it is essential to consider both

types of uncertainties. Bayesian statistics allow treatment of both types of uncertain-

ties together.

5.2.2 Proposed method

The proposed method for efficient estimation of collapse risk is illustrated in Figure

5.1 and the steps are summarized below. The details and calibration of the analysis

method are discussed in the subsequent sections.

1. Obtain an initial estimate of the collapse fragility curve. Quantify the un-

certainty in estimated median collapse capacity considering the limitations in

structural idealizations, calibration of model parameters, number of analyses

and software used for structural analyses. (Figure 5.1.a).

2. Select two (or more) IM levels. Select IM1 corresponding to probability of

collapse levels below 10% on the initial fragility curve. Select IM2 corresponding

to probability of collapse between 25% and 75% on the initial fragility curve

depending on the level of uncertainty in estimated median collapse capacity.

Obtain prior distributions at these IMs (Figure 5.1.b).

3. Select ground motions at the two chosen IM levels (Figure 5.1.c).

4. Conduct nonlinear time history analyses using the selected ground motions at

the two chosen IM levels. (Figure 5.1.d).

5. Update the prior distributions using structural analysis results and obtain pos-

terior distributions. (Figure 5.1.e).

6. Using posterior distributions, obtain a final estimate of collapse fragility func-

tion. (Figure 5.1.f).

5.2.3 Analysis Rule

Step 1

The analysis requires the users to start with an initial estimate of a collapse fragility

curve. To treat epistemic and aleatory uncertainties in fragility functions, we denote

the median collapse capacity (Θ) as a Bayesian random variable. Θ is modeled using

a lognormal distribution with median θ and a logarithmic standard deviation βθ.

In cases where information on βθ is not available, users can make judgment-based

assumptions for estimating βθ considering the limitations in structural idealizations,

calibration of model parameters, number of analyses, and software used for structural

analyses such as: “Median collapse capacity is estimated as θ within ±∆% certainty

with (1− α)% confidence.” This leads to the median collapse capacity being defined

using a lognormal distribution having median at θ and dispersion as given in the

Equation 5.3.

124


(a)

Sa (T1, 5%)

Pco

llap

se

0

0.5

1

mBound

5%θ

mBound

95%

Initial Fragility

Uncertainty in Median

Bound 5%

Bound 95%

(b)

Sa (T1, 5%)

Pco

llap

se

0

0.2

0.4

0.6

0.8

1

IM1

IM2

Initial Fragility

Prior Dist.

(c)

10−1

100

101

10−2

10−1

100

T (s)

Sa (

g)

(d)

Sa (T1, 5%)

Pco

llap

se

0

0.2

0.4

0.6

0.8

1

IM1

IM2

Initial Fragility

Prior Dist.

Analysis Data

(e)

Sa (T1, 5%)

Pco

llap

se

0

0.2

0.4

0.6

0.8

1

IM1

IM2

Initial Fragility

Prior Dist.

Analysis Data

Posterior Dist.

(f)

Sa (T1, 5%)

Pco

llap

se

0

0.2

0.4

0.6

0.8

1

IM1

IM2

Initial Fragility

Prior Dist.

Analysis Data

Posterior Dist.

Posterior Fragility

Figure 5.1: Steps of the proposed method: a) Initial collapse fragility curve and uncer-tainty in median capacity, b) Selected IM levels and prior distributions at these levels, c)Ground motions selected to match conditional spectra, d) Data from nonlinear time historyanalyses at selected IM levels, e) Posterior distributions at selected IM levels, and f) Finalestimate of the collapse fragility function

βθ =

√√√√ln

([∆

Φ−1(1− α/2)

]2+ 1

)(5.3)

An example of such an assumption is as follows: “Median collapse capacity is

estimated as 1 g within ±50% certainty with 90% confidence.” This statement is

translated into a lognormal distribution of the median collapse capacity having a

median of 1 g (θ = 1) and logarithmic standard deviation of approximately 0.3

(βθ = 0.3) (Ellingwood & Kinali, 2009).

In summary, as the first step, users are expected to provide an initial estimate

of the collapse fragility curve and quantify its uncertainty in Θ with θ and βθ. Note

that the information provided by the users in Step 1 is obtained before conducting

any response history analyses.

Step 2

The analysis requires selection of at least two IM levels for Step 2. Recommendations

about selecting IM levels are provided in the next section. The initial data provided

in Step 1 defines prior distributions at the IM levels of interest. These prior distri-

butions are probability distributions defining P (C|IM = imi) and they reflect initial

knowledge. Prior distributions at IM = imi are defined to have beta distributions

and are denoted by P (τi).

P (τi) =Γ(αi + βi)

Γ(αi)Γ(βi)τ αi−1i (1− τi)βi−1 (5.4)

where Γ(·) is the gamma function and αi and βi are the parameters defining the

beta distribution. The prior probability density function of probability of collapse at

IM = imi is given in Equation 5.4 as a function of the variable τi, where 0 ≤ τi ≤ 1.

αi and βi are computed as follows: Define two lognormal distributions, whose cu-

mulative distribution functions are called Bound5% and Bound95% (See Figure 5.1a).

Dispersions of these curve equal to β. Median values of Bound5% and Bound95%

are corresponding to 5% and 95% quantiles of Θ, and are denoted as mBound5% and

mBound95% , respectively.

126


To find the prior distribution at IM1, first obtain Pcol,1 values from Bound5% and

Bound95% curves corresponding to IM1. These two points characterize 5% and 95%

quantiles of a beta distribution.

P

(τi < Φ

(ln IMi − lnmBound5%

β

))= 0.05

P

(τi < Φ

(ln IMi − lnmBound95%

β

))= 0.95

(5.5)

Second, obtain Pcol,1 from the initial fragility curve, which is used as the mode of

the beta distribution.

Pcol,i = Φ

(ln IMi − ln θ

β

)(5.6)

Lastly, find the parameters of a beta distribution, (α1, β1) that satisfies the afore-

mentioned constraints. Beta(α1, β1) defines the prior distribution at IM1 (P (τ1)).

The solution can be obtained iteratively finding the parameters of the distribution

satisfying the quantile constraints.

To find the prior distribution at IM2, repeat the same procedure, this time for

IM2, and obtain Beta(α2, β2) as the prior distribution at IM2 (P (τ2)).

Step 3

Select hazard consistent ground motions at all IM levels. The ground motions should

be selected matching the conditional spectra at all levels. For more details about

ground motion selection, refer to (Jayaram et al. , 2011).

Step 4

Conduct nonlinear time history analyses using the selected ground motions at the IM

levels of interest and report the count of cases in which collapse response is obtained.

We also recommend incorporating modeling uncertainty by sampling structural model

realizations.

Step 5

Bayes theorem enables updating the prior knowledge when data is available. In this

method, data is collected using structural analyses.

At each IM level, the number of collapses and no-collapses obtained from analyses

using different ground motions and structural model realizations define a binomial

distribution. This distribution results from the assumption that each observation

obtained using a ground motion and structural model realization pair is independent

from all the others. The probability of observing the likelihood data, Xi, which is

characterized by xi collapses out of ni analyses at IMi level is given by

P (Xi|τi) = P (xi collapses in ni analyses) =

(xini

)τxii (1− τi)ni−xi (5.7)

where τi defines the probability of collapse at IM = imi. The data, Xi, given the

prior knowledge is termed likelihood. We denote the likelihood function by P (Xi|τi).Under Bayes theorem, the conditional distribution of parameters given the data

is defined as follows:

P (τi|Xi) = cP (τi)P (Xi|τi) (5.8)

where c =∫τiP (Xi|τi)P (τi)dτi is a normalizing constant. In Equation 5.8, the prior

P (τi) is defined to have a beta distribution (See Equation 5.4) and the likelihood

is binomial (See Equation 5.7). Beta and binomial distributions form a conjugate

pair, and the resulting posterior distribution P (τi|Xi) stays in the same family as the

prior distribution (Gelman et al. , 2013). Conjugacy provides significant mathematical

convenience in computing the posterior distributions. Using the beta-binomial model,

the posterior distribution defining probability of collapse at IMi given the data is

τi|Xi ∼ Beta(αi + xi, βi + ni − xi)

P (τi|Xi) ∝ τ αi+xi−1i (1− τi)βi+ni−xi−1

(5.9)

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Step 6

To obtain the final fragility function, we use the maximum likelihood method with

the posterior distributions characterizing the probability of collapse at IMi. The pa-

rameters defining the collapse fragility function, namely, the median collapse capacity

and dispersion, are obtained by maximizing the likelihood function given in Equation

5.10.

{θ, β} = arg maxθ,β

m∑i=1

ln P (τi|Xi) (5.10)

The resulting collapse fragility function is characterized by the parameters θ and

β. Note that if used with 2 IM levels, Equation 5.10 results in the final fragility

function passing through the modes of the posterior distributions.

5.3 Selection of IM Levels

In this section we investigate the efficiency of collapse risk estimates due to IM selec-

tion. We also provide recommendations regarding optimal selections of these levels

in order to obtain efficient estimates of λc.

We focus our efforts on selecting two optimal IM levels. To investigate the effects of

selecting different combinations of IM levels, we search among possible combinations

of IM1 and IM2 using different hazard curves and possible values for ∆, with an

assumed value of α = 10%. In other words, it is assumed that users start with the

statement “Median collapse capacity is estimated as θ within ±∆% certainty with

90% confidence” and we examine the estimates obtained using various values of the

parameter ∆ as well as different hazard curves.

For calibration purposes, an initial collapse fragility curve is assumed, which is

characterized by θ = 1 g and β = 0.4. This initial curve is used as a user-defined

prior estimate of collapse fragility function. It is noted that any assumed values of θ

and β would have been valid. We choose to assume the aforementioned values for two

reasons. The first one is that β = 0.4 is a common value for collapse fragility functions

that incorporates record-to-record variability when using IM as Sa(T1). The second

reason is that any result displayed in the format of IM can be interpreted as a fraction

of median IM, IM/θ, and the results can be easily translated to any fragility function

with β = 0.4.

IM is used as 5% damped spectral acceleration at the fundamental-mode period

of the structure (Sa(T1, 5%)) in this study. Three hazard curves are used. The

first one is a seismic hazard curve for downtown Los Angeles for Sa(1s, 5%) on a

NEHRP B/C boundary site. The other two hazard curves are obtained using idealized

power-law hazard curves having the form λIM(im) = k0(im)k. The second hazard

curve is characterized by k = 2 and k0 = 0.0002, whereas the third hazard curve

is characterized by k = 3 and k0 = 0.00012. These parameters are calibrated to

approximately match the Los Angeles hazard curve (Baker, 2015), where the fitted

curves differ from the Los Angeles hazard curve especially for large ground motion

intensities. The initial collapse fragility function combined with these hazard curves

result in mean annual frequency of collapse of 2.3E-4, 2.5E-4, and 2.4E-4, respectively.

The calibration approach presented in this section tries to answer the question

“Given a prior estimate of the fragility curve and the hazard at the site, what are the

most informative IM values at which to conduct structural analyses for computing

λc?” For this purpose, we assume that the target fragility curve is unknown.

The user starts the analysis with an initial fragility function as follows: θ = 1,

β = 0.4. The uncertainty in the median collapse capacity is quantified using the ∆

parameter and βθ is computed using Equation 5.3. It is assumed that users make

a correct quantification of the uncertainty, i.e., the median collapse capacity has a

lognormal distribution which is unbiased and has a coefficient of variation as calcu-

lated. Using the probability density function of the median collapse capacity, we draw

simulations of possible target median collapse capacities. For each sampled median

collapse capacity, we obtain a sampled target collapse fragility curve. Target collapse

fragility curve defines the probability of collapse at any IM level. Using a binomial

distribution, the structural analysis data, in the form of collapses and no-collapses,

can be simulated using this probability. Therefore, we use the sampled target fragility

130


curve to obtain simulated structural analysis data. Simulated structural analysis data

will be referred as pseudo structural analysis data herein. Using the pseudo structural

analysis data, we update the initial fragility function using the Bayesian approach and

obtain the estimated fragility function and estimated λc, which will be represented

with λc. We draw N = 10, 000 simulations of median collapse capacity in order to

obtain N estimates of λc. We study the error in estimating λc, i.e. estimated λc

versus target λc.

The problem can be stated as a minimization problem, where mean squared error

(MSE) is used as the optimality criterion. We aim to find the IM pair, corresponding

to Pcol values on the initial fragility curve, which minimizes the MSE of λc. For the

generalization of results, MSE is evaluated for different values of ∆ and for different

hazard curves.

minimizePcol,1,Pcol,2

MSE =

(1

N

N∑i=1

(λci − λ∗c,i

)2)subject to 0.01 ≤ Pcol,1 ≤ 0.99

0.01 ≤ Pcol,2 ≤ 0.99

|Pcol,2 − Pcol,1| ≥ 0.1

(5.11)

where λc,i and λci represent the estimated and target values of λc for the ith simulation.

We also put a constraint that the two selected Pcol values should be at least 0.1 apart to

ensure that the ground motion intensities have enough separation to give information

on different IM levels.

To illustrate the results, MSE values are shown in Figure 5.3 using contours as

a function of pairs of IM values for the LA site with ∆ = 0.4. The pairs of IM

values resulting in minimum error are indicated with red stars. It is observed that

having the points in the lower tail of the fragility curve (selected points correspond

to probabilities of 0.03 and 0.35) gives minimal error for this exercise.

Pcol

at IM1

0 0.2 0.4 0.6 0.8 1

Pcol a

t IM

2

0

0.2

0.4

0.6

0.8

1×10

-5

1

2

3

4

5

6

Figure 5.2: Contours showing the mean square error corresponding to different pairs of IMvalues. ∆ is used as 0.4. The pairs of IM values resulting in minimum MSE are indicatedwith red stars. IM values are computed corresponding to probabilities of collapse on theinitial fragility curve.

In Figure 5.3, pairs of IM values resulting in minimal MSE for the three sites

considered are plotted in terms of their collapse probabilities on the initial fragility

curve. Blue and red colors show IM1 and IM2, respectively. IM values resulting in

minimum MSE are shown with circles, whereas IM values resulting in MSE values

within 10% of the minimum MSE are shown with dots. Also shown is the best fit

line to the IM values resulting in minimum MSE.

132


∆

0.1 0.2 0.3 0.4 0.5 0.6

Pcolla

pse

0

0.2

0.4

0.6

0.8

1

MinMSE

Pcol,1

MinMSE

Pcol,2

Best Fit Line to MinMSE

MinMSE

±10%*MinMSE

Pcol,1

Figure 5.3: Pairs of IM values giving minimal MSE values. IM values are computedcorresponding to probabilities of collapse on the initial fragility curve. IM values resultingin minimum MSE are shown with circles, whereas IM values resulting in MSE values within10% of the minimum MSE are shown with dots. Also shown is the best fit line to the IMvalues resulting in minimum MSE.

From Figure 5.3, we see that the difference between IM1 and IM2 grows as ∆ gets

bigger. Bigger ∆ values correspond to users having more uncertainty on the initial

collapse fragility curve. As the uncertainty grows, to better predict the fragility

function, one needs to cover a large portion of ground motion intensities. Baker

(2015) recommends choosing IM levels near the lower tail up to slightly above median.

Selection of IM levels in the Bayesian approach agrees with his recommendation

as the uncertainty in the initial collapse fragility functions increase. However, for

small values of ∆, we see that the IM pairs are concentrated in the lower tail of the

fragility function. In other words, as the users become more confident in the initial

information, it is recommended to constrain the lower tail of the fragility function by

conducting analysis using ground motions characterizing this portion. This is inline

with the previous research by Eads et al. (2013) showed that collapse response is

governed by the lower tail of the collapse fragility functions. We also observe that

IM1 selected on the lower tail of the fragility function, i.e. Pcollapse ≤ 0.1, gives

minimal errors.

Based on Figure 5.3, we recommend that IM1 is selected corresponding to prob-

ability of collapse of 10% or lower on the initial collapse fragility curve. Select IM2

such that it corresponds to probability of collapse between 25% and 75% on the initial

fragility curve depending on ∆. We also note that selection of IM levels depends on

various factors. One factor is related to selecting ground motions with high intensi-

ties. Ground motions with high intensities are rarer in databases, which makes their

selection harder. Another factor is related to seismic hazard deaggregations. USGS

provides seismic hazard deaggregations for sites in conterminous US (USGS, 2015a)

and these deaggregations are provided for distinct values of exceedance probabili-

ties. The deaggregations combined with the reliability of seismic hazard analysis for

ground motion intensities corresponding to smaller frequencies (i.e. annual frequency

of exceedance smaller than 1x10−5) might play a role in the selection of hazard con-

sistent ground motions. Note that our recommendation should be adapted to the

problem of interest given the aforementioned factors.

5.4 Sensitivity to Number of Structural Analyses

Collection of likelihood data to update prior belief is essential to obtain robust es-

timations in Bayesian data analysis. In this section, we present the sensitivity of

posterior estimates to the number of structural analyses.

For the sensitivity analysis, an initial collapse fragility curve characterized by θ = 1

g and β = 0.4 is assumed. The IM values, at which structural analyses are conducted,

are selected as follows: IM1 is selected as corresponding to 5% on the initial fragility

curve. IM2 is chosen such that for ∆ = 0.2 and ∆ = 0.6, it corresponds to collapse

probabilities of 35% and 75% on the initial fragility curve, respectively. For ∆ values

between 0.2 and 0.6, the aforementioned two points are linearly interpolated.

As in the case of calibration of IM values, we assume that the target fragility

134


curve is unknown but that users provide correct quantification of their uncertainty,

i.e., median collapse capacity has a lognormal distribution that is unbiased and has a

coefficient of variation as a function of ∆. Using the distribution of median collapse

capacity, which is quantified by the user, we draw N = 10, 000 simulations of possible

target median collapse capacities. For each sampled median collapse capacity, we

obtain a sampled target collapse fragility curve using β = 0.4. Then, we sample

realizations of structural analyses from this curve. We update the initial fragility

function and study the error in λc.

The sensitivity of the number of structural analyses is assessed using mean squared

error (MSE) in λc. In other words, for different values of the number of structural

analyses, MSE of N = 10, 000 λc estimates are obtained. We use the hazard curve at

downtown LA and ∆ values of 0.2, 0.3, 0.4 and 0.6. It is assumed that α = 10%.

Figure 5.4 shows the improvement in MSE of λc with the number of structural

analyses (n). The improvement in MSE is quantified using a ratio of posterior MSE

to initial MSE. Initial MSE corresponds to the case in which no structural analysis is

conducted (n = 0). In other words, this is the MSE obtained using the initial fragility

function.

From Figure 5.4 , we see that with n = 30, MSE is reduced to 0.2 to 0.4 of

the original MSE for different values of ∆. Around n=30, we also observe that the

improvement in MSE saturates for ∆ ≥ 0.4.

5.5 Robustness Analyses of Proposed Method

In this section, we present a robustness analyses of the Bayesian method by studying

statistical properties of the estimated parameters. We perform various tests to study

the accuracy and variability of λc estimates under different user choices. These choices

are in terms of ∆ and initial collapse fragility functions.

Before performing any tests, we first select a target fragility function. For consis-

tency with the previous sections, we choose a target fragility function characterized

by θ∗ = 1 g and β∗ = 0.4. We adopt a Monte Carlo approach and simulate pseudo

structural analysis data at the IM levels of interest. IM1 is selected as corresponding

# of Structural Analyses at each IM

0 30 50 100 150

MS

EP

oste

rio

r/MS

EIn

itia

l

0

0.2

0.4

0.6

0.8

1

∆=0.2∆=0.3

∆=0.4

∆=0.5∆=0.6

Figure 5.4: Sensitivity of number of structural analyses for different values of ∆ for thesite at LA

136


to 5% on the initial fragility curve. IM2 is chosen such that for ∆ = 0.2 and ∆ = 0.6,

it corresponds to collapse probabilities of 35% and 75% on the initial fragility curve,

respectively. For ∆ values between 0.2 and 0.6, the aforementioned two points are lin-

early interpolated as used in the previous section. Note that for different user choices,

ground motion intensities IM1 and IM2 vary, and we sample structural analysis data

at the corresponding IM levels. We use the proposed method to find an estimate of

λ∗c and study the accuracy and efficiency of these estimates. The tests are conducted

using a hazard curve at downtown LA. It is assumed that α = 10%.

In the first part of the robustness analyses, we assume that users start with an

unbiased estimate of the collapse fragility curve. In other words, for the given setting

of the problem, the initial fragility function is characterized by θ = 1 and β = 0.4. We

assess the statistical efficiency of λc, using different uncertainty quantifications, i.e.

∆ values. 30 structural analyses are used at each IM level. We simulate N = 10, 000

possible realizations of structural analyses data at the two chosen IM levels. Bayesian

method is applied to the initial fragility functions using the pseudo structural analysis

data resulting in N = 10, 000 estimates of collapse fragility function and λc. Figure

5.5 shows the boxplots of λc for ∆ values of 0.2, 0.3, 0.4, 0.5 and 0.6. Also shown in the

figure is the boxplot of the N = 10, 000 estimates of λc obtained using a traditional

maximum likelihood method (MLE) using the same structural analysis data with the

case indicated with ∆ = 0.6.

λc

×10-4

0

1

2

3

4

5

Bayesian ∆=0.2 ∆=0.3 ∆=0.4 ∆=0.5 ∆=0.6 MLE

Target λc

Figure 5.5: Distribution of λc estimates obtained using Bayesian Method with different ∆values and MLE method. For initial fragility function, an unbiased median collapse capacityθ = 1 g and β = 0.4 are assumed. Target λc is shown using a blue solid line.

Table 5.1: Median and coefficient of variation in estimated values of λc with different ∆values and MLE method. For initial fragility function, an unbiased median collapse capacityθ = 1 g and β = 0.4 are assumed.

∆ = 0.2 ∆ = 0.3 ∆ = 0.4 ∆ = 0.5 ∆ = 0.6 MLE

Median λc/λ∗c 1.01 1 0.98 0.99 0.99 0.98

Cov of λc 0.12 0.17 0.22 0.26 0.29 0.46

From Figure 5.5 and Table 5.1, we see that if used with an initial unbiased estimate

of the collapse fragility function, Bayesian method provides an unbiased estimate of

λc. We see that the variability in the λc is reduced significantly using the Bayesian

approach compared to MLE. We obtain coefficient of variation values ranging between

0.1 and 0.3 using the Bayesian method for ∆ values between 0.2 and 0.6. Whereas,

using the same structural analyses data, MLE produces a coefficient of variation of

0.44. The variability in MLE estimates is improved if one uses the IM levels recom-

mended by Baker (2015), i.e. IM values corresponding to 10% and 70% probabilities

138


of collapse. Using these IM levels on the unbiased initial guess, we obtain a coefficient

of variation of 0.26 in 10000 λc estimates obtained using MLE with 200 structural

analyses in total. Using a ∆ of 0.5, we obtain similar variability in λc estimates using

the Bayesian method with 60 structural analyses in total. In other words, similar

variability is obtained with 70% reduction in the total number of structural analyses

using the Bayesian method.

In the second part of the robustness analyses, we assume that users start with a

biased estimate of the collapse fragility curve. In other words, for the given setting

of the problem, the initial median collapse capacity estimate is not equal to θ∗ = 1.

We further assume β = 0.4 and ∆ = 0.5. We assess the statistical efficiency of

λc estimator using different θ values. We again simulate 10000 possible realizations

of structural analyses data at the two chosen IM levels. Bayesian method is applied

using the pseudo structural analysis data to obtain 10000 estimates of collapse fragility

function and λc. Figure 5.6 shows the boxplots of λc estimates using θ values of 0.7,

0.9, 1.1 and 1.3. Also shown in the figure is the boxplot of the 10000 estimates of λc

obtained using an unbiased initial guess with θ = 1, for comparison.

λc

×10-4

0

1

2

3

4

5

θ=0.7 θ=0.9 θ=1 θ=1.1 θ=1.3

Target λc

Figure 5.6: Distribution of λc estimates obtained using Bayesian Method with different θvalues. For initial fragility function, ∆ = 0.5 and β = 0.4 are assumed. Target λc is shownusing a blue solid line.

Table 5.2: Median and coefficient of variation of estimated values of λc with different θvalues. For initial fragility function, ∆ = 0.5 and β = 0.4 are assumed.

θ = 0.7 θ = 0.9 θ = 1 θ = 1.1 θ = 1.3

Median λc/λ∗c 1.19 1.03 0.99 0.96 0.90

Cov of λc 0.23 0.24 0.26 0.26 0.26

From Figure 5.6 and Table 5.2, we see that Bayesian method introduces bias to λc

estimates if used with an initial biased estimate of the collapse fragility function. We

see that the variability in λc estimates is similar for different θ values; the coefficient

of variation is observed to be on the order of 0.25. We also observe that the amount

of bias introduced in the Bayesian method affects the amount of bias observed in

λc. We see that median of λc estimates differ by 3-4% for an initial estimate that

is biased with 10%. The bias in λc estimates range between 10% to 19% for a 30%

biased initial guess.

In general, models aim to balance complexity and prediction accuracy. This results

in the common phenomena of bias-variance trade-off. Bias measures the ability of

our models to capture reality and is a result of our lack of knowledge. Variance is a

measure of consistency in the estimated parameters. High bias/low variance methods

are commonly used in machine learning and they are especially preferable for small

data sets. One common biased machine learning technique is regularized regression. It

incorporates a prior belief about the estimated parameters, which improves prediction

accuracy by reducing the variance of the classifier with a price paid of introducing

bias (Hastie et al. , 2009).

5.6 Model Checking

Bayesian data analysis starts with a probability model to incorporate prior knowledge

of the problem and collects data to resolve a full posterior probability model. The

assumptions in the analysis are critical: any poor assumption can lead to inaccurate

inferences. Therefore, diagnosis of the model and identification of the inadequacies,

140


if any, is an important aspect of statistical modeling. In the next section, we present

a model checking method for assessing the plausibility of the posterior distributions.

5.6.1 Posterior Predictive Checking

Posterior predictive checking involves comparing the observed structural analyses

data to predictions obtained using the posterior distribution. The goal is to identify

any systematic discrepancies and evaluate the adequacy of the fit of the posterior

distribution.

The predictions used in the checking of the model are obtained from a posterior

predictive distribution, which is defined as follows:

P (Xrepi |Xi) =

∫P (Xrep

i |τi)P (τi|Xi)dτi (5.12)

where Xi are the observed data and P (τi|Xi) is the posterior distribution that is

obtained by updating the prior distribution using Equation 5.8. Xrepi are the repli-

cated values of Xi and are obtained as follows: First simulate τi from the posterior

distribution, P (τi|Xi). Then, use τi to simulate Xrepi from the likelihood function

P (Xi|τi).Replicated values are used to represent the data that would be observed if the

experiment were repeated. Thus, P (Xrepi |τi), in fact, defines a sampling distribution

for Xi given τi. The uncertainty in sampling Xi values given τi and parametric uncer-

tainty of τi are propagated to obtain a posterior predictive distribution, P (Xrepi |Xi)

(Lynch & Western, 2004).

Random samples are drawn from the posterior predictive distribution, which is

given in Equation 5.12, and they are compared to observed data using histograms and

measures of statistical significance using test quantities. Test quantities are defined

in order to make formal comparisons of observed (Xi) and replicated data (Xrepi )

in terms of scalar measures. Some test quantities, T (Xi, τi), that can be used for

posterior predictive checking include mean, standard deviation and skewness.

The adequacy of the fit is measured in terms of tail-area probabilities called

Bayesian p-values. Bayesian p-values (pB) can be obtained using test quantities using

Equation 5.13:

pB = P (T (Xrep,si , τi) ≥ T (Xi, τi)|Xi) (5.13)

Bayesian p-values define the plausibility of the replicated data given the observed

data. pB near 0 or 1 are statistically significant. If extreme tail probabilities (i.e. pB

of less of 0.05 or greater than 0.95) are observed, the model is less likely to capture the

feature that is observed in the data and being measured by the test statistics. Note

that statistically significant p-values might be due to model inadequacy or random

chance (Gelman et al. , 2013).

It is common practice to use multiple test statistics and assess the capability of

the model to capture different features present in the data. If multiple test statistics

are used, the resulting pB values should be adjusted with a multiple comparisons

adjustment factor.

A pseudo-algorithm is provided below for posterior predictive checking.

for each IMi do

for s=1,...,N=10,000 doDraw τ si = P (C|IMi) from the posterior distributions that are defined

by P (τi|Xi) ∼ Beta(αi∗, βi

∗) ;

Draw Xrep,si = {Xrep,s

i1, Xrep,s

i2, ..., Xrep,s

ini} as independent Bernoulli

random variable with probability τ si = P (C|IMi) ;

Compute T (Xrep,si , τi) ;

end

end

compute observed T (Xi) ;

compare the histogram of T (Xrep,si , τi) with respect to T (Xi) ;

compute Bayesian p-value ;Algorithm 1: Posterior predictive checking for the Bayesian approach

5.6.2 Numerical Example

In this section, we apply the posterior predictive checking method to an example case.

We assume that the target fragility function is characterized by θ∗ and β∗ of 1 g and

142


0.4, respectively. We apply the Bayesian method to two different scenarios. In the

first scenario, ∆ = 0.2 and initial guess for θ is a very good guess with θ = 1 g. In

other words, the user’s fragility function estimate is unbiased with a high confidence.

In the second scenario, the user makes a very biased initial guess but again has high

confidence; for Scenario 2 we assume θ = 0.5 g with ∆ = 0.2. The fragility curves for

the two Scenarios are shown in Figure 5.7 in comparison to the target fragility curve.

We generate pseudo structural analysis from the target fragility function and use

that apply the Bayesian method to both cases. Later, posterior predictive checking

is applied to check the plausibility of final estimates.

Sa (T1, 5%)

0 0.5 1 1.5 2

Pco

llap

se

0

0.2

0.4

0.6

0.8

1

Target Fragility

Scenario 1

Scenario 2

Figure 5.7: Fragility curves and uncertainty in median estimates for Scenarios 1 and 2 areplotted with respect to target fragility curve.

Figures 5.8 and 5.9 show the histograms of mean, standard deviation and skewness

values of τ2 values, which are obtained for IM2 using posterior predictive distribution

under Scenarios 1 and 2, respectively. The mean, standard deviation and skewness

of the observed structural analysis data at IM2 are indicated with the red solid lines.

Table 5.3 lists the Bayesian p-values for this exercise.

Mean of τ2

0 5 10 15 200

1000

2000(a)

Standard Deviation of τ2

0 0.5 1 1.5 2 2.5 3Fre

quency

0

1000

2000(b)

Skewness of τ2

0 0.2 0.4 0.6 0.80

1000

2000(c)

Figure 5.8: Histograms of test statistics obtained using posterior predictive distributionfor Scenario 1. The mean, standard deviation and skewness of the observed structuralanalysis data at IM2 are indicated with the red solid lines.

Mean of τ2

0 2 4 6 8 10 12 14 16 180

1000

2000(a)


0 0.5 1 1.5 2 2.5Fre

quency

0

1000

2000(b)

Skewness of τ2

0 0.2 0.4 0.6 0.80

1000

2000(c)

Figure 5.9: Histograms of test statistics obtained using posterior predictive distributionfor Scenario 2. The mean, standard deviation and skewness of the observed structuralanalysis data at IM2 are indicated with the red solid lines.

144


Table 5.3: Bayesian p-values obtained using different scenarios

Bayesian p-values

Scenario θ Mean of τ2 Standard Deviation of τ2 Skewness of τ2

1 1 g 0.63 0.63 0.49

2 0.5 g 0.98 0.98 0.07

We observe from Figure 5.8 that the observed data is plausible under the posterior

predictive distribution for Scenario 1. The same observation does not hold true for

Scenario 2. There is significant discrepancy between the observed value and the

posterior predictive distribution in Figure 5.9. From Table 5.3, we also observe that

Scenario 2 has significant pB values for mean and standard deviation. These are

indicators that the fit of the model obtained for Scenario 2 is insufficient.

5.7 Discussion on Obtaining Prior Information

There are various simplified methods to compute collapse capacity of structures.

A major portion of these simplified methods estimate collapse capacity based on

pushover analysis on the MDOF structure (Vamvatsikos & Cornell, 2005; FEMA,

2009a; Han et al. , 2010; Shafei et al. , 2011; Fajfar & Dolsek, 2014). Pushover analy-

sis is generally conducted before conducting any nonlinear dynamic history analyses

and therefore initial collapse capacity estimates based on such methods are assumed

not to add to the computational demand. Furthermore, Perus et al. (2013) provide a

database of approximate IDA curves based on pushover response of SDOF structures.

This database can be used to obtain an initial estimate of IDA behavior.

There are also empirical collapse fragility functions based on building typologies,

which can be used to obtain an initial estimate of the collapse capacity of a building

(Jaiswal et al. , 2011; FEMA, 2012). Furthermore, for seismic evaluation purposes,

the information on code-conformability and the corresponding code requirements can

also be used as prior information to obtain an initial estimate.

Azarbakht & Dolsek (2007, 2011); Eads (2013) provide methods to limit the com-

putational demand of incremental dynamic analysis on the MDOF structure. Han &

Chopra (2006) also provides a method to estimate collapse capacity based on non-

linear dynamic history analyses on the SDOF structure. These methods can also

be used to estimate initial fragility curve estimates; however, they involve nonlinear

dynamic analyses and add to the computational demand.

5.8 Collapse Risk Assessment of a 4-Story Rein-

forced Concrete Moment Frame Building

In this section, we illustrate the Bayesian method to assess the collapse risk of a

4-story reinforced concrete moment frame building.

5.8.1 Building Site and Seismic Hazard Characterization

The structure is located south of downtown Los Angeles at 33.996◦N , 118.162◦W . The

site is surrounded by seven faults within a distance of 20 km. Shear wave velocity

(Vs,30) is estimated to be 285 m/s (Goulet et al. , 2007). The seismic hazard curve is

estimated using USGS seismic hazard analysis tools (USGS, 2015b).

5.8.2 Structural Modeling and Analysis

The case study structure is a 4-story reinforced concrete (R/C) special moment frame

structure designed by Haselton (2006). The design of the structural system is in

compliance with the 2003 International Building Code (IBC, 2003) and ASCE 7-02

provisions (American Society of Civil Engineers, 2002).

The structural system is modeled using a concentrated plasticity idealization.

Frame elements are modeled as elastic members having rotational springs at the

ends, whose hysteretic behavior is governed by a trilinear backbone curve and cyclic

and in-cycle degradation rules (Ibarra et al. , 2005). The parameters defining this

model are determined using empirical equations that are calibrated using a dataset

consisting of experimental test results of reinforced concrete columns (Haselton et al.

, 2008). A Rayleigh damping of 3% is applied to the first and third modes of the

146


structure. P-∆ effects are modeled using a leaning-column. The fundamental modal

period of the structure is 0.94 s. The system is analyzed using Open System for

Earthquake Engineering Simulation Platform (OpenSEES, 2015).

Nonlinear static analysis is a state-of-the-art procedure generally recommended

before conducting dynamic analysis to check the nonlinear analysis model and/or

augment dynamic analysis results to better understand structural response (Deier-

lein et al. , 2010). Vamvatsikos & Cornell (2005) propose an efficient method for

estimating collapse fragility functions. This method is called SPO2IDA and the re-

sults from this method can be used as an initial estimate for the Bayesian method.

SPO2IDA uses the results from nonlinear static pushover (SPO) analysis to make

inference about dynamic analysis results by providing a connection between SPO

curves and incremental dynamic analysis (IDA) curves. SPO2IDA summarizes the

results in terms of 16%, 50% and 84% fractile IDA curves. The behavior tracked in

IDA curves can be linked to collapse response and the fractals of IDA curves can be

used to obtain an initial estimate of the collapse fragility function.

In this study, we use SPO2IDA to obtain an initial estimate of the collapse fragility

curve. Since SPO analysis is conducted before any dynamic analysis, we assume that

it does not add to the computational demand of structural analyses required to obtain

collapse risk estimates.

The pushover curve of the case study structure are shown in Figure 5.10 along

with a fitted trilinear model. Also shown in Figure 5.10 are the 16%, 50% and 84%

fractile IDA curves obtained using SPO2IDA for the 4-story R/C structure.

0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

Sa

(T1=

0.9

4 s

,5%

)

µ=δ/δyield

IDA−16%

IDA−50%

IDA−84%

Static Pushover

Fitted Trilinear

Collapse Fragility

Figure 5.10: 16, 50 and 84% fractile IDA curves from SPO2IDA along with an estimateof collapse fragility curve for the 4-story R/C structure

IDA curves suggest dynamic instability when their slope approaches 0. We see

that the median IDA curve reaches a slope of 0 at a capacity of 2.19 g. Using the

estimated fractal IDA curves, a collapse fragility curve having a median collapse

capacity of 2.19 g and dispersion of 0.43 is obtained. This estimate is also shown in

Figure 5.10. Since SPO2IDA does not include modeling uncertainty, the dispersion of

0.43 is inflated using a square root of sum of squares approach with dispersion due to

modeling uncertainty of 0.35. This value is determined from the analysis of reinforced

concrete frame structures in Chapter 6, i.e. we recommend using a dispersion due to

modeling uncertainty as 0.35. A final dispersion value of 0.55 is obtained. We use

∆ = 0.4 for quantifying the uncertainty in median collapse capacity estimate using a

90% confidence interval.

5.8.3 Ground Motion Selection

Thirty ground motions are selected to match the conditional spectra at 2 different

IM levels (Jayaram et al. , 2011). 5%-damped spectral acceleration at the funda-

mental mode period of the structure Sa(T1 = 0.94s, 5%) is used for IM. As noted

148


previously, IM1 is recommended to be at or below 10% probability of collapse on

the initial fragility curve. For the site where the structure is located, the ground

motion intensity with a probability of exceedance of 2% in 50 years is selected for

IM1 , which has a probability of collapse of 9% on the initial fragility function and

has Sa(T1 = 0.94s, 5%) of 1.05 g. IM2 is recommended to be selected such that it

corresponds to probability of collapse between 25% and 75% on the initial fragility

curve depending on ∆. The highest ground motion intensity that can be obtained

from USGS Interactive deaggregation tool (USGS, 2015a) corresponds to a probabil-

ity of exceedance of 1% in 200 years. This intensity is 1.96 g for the given site and

corresponds to 42% on the initial fragility curve. We select this intensity as IM2.

5.8.4 Modeling Parameter Uncertainty

Six parameters defining the behavior of a component (Ibarra et al. , 2005) are treated

as random variables. These parameters are flexural strength (My), ratio of maxi-

mum moment and yield moment capacity (Mc/My), effective initial stiffness which is

defined by the secant stiffness to 40% of yield force (EIstf,40/EIg), plastic rotation

capacity (θcap,pl), post-capping rotation capacity (θpc) and energy dissipation capac-

ity for cyclic stiffness and strength deterioration (γ). The variability in the modeling

parameters is represented by the following: logarithmic standard deviations for θpc,

θcap,pl, γ, My, EIstf,40/EIg and Mc/My are equal to 0.73, 0.59, 0.5, 0.31, 0.27 and

0.1, respectively (Haselton et al. , 2008; Ugurhan et al. , 2014). Equivalent viscous

damping ratio, column footing rotational stiffness and joint shear strength are also

treated as random having logarithmic standard deviations of 0.6, 03 and 0.1 (Hasel-

ton, 2006), respectively. The correlation structure among the random variables are

obtained from Ugurhan et al. (2014). In this model, beam-to-beam and column-

to-column correlations are idealized as perfect correlation; whereas beam-to-column

and within component correlations are idealized by correlation coefficients that are

derived using random effects regression. Thirty realizations of model parameters are

obtained using Latin Hypercube sampling.

5.8.5 Results

Ground motions in each suite are scaled to the corresponding IM levels and are

matched with model realizations in order to conduct nonlinear time history analyses.

The number of collapses observed are 5 and 13 out of 30 analyses at IM1 = 1.05 g

and IM2 = 1.96 g, respectively.

At IM1, prior information leads to a distribution of Beta(α1 = 2.35, β1 = 14.25)

using equation 5.4. Using the Bayesian approach along with the observed data of

5 collapses out of 30 analyses, this distribution is updated to Beta(α1 = 7.35, β1 =

39.25) using 5.9.

Similarly, at IM2, prior information leads to a distribution of Beta(α2 = 3.98, β2 =

5.09). Using Bayesian approach along with the observed data of 13 collapses out of

30 analyses, this distribution is updated to Beta(α2 = 16.98, β2 = 22.09).

Using the maximum likelihood estimation method, where the posterior distribu-

tions at the two IM levels are used as probability distributions defining the likelihood

function, the final collapse fragility curve is obtained to have a median collapse ca-

pacity of 2.22 g and dispersion of 0.7.

Figure 5.11 shows the initial and posterior collapse fragility functions along with a

collapse fragility function obtained using maximum likelihood estimation. Also shown

are the analysis data, prior and posterior distributions at the considered IM levels.

150


Sa (T1, 5%)

0 0.5 1 1.5 2 2.5 3 3.5

Pcolla

pse

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Initial Fragility

Prior Dist.

Analysis Data

Posterior Dist.

Posterior Fragility

MLE Fragility

Figure 5.11: Bayesian method illustrated on a 4-story reinforced concrete frame structurein comparison with maximum likelihood estimation.

From Figure 5.11, the following observations are made for this exercise: Initial

fragility function and the analysis data agree well at IM2. This resulted in prior

and posterior distributions at IM2 centered at similar points, and the variability

in the posterior distribution is reduced because of the good agreement between the

analysis data and the initial information. At IM1, we see that the analysis data

shifted the posterior distribution towards higher probabilities. As a result, the initial

estimate of the dispersion and median collapse capacity is increased by 27% and 1.4%,

respectively.

To check the plausibility of the posterior distributions, posterior predictive check-

ing, which is discussed in Section 5.6.1, is applied. The histograms of the test statistics

at IM1 and IM2 are shown in Figure 5.12 along with the observed values of the struc-

tural analysis data. From Figure 5.12, no significant discrepancy is observed between

the posterior predictive distribution and the analysis data, and no additional analysis

is needed.

(a)

Mean of τ1

0 5 10 150

1000

2000(a)


0 0.5 1 1.5 2 2.5Fre

quency

0

1000

2000(b)

Skewness of τ1

0 0.2 0.4 0.6 0.80

1000

2000(c)

(b)

Mean of τ2

5 10 15 20 250

1000

2000(a)


1.5 2 2.5 3Fre

quency

0

2000

4000(b)

Skewness of τ2

-0.4 -0.2 0 0.2 0.4 0.60

1000

2000(c)

Figure 5.12: Histograms of test statistics obtained using posterior predictive distributionfor a) IM1 b) IM2. The mean, standard deviation and skewness of the observed structuralanalysis data at IM2 are indicated with the red solid lines.

Table 5.4 shows the Bayesian p-values. None of the p-values are statistically

significant, which confirms model adequacy.

Table 5.4: Bayesian p-values obtained for IM1 and IM2

Bayesian p-values

Mean of τi Standard Deviation of τi Skewness of τi

IM1 0.49 0.49 0.65

IM2 0.55 0.44 0.55

5.8.6 Discussions

To benchmark the collapse response of the case study structure, we conduct an ex-

tensive collapse response analysis of the structure incorporating ground motion and

modeling uncertainties. We select 200 ground motions consistent with conditional

spectra at five IM levels. Table 5.5 summarizes the ground motion intensity levels

used for benchmarking collapse response of the frame structure.

152


Table 5.5: IM levels that are used for benchmarking collapse response of the 4-Story R/Cframe structure

Probability of Exceedance Sa (T1 = 0.94s, 5%) (g) Fraction of Collapses

5% in 50 years 0.74 0.03

2% in 50 years 1.05 0.13

1% in 50 years 1.32 0.27

1% in 100 years 1.63 0.37

1% in 200 years 1.98 0.41

200 model realization are obtained using Latin Hypercube sampling. Each ground

motion is matched with a model realization, and in total 1000 nonlinear time history

analyses are conducted. Using a maximum likelihood estimator, we obtain the col-

lapse fragility curve of the structure having a median of 2.09 g and a dispersion of

0.61.

Figure 5.13 shows the target fragility function along with its data. Also shown

in red is the collapse fragility function obtained using the Bayesian approach. Table

5.6 summarizes the results obtained using the aforementioned approaches in terms of

different collapse metrics. Listed in Table 5.6 are the collapse fragility parameters θ

and β along with λc, and the probability of collapse in 50 years, which is obtained

using a Poisson assumption.

Sa (T1, 5%)

0 0.5 1 1.5 2 2.5 3 3.5

Pcolla

pse

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Target Fragility

Target Data

Posterior Fragility

Figure 5.13: Target collapse fragility curve along with the curve estimated using aBayesian approach for the 4-story reinforced concrete frame structure.

Table 5.6: Collapse risk estimates

Estimator θ (g) β λc(10−5) Pcol50years

Initial 2.19 0.55 17.72 0.0088

Bayesian 2.22 0.70 31.31 0.0155

MLE 2.24 0.78 43.11 0.0213

Target 2.09 0.61 25.37 0.0126

Table 5.6 shows that the Bayesian method slightly shifts the median collapse

capacity away from the target value. Although the initial estimate overestimates the

median collapse capacity by 5%, updating the initial curve increases this difference

to 6%. We also see that dispersion of the target fragility function is overestimated by

15% by the Bayesian approach.

Although median collapse capacities and dispersions of the target and the Bayesian

estimate differ, Figure 5.13 shows that the lower portion of the fragility function, up

154


until 40%, is well-constrained by the Bayesian method. Table 5.6 shows that the

Bayesian method provides a good estimate of λc. This is because of the good match

in the lower portion of the fragility function, since the lower tail of the fragility

function is an important contributor to λc as previously noted by Eads et al. (2013).

For the case study, the Bayesian method starts with an initial estimate of λc

which underestimates the target λc by 30%. After applying by the method, the final

estimate of λc differs from the target response by 25%. This difference is 70% using

an MLE approach.

5.9 Conclusions

In this paper, we have presented an efficient method for collapse risk assessment.

The method uses Bayesian updating and incorporates prior information on structural

collapse capacity. The prior information on collapse response can be informed by

a variety of sources, including information on the building design criteria, empirical

fragility functions or nonlinear static (pushover) analysis. This information is later

updated using the results from a few structural analyses, and posterior estimates of

collapse risk are obtained. The method is tested using simulated structural analysis

data and statistical properties of the estimators are assessed. Further, we have illus-

trated the method using 4-story reinforced concrete frame structure for assessing its

collapse risk.

It is observed that Bayesian method results in considerable reduction in variabil-

ity in λc estimates compared to the conventional method of maximum likelihood.

Using the Bayesian method, 70% reduction in the computational demand, which is

measured in terms the total number of structural analyses conducted, is obtained to

achieve similar variability in λc estimates. Initial guesses play a crucial role in the

accuracy of the estimates. It is observed that biased initial guesses result in biased

estimates of collapse risk and the amount of bias depends on the bias introduced

in the initial guess. We have also provided a model checking method, called poste-

rior predictive checking, for assessing the plausibility of posterior distributions. The

approach also enables propagating ground motion variability and model uncertainty

through efficient sampling of model realizations. The Bayesian method is observed

to significantly improve statistical efficiency of collapse risk predictions compared to

alternative methods, and provide considerable reduction in computational demand

for probabilistic collapse risk assessment of structures.

156

Chapter 6

Quantifying the Impacts of

Modeling Uncertainties on the

Seismic Drift Demands and

Collapse Risk of Buildings with

Implications on Seismic Design

Checks

This chapter is based on:

Gokkaya, B.U., J.W. Baker, and G.G. Deierlein (2015). Quantifying the Impacts of

Modeling Uncertainties on the Seismic Drift Demands and Collapse Risk of Buildings

with Implications on Seismic Design Checks, Earthquake Engineering & Structural

Dynamics (accepted for publication).

6.1 Abstract

Robust estimation of structural collapse risk using nonlinear structural analysis should

consider the uncertainty in modeling parameters as well as variability in earthquake

157

ground motions. In this paper, we illustrate incorporation of the uncertainty in

structural model parameters in nonlinear dynamic analyses to probabilistically as-

sess story drifts and collapse of buildings under earthquake ground motions. Monte

Carlo simulations with Latin Hypercube sampling are performed on a set of ductile

and non-ductile reinforced concrete building archetypes to quantify the influence of

modeling uncertainties and how it is affected by the ductility and collapse modes of

the structures. The uncertainties considered include variability in model parameters

for structural component stiffness, strength, ductility and energy dissipation, which

are derived from test data and include correlations among the parameters. Inclusion

of modeling uncertainty is shown to increase the mean annual frequency of collapse

by about 1.7 times, as compared to analyses based on median model parameters,

for a high-seismic site in California. Modeling uncertainty has a smaller effect on

drift demands at levels usually considered in building codes. For example, for the

same set of buildings, modeling uncertainty increases the mean annual frequency of

exceeding story drift ratios of 0.03 by about 1.1 times. A novel method is introduced

to relate drift demands at maximum considered earthquake intensities are related

to collapse safety through a joint distribution of deformation demand and capacity,

taking into account simulated instances of collapse and no-collapse. This framework

enables linking seismic performance goals specified in building codes to drift limits

and other acceptance criteria. The distributions of drift demand at maximum con-

sidered earthquake level and drift capacity of selected archetype structures enable

comparisons with drift limits as specified in the proposed seismic criteria for the next

edition (2016) of ASCE 7. Based on the results of this study, the proposed drift limits

are found to be unconservative, relative to the target collapse safety in ASCE 7.

158

CHAPTER 6. IMPACTS OF MODELING UNCERTAINTIES 159

6.2 Introduction

Owing to the variability in earthquake ground motions and the uncertainty in struc-

tural model idealizations, any efforts to rigorously assess collapse safety should con-

sider these effects. For collapse assessment by nonlinear dynamic analysis, this in-

cludes reliable characterization and propagation of the aforementioned sources of un-

certainty through the analysis. Ground motion variability and its impacts on seismic

response assessment has been studied by many researchers who have proposed ground

motion selection and analysis strategies for taking into account record-to-record vari-

ability (e.g. Bradley, 2010; Jayaram et al. , 2011, and others). Another important

contributor to the robustness of seismic response predictions is modeling uncertainty

(Bradley, 2013). A blind prediction contest to analyze the response of a single bridge

column (Terzic et al. , 2015) highlighted the significance of modeling uncertainty in

structural seismic response predictions. The variability in the engineering parameters

submitted by the contestants were remarkable for this highly constrained experiment.

Maximum displacements were predicted under different earthquake ground motions

with an average coefficient of variation of 0.4 and the median bias of the predictions

corresponding to different ground motions ranged from 5% to 35%.

One of the major challenges in incorporating modeling uncertainty into seismic re-

sponse assessment is balancing computational demand and accuracy. Sensitivity anal-

ysis provides insight regarding relative importance of modeling parameters (Porter

et al. , 2002b; Ibarra & Krawinkler, 2005; Celik & Ellingwood, 2010), and First-

Order Second-Moment (FOSM) is often used to propagate modeling uncertainty in

seismic performance assessment (i.e. Lee & Mosalam, 2005; Haselton, 2006). How-

ever, FOSM can loose accuracy when the relationship between input and response

variables is highly nonlinear, which is a concern when modeling collapse. Methods

such as nonlinear response surface Liel et al. (2009) and Artificial Neural Network

methods have been used to incorporate nonlinearities, although most of these methods

do not scale efficiently or accurately when the number of random variables increase.

Among the available methods, Monte-Carlo based methods remain as the most flex-

ible and scalable methods to account for modeling uncertainty, albeit sometimes at

large computational expense (i.e. Dolsek, 2009; Vamvatsikos & Fragiadakis, 2010; Yin

& Li, 2010).

In this paper, Monte-Carlo based simulations are used to incorporate the vari-

ability of structural model parameters in nonlinear dynamic analyses for assessing

deformations and collapse risk of buildings under earthquake ground motions. A set

of archetype buildings are analyzed to quantify the significance of modeling uncer-

tainty and to examine its implication on design approaches that utilize nonlinear

dynamic analysis. The effects of modeling uncertainty are evaluated for both duc-

tile and non-ductile reinforced concrete structures to investigate whether the extent

to which ductility and strength irregularities influence the significance of modeling

uncertainties. The procedures illustrate and contrast so-called multiple stripe anal-

yses (MSA) versus incremental dynamic analyses (IDA) for ground motion selection

and scaling, each of which has its advantages and limitations. MSA conducted with

hazard consistent ground motions, are used to directly evaluate collapse resistance in

terms of spectral ground motion intensity, whereas IDA are used to establish drift ca-

pacities of the buildings, which can then be used together with drift demands to assess

collapse safety. In addition to outlining consistent procedures for evaluating modeling

uncertainties for collapse risk, results of the study provide benchmark data to help

establish guidelines to account for modeling uncertainties in design and assessment.

The approach to evaluating collapse risk through drift demands is motivated in

part by emerging methods to validate building designs using nonlinear dynamic anal-

ysis. For example, an important global-level safety check in helping ensure that

buildings meet minimum collapse safety targets is through story drifts, which are

determined from nonlinear dynamic analyses, in the guidelines for the seismic design

of tall buildings (PEER, 2010; LATBSDC, 2015) and a recently proposed update for

the next edition of ASCE 7 (American Society of Civil Engineers Standards Com-

mittee, 2015). To investigate a structure’s collapse capacity in terms of drift and

its relationship to calculated drift demands, a probabilistic framework is proposed to

jointly quantify joint drift demands and capacities, including uncertainties associated

with both ground motions and structural modeling parameters. This framework is

employed in nonlinear analyses to characterize the joint distribution of peak drift at

160


maximum considered earthquake level and capacity for selected archetype structures.

The resulting distributions are used to assess the proposed acceptance criteria for new

provisions in ASCE 7 (American Society of Civil Engineers Standards Committee,

2015) and provide a strategy for improved calibration of such acceptance criteria.

6.3 Seismic Performance Assessment

A risk-based seismic assessment strategy is adopted for using nonlinear dynamic anal-

yses to assess the impacts of both ground motion and modeling uncertainty on drift

demands and collapse. Illustrated in Figure 6.1 are key aspects of the assessment

procedure. Referring to Figure 6.1.a, a so-called multiple stripes (MSA) approach

is used, wherein nonlinear dynamic analyses are conducted at multiple ground mo-

tion intensities using ground motions at selected and scaled to multiple intensities

(Jalayer, 2003). We conduct MSA using suites of ground motions that are selected

and scaled to match the unique seismic hazard at a building site in order to ensure

hazard consistency at a site. MSA involves conducting nonlinear time history analyses

at multiple levels of ground motion intensity, where at each intensity the distribution

of peak story drift ratio (SDR) and other engineering demand parameters (EDPs) of

interest are recorded. In this study, the ground motions are selected and scaled using

a conditional spectra (CS) approach, where the ground motion intensity is defined

based on the spectral acceleration at the fundamental period of structure, Sa(T1),

and the CS characterizes the target response spectra, including mean and variability,

that are derived from probabilistic seismic hazard analysis (Lin et al. , 2013b). The

details of ground motion selection are explained in the following sections. Hazard

consistent ground motions are selected at different intensity levels. For the code-

conforming buildings considered in this study, the ground motions are evaluated at

five intensities with frequencies of exceedance ranging from 5% in 50 years to 1% in

200 years.

In Figure 6.1.a, the points at each intensity strip provide data to characterize the

maximum story drift demands and the frequency of collapse. Since the exact onset of

collapse is not detected in the typical MSA procedure, a SDR in excess of 10% is used

sdr 10%

Sa

(T

1)

SDR

(a)

Sa (T1)

P (b)

Sa (T1)

λS

a (

T1)

(c)

Pc ⋅ |d

λS

a (

T1)|

Sa (T1)

(d)

Collapse PDF

βc,T

; Pc,T

βc,RTR

; Pc,RTR

θc,T

θc,RTR

PSDR>sdr

PSDR>sdr

k

Sa2/50

λc,T

λc,RTR

0.5

10/50 yr

2/50 yr

Figure 6.1: Illustration of seismic performance assessment a) Multiple stripes analyses atdifferent ground motion intensity levels b) Collapse and drift-exceedance fragility functionsc) Seismic hazard curve d) Collapse risk deaggregation curves

as indicator of structural collapse because at this level most engineering structures are

identified to have no or insufficient stiffness (Vamvatsikos & Cornell, 2002). Thus, the

incidences of collapse at each intensity can be used to compile vertical statistics to fit

a lognormal distribution of collapse probability, in terms of Sa(T1). This distribution

is illustrated by the collapse probability density function (PDF) in Figure 6.1.a and

the corresponding collapse cumulative distribution function (CDF) in Figure 6.1.b.

Data at each ground motion intensity strip are used to determine horizontal statistics

of drift exceedance probability distributions, shown by the drift PDF in Figure 6.1.a.

The details are explained in Section 6.6. Integrating regions of the drift PDF at

162


each intensity level, such as the P (SDR > sdr) shown in Figure 6.1.a, can then be

used to develop drift exceedance CDF’s, such as shown in Figure 6.1.b. As described

later in Sections 5 and 6, horizontal statistics of the drift data become relevant when

evaluating drift demands at specified ground motion intensities.

Collapse fragility curves obtained using MSA data are shown by the blue lines

in Figure 6.1.b, where two estimates of collapse fragility curves are provided. One

estimate, shown by the dashed line, is obtained by analyses using median model pa-

rameters. As such, this estimate incorporates only record-to-record variability (RTR)

in the ground motions and, hence, carries the subscript RTR. The second estimate,

shown in the solid line, is from analyses that incorporate both record-to-record vari-

ability as well as modeling uncertainty and carries the subscript T , indicating it

includes a “total” estimate of uncertainty. Limit states corresponding to collapse and

drift exceedance are indicated with c and SDR ≥ sdr, respectively. An example

drift-exceedance fragility function for SDR ≥ sdr is shown by the gray curve in Fig-

ure 6.1.b, which could in concept also be calculated from analyses that may or may

not include modeling uncertainties.

The drift and collapse fragility curves in Figure 6.1.b are represented by lognormal

distributions, which are described by a median (θ) and logarithmic standard deviation

(also known as dispersion, β). All parameters of interest are used with appropriate

subscripts, wherever applicable, corresponding to their limit states and the type of

uncertainty they include, i.e. θc,T vs. θc,RTR.

Risk-based assessment of structural response provides the mean annual frequency

of exceedance (λ) of certain limit performance states (collapse or drift exceedance)

given the seismic hazard at the design site of the structure. Figure 6.1.c illustrates a

seismic hazard curve. The λ for each limit state can be obtained by integrating the

corresponding fragility curve with the seismic hazard curve as follows:

λLS =

∫ ∞0

P (LS|IM = im)


d(im)

∣∣∣∣ d(im) (6.1)

where LS refers to the limit state of interest, P (LS|IM = im) is the probability of

LS given the ground motion intensity IM = im, and∣∣∣dλIM (im)

d(im)

∣∣∣ defines the absolute

value of the slope of the hazard curve at im.

Deaggregation of λLS helps identify the ground motion intensities that contribute

most in estimating λLS (Eads et al. , 2013). These curves plot the terms of in the

integral in Equation 6.1 with respect to IM . Deaggregation curves of λc with and

without modeling uncertainty are provided in Figure 6.1.d, where the areas under

these curves yield λc.

We see from Figure 6.1.b that, in the presence of modeling uncertainty, collapse

fragility curve has smaller median collapse capacity (i.e. θc,T < θc,RTR) and higher

dispersion (i.e. βc,T > βc,RTR) for the given case. This results in ground motions

having smaller intensities and higher rates of occurrence contributing more to λc as

shown in Figure 6.1.d.

In the following sections, we use the aforementioned seismic performance assess-

ment strategy along with the presented performance metrics to quantify the impacts

of modeling uncertainty for an extensive set of archetype structures.

6.4 Case Study Seismic Response Analysis with

Modeling Uncertainty

6.4.1 Structural Models

Thirty ductile reinforced concrete frame buildings were designed by Haselton (2006)

using modern building code standards (IBC, 2003; American Society of Civil Engi-

neers, 2002) for a high-seismicity site at Los Angeles (LA), California. These buildings

range in height from 1 to 20 stories, including a few designs that have varying amounts

of strength irregularity up the building height. The structural systems are analyzed

using Open System for Earthquake Engineering Simulation Platform (OpenSEES,

2015) using concentrated plasticity models. An illustrative structural idealization is

provided in Figure 6.2.a and the detail of the beam-column connections is provided

corresponding to ductile frames. Frame elements are modeled as elastic members

having rotational springs at the ends, whose hysteretic behavior is governed by a tri-

linear backbone curve (Figure 6.2.b) and cyclic and in-cycle degradation rules (Ibarra

164


M-θ

Ductile Frames: Non-Ductile Frames:

i

j M-θ

V, Δs

P, Δa

i

j

(a)

My

Mc

θ

M

θpc

θcap,pl

EIstf

(b)

V

Vn

δs/L δ

a/L

P

δhor

/L

shear

drift

capacity

shear

strength

axial

failure

(c)

Figure 6.2: a) Illustrative structural idealization of the models used in this study. Detailsof beam-column connections for ductile and non-ductile frames are provided. b) Backbonecurve for concentrated plasticity model c) Illustration of the shear and axial failure models

et al. , 2005). The parameters defining the M-θ hinge model are determined using

empirical equations that are calibrated using a dataset consisting of experimental

test results of over two hundred reinforced concrete columns (Haselton et al. , 2008).

Compatible with the design, Rayleigh damping is used with 3% equivalent critical

damping in the first and third modes of the structure, and P-∆ effects are modeled

using a leaning-column.

Raghunandan et al. (2015) designed non-ductile reinforced concrete frame build-

ing according to the 1967 Uniform Building Code (UBC, 1967) for three sites in Pacific

Northwest region, namely Los Angeles, Portland and Seattle. In this study, we use

three of their non-ductile reinforced concrete frame buildings of 2, 4 and 8 stories in

height designed for a high-seismicity site at LA, CA. Similar to the ductile frames, the

flexural response of beams and columns are modeled using the concentrated plasticity

model as in Figure 6.2.b, but with less ductile properties. In addition, the column

models include zero-length shear and axial springs (Figure 6.2.c) to idealize shear

and axial failure, respectively. Axial springs have limit surfaces defined by the force-

displacement relationship given by Elwood (2004). Shear springs have limit strengths

defined by Sezen (2002) in the small displacement range, which is triggered in the

case of a brittle shear failure; and they have deformation limits defined by Elwood

(2004) to characterize the behavior in the large displacement range. Detailed infor-

mation about the design of buildings can be found in Raghunandan et al. (2015).

Compatible with the design, Rayleigh damping is used with 5% equivalent critical

damping in the first and third modes of the structure, and P-∆ effects are modeled

using a leaning-column.

6.4.2 Seismic Hazard Analysis and Ground Motion Selection

Each of the archetype structures are analyzed based on the seismic hazard at their cor-

responding design sites in Los Angeles, CA (ductile structures at 33.996◦N, 118.162◦W;

non-ductile structures at 34.05◦N, 118.25◦W), where both sites have NEHRP Site

Class D soil. The hazard curves and deaggregation information at all sites are ob-

tained from the USGS hazard curve application (USGS, 2015b) and the 2008 USGS

interactive deaggregations web tool (USGS, 2015a), respectively. Interpolation is

used for periods that are not directly available in the tools. The deaggregation in-

formation include magnitude, distance and ε at given ground motion intensity levels.

Five intensities are used corresponding to exceedance probabilities of 5% in 50, 2%

in 50, 1% in 50, 1% in 100 and 1% in 200 years. Target ground motion spectra are

approximated using this information with the ground motion prediction equation by

Campbell & Bozorgnia (2008) with a conditional spectra approach outlined in Lin

et al. (2013a). Ground motions are selected using an algorithm by Jayaram et al.

(2011) from PEER-NGA database. Suites of 200 ground motions are selected at each

166


intensity level to provide a different motion for each structural realization that will

be considered in the modeling uncertainty analyses.

6.4.3 Modeling Uncertainty

Characterization of Modeling Uncertainty

The six parameters that define the M-θ hysteretic hinge model are treated as random

variables, including the five parameters that define the backbone curve in Figure 6.2.b

and a sixth parameter that defines the cyclic energy dissipation capacity. Following

(Ibarra et al. , 2005), these parameters are the flexural strength (My), ratio of max-

imum moment and yield moment capacity (Mc/My), effective initial stiffness which

is defined by the secant stiffness to 40% of yield force (EIstf,40/EIg), plastic rotation

capacity (θcap,pl), post-capping rotation capacity (θpc) and energy dissipation capacity

for cyclic stiffness and strength deterioration (γ).

For non-ductile frames, the uncertainty in the shear and axial parameters, Vn,

δs/L, δa/L (Figure 6.2.c) are likewise treated as random variables. These parameters

were empirically calibrated and summary statistics of the predictive capacity models

are reported in each respective studies (Sezen, 2002; Elwood, 2004). In addition to

reporting the variability in the parameters, these studies also report small biases in

the model parameters (measured to calculated values of 1.05 for Vn and 0.97 for δs/L,

δa/L) that are incorporated in the uncertainty analyses.

In addition to the parameters defining the beam and column component models,

equivalent viscous damping ratio (ξ), column footing rotational stiffness (Kf ) and

joint shear strength (Vj) are also treated as random. The modeling parameters are

assumed to have lognormal distributions and the variability in these parameters are

represented using logarithmic standard deviations given in Table 6.1.

In addition to variability of the modeling parameters themselves, previous studies

have shown that the assumed correlation between parameters can significantly affect

the calculated collapse behavior (Haselton et al. , 2007). In a previous study, we

use random effects regression models on a database of reinforced concrete column

tests (Haselton et al. , 2008) to quantify the correlations of random model param-

eters within and between the structural components (Ugurhan et al. , 2014). As

described by (Ugurhan et al. , 2014), correlation of parameters within components

(e.g., the relationship of strength to ductility, My to θcap,pl, within a member) were

determined using each of the over two hundred column tests, and the correlation of

parameters between components (e.g., the relationship of parameters for beams ver-

sus columns within a building) were determined by comparing results of specimens

that were constructed and tested at different labs. For the uncertainty analyses in

this study, we assumed parameters between beams within a building (beam-to-beam)

and parameters between columns within a building (column-to-column) to be fully

correlated, the reasoning being that they have the same details and are built by the

same contractor. On the other hand, parameters between beams and columns within

a building (beam-to-column) and within each component model are assumed to be

partially correlated. The correlation coefficients are shown in Table 6.2, following

(Ugurhan et al. , 2014), where the coefficients in the left matrix are for parameters

within components (component i to i) and the right matrix is for parameters between

components (component i to j). These coefficients define the correlations using log-

arithms of the parameters, so that the natural logarithms of the parameters follow

a multivariate normal distribution. Other than the parameters for the beam and

column hinges, all other parameters are assumed to be uncorrelated.

Table 6.1: Logarithmic standard deviations of random variables

Random Variables & Dispersion Values

θcap,pl 0.59EIstfEIg

0.27 My 0.31 Mc

My0.10 θpc 0.73 γ 0.50

ξ 0.60 Kf 0.30 Vj 0.10 Vn 0.15 δs/L 0.34 δa/L 0.26

Propagation of Modeling Uncertainty

Monte-Carlo simulation-based uncertainty propagation methods are intuitive and

straightforward to implement, and aside from the computational expense are fairly

scalable to problems with many random variables. These methods involve drawing

168


Table 6.2: Correlation of random variables defining backbone curve and hysteretic behav-ior of a concentrated plasticity model

Component i Component j

θcap,pliEIstfEIg i

Myi

Mc

My iθpci γi θcap,plj

EIstfEIg j

Myj

Mc

My jθpcj γj

Com

pon

ent

i

θcap,pli 1.0 0.0 0.1 0.3 0.2 0.0 0.7 0.0 0.0 0.1 0.1 0.0EIstfEIg i

1.0 0.1 -0.1 0.0 0.1 0.7 0.1 -0.1 0.0 0.0

Myi1.0 0.3 0.1 0.1 0.9 0.2 0.1 0.1

Mc

My i1.0 0.0 0.2 0.7 0.0 0.0

θpci (sym.) 1.0 0.2 (sym.) 0.3 0.0

γi 1.0 0.4

random realizations of the variables from the joint probability distributions and con-

ducting analyses with these realizations. Latin hypercube sampling is shown to be an

effective sampling method for seismic response assessment (Dolsek, 2009; Vamvatsikos

& Fragiadakis, 2010; Ugurhan et al. , 2014). The effectiveness of the method results

from the stratification of the probability distribution. It operates by drawing random

realizations of model parameters from equal probability disjoint intervals of the range

of these parameters (Helton & Davis, 2003). Considering the trade off of accuracy

versus computational time, we ran 200 realizations of the model parameters using

Latin Hypercube sampling at each ground motion intensity. Stochastic optimization

using simulated annealing is applied to preserve the correlation structure among the

random variables, which are obtained using using Latin Hypercube sampling (Dolsek,

2009). The sampled model realizations are each matched randomly with one of the

200 ground motions at each stripe, yielding 1000 analyses per structure. In addition,

to distinguish the influence of modeling uncertainty from record-to-record variability,

we re-analyze the structures using median model parameters with the same ground

motions.

6.5 Impacts of Modeling Uncertainty on Collapse

Response

6.5.1 Effect of Modeling Uncertainty on Collapse Fragility

Parameters

Following the approach outlined previously in Figure 6.1, collapse fragility curves are

computed for each structure and the impacts of modeling uncertainty are investigated

by comparing ratios of the fragility curve parameters (βc, θc) and the collapse rates

(λc) computed using models with and without modeling uncertainty, i.e. βc,T/βc,RTR.

In Figure 6.3, the change in median collapse capacity due to modeling uncertainty

is plotted with respect to the change in the dispersion for fragility curves computed

for each building archetype. Different markers are used to distinguished between the

ductile and nonductile frames, ductile frames with story strength irregularities, and

the number of stories. The buildings with strength irregularities include 8 and 12

story ductile frames, where the first stories have lateral strengths that are 30% less

than the stories above.

As observed in Figure 6.3.a, modeling uncertainty, in general, tends to reduce

the median collapse capacity and increase the dispersion, with a maximum reduction

of 21% in θc,T/θc,RTR and increase of 10% to 70% in βc,T/βc,RTR. Interestingly the

structures with smaller change in dispersion generally exhibit more reduction in the

median collapse capacity, and vice versa. Referring to the collapse fragility and

deaggregation curves in Figure 6.1, both of these effects will each tend to increase the

collapse risk. To the extent that the change in median and dispersion tend to have

opposite trends, they tend to be partially compensating (i.e., one structure’s collapse

risk may increase due to a large increase in βc, whereas the others will increase due

to a reduction in θc). Non-ductile frames and ductile frames with soft stories are

observed to display more shift in median collapse capacity, compared to the other

structures. This may be explained by the observation that these systems are more

sensitive to weakest link type behavior (i.e., system level response can be idealized as

a series of collapse mechanisms, where the weakest will generally govern).

170


1 1.2 1.4 1.6 1.80.7

0.8

0.9

1

1.1

βT/β

RTR

θT/θ

RT

R

(a)

Ductile

Soft Story Ductile

Nonductile

1 1.2 1.4 1.6 1.80.7

0.8

0.9

1

1.1

βT/β

RTR

θT/θ

RT

R

(b)

1 Story−Ductile

2 Stories−Ductile

4 Stories−Ductile

8−20 Stories−Ductile

8−20 Stories−SS Ductile

2 Stories−NonDuctile



Figure 6.3: Change in median collapse capacity with respect to change in the dispersionvalues of collapse fragility curves a) With respect to structural characteristics b) Withrespect to number of stories and structural characteristics

In Figure 6.3.b, the same data are grouped with respect to the number of stories

and structural characteristics. Here it is interesting to observe that the single-story

buildings and a couple of the two-story buildings are the only ones to experience a

large increase in dispersion, βc,T/βc,RTR, with little change in median θc,T/θc,RTR. The

slight increase in median to θc,T/θc,RTR values above 1 seems to be a result of the large

change in dispersion that flattens out the fragility curve. This is expected since short

period structures exhibit more variability in response compared to moderate to large

period structures. Furthermore, these buildings tend to have only one collapse mode,

such that the variability in model parameters is more directly linked to the variability

in response. This is in contrast to the other mid to high-rise buildings that generally

display multiple failure modes, where the collapse mechanism can be idealized as a

series of collapse mechanisms, where the weakest will generally govern. In the latter

cases, while a particular perturbation of model parameters result in the increase of

the capacity of some mechanisms, the capacity in another mechanism might decrease

and can govern the response. Therefore, in general, there is a decrease in median

collapse capacities for structures having more than one collapse mechanism.

6.5.2 Net Effect of Modeling Uncertainty on Collapse Rates

The net effect of the change in median and dispersion due to modeling uncertainty can

be quantified by integrating the resulting fragility curves with a seismic hazard curve

to determine the change in mean annual frequency of collapse, λc. Using the hazard

curves for the building sites in Los Angeles, we calculated the ratios of λc,T/λc,RTR for

the 33 building archetypes. The average collapse rate ratio is calculated to be about

1.8 with a coefficient of variation of 0.2, indicating that the modeling uncertainty

increases the collapse risk by about 80%, relative to the risk for the median model.

To further investigate how sensitive this change is to the ground motion hazard curve,

we integrated the collapse fragilities with idealized hazard curves that are modeled

with a power-law hazard curve of the form λ = k0IM−k, where the hazard curve slope

k is varied and k0 is fit over the range between Sa(T1) values corresponding to 10% to

2% in 50 yr exceedance probabilities (see Figure 6.1.c). Note that λc values are not

realistic for sites except LA since the structures are not designed for those sites and the

ground motions used in the derivation of fragility curves are not compatible with the

site-specific hazard and a rough extension of the results to different site characteristics

is provided through ratios of λc,T/λc,RTR. The results of these analyses are shown in

Figure 6.4, where each of the boxplots corresponds to the change in λc values for the

set of archetype buildings for sites with varying hazard curve slopes. For comparison,

the red boxplot (close to k equal to 3) are data from the collapse rates determined

by the USGS hazard curves for the Los Angeles building sites. Lower values of k

are more representative of sites in the central and eastern United States, where there

are larger differences in the spectral intensities (Sa) between the 10% to 2% in 50 yr

exceedance probabilities.

As indicated in Figure 6.4, for hazard curves with smaller slopes, the modeling

uncertainties tend to have a smaller effect on collapse risk, as compared to the Los

Angeles site with k ∼ 3. Referring back to Figure 6.1, the reason for this relates

to the relationship of the slope of the hazard curve relative to the lower tail of the

collapse fragility curve. Deaggregation of λc help identify the ground motion inten-

sities that contribute most to λc estimates. In the presence of modeling uncertainty,

the dominant ground motion contributors in computing λc is shifted toward smaller

172


intensities (See Figure 6.1.d). The magnitude of the shift in dominant ground mo-

tion intensities depends on the change in fragility curve characteristics as well as the

change in the instantaneous slope of the true hazard curve. The impacts of modeling

uncertainty on λc would become more pronounced when the collapse fragility curve

is pushed toward the regions of hazard curves having higher instantaneous slopes and

this might result in different impacts on λc at different sites.

1

1.5

2

2.5

3

3.5

k1 1.5 2 2.5 3 3.5 4

λc,T

/λc,R

TR

k

0IM

−k

LA Hazard Curve

0.95−k

exp(1/2k20.33

2)

Figure 6.4: The distribution of the change in λc due to modeling uncertainty is shown forsites characterized power-law hazard curve and the site at LA using blue and red boxplots,respectively. Also shown in green is the change in λc due to modeling uncertainty using anapproximate closed form expression.

The collapse rate ratios calculated for the archetype models can be described by

a closed-form expression, shown by the green curve in Figure 6.4. The expression for

this curve is derived using a closed form expression for collapse rate λc by Jalayer

(2003) as a function of median and dispersion of collapse fragility and the slope of

the idealized hazard curve. Taking ratios of the rates, the change in collapse rate due

to modeling uncertainty in λc is obtained as follows:

λc,Tλc,RTR

=

[θc,Tθc,RTR

]−ke

12k2β2

c,M (6.2)

where βc,M represents the additional dispersion due to modeling uncertainty and

the other terms are as defined previously. Assuming that the modeling and record-

to-record effects are statistically independent, the modeling dispersion βc,M can be

approximated using a square root of sum of squares (SRSS) approach through the

equation given below:

βc,T =√β2c,RTR + β2

c,M (6.3)

As shown in Figure 6.4, the predictions by Equation 6.2 give good agreement using

the average values from the building archetype studies, where the average median shift

is 0.95 (with a coefficient of variation of 0.12) and the average βc,M is 0.33 (with a

coefficient of variation of 0.18).

6.5.3 Equivalent Value of Modeling Uncertainty

To the extent that the results from the archetype studies are representative other

framed buildings, it is useful to examine how the effects of modeling uncertainties

can be generalized. This would inform, for example recent performance-based guide-

lines (FEMA, 2009b, 2012) that incorporate judgement-based modeling uncertainty

factors that are combined, using an SRSS rule, with other sources of variability. Fig-

ure 6.5.a provides an illustration of how modeling uncertainty parameters can be used

to approximate the calculated collapse fragility for the 12-story building archetype

(ID 2067). Shown in Figure 6.5.a are the collapse fragilities obtained from simulations

for the median model (Pc,RTR) and for the models that include modeling uncertainty

(Pc,T ). In between are two approximate fragility curves, one that has an adjustment

to the median and dispersion and the second that has only a change to the disper-

sion. The curve with the median shift and added dispersion is based on the average

values from the archetype study (average median shift of 0.95 and average modeling

uncertainty (βM) of 0.33). The other curve includes an equivalent value of modeling

uncertainty of β∗c,M equal to 0.4, which is calibrated so as to match, on average, the

collapse rates that are determined using the simulated collapse fragility curves (Pc,T )

and the hazard curves for the archetype building studies.

The approach to calibrate the equivalent modeling uncertainty β∗c,M is as follows:

174


(1) determine the λc,T for each archetype that are obtained incorporating modeling un-

certainty for that archetype, (2) develop collapse fragility functions for each archetype

that are obtained using median model parameters for that archetype, i.e θc,RTR and

βc,RTR, (3) combine βc,RTR along with an estimate of β∗c,M for each archetype using

the SRSS approach using Equation 6.3, (4) integrate the estimated collapse fragility

curve with the hazard curve to find an estimate of ˆλc,T for each archetype, (5) repeat

steps 3 and 4, selecting updated estimates of β∗c,M to minimize the error between the

correct and estimated values of λc,T , until the desired accuracy is achieved. When

run for the archetype buildings reported earlier, the resulting mean value of β∗c,M is

0.4. The resulting estimates of λc,T that are obtained using an SRSS rule with β∗c,M

of 0.4 versus target values are given in Figure 6.5.b using black squares. With this

approach, mean measured to computed λc,T is obtained as 1.03 with a coefficient of

variation of 0.18. Also shown in Figure 6.5.b are (1) red points determined based

on estimated fragility curves that employ the average median shift of 0.95 and mod-

eling dispersion of 0.33, and (2) blue points from the median model fragility curves

λc,RTR. This latter case corresponds to neglecting modeling uncertainty and leads to

a significant (on average 58%) underestimation of λc.

From Figure 6.5.b the two approximate fragility curves are shown to provide

similar estimations of λc,T . This is in spite of the apparent differences in the curves

shown in 6.5.a and occurs because the two methods yield similar estimates in the lower

tail region of the collapse fragility function, which dominates the contribution to λc.

The approximate model with the median shift and modeling dispersion approach

provides a slightly better alternative compared to the sole use of β∗c,M alone, but

overall, both give estimates with minimal bias (less than 3% bias) in λc,T with a

coefficient of variation close to the value of 0.18 in the original simulated results.

0 0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Sa(T1=2s, 5%)

Pc

(a)

10−4

10−3

10−4

10−3

λc,T

Estim

ate

d λ

c,T

(b)

PC,RTR

Pc,T

with βc,M

=0.4

Pc,T

with βc,M

=0.33,

θc,T

/θc,RTR

=0.95

+/− 20\%

+/− 50\%

Pc,RTR

Pc,T

with βc,M

=0.4

Pc,T

with βc,M

=0.33,

θc,T

/θc,RTR

=0.95

Pc,T

Figure 6.5: a) Illustration of practical approaches to incorporate modeling uncertaintyusing 12-story structure with ID 2067 b) Estimated λc using the practical approaches

6.6 Impacts of Modeling Uncertainty on Drift De-

mands

In this section, impact of modeling uncertainty on drift demands is examined, first,

considering drift exceedance (fragility) curves, and then evaluating the drift demands

at specified intensity levels.

6.6.1 Influence of Modeling Uncertainty on Drift Exceedance

Fragility Curves

Referring back again to Figure 6.1.b, the drift-exceedance fragility curves are quan-

tified in a similar manner to collapse fragilities, except that the limit state is defined

based on the SDR exceeding a specified sdr value, i.e SDR ≥ sdr. Shown in Fig-

ure 6.6.a are drift-exceedance fragility curves for one of the 8-story archetypes (ID

1012), plotted for results from the median model (dashed lines, representing RTR un-

certainties) and for the simulations that include modeling uncertainties (solid lines,

representing the “total” uncertainties). Drift fragilities are shown for five sdr val-

ues, ranging between 0.03 to 0.10. Note that the drift fragility for sdr equal to 0.10

176


is identical to the collapse fragility curve. Figure 6.6.b shows the change due to

modeling uncertainty in the medians (θSDR>sdr,T/θSDR>sdr,RTR) and the dispersions

(βSDR>sdr,T/βSDR>sdr,RTR) of the drift fragility curves PSDR>sdr, similar to the col-

lapse fragility curve ratios shown previously in 6.3. The average changes for the 33

archetypes are shown by squares, and the lines around the squares extend to 16%

and 84% fractals of the change in the corresponding parameters. In Figure 6.6.c,

the change in mean annual exceedance rates (λSDR>sdr,T/λSDR>sdr,RTR) are shown,

obtained from integrating the drift fragility curves with the ground motion hazard

for the building site in Los Angeles.

Based on the results shown in Figure 6.6, for story drift ratios below 0.03, mod-

eling uncertainties have a relatively small but measurable impact on the results. For

example, at the 0.03 drift limit, modeling uncertainties increase the mean annual

frequency of exceedance about on average by 20%. Compared to the reliability of

nonlinear analyses, which are usually run with ten or fewer ground motions, the in-

fluence of uncertainties would be difficult to assess. However, as there is a measurable

increase in the median drift, dispersion and annual frequency, one could envision ap-

plying a reliability factor to account for the modeling uncertainties, even at these

comparatively low drift levels. At peak story drifts of 0.04 and beyond the modeling

uncertainties have a greater effect and may be significant. In the limit of sdr about

0.07 to 0.10, the modeling uncertainty is on roughly the same order as the uncer-

tainties in the collapse fragility, where the dispersion ratio (βSDR>sdr,T/βSDR>sdr,RTR)

increase to about 1.3 and the exceedance rate (λSDR>sdr,T/λSDR>sdr,RTR) increases by

a factor of about 1.8. As described previously for the collapse fragility, at large drifts

in excess of sdr of 0.07, the equivalent βc,M is about 0.33.

6.6.2 Influence of Modeling Uncertainty on Drift Demands

We next investigate the impacts of modeling uncertainty on story drift demands at

different ground motion intensity levels. This evaluation of drift demands at a spec-

ified intensity is particularly relevant to the manner in which drifts are assessed in

design practice. As with the collapse fragilities, ground motion intensities evaluated

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

Sa (T1=1.9s, 5%)

PS

DR

>sdr

(a)

sdr=0.03; T

sdr=0.03; RTR

sdr=0.04

sdr=0.06

sdr=0.08

sdr=0.10

1 1.1 1.2 1.3 1.4 1.50.9

0.95

1

1.05

βSDR>sdr,T

/βSDR>sdr,RTR

θS

DR

>sdr,

T/θ

SD

R>

sdr,

RT

R

(b) sdr=0.03

sdr=0.04

sdr=0.06

sdr=0.08

sdr=0.10

0.1

0.2

0.3

0.4

0.03 0.04 0.06 0.08 0.10sdr

βS

DR

>sdr,

M

(c)

1

1.5

2

2.5

0.03 0.04 0.06 0.08 0.10sdr

λS

DR

>sdr,

T/λ

SD

R>

sdr,

RT

R

(d)

Figure 6.6: a) Example drift-exceedance fragility curves for an 8 story structure with ID1012 b) Change in median collapse capacity with respect to change in the dispersion valuesof fragility curves defining drift-exceedance limit states c) Boxplots showing the change inλSDR>sdr as a function of sdr

in this study correspond to exceedance probabilities of 5%, 2% and 1% in 50 years,

1% in 100 years and 1% in 200 years. The values of SDR observed at each IM are

recorded for each structure, and empirical cumulative distribution functions (ECDFs)

corresponding to PSDR≤sdr are obtained. Figure 6.7.a compares the ECDFs of drift

demands obtained using a median model (dashed lines, corresponding to RTR uncer-

tainties) and models incorporating modeling uncertainty (solid lines, corresponding

to “total” uncertainties) for the 8-story building archetype (ID 1012).

178


0 0.02 0.04 0.06 0.08 0.10

0.25

0.5

0.75

1

SDR

P(S

DR

<sdr)

(a)

0.5 1 1.5 20.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Sa (T1,5%)/Sa

2/50 yr

θsdr

T/θ

sdr

RT

R

(b) y=0.15x+0.90

0.5 1 1.5 2

1

1.2

1.4

1.6

1.8

Sa (T1,5%)/Sa

2/50 yr

βsdr

T/β

sdr

RT

R

(c)

y=1.05

0.5 1 1.5 2

1

1.2

1.4

1.6

1.8

Sa (T1,5%)/Sa

2/50 yr

βsdr

T/β

sdr

RT

R

(d)

y=1.14

Sa5/50 yr

=0.9g

Sa2/50 yr

=1.3g

Sa1/50 yr

=1.6g

Sa1/100 yr

=1.9g

Sa1/200 yr

=2.3g

Figure 6.7: a) ECDF of drift demands using median model and with modeling uncertaintyfor the 8-story structure with ID 1012 b) Change in counted median of drift demand distri-butions w.r.t. drifts for different structures c) Change in dispersion of the fitted lognormaldistributions to sdr values of no-collapses d) Change in dispersion of the fitted lognormaldistributions to sdr values up to median

It is observed from Figure 6.7.a that the difference in deformation demands be-

tween the median model and the model incorporating modeling uncertainty is neg-

ligible for drift ratios smaller than about 0.025. At the ground motion intensity of

2% probability of exceedance in 50 years (2/50 yr), the probability that SDR will be

smaller than 0.03 is 47% using a median model, increasing to 53% in the presence

of modeling uncertainty. On average for all of the building archetypes, at the 2/50

yr ground motion, the probability that SDR will be smaller than 0.03 is about 4%

larger in the presence of modeling uncertainty. This increases to 8% at SDR of 0.06,

following the trend discussed previously of increasing impacts of modeling uncertainty

with increasing drifts.

Figure 6.7.b shows the change in counted medians in the presence of modeling

uncertainty as a function of ground motion intensities that are normalized with respect

to Sa2/50yr. Counted medians correspond to SDR values at the 50% probability level

of Figure 6.7.a. For each IM level, sdr values of the solid lines divided by that of

dashed lines quantify the change in counted medians of drifts. At the 5/50 yr ground

motion, there is no change in median drifts, but at larger intensities the ratio increases

linearly, with a ratio of 1.05 at the 2/50 yr intensity and up to about 1.2 at the 1/200

yr intensity. For some of the building archetypes and ground motion intensity levels,

more than half of the analyses results in collapse response (i.e. Figure 6.7.a, 1/200 yr).

While these collapse data points are not shown in Figure 6.7.b, they are incorporated

in the computation of the fitted line using a maximum likelihood approach with a

normal distribution by right-censoring these values at 1.

Figure 6.7.c and 6.7.d show the change due to modeling uncertainty in dispersion

values of the fitted lognormal distributions to sdr values as a function of ground mo-

tion intensities that are normalized with respect to Sa2/50yr. Lognormal distributions

are fitted to the entire set of no-collapse data in Figure 6.7.c, whereas, in Figure

6.7.d, the distributions are fit to the sdr values that are smaller than their respective

medians. This latter approach was done to better fit the lower tail of the ECDFs,

since the upper portion seems to be more affected by collapse occurrences than by

the non-collapse drifts.

Plots of dispersion ratios using either of the approaches (Figure 6.7.c or Figure

6.7.d) indicate rather modest average changes of about 1.05 to 1.14 in the dispersion

due to modeling. In addition, the plots do not reveal much if any trend between

the change in dispersion with ground motion intensity. At first glance, this result

appears to be inconsistent with the impacts observed at collapse (e.g. Figure 6.3),

however, the apparent difference is related to the consistent treatment of collapse

and non-collapse cases, which is addressed in the following section. Overall, however,

the change in dispersion of drift demands due to modeling uncertainty is practically

180


negligible, whereas the change in median drift (Figure 6.7.b) is more significant.

6.7 MCE Level Drift Demand and Acceptance Cri-

teria

In the previous sections, we have examined the influence of modeling uncertainties on

collapse risk and on drift demands, treating the two as related but distinct measures

of performance. However, to ensure building safety, an important global-level check

is through story drift limits in emerging building code design requirements that em-

ploy the use of nonlinear dynamic analysis, such as applied to tall buildings (PEER,

2010; LATBSDC, 2015) or in the recently proposed ASCE 7 requirements American

Society of Civil Engineers Standards Committee (2015). Typically, these drift limits

are checked under maximum considered earthquake (MCE) ground motions. In this

section, we propose a framework for relating story drift demands and acceptance cri-

teria to collapse safety goals, considering how variability due to modeling uncertainty

affects these criteria.

An essential first step in establishing drift-based acceptance criteria is to determine

story drift limits for collapse. This can be done using incremental dynamic analysis

(IDA), using an approach formalized by (Vamvatsikos & Cornell, 2002). IDA methods

involve the scaling of ground motions through a range of increasing ground motion

intensity levels until the structure displays dynamic instability. This enables quantifi-

cation of drift capacities, which can then be related to induced story drift demands

at any ground motion intensity level. An example IDA curve for one ground motion

is shown by the black line in Figure 6.8, where the collapse capacity corresponds to

the point where the curve flattens out, indicating the onset of instability.

As with MSA procedures (described previously in Figure 6.1), IDA procedures

are usually performed with multiple ground motions to obtain statistical values of

drift demands and collapse capacity. So, for example, the row of drift demands

shown by the red points at the MCE intensity in Figure 6.8 could be one stripe of an

MSA or IDA. However, an important distinction between MSA and IDA procedures

Sa (

T1, 5%

)

SDR

CollapsePDF

MCE Drift PDF

MCE

SDRMCE

SDRCapacity

10%

Figure 6.8: An example IDA curve in com-parison with MSA data

Sa

(T

1,

5%

)

SDR

di’>c

i’

ci>d

i

MCE

di

ci’ c

i

Figure 6.9: Two example IDA curves areprovided with red and blue lines having driftcapacities above and below MCE level, re-spectively.

is that IDA approaches usually involve scaling a single set of ground motions to

larger and larger values, whereas MSA procedures enable the use of different ground

motions at each intensity, which can be selected and scaled to match the target

conditional spectra that varies with ground motion intensity. Because IDA procedures

do not make this distinction, the IDA results may require adjustment of the ground

motion intensities (the vertical statistics in Figure 6.8). This is why, for example, the

FEMA P695 (FEMA, 2009b) procedures apply a spectral shape factor adjustment

to the spectral acceleration collapse intensity. For the same reason, MSA and IDA

procedures run with different record sets may have different drift demands at any

ground motion intensity level. However, the IDA procedure is assumed to provide

a reliable measure of the drift at collapse (using horizontal statistics in Figure 6.8),

which is a measure that is not readily obtained using MSA procedures.

Ultimately, it is proposed to use IDA procedures to determine the drift at collapse,

which then can be compared to the earthquake-induced drift demands determined

using MSA with hazard-consistent ground motions. However, as a first step, IDA

procedures will be used to characterize the joint distribution of drift demands under

MCE intensity ground motions and drift capacity at the onset of collapse.

182


We first present the proposed framework for obtaining distributions of drift capac-

ity and demand at MCE. Next, we apply this framework to selected archetypes and

discuss the results. Finally, we examine the implication of these results on the risk-

based acceptance criteria for use of nonlinear dynamic analysis in the proposed next

update ASCE 7 American Society of Civil Engineers Standards Committee (2015).

6.7.1 Framework for Obtaining Distributions of Building Drift

Capacity and Demand at MCE

Using distributions of drift capacity and demand at MCE, the probability of collapse

at the MCE intensity, P (Collapse), can be obtained using the equation below:

P (Collapse) = P (D > C) (6.4)

where D and C represent the distributions of drift demand at MCE and drift capacity,

respectively (Cornell et al. , 2002; Jalayer, 2003). To evaluate Equation 6.4, one needs

the parameters defining the joint distribution of D and C. In this study, we model

the distribution of demands and capacities using a bivariate lognormal distribution,

whose parameters are obtained using maximum likelihood estimation.

IDA is employed to obtain structural drift capacities and to relate these to MCE

level demands. In doing so, consideration must be given to cases where collapse

occurs below the MCE demand level. This is illustrated in Figure 6.9, which shows

two example IDA curves along with the ground motion intensity corresponding to

MCE. For IDA curves whose capacities are below MCE level (blue curve in Figure

6.9), demand at MCE can not be exactly quantified, but the available information

is that demand at this level will be greater than the capacity associated with these

curves. Otherwise, for IDA curves whose capacities are above MCE level (red curve

in Figure 6.9), both the drift capacity and demand at MCE can be obtained from the

associated IDA curve.

For ground motions that cause collapse below the MCE hazard level, their MCE

drift demand values are capped at their drift capacity values, thus resulting in right-

censored demand values. Using a maximum likelihood estimation method with right

censored data, one can then determine the parameters of the joint distribution char-

acterizing the demand at MCE and the capacity. These parameters are medians and

dispersions of drift demand and drift capacity, along with the correlation between

logarithms of demand and capacity, which are indicated as mD, βD, mC , βC and

ρD,C , respectively.

For IDA curves that do not exhibit collapse response at MCE (ci > di), one can

obtain the demand and capacity values directly (Figure 6.9). This means that demand

and capacity values are not censored. The likelihood of observing C = ci and D = di

is computed using the bivariate lognormal distribution of demand and capacity:

Likelihood = P (C = ci, D = di)

= fC,D(C = ci, D = di)(6.5)

where f(·) represents a lognormal probability density function.

For IDA curves that exhibit collapse response below MCE (di′ > ci′), drift demand

at MCE cannot be determined, and is thus censored at its respective capacity. The

likelihood of observing C = ci′ and D > ci′ is computed using the equation below.

Likelihood = P (D > ci′|C = ci′)fC(ci′)

=

(1− Φ

(ln(ci′)− ln(mD|C)

βD|C

))fC(ci′)

(6.6)

where Φ(·) is the standard normal cumulative distribution function, and mD|C and

βD|C are the conditional median and dispersion of demand given capacity, respectively.

Since the joint distribution of demand and capacity are bivariate lognormal, the

marginal probability density function of capacity fC(c) has a lognormal distribution.

Assuming the observations are independent, the overall likelihood can be obtained

as a multiplication of likelihood functions of individual data points:

184


Likelihood =

(m∏i=1

fC,D(C = ci, D = di)

)(n∏

i′=1

(1− Φ


βD|C

))fC(ci′)

)(6.7)

where m and n refer to the total number of cases corresponding to no-collapse and

collapse at MCE, respectively, and∏m

i=1 denotes a product over i values from 1 to m.

The parameters of interest can be obtained by maximizing the overall likelihood:

{mD, βD, mC , βC , ρD,C} =

arg max

(m∏i=1

fC,D(C = ci, D = di)

)(n∏

i′=1

(1− Φ


βD|C

))fC(ci′)

)(6.8)

These parameters can be used with Equation 6.4 to estimate P (Collapse) as

follows:

P (Collapse) = Φ

− ln(mC)ln(mD)√

β2C + β2

D − 2ρD,C βC βD

(6.9)

6.7.2 Distributions of Building Drift Capacity and Demand

at MCE for Selected Archetype Structures

To characterize the joint distribution of demand and capacity, IDAs are conducted

for selected archetype structures including modeling uncertainty. The generic FEMA

P695 far-field ground motion set of FEMA (2009b) of 44 records is used for this pur-

pose. Each of the records is used with an IDA approach with 4400 random structural

realizations, resulting in a total of 4400 IDA curves. Drift capacity, determined from

each IDA curve, is defined as the drift value corresponding to the point where the

IDA curve flattens to 10% of its initial slope, beyond which drift demands tend to

increase very rapidly (Vamvatsikos & Cornell, 2002). Note that this criteria is nec-

essary for establishing robust horizontal statistics for drift capacity, whereas collapse

ground motion intensities evaluated by vertical statistics can be determined using the

less discriminating fixed drift limit of 0.10 (see Figure 6.1). We have selected to use

10% of its initial slope due to its numerical stability in comparison to a drift capacity

determined using zero slope. Different characterizations of collapse response has been

explored by Deniz (2015) and different strategies can be easily adopted for charac-

terizing drift capacity. Future work is recommended to characterize the sensitivity of

the drift capacities to alternative definitions of collapse.

As noted previously, the ductile reinforced concrete moment frame archetypes

were designed according to 2003 IBC, using the 2002 USGS seismic design value

maps. Therefore, these design value maps are used to set the intensity level for

determining the MCE drift demands, which for the soil class Sd site correspond to

MCE intensities of Sms = 1.5g, and Sm1 = 0.9g (Haselton et al. , 2011).

Shown in Figure 6.10.a are the 16%, 50% and 84% fractal IDA curves for the

12 story archetype (ID 1019), the fitted collapse fragility function as a function of

Sa(T1) (vertical statistics) along with the histograms of the drift demand at MCE

and the drift capacity (horizontal statistics). The joint occurrence of drift demand

and drift capacity values are used to generate the scatterplot shown in Figure 6.10.b.

The uncensored data, which corresponds to the cases that collapse above the MCE

intensity, are indicated using blue dots, whereas censored data, which corresponds to

cases that collapse below MCE, are indicated using red dots. The aforementioned

framework equations are applied to develop the estimated probability distribution

contours for drift demand and capacity shown in Figure 6.10.b. As indicated, this

approach allows the development of a full distribution of demand versus capacity over

the censored and uncensored region.

The fitted bivariate lognormal parameters of drift demands and capacities for the

12 Story structure (ID 1019) and four other archetype buildings are summarized in

Table 6.3. In addition, the first row lists the MCE level ground motion intensity,

Sa(T1), and the bottom two rows provide P (Collapse) computed using Equation 6.9

(from horizontal drift statistics) along with the P (Collapse) values obtained using

the collapse fragility function from IDA (vertical statistics). Ideally, the two bottom

rows will agree, given that the drift demands, drift capacities, and collapse Sa(T1)

186


0 0.05 0.1 0.150

0.5

1

1.5

2

Sa

(T1=

2.1

s, 5

%)

SDR

(a)

Demand SDR

Ca

pa

city

SD

R

(b)

10−2

10−1

10−2

10−1

Fitted Probability Dist. Contours

Right Censored Data

Uncensored Data

16% fractile

50% fractile

84% fractile

Collapse Fragility

Drift Dist. @ MCE

Capacity Dist.

Figure 6.10: a) IDA curves and drift demand at MCE and capacity histograms for ID:1019along with collapse fragility curve b) Probability distribution of demand at MCE and ca-pacity for ID:1019

values are all based on the same set of ground motions.

Based on comparisons of the calculated collapse probabilities, P (Collapse), in the

bottom rows of Table 6.3, the proposed approach is validated to characterize collapse

using story drift criteria. The estimated P (Collapse) value from drift distributions

and target P (Collapse) value from ground motion intensity capacities are fairly close.

Other significant observations from the data are:

• The median demands at MCE are about 0.03 with dispersion values ranging

between 0.44 to 0.62, both of which are in line with expectations based on

design level drift limits in building codes and results of other dynamic analysis

studies

• The median drift capacities range from about 0.05 to 0.06 with dispersions of

about 0.40 to 0.45. The median drift capacities tend to coincide with when

beam and column plastic hinges tend to reach their peak point.

• The drift demands and capacities tend to be negatively correlated, where the

correlation coefficients range from -0.04 to -0.24 with an average value of -0.14.

Table 6.3: Parameters of the joint distribution of demand and capacity and the estimatedP (Collapse) along with target P (Collapse) that is obtained using IDA results

4 Story ID:

1008

8 Story: ID

2065

12 Story: ID

1019

20 Story: ID

1021

MCE (g) 0.95 0.56 0.44 0.35

mD 0.030 0.034 0.030 0.028

βD 0.44 0.57 0.52 0.62

mC 0.061 0.061 0.057 0.052

βC 0.46 0.41 0.39 0.44

ρD,C -0.11 -0.24 -0.18 -0.04

P (Collapse) 0.14 0.22 0.18 0.22

P (Collapse) 0.13 0.22 0.18 0.19

This implies that in a given simulation, if the drift demand is larger than aver-

age, then the drift capacity will tend to be less than average. This is plausible

from the standpoint that weaknesses (lower stiffness and strength) that tend to

produce larger drifts in a given realization will likewise tend to reduce the drift

at the onset of collapse. The SAC/FEMA guidelines (Cornell et al. , 2002) also

noted the tendency for negative correlations between drift demands and capaci-

ties, and they adopt a perfect negative correlation for their computations. Drift

demands and capacities are also discussed by Kazantzi et al. (2014); Jalayer

et al. (2007) to be negatively correlated, and the assumption of uncorrelated

demand and capacity will result in underestimation of P (Collapse).

6.7.3 Impacts of Modeling Uncertainty on the Distributions

of Building Drift Capacity and Demand at MCE

To evaluate the contribution of modeling uncertainty on the distributions of drift

demands and capacities, we repeated the analyses just described, except using only

188


median model parameters. This is similar to how the modeling effects were distin-

guished previously, except in this case the RTR variability is associated the generic

FEMA P695 ground motions (FEMA, 2009b) that are used with an IDA approach.

In Figure 6.11, the 16%, 50% and 84% fractal IDA curves and other collapse and drift

distributions are shown for the 4 story archetype (ID 1008) without (Figure 6.11.a)

and with modeling uncertainties (Figure 6.11.b). Comparing the two reveals that

the modeling uncertainty significantly increases the variability in the drift capacity

(magenta colored histogram) but otherwise results in relatively small changes to the

other demands and capacities.

0 0.05 0.1 0.150

0.5

1

1.5

2

2.5

3

Sa

(T1=

0.9

s, 5

%)

SDR

(a)

0 0.05 0.1 0.150

0.5

1

1.5

2

2.5

3

Sa

(T1=

0.9

s, 5

%)

SDR

(b)

16% fractile

50% fractile

84% fractile

Collapse Fragility

Drift Dist. @ MCE

Capacity Dist.

Figure 6.11: IDA curves and drift demand and capacity histograms in terms of drifts forID:1008 a) using median model b) with modeling uncertainty

The change in the fitted bivariate lognormal parameters due to modeling uncer-

tainty for the four building archetypes are shown in Table 6.4. As observed in Figure

6.11, the dispersion of drift capacity increases significantly, ranging from 36% to 84%

with an average of 65%. In addition, the median drift capacity decreases by 5% to 9%.

The MCE level median drift demand increases by about 3% to 8%, which parallels

the trend noted in the counted median demands using MSA (See Figure 6.7.b). The

reverse trends in median drift capacity and drift demands also follows from the neg-

ative correlations noted previously. The change in dispersion of MCE drift demands

ranges from -6% to 5%, which is almost insignificant and consistent with the results

shown previously using MSA (See Figure 6.7.c and 6.7.d).

In summary, the major effects of modeling uncertainty are to (1) significantly

increase the dispersion in the drift capacity by about 65%, and (2) reduce the margin

between the median drift demands and capacities by about 12%.

Table 6.4: Change in the parameters of the joint distribution of demand and capacity dueto modeling uncertainty

4 Story ID:

1008

8 Story: ID

2065

12 Story: ID

1019

20 Story: ID

1021

mD,T/mD,RTR 1.05 1.04 1.08 1.03

βD,T/βD,RTR 1.05 0.99 1.03 0.94

mC,T/mC,RTR 0.95 0.91 0.93 0.94

βC,T/βC,RTR 1.59 1.84 1.36 1.80

6.7.4 Evaluation of ASCE 7 Story Drift Acceptance Criteria

- Building Archetype Results

The ASCE 7 provisions American Society of Civil Engineers Standards Commit-

tee (2015) indicate that for typical (risk category 1&2) buildings the probability of

collapse at MCE should be less than 10%. The provisions further formalize these

risk targets to limit the probabilities of collapse to 10%, 6% and 3% under MCE

intensities for buildings in risk categories 1&2, 3 and 4, respectively. Presumably,

the design acceptance criteria should be based on these collapse safety targets, how-

ever, in explaining the newly proposed provisions for design verification by nonlinear

dynamic analysis the authors (Haselton et al. , 2014) note that the specified MCE

drift limits are based primarily on calibration to existing story drift limits, which

were established long before formal risk targets were proposed. With the aim to-

wards providing a sounder basis for the drift limits in ASCE 7 (American Society

of Civil Engineers Standards Committee, 2015) and similar standards, these limits

190


are evaluated through the probabilistic framework proposed in this paper for relating

earthquake drift demands to collapse capacities.

The archetype buildings described previously are considered to represent buildings

in risk category 1&2, for which the proposed new requirements in ASCE 7 American

Society of Civil Engineers Standards Committee (2015) for design acceptance by

nonlinear dynamic analysis stipulate a story drift ratio limit of 0.05 for buildings of

4 or less stories and 0.04 for taller buildings. The limit is to be compared to the

average of the peak story drift demands when the building subjected to eleven pairs

of ground motions that are scaled to the MCE intensity.

In Table 6.5, the mean and median MCE drift demands (mD and µD) are re-

ported for four archetype buildings along with the computed probability of collapse,

P (collapse), which is determined using the collapse assessment framework. Values of

P (Collapse) are also reported, which are obtained using the collapse fragility func-

tions of the respective structure from the IDA. Since the drift demands and resulting

collapse probabilities from the IDA are based on ground motions whose spectral

shapes may not be consistent with the actual hazard, we also provide in Table 6.3

the median MCE drift demands and P (Collapse) using hazard consistent ground mo-

tions from the MSA procedure. We similarly computed and report in Table 6.3 the

resulting P (collapse) and mean drift demand (µD) using the hazard consistent MCE

median drift demands along with the other parameters of demand and capacity that

are given in Table 6.3 for each structure.

For the 4-story archetype (ID:1008), the mean demand from the risk consistent

MSA motions is 0.029, which is considerably less than the ASCE 7 MCE drift limit

of 0.05. Thus, this would imply that the building meets the ASCE 7 safety criterion

by a considerable margin. Yet, the P (Collapse) is calculated as 10% (bottom row

of table) which is exactly equal to the maximum collapse risk target in ASCE 7.

The approximate procedure based on relating drift demands to capacity would give

P (collapse) equal to 13%. Thus, one can imagine that if the building were less stiff or

strong, such that it just met the 0.05 drift limit that the structure would not meet the

safety requirement. For comparison, though not directly relevant to this argument,

the MCE drift demand (0.033) and probabilities of collapse (0.13 and 0.14) from the

Table 6.5: Probability of collapse and mean and median drift demand at MCE obtainedusing generic and hazard consistent ground motions

4 Story

ID: 1008

8 Story:ID

2065

12

Story:ID

1019

20

Story:ID

1021

Generic

GMs

mD 0.030 0.034 0.030 0.028

µD 0.033 0.040 0.034 0.034

P (Collapse) 0.14 0.22 0.18 0.22

P (Collapse) 0.13 0.22 0.18 0.19

Hazard

Consistent

GMs

mD 0.026 0.029 0.024 0.020

µD 0.029 0.034 0.027 0.024

P (Collapse) 0.13 0.16 0.14 0.17

P (Collapse) 0.10 0.18 0.13 0.16

IDA procedure are slightly higher than those of the hazard consistent MSA procedure.

Similar results to the 4-story archetype are reported in Table 6.3 for the 8-, 12-

, and 20-story archetypes, all of which meet by a reasonable margin the ASCE 7

MCE drift limit of 0.04 drift (with µD values for the MSA ground motions of 0.034,

0.027 and 0.024). However, all of the buildings fail to meet the maximum collapse

probability limit of 10% at MCE. As indicated in the bottom row of the table, the

calculated P (Collapse) values range from 13% to 18%.

It should be noted that similar nonlinear analysis requirements for tall buildings

(PEER, 2010; LATBSDC, 2015) specify more stringent MCE drift limits of 0.03, as

compared to the values of 0.04 to 0.05 in ASCE 7. In looking at the drift demands

and collapse probabilities for the four building archetypes in Table 6.3, the 0.03 drift

limit would appear to be more consistent with the target collapse probability limit of

10% at MCE.

192


6.7.5 Evaluation of ASCE 7 Story Drift Acceptance Criteria

- Parametric Analysis

To further investigate the proposed ASCE 7 Chapter 16 drift requirements, we used

the dispersions in the drift demands and drift capacities from the building archetype

studies to parametrically evaluate the relationship between the drift limits and the

implied collapse probabilities. Based on the data shown previously in Table 6.3,

the dispersions in MCE drift demand and drift collapse capacity, βD and βC , are

assumed equal to 0.55 and 0.45, respectively. Using these values, the ratio of median

(and mean) drift demands and capacities can be related to the probability of collapse

using Equation 6.9.

As a first exercise, using the specified drift limits and target probabilities from

ASCE 7 Chapter 16 (American Society of Civil Engineers Standards Committee,

2015), the implied drift capacities can be back-calculated. For buildings in risk cate-

gories 1&2, 3 and 4, the specified drift limits are 0.04, 0.03 and 0.02, and the target

probabilities of collapse are 10%, 6% and 3%. Using the assumed dispersions with no

correlation between drift demand and capacities, the resulting probability distribu-

tions are determined and plotted in the left column of Figure 6.12. From these, the

implied mean story drift capacities are 0.095, 0.086 and 0.072, for risk risk categories

1&2, 3 and 4, respectively. A few observations of these values: (1) the values are all

larger than the collapse mean drift ratios of 0.057 to 0.068 (obtained from the median

drift capacities in Table 6.3), and (2) it is counter intuitive that the drift capacities

would be smaller for the higher risk categories, suggesting that there is a relative

inconsistency between the drift limits and failure probabilities for the different risk

categories.

As a second exercise, constant values of drift capacities will be assumed for build-

ings in all three risk categories, and allowable drift limits will then be calculated.

For this analysis, the mean drift capacities are assumed to range between 0.06 and

0.10, where the 0.06 estimate is based on the results of our archetype study and the

0.10 is an upper bound estimate. Similar values are assumed for dispersions with a

correlation between drift demand and capacities of -0.15, which is based on the data

shown previously in Table 6.3. Shown in the right side of Figure 6.12 are probability

distributions for capacity and the corresponding drift limits assuming an intermediate

collapse drift capacity of 0.08. In this case the allowable drift limits would be 0.032,

0.026 and 0.020 for risk categories 1&2, 3 and 4, respectively. Interestingly, the cal-

culated drift limit of 0.032 for risk categories 1&2 is close to the limit of 0.03 specified

in the tall building guidelines (PEER, 2010), and the calculated drift limit of 0.02 for

risk category 4 is equal to the current limit specified in ASCE 7 (American Society of

Civil Engineers Standards Committee, 2015). These values, along with the calculated

drift limits corresponding to drift capacities of 0.06 and 0.10 are summarized in Table

6.6. Comparing these to current standards suggest that the current ASCE 7 drift

limits would meet the target collapse risk if the story drift capacity is 0.10. However,

to the extent that the upper bound drift capacity of 0.10 may be too optimistic (and

well in excess of the values of 0.06 calculated in the building archetype study), this

would suggest that the ASCE 7 drift limits should be reduced.

Table 6.6: Story drift limits for global acceptance criteria corresponding to drift capacitiesof 0.06 to 0.10

Risk Categories

µC 1 & 2 3 4

0.06 0.024 0.019 0.015

0.08 0.032 0.026 0.020

0.10 0.04 0.032 0.025

6.8 Conclusions

In this study, we characterize and propagate modeling uncertainty and quantify its

impacts on the seismic response assessment of structures. The uncertainty in param-

eters defining analysis models and limit state surfaces are incorporated into nonlinear

dynamic analysis for ductile and non-ductile reinforced concrete frame structures.

194


0.04 0.0950

0.5

1

0.032 0.080

0.5

1

0.03 0.0860

0.5

1

Pro

ba

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ty D

en

sity

0.026 0.080

0.5

1

0.02 0.0720

0.5

1

SDR0.02 0.08

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SDR

(a) Risk Category 1 & 2 − P(Collapse) = 0.1

(b) Risk Category 3 − P(Collapse) = 0.06

(c) Risk Category 4 − P(collapse) = 0.03

Figure 6.12: Back-calculated (left) and proposed (right) distributions of drift demandsand capacities for buildings having more than 4 stories a) Risk Categories 1 and 2 b) RiskCategory 3 and c) Risk Category 4. Red and blue lines represent the distributions of demandand capacity, respectively. Mean values are indicated with dashed black lines.

Variability and correlations in structural model parameters are propagated in the dy-

namic analyses using Monte-Carlo simulation with Latin Hypercube sampling. Earth-

quake ground motions are incorporated through multiple stripe analyses with hazard

consistent ground motions that are selected and scaled using a conditional spectra

approach. Impacts of modeling uncertainty are evaluated for fragility functions and

mean annual exceedance rates for drift limits and collapse as well as drift (deforma-

tion) demands and capacities of thirty-three archetype building configurations.

In analyses of the archetype concrete moment frame buildings the modeling un-

certainty both shifted the medians and increased dispersion in the collapse fragility

curves. The median collapse intensity (spectral accelerations) shifted by up to -20%

downward or +10% upward, with an average shift of downward of about -5%. Dis-

persion (variability) in the collapse fragilities increased by about 10% to 70% with

an average increase of about 30%. Cases with small upward (positive) shifts in me-

dians were usually accompanied by the largest increases in dispersion, which tends

to flatten out the fragility curve. The largest downward (negative) shifts in median

tend to occur in structures having weak story mechanisms and non-ductile frames,

where weakest link failure mechanisms are more prevalent. When integrated with

earthquake hazard curves for a high-seismic region of California, the modeling un-

certainty increased the mean annual frequencies of collapse, λc, by about 1.4 to 2.6

times, compared to the analyses that only include ground motion record-to-record

variability. Subject to the same assumptions in modeling parameters, investigation

of sites with hazard curves more representative of the mid and eastern United States

suggest that the influence of modeling uncertainties will be smaller for these regions.

Equivalent parameters to represent modeling uncertainties in collapse fragilities

are back calculated using an SRSS approach to combine variabilities due to ground

motions (record-to-record) and structural modeling and by calibration to achieve

consistent mean annual frequency rates for the high-seismic site. From these analyses,

the average effect of modeling uncertainty on collapse fragility can be represented by

either (1) a shift in median by -5% combined with an added modeling dispersion βc,M

of 0.33, or (2) no shift in median and an added modeling dispersion of βc,M of 0.40.

These general values are, of course, subject to the limitations of the study of concrete

moment frames designed for a high-seismic site.

The impacts of modeling uncertainty on drift-exceedance limit states are observed

to generally be smaller than for collapse and vary depending on the magnitude of the

drifts (amount of inelasticity). For drift-exceedance fragility curves at drift ratios

of 0.03, the modeling uncertainty resulted in a small negative shift in median and

increase in dispersion, which increased the mean annual frequency of drift exceedance

up to 1.3 times with an average increase of about 1.2, compared to the models with

only ground motion variability. At drift-exceedance ratios of 0.07, the changes in

drift-fragility curves and mean annual exceedance rates were about the same as for

the changes in collapse fragilities, with increases in mean annual frequencies of 1.4 to

196


2.6 times with an average increase of about 1.8 times, compared to cases with only

ground motion variability.

To help inform the calibration of drift limits for design using nonlinear dynamic

analysis, we developed a framework for quantifying the joint probability distributions

of story drift demands and story drift collapse capacities. This framework considers

joint treatment of analysis causing collapses and no-collapses to develop consistent

estimates of story drift collapse capacities from incremental dynamic analyses that

include both ground motion and modeling uncertainties. Applying this framework

to four archetype buildings provides estimates of drift demands at MCE-level ground

motion intensity and drift collapse capacities. The resulting median drift capacities

range from story drift ratios of about 0.052 to 0.061, with and average dispersion of

0.45. Drift demands of the archetypes, calculated at MCE level conditional spectra

intensities, range from 0.028 to 0.034, with an average dispersion of 0.55. A small

negative correlation of -0.14 is obtained between drift demands and capacities. The

analyses further indicate that modeling uncertainty (1) increases the median MCE

drift demands by about 5%, (2) increases the dispersion in MCE demand by a negli-

gible amount, (3) reduces the drift collapse capacity by about 7%, and (4) increases

the dispersion in drift collapse capacity by about 65%. Thus, the combined effects of

median shifts and increased dispersion on collapse drift capacity can have a significant

effect on increasing the probability of drift demands exceeding drift capacities.

Finally, the procedure and data for evaluating MCE level drift demands and drift

capacities are used to assess the nonlinear analysis procedures and drift limits in a

recently proposed new chapter 16 in ASCE 7 (American Society of Civil Engineers

Standards Committee, 2015). Subject to the limited scope of our study, the proposed

MCE level drift limits are found to be unconservative and inconsistent with the stated

acceptable collapse risk targets in ASCE 7. For example, whereas the proposed drift

limits for risk category 1&2 buildings range from 0.04 to 0.05, our analyses suggest

that the limits should be about 0.03, which coincidentally is the specified drift limit

in tall building guidelines for design by nonlinear analysis (PEER, 2010; LATBSDC,

2015) .

The methods and strategies employed in this paper to characterize and propagate

modeling uncertainties are based on fairly well established approaches. The main

challenge has been in characterizing the model parameter uncertainties, developing

and verifying the nonlinear analysis models, and setting up procedures to conduct

the simulations and process the output. The tools and methods presented in this

paper can be used to explore the impacts of modeling for other building archetypes

and performance metrics. As shown in these examples, the importance of consid-

ering modeling uncertainty depends, in part, on the particular demand parameters

or performance metrics of interest. More importantly, these examples demonstrate

the feasibility, through modern information and computing technologies, of rigorously

propagating model uncertainty to develop more reliable and risk-consistent methods

for seismic assessment and design of buildings and other structures.

198

Chapter 7

Conclusions

7.1 Summary of Findings and Contributions

This dissertation focuses on modeling uncertainty in seismic performance assessment

of structures. It is based on three main objectives: (1) reliable characterization

of modeling uncertainty by developing strategies for statistically characterizing the

variability in structural component modeling parameters and providing efficient and

reliable procedures to incorporate modeling uncertainties in the probabilistic seismic

response assessment of structures; (2) robust examination of the impacts of modeling

uncertainty on structural seismic response including collapse behavior; (3) provision

of a comprehensive framework for linking seismic response of structures with explicit

consideration of modeling uncertainty and seismic performance goals in building codes

for improved seismic safety. The following sections elaborate important findings and

contributions of this dissertation.

7.1.1 Methods for Incorporating Modeling Uncertainty in

Probabilistic Seismic Response Assessments

Probabilistic seismic response assessment of structures requires (1) characterization of

uncertainty in the random variables and (2) methods for propagating the uncertainties

to achieve robust estimates of collapse risk. This dissertation contributes to both with

199

illustrative applications to the seismic performance assessment of reinforced concrete

moment frame buildings.

Chapter 3 provides a framework for quantifying the correlation in the variability

of structural component model parameters from laboratory tests. The framework

utilizes random effects regression models to differentiate the variability in test results

within and between different laboratories. This enables quantification of the correla-

tions between model parameters at a component level and the interactions of multiple

components associated with a system’s response. The framework is demonstrated us-

ing a database that is composed of cyclic tests of reinforced concrete columns to

quantify the correlations of the parameters for a concentrated plasticity model. The

method considers both correlation of parameters within a component and between

different components. Like parameters between components are observed to have

significant correlations; whereas correlations of different model parameters between

components are rather small. Within a component, parameters are observed to have

little to no correlation.

Chapter 4 explores uncertainty propagation methods for probabilistic prediction

of collapse response of structures. A major challenge in selecting a structural reliabil-

ity method is balancing computational efficiency and accuracy. Chapter 4 provides

insights on the relative performance of the considered methods for collapse response

assessments. The methods are categorized as moment-based, simulation-based and

surrogate methods. The methods are compared based on their following properties:

(1) computational demand, (2) statistical efficiency of estimators, (3) characterization

of uncertainty, (4) behavior under high-dimensional problems, (5) ability to capture

nonlinearity of the limit state functions, and (6) ease of implementation. Based on

these, Latin Hypercube sampling (LHS) is found to provide superior performance

in terms of computational and statistical efficiency and scalability for collapse risk

estimation. For a similar computational effort of approximately 1300 incremental

dynamic analyses, different methods compare as follows: FOSM yielded collapse rate

(λc) estimates that differed from the benchmark response, which is determined using

random-sampling based Monte Carlo simulations, up to 100%. If calibrated to pre-

dict the lower tail, the response surface method closely estimates the important lower

200

CHAPTER 7. CONCLUSIONS 201

tail of the collapse fragility curve and yields good estimations of λc. Also discussed

in Chapter 4 is a neural-network-based collapse response assessment. This method

performs fairly well estimating the overall fragility curve as well as λc (estimates differ

from the benchmark by only 4-7%). However, the calibration and training of neu-

ral networks are comparably challenging, which makes their implementation difficult.

Good estimates of collapse fragility and λc are also obtained with LHS; λc estimates

differ from the benchmarked responses by 1% with a coefficient of variation of 3-6%.

The computational effort of LHS is further reduced by a factor of 3.3 using an MSA

approach compared to an IDA approach, while still yielding similar variability in the

estimators.

To dramatically reduce the computational demand of uncertainty propagation,

Chapter 5 proposes a Bayesian approach for collapse risk assessments. The method is

suggested to be used with 30 structural analyses at two or more intensity levels. The

method incorporates prior information on structural collapse capacity, which can be

informed by a variety of sources, including information on the building design criteria,

empirical fragility functions or nonlinear static (pushover) analysis. This information

is later updated in a Bayesian framework using the results from a few structural

analyses and posterior estimates of collapse risk are obtained. Sensitivity analysis

and robustness tests of the method are performed using simulated structural analysis

data and statistical properties of the estimators are assessed. The Bayesian method

is observed to significantly improve statistical efficiency of collapse risk predictions

compared to an alternative maximum likelihood method, and it provides considerable

reduction in computational demand for probabilistic collapse risk assessment of struc-

tures. To obtain a coefficient of variation of 0.25 in λc, maximum likelihood requires

200 structural analyses; however, using the Bayesian method, similar variability can

be achieved with only 60 structural analyses. Initial estimates of the collapse fragility

(i.e., prior distributions) play a crucial role in the accuracy of the Bayesian estimates.

It is observed that biased initial guesses result in biased estimates of collapse risk,

where the amount of bias depends on both the bias and confidence introduced in the

initial guess. A model checking method, called posterior predictive checking, is pro-

vided for assessing the plausibility of posterior distributions. Further, an illustrative

collapse risk assessment of a 4-story reinforced concrete frame structure is provided.

7.1.2 Impacts of Modeling Uncertainty on Structural Seismic

Performance

Benchmark estimates are provided for the impacts of modeling uncertainty for re-

inforced concrete structures, to further develop recommendations and guidelines to

incorporate their effects.

First, Chapter 3 explored the impacts of correlations among random variables

on seismic performance. Two extreme cases of uncorrelated and perfectly correlated

models resulted in significant differences in collapse risk estimates. A factor of 2.4

difference in λc estimates was observed at a seismically active site in Los Angeles

for a 4-story reinforced concrete frame structure. A realistic representation of corre-

lation structure among random variables defining concentrated plasticity models is

then considered. The collapse fragility function using these correlation coefficients

is similar to that from an uncorrelated model, especially at smaller ground motion

intensities, which contribute most to λc. Finally, by assuming perfect correlation for

beam-to-beam and column-to-column interactions, a collapse response similar to that

observed with all members are partially correlated are obtained. This assumption re-

sults in significant reduction in the number of random variables needed for modeling

uncertainty propagation studies.

Second, the uncertainty in parameters defining analysis models and limit state

surfaces are incorporated into nonlinear dynamic analysis for ductile and non-ductile

archetype reinforced concrete frame structures in Chapter 6 to evaluate impacts of

modeling uncertainty on seismic performance using hazard-consistent ground motions.

In analyses of the archetype concrete moment frame buildings, it is observed that

impacts of modeling uncertainty is more dominant for collapse response compared

to other limit states. For drift-exceedance fragility curves at drift ratios of 0.03, the

modeling uncertainty resulted in a small negative shift in median and increase in

dispersion, which increased the mean annual frequency of drift exceedance up to 1.3

times with an average increase of about 1.2, compared to the models with only ground

202


motion variability. At story drift ratios of 0.04 or greater, modeling uncertainties

have more significant effects. At story drift ratios of 0.04, modeling uncertainty

shifted the median collapse capacity on average by -5% and increased the dispersion

(variability) by about 20%. For collapse response, median collapse capacity shift

and dispersion increase, on average, are -5% and 30%, respectively. More significant

reduction in median collapse capacity is obtained for structures that are prone to

weakest link failure mechanisms (i.e. structures having weak story mechanisms and

non-ductile frames). The combined effects of modeling uncertainty increased λc, by

about 1.4 to 2.6 times, compared to the analyses that only include ground motion

record-to-record variability for a high seismicity site at Los Angeles. Subject to the

same assumptions in modeling parameters, investigation of sites with steeper hazard

curves more representative of the central and eastern United States suggest that the

influence of modeling uncertainties will be smaller for these regions.

Finally, the impacts of modeling uncertainty are investigated in terms of story drift

ratios. To ensure building safety, an important global-level check is through story drift

limits in emerging building code design requirements that employ the use of nonlinear

dynamic analysis, such as applied to tall buildings (PEER, 2010; LATBSDC, 2015)

or in the recently proposed ASCE 7 requirements American Society of Civil Engi-

neers Standards Committee (2015). Typically, these drift limits are checked under

maximum considered earthquake (MCE) ground motions. The analyses indicate that

modeling uncertainty (1) increases the median drift demands at maximum considered

earthquake (MCE) level by about 5%, (2) increases the dispersion in MCE demand

by a negligible amount, (3) reduces the median drift collapse capacity by about 7%,

and (4) increases the dispersion in drift collapse capacity by about 65%. Thus, the

combined effects of median shifts and increased dispersion on collapse drift capacity

can have a significant effect on increasing the probability of drift demands exceeding

drift capacities. As there is a quantifiable increase in the median drift, dispersion

and annual frequency under MCE ground motions due to modeling uncertainty, one

could envision applying a reliability factor to account for the modeling uncertainties,

even at comparatively low drift levels.

7.1.3 Framework for Linking Seismic Response to Seismic

Performance Goals Specified in Building Codes

An important premise of performance-based methodologies is to quantify building

performance in terms of decision variables, including casualties, repair and replace-

ment costs, downtime. These decision variables are affected by the estimates in

structural collapse probabilities. For instance, human loss is largely attributable to

building collapse. FEMA-P58 (FEMA, 2012) provides several alternative procedures

for developing collapse fragilities. Among the procedures are static pushover to in-

cremental dynamic analysis (SPO2IDA) method (Vamvatsikos & Cornell, 2005) and

a method for developing fragility functions using engineering judgement. In Chap-

ter 5, I propose a Bayesian approach to obtain practical estimates of collapse risk.

The proposed approach can be used to augment the collapse fragility procedures in

FEMA-P58 (FEMA, 2012). An illustration of the method is provided in Chapter 5,

in which the prior information is obtained from SPO2IDA. It is also possible to obtain

an initial estimate using engineering judgment and supplement it using few analyses

with the Bayesian approach.

Risk-based assessment of structural response provides time-based estimates of

structural response (i.e. the mean annual frequency of exceedance of certain limit

performance state), which can then be compared to seismic performance goals in

building codes. The risk-based assessment strategy adopted in this dissertation is

applied as follows:

• To achieve hazard consistency, a conditional spectra approach is used. Ground

motions are selected to match conditional spectra at various intensity levels and

record-to-record variability is propagated.

• Structural idealizations that are capable of simulating collapse response is used

and structural response is assessed up to collapse.

• The uncertainty in the structural idealizations are propagated using Monte-

Carlo simulations with Latin Hypercube sampling with realistic characterization

of variability and correlations among random variables.

204


• The record-to-record variability and modeling uncertainty are combined and a

multiple-stripes analysis strategy adopted to predict collapse risk of structures.

A uniform collapse risk of 1% in 50 years is targeted in MCE seismic design

maps ??. In the presence of modeling uncertainty, for the 30 code-conforming ductile

reinforced-concrete structures, an average mean annual frequency of collapse, λc of

4.04E-4 is obtained at a high-seismicity site at LA, which corresponds to a probability

of collapse of 2% in 50 years. Given this prediction, the compliance of the collapse

risk with the intended performance should be investigated in the future. An average

λc of 4.02E-3 is obtained for the three non-ductile reinforced concrete structures at a

high-seismicity site, corresponding to a probability of collapse of 18% in 50 years.

An important contribution of this dissertation is a framework proposed in Chap-

ter 6 that links structural response predictions to collapse safety goals specified in

building codes. The essence of the method is based on the quantification of a joint

probability distribution of demand and capacity in terms of engineering demand pa-

rameters. The framework is unique in that it enables joint treatment of collapsed and

non-collapsed structural analyses results resulting in consistent treatment of struc-

tural collapse capacity and demand at particular ground motion intensity levels. A

demonstration of the method is performed for selected archetype structures and es-

timates of drift demands at MCE-level ground motion intensity and drift collapse

capacities are obtained. This data, combined with the proposed framework for eval-

uating MCE level drift demands and drift capacities, are used to assess the nonlinear

analysis procedures and drift limits in a recently proposed new chapter 16 in ASCE

7 (American Society of Civil Engineers Standards Committee, 2015). Subject to the

limited scope of our study, the proposed MCE level drift limits are found to be uncon-

servative and inconsistent with the stated acceptable collapse risk targets in ASCE 7

(i.e. 10% probability of collapse at MCE). For example, whereas the proposed drift

limits for risk category 1&2 buildings range from 0.04 to 0.05, our analyses suggest

that the limits should be about 0.03, which coincidentally is the specified drift limit

in tall building guidelines for design by nonlinear analysis (PEER, 2010; LATBSDC,

2015).

7.2 Limitations and Future Research

This dissertation investigated issues related to probabilistic seismic risk assessment of

structures and the impacts of modeling uncertainty on seismic collapse safety. This

work is subjected to some limitations and suggests opportunities for future work.

7.2.1 Structural Models and Analyses Strategies

This dissertation quantified the impacts of modeling uncertainty on a reinforced con-

crete bridge column, 30 ductile and 3 non-ductile reinforced concrete moment frames.

Assessment of the impacts of modeling uncertainty on additional types of structural

systems would be a natural extension of this dissertation to generalize the results for

different structural systems.

This dissertation investigated the impacts of modeling uncertainty on story drift

ratios and limit states that are quantified using drift ratios. The influence of mod-

eling uncertainty on other demand parameters, including floor accelerations, global

and local forces, should be explored in the future. In addition, the change in statisti-

cal properties of the engineering demand parameters with respect to ground motion

intensity levels should further be investigated.

7.2.2 Types of Uncertainties

In this dissertation, the statistical distributions of random variables defining con-

centrated plasticity models are characterized using their variability estimates and

correlation structure. These model parameters are assumed to be unbiased; i.e. the

analysis model is assumed to characterize median values of model parameters. For

non-ductile frames, the uncertainty in the shear and axial parameters is incorpo-

rated using the variability in the parameters as well as biases reported in previous

studies. Bias in random variables can be an important factor affecting the accuracy

in structural response estimation. It can be addressed incorporating bias in random

variables through uncertainty propagation, as done for non-ductile frames, or through

post-processing of results via bias factors. Future work is recommended to investigate

206


the importance and the need for bias factors in collapse risk estimation.

Sources of modeling uncertainties include variability in physical and phenomeno-

logical quantities and uncertainty in structural idealizations and analysis strategies.

This dissertation mainly addresses the former two sources of uncertainty. The lat-

ter one was briefly discussed in Chapter 2, using blind prediction contest data, and

compared modeling uncertainty predictions of a bridge column to blind prediction

analyses and shake table test results. Future work is needed to explore blind predic-

tion submissions for quantifying modeling uncertainty in structural response with the

aim of validating current strategies to incorporate their effects.

In this dissertation, case study structures are idealized using two-dimensional

models that employ a concentrated plasticity approach for modeling flexural fail-

ure behavior, following the previous research of Haselton (2006) who calibrated and

applied the models for collapse analysis. Future work should explore the impacts

of modeling uncertainty using different idealization strategies and refine the added

uncertainty due to a particular choice of structural idealization and analysis model.

7.2.3 Reliability Analyses

Machine learning is an extensive and continuously developing field, offering a wide-

range of regression and classification-based techniques that can be used with proba-

bilistic seismic collapse assessment. In this dissertation, we explored regression-based

techniques and used neural networks to predict quantiles of collapse capacity of struc-

tures, in terms of ground motion intensities. Classification-based techniques can also

be used for probabilistic seismic collapse assessment to predict probabilities of collapse

at alternative ground motion intensity levels. It is the author’s opinion that classifi-

cation methods will increase efficiency of probabilistic collapse assessment. However,

a major challenge would be feature selection owing to characterization of features

representing ground motion variability.

An efficient method adopting a Bayesian approach is proposed for seismic collapse

assessments. Sensitivity and robustness analyses of the method is performed using

simulated analysis data. A demonstration is provided using a 4-story reinforced

concrete frame, prior information of which is obtained using pushover analyses. For

generalization of the methods, future research should explore the effectiveness of

different strategies for obtaining prior estimates and extend the robustness analyses

using different structures.

A framework linking global story drift acceptance criteria to seismic performance

goals is proposed. This framework is applied to a limited number of structures and

the results are compared with the specified drift limits and target probabilities from

ASCE 7 Chapter 16 (American Society of Civil Engineers Standards Committee,

2015). Future research is needed to generalize the results for other types of systems

to affirm and refine our recommendations for MCE level drift limits using refined

strategies for criteria for determining drift capacities.

7.2.4 Other Limitations

This dissertation explored the impacts of modeling uncertainty on axial and shear

failure of non-ductile frames in addition to side-sway failure. The characterization of

the uncertainty in axial and shear failure models is obtained from previous studies

and the correlation of the model parameters characterizing axial and shear failures is

neglected.

The impacts of modeling uncertainty are mainly assessed using risk predictions

at a high-seismicity region in Los Angeles, CA. To a limited extent, the results are

generalized to sites having different seismic hazard characteristics through idealized

power-law hazard curves and a site at Memphis.

Collapse behavior is identified using thresholds on engineering demand parame-

ters. Refined characterization of collapse response has been explored by Deniz (2015)

(i.e. rules characterizing collapse response based on damage measures, intensity mea-

sures or energy) and different strategies can easily be adopted for future studies.

This dissertation proposes a framework for characterization of model parameter

correlation from components tests. This framework is demonstrated using a limited

database of reinforced-concrete column tests, which is composed of 255 tests. The

robustness of the correlation coefficients depends on the quality of the fit of the

208


empirical models that are used in this framework as well as the protocol and the fit of

the empirical equations by Haselton et al. (2008) and Panagiotakos & Fardis (2001),

that predicts the model parameters.

The uncertainty due to construction and human errors are out of scope of this

dissertation.

7.3 Concluding Remarks

This dissertation contributes to the field of probabilistic seismic performance assess-

ment of structures by examining the impacts of modeling uncertainty, providing reli-

ability analysis methods that are tailored to seismic collapse response assessment and

enabling a refined link of global structural response predictions and seismic collapse

safety of structures. The main contributions are the tools and the frameworks to

assess the failure of a complex system in the presence of modeling uncertainty and

understand the potential effects of this uncertainty in structural design and analy-

sis. Understanding the effects of modeling uncertainties, and accounting for their

effects in design codes, will ultimately lead to safer, more efficient and more econom-

ical structural designs. The results presented in this dissertation will ultimately help

solve problems being faced by technology transfer projects to link performance-based

engineering approaches with building design standards.

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