seismic vulnerability assessment of a family of

The Pennsylvania State University

The Graduate School

College of Engineering

SEISMIC VULNERABILITY ASSESSMENT OF A FAMILY OF HORIZONTALLY

CURVED STEEL BRIDGES USING RESPONSE SURFACE METAMODELS

A Dissertation in

Civil Engineering

by

Junwon Seo

© 2009 Junwon Seo

Submitted in Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

August 2009

The dissertation of Junwon Seo was reviewed and approved* by the following:

Daniel G. Linzell Associate Professor of Civil and Environmental Engineering Dissertation Advisor Chair of Committee

Jeffrey A. Laman Associate Professor of Civil and Environmental Engineering

Andrew Scanlon Professor of Civil and Environmental Engineering

Linda M. Hanagan Associate Professor of Architectural Engineering

Peggy A. Johnson Head of the Department of Civil and Environmental Engineering

*Signatures are on file in the Graduate School

iii

ABSTRACT

Civil infrastructure systems must be designed and constructed to resist the effects of

natural and manmade hazards to ensure public safety and to support the socio-economic goals and

needs of society. In recent decades, earthquake hazards have been viewed as extremely important

among the natural hazards impacting civil infrastructure systems across certain regions in the

United States. The occurrence of three major earthquakes during that period (San Fernando in

1971, Loma Prieta in 1989, and Northridge in 1994) demonstrated the possible seismic

vulnerabilities that existing bridges may contain. These major seismic events also have provided

the impetus for significant improvements in engineering practices for bridge seismic design,

analysis, and vulnerability assessment.

Bridge seismic risk assessment tools have been proposed and utilized by many engineers

and researchers since the inception of earthquake engineering in the 1970’s. These tools have

predominately used fragility curves, which are conditional probability statements that give the

probability of a bridge reaching or exceeding a particular damage level for an earthquake of a

given intensity level, for examining straight bridge structures. Fragility curves for the bridge

components and system are essential inputs into the final damage estimation algorithm for a given

earthquake event.

Since these tools were developed for evaluating the seismic vulnerability of straight

bridges, they cannot be applied to curved bridges. There has been a steady growth in the use of

horizontally curved steel bridges since approximately 1970, which coincides with the initiation

period of the earthquake engineering field. Effects of various curved bridge parameters,

including radius of curvature, on the fragility of bridges across geographic regions must be

investigated for their seismic assessment.

In this study the characteristics of structures in a target inventory were used to estimate

fragilities for a family of horizontally curved steel bridges. Consideration was restricted to the

iv

class of bridge structures consisting of horizontally curved steel I-girder bridges. Statistically

significant predictors for seismic vulnerability assessment were identified using Design of

Experiments (DOE) and other statistical tools, and appropriate seismic response surface

metamodels (RSMs) were developed to rapidly predict seismic response for a family of

horizontally curved steel bridges. Fragility curves for horizontally curved steel I-girder bridges

were estimated using the metamodels with Monte Carlo simulation. The use of metamodels

reduced the required computations and made it practical to carry out probabilistic response

calculations in an efficient manner. Various sources of structural uncertainty were considered

and tracked throughout the study, including radius of curvature, number of spans, cross-frame

spacing, girder spacing, span length. This approach allowed for the direct implementation of

findings into existing seismic risk assessment packages (e.g., FEMA Hazards U.S. Multi-Hazard

loss assessment package, etc.). Findings from the study show that this approach provides reliable

fragility curves for a family of horizontally curved steel I-girder bridges in the target region.

v

TABLE OF CONTENTS

LIST OF FIGURES ......................................................................................................................viii

LIST OF TABLES.........................................................................................................................xii

ACKNOWLEDGEMENTS..........................................................................................................xiv

Chapter 1 INTRODUCTION.......................................................................................................... 1

1.1 Background................................................................................................................................ 1 1.2 Problem Statement ..................................................................................................................... 2 1.3 Objective, Scope and Organization............................................................................................ 3

Chapter 2 LITERATURE SEARCH .............................................................................................. 5

2.1 Historical Review....................................................................................................................... 5 2.1.1 Horizontally Curved Steel Bridges ........................................................................... 6

2.2 Fragility Curves ......................................................................................................................... 8 2.2.1 Expert Based Fragility Curves.................................................................................. 9 2.2.2 Empirical Fragility Curves ..................................................................................... 10 2.2.3 Analytical Fragility Curves .................................................................................... 13 2.2.4 Analytical Fragility Curve Development Using Metamodels .................................... 17

2.3 Conclusion ............................................................................................................................... 18

Chapter 3 RESPONSE SURFACE METAMODEL METHODOLOGY FOR GENERATION OF BRIDGE FRAGILITY CURVE .................................................................................................... 19

3.1 Metamodels.............................................................................................................................. 19 3.2 Experimental Designs .............................................................................................................. 21

3.2.1 Full Factorial Design ............................................................................................. 22 3.2.2 Central Composite Design ..................................................................................... 23

3.3 Response Surface Metamodels ................................................................................................ 25 3.4 Response Surface Metamodels for Seismic Fragility Assessment .......................................... 31 3.5 Conclusion ............................................................................................................................... 34

Chapter 4 3-D ANALYTICAL MODELING APPROACH OF HORIZONTALLY CURVED STEEL BRIDGE............................................................................................................................ 35

4.1 Modeling Approach ................................................................................................................. 36 4.1.1 Superstructure....................................................................................................... 38 4.1.2 Substructure.......................................................................................................... 39

4.2 Model Validation ..................................................................................................................... 42 4.2.1 Examined Bridge Description ................................................................................ 43

vi

4.2.2 3-D Analytical Model ............................................................................................ 48 4.2.3 Validation Procedure ............................................................................................. 51

4.2.3.1 Field Testing ............................................................................................. 51 4.2.3.2 Static 1 ..................................................................................................... 53 4.2.3.3 Static 2 ..................................................................................................... 56 4.2.3.4 Static 3 ..................................................................................................... 58 4.2.3.5 Static 4 ..................................................................................................... 60 4.2.3.6 Discussion ................................................................................................ 62

4.3 Seismic Response Methodology .............................................................................................. 63 4.3.1 Mode shapes ......................................................................................................... 65 4.3.2 Seismic Response.................................................................................................. 67

4.4 Conclusions.............................................................................................................................. 72

Chapter 5 HORIZONTALLY CURVED STEEL BRIDGE INVENTORY AND GROUND MOTION DEVELOPMENT ......................................................................................................... 74

5.1 Horizontally Curved Steel Bridge Inventory Analysis ............................................................ 75 5.2 Potential Key Parameters for Horizontally Curved Steel Bridge............................................. 77

5.2.1 Macro-Level Parameters ........................................................................................ 78 5.2.1.1 Number of Spans ....................................................................................... 78 5.2.1.2 Maximum Span Length .............................................................................. 80 5.2.1.3 Deck Width ............................................................................................... 81 5.2.1.4 Column Height .......................................................................................... 82 5.2.1.5 Radius of Curvature................................................................................... 84 5.2.1.6 Girder Spacing.......................................................................................... 85 5.2.1.7 Cross-Frame Spacing ................................................................................ 86

5.2.2 Micro-Level Parameters......................................................................................... 87 5.3 Synthetic Ground Motions....................................................................................................... 88 5.4 Conclusions.............................................................................................................................. 92

Chapter 6 SCREENING OF HORIZONTALLY CURVED STEEL BRIDGE PARAMETERS 93

6.1 Screening Experiments for Inputs............................................................................................ 95 6.2 Screen Experiments for Outputs: Seismic Response ............................................................... 98 6.3 Parameter Screening ................................................................................................................ 99 6.4 Conclusions............................................................................................................................ 108

Chapter 7 SEISMIC FRAGILITY CURVES FOR HORIZONTALLY CURVED STEEL BRIDGES .................................................................................................................................... 109

7.1 RSMs Construction................................................................................................................ 110 7.2 Seismic Performance Levels.................................................................................................. 126 7.3 Seismic Fragility Curve Generation....................................................................................... 129

7.3.1 Seismic Fragility Curves of Bridge Component ..................................................... 131 7.3.2 Holistic Seismic Fragility Curves ......................................................................... 138 7.3.3 Fragility Curve Case Studies ................................................................................ 141

vii

7.4 Conclusions............................................................................................................................ 149

Chapter 8 CONCLUSIONS, IMPACT AND FUTURE RESEARCH....................................... 150

8.1 Summary and Conclusions .................................................................................................... 150 8.2 Impact .................................................................................................................................... 154 8.3 Areas for Future Research ..................................................................................................... 155

Bibliography ................................................................................................................................ 156

Appendix A PLACKETT-BURMAN DESIGN......................................................................... 166

Appendix B CENTRAL COMPOSITE DESIGN ...................................................................... 171

viii

LIST OF FIGURES

Figure 2-1: Sample Set of Fragility Curves. .................................................................................... 9

Figure 2-2: Empirical Fragility Curves of Multi-Span Bridges for PGA (Basoz and Kiremidjian, 1997). .................................................................................................................................... 12

Figure 2-3: Empirical Fragility Curves of Bridge Structures for PGA (Yamazaki, 2000). ........... 13

Figure 3-1: Steps for Constructing Metamodels (Simpson et al., 2001)........................................ 21

Figure 3-2: Schematic Representation of (a) Full Factorial Design (FFD) and (b) Central Composite Design (CCD) for Three Variables (Simpson et al., 2001)................................. 22

Figure 3-3: Central Composite Design (CCD) for Three Dimensions. ......................................... 25

Figure 3-4: Fragility Construction Flowchart using RSMs............................................................ 33

Figure 4-1: Illustration of Typical Curved Steel Bridge Components. .......................................... 37

Figure 4-2: 3-D Analytical Model of Typical Curved Steel Bridge. ............................................. 37

Figure 4-3: Typical Curved Steel Bridge Superstructure Cross Section........................................ 38

Figure 4-4: Circular Reinforced Concrete Column........................................................................ 40

Figure 4-5: Discretization of Typical Pier. ....................................................................................41

Figure 4-6: Picture of Examined Curved Steel I-Girder Bridge. ................................................... 43

Figure 4-7: Examined curved steel I-girder bridge (Nevling, 2003). ............................................ 44

Figure 4-8: Curved Steel I-Girder Bridge Configuration............................................................... 46

Figure 4-9: Photos of Spherical Bearing System. .......................................................................... 48

Figure 4-10: Analytical Modeling of Spherical Bearing................................................................ 50

Figure 4-11: 3-D Analytical Model of Horizontally Curved Steel Bridge. ................................... 51

Figure 4-12: Static Test Truck Positions (Nevling, 2003). ............................................................ 53

Figure 4-13: Girder Instrument Locations Over Pier – Selected for Validation (Nevling, 2003). 53

Figure 4-14: Comparison Graph – Bending Moments; Static Testing 1. ...................................... 55

Figure 4-15: Percent Difference Histogram – Bending Moments; Static Testing 1. ..................... 55



ix





Figure 4-22: Seismic Loading Direction........................................................................................ 63

Figure 4-23: Ground Motion Used for Illustration of Seismic Responses (Chopra, 2000): .......... 64

Figure 4-24: Mode Shapes for Curved Bridge; First to Fourth...................................................... 66

Figure 4-25: Deck Displacement Time Histories for the Curved Steel I-Girder Bridge under (a) Global Longitudinal Loading (b) Global Transverse Loading.............................................. 68

Figure 4-26: Column 1 & 2 of the Curved Steel I-Girder Bridge Under ....................................... 70

Figure 5-1: Maryland, New York, Pennsylvania Considered in the Inventory Study with Hazard Map (USGS, 2002)................................................................................................................ 75

Figure 5-2: Horizontally Curved Steel I-Girder Bridge Counts with respect to the Number of Curvatures. ............................................................................................................................ 76

Figure 5-3: Frequency Histograms for Number of Spans.............................................................. 79

Figure 5-4: Probability Density Function for Number of Spans.................................................... 80

Figure 5-5: Cumulative Distribution Function for Maximum Span Length. ................................. 81

Figure 5-6: Cumulative Distribution Function for Deck Width..................................................... 82

Figure 5-7: Cumulative Distribution Function for Column Height. .............................................. 84

Figure 5-8: Cumulative Distribution Function for Radius of Curvature........................................ 85

Figure 5-9: Cumulative Distribution Function for Girder Spacing................................................ 86

Figure 5-10: Cumulative Distribution Function for Cross-Frame Spacing.................................... 87

Figure 5-11: Histogram of PGA Values of Rix and Fernandez Ground Motion Suite. (Rix and Fernandez-Leon 2004). ......................................................................................................... 90

Figure 5-12: Each Representative Rix and Fernandez Ground Motion at PGA Zone of 0.01g, 0.24g, and 0.44g. ................................................................................................................... 91

Figure 5-13: Mean and Mean ± One Standard Deviation of Response Spectra - Rix and Fernandez (Rix and Fernandez-Leon 2004). .......................................................................................... 91

Figure 6-1: Screening Procedure. .................................................................................................. 94

x

Figure 6-2: Specified Analytical Model between Bearing and Abutment. .................................... 99

Figure 6-3: Results of Maximum Tangential Deformation at Abutments. .................................. 103

Figure 6-4: Results of Maximum Radial Deformation at Abutments.......................................... 104

Figure 6-5: Results of Maximum Column Curvature Ductility. .................................................. 105

Figure 6-6: Results of Maximum Tangential Deformation at Bearings. ..................................... 106

Figure 6-7: Results of Maximum Radial Deformation at Bearings. ............................................ 107

Figure 7-1: RSMs Construction for Horizontally Curved Steel Bridges. .................................... 111

Figure 7-2: Responses Surface Plots for Mean of Maximum Tangential Deformation at Abutment of Horizontally Curved Steel Bridges. ................................................................................ 116

Figure 7-3: Responses Surface Plots for Mean of Maximum Radial Deformation at Abutment of Horizontally Curved Steel Bridges. .................................................................................... 118

Figure 7-4: Responses Surface Plots for Mean of Curvature Ductility at Column or Abutment of Horizontally Curved Steel Bridges. .................................................................................... 120

Figure 7-5: Responses Surface Plots for Mean of Maximum Tangential Deformation at Bearing of Horizontally Curved Steel Bridges. ................................................................................ 122

Figure 7-6: Responses Surface Plots for Mean of Maximum Radial Deformation at Bearing of Horizontally Curved Steel Bridges. .................................................................................... 124

Figure 7-7: Development of Seismic Fragility Curves. ............................................................... 130

Figure 7-8: Probability Density Functions for Significant Curved Bridge Parameters. .............. 131

Figure 7-9: Seismic Fragility Curves of Column Curvature Ductility at Horizontally Curved Steel Bridges. ............................................................................................................................... 132

Figure 7-10: Seismic Fragility Curves for Bearing (Tangential Deformation) at Horizontally Curved Steel Bridges........................................................................................................... 134

Figure 7-11: Seismic Fragility Curves for Bearing (Radial Deformation) at Horizontally Curved Steel Bridges. ...................................................................................................................... 135

Figure 7-12: Seismic Fragility Curves for Abutment (Passive-Tangential Deformation) at Horizontally Curved Steel Bridges. .................................................................................... 135

Figure 7-13: Seismic Fragility Curves for Abutment (Passive-Radial Deformation) at Horizontally Curved Steel Bridges. .................................................................................... 136

Figure 7-14: Seismic Fragility Curves for Abutment (Active-Tangential Deformation) at Horizontally Curved Steel Bridges. .................................................................................... 137

xi

Figure 7-15: Seismic Fragility Curves for Abutment (Active-Radial Deformation) at Horizontally Curved Steel Bridges........................................................................................................... 138

Figure 7-16: Bi-variate Joint PDF Integrated Over System Failure Domains (Ang and Tang, 1975). .................................................................................................................................. 139

Figure 7-17: Holistic Seismic Fragility Curves of A Family of Horizontally Curved Steel I-Girder Bridges. ............................................................................................................................... 141

Figure 7-18: Seismic Fragility Curves for B-1. ........................................................................... 143









Figure 7-27: Seismic Fragility Curves for B-10. ......................................................................... 148

xii

LIST OF TABLES

Table 2-1: Damage Probability Matrix for Multi-Span Bridges (%) (Basoz and Kiremidjian, 1997). .................................................................................................................................... 11

Table 4-1: Girder Dimensions. ...................................................................................................... 47

Table 4-2: Radius and Span Lengths. ............................................................................................ 47

Table 4-3: Test Truck Parameters (Nevling, 2003). ...................................................................... 52

Table 4-4: Summary of Single Truck Load Cases (Nevling, 2003). ............................................. 52

Table 4-5: Maximum Displacements at bearing and abutment. .................................................... 72

Table 5-1: Number of Span Statistics for Horizontally Curved Steel Bridges. ............................. 79

Table 5-2: Span Length Statistics for Curved Steel I-Girder Bridges. .......................................... 80

Table 5-3: Deck Width Statistics. .................................................................................................. 82

Table 5-4: Column Height Statistics. ............................................................................................. 83

Table 5-5: Radius of Curvature Statistics. ..................................................................................... 84

Table 5-6: Girder Spacing Statistics. ............................................................................................. 86

Table 5-7: Cross-Frame Spacing Statistics. ................................................................................... 87

Table 5-8: Potential Micro-Level Parameters for Response Surface Model Generation............... 88

Table 5-9: Sets of Ground Motion Records (Rix and Fernandez-Leon 2004); The presence of A. in a field means “Available”, while N.A. means “Not Available”........................................ 90

Table 6-1: Sample Full Factorial Experimental Design................................................................. 95

Table 6-2: Selected Horizontally Curved Steel I-Girder Bridge Parameters. ................................ 96

Table 6-3: Two Level Predictor Parameters for Plackett-Burman Experimental Design.............. 97

Table 6-4: Summary of Most Significant Horizontally Curved Steel I-Girder Bridge Parameters.............................................................................................................................................. 108

Table 7-1: Screened Most Significant Parameters for RSMs models.......................................... 112

Table 7-2: Statistical Performance Measures for RSMs. ............................................................. 125

Table 7-3: HAZUS Qualitative Performance Levels (FEMA, 2003). ......................................... 126

Table 7-4: Performance Levels for Bridge Components (Nielson, 2005). .................................. 127

xiii

Table 7-5: Performance Levels for Horizontally Curved Bridge................................................. 128

Table 7-6: Characteristics of Ten Existing Curved Steel Bridges. .............................................. 142

xiv

ACKNOWLEDGEMENTS

As with any significant work of an individual, this work would not have been possible

without the intellectual contribution of others. I would like to thank all those people who have

made it so.

Especially, I would like to thank Dr. Daniel G. Linzell. He has offered the guidance,

knowledge, support, opportunity, and camaraderie which have made my graduate experience so

successful at The Pennsylvania State University. I also would like to thank to the members of my

doctoral guidance committee who provided critical evaluation of this work: Drs. Jeffrey A.

Laman, Andrew Scanlon, and Linda M. Hanagan. I would also like to acknowledge Drs. Barry J.

Goodno, Laurence Jacobs, Reginald DesRoches and Jin-Yeon Kim for their advice and

encouragement at The Georgia Institute of Technology.

This work was funded through the Thomas D. Larson Pennsylvania Transportation

Institute (LTI) by the Pennsylvania Department of Transportation (PennDOT). Their financial

and technical support is acknowledged. For additional financial support provided by the Korean

Government, I would also like to thank Korea Institute of Energy and Resources Technology

Evaluation and Planning Foundation (KETEP). I would like to extend my thankfulness to my

former advisors at Yonsei and Konyang University in South Korea: Drs. Sang-Hyo Kim and

Gwang-Hee Heo.

No graduate experience is complete without the friendship of my fellow graduate

students. At The Pennsylvania State University, I have been fortunate to be surrounded by an

excellent group of graduate students who have offered technical knowledge, lively discussions,

and friendship. I would like to give thanks to Chien-Chung, Javier, Gaby, Tanit, Jared, Zi,

Richard, Deanna, Lynsey, Mohammad, Andrew Coughlin, and Venkata. I would like to give

many thanks to my former officemates and friends at The Georgia Institute of Technology:

Joonam, Jongwan, Chiwon, Matthew, Bryant, Leonardo, Pier, Davide, Benoit, Bassem, Jason,

xv

Jamie, Peeranan, Susendar, Masahiro and the many other students with whom I’ve been

associated. Also, many thanks go to the Korean students at Penn State and Georgia Tech.

I am truly grateful for the unconditional support of my family in South Korea. My

parents and my sisters have offered endless support, confidence in me, wise advice, and love. I

extend my sincere appreciation and love to them.

xvi

Dedication

To

My Parents Myongseok and Hyangsook, and My Sisters Wonmi and Wonhee

Chapter 1

INTRODUCTION

1.1 Background

Transportation networks are spatially distributed systems whereby components are

exposed to natural and man-made hazards. Transportation networks significantly affect world-

wide social and economic stability due to the dependence on the reliable supply of goods and

services (Dueñas-Osorio, 2005). Highways, railroads, airports and harbors represent critical

elements of the social infrastructure needed to balance supply and demand activities on national

and international scales.

If an earthquake strikes near urban regions, it is essential that the transportation systems

remain operational. Past earthquakes have demonstrated that loss of critical highway components

(e.g., bridges, roadways, etc.) can severely impact the economy of the regions and recovery

activities (Murachi et al., 2003).

Bridges are one of the most vulnerable highway components during an earthquake

(Shinozuka et al., 2000; Choi et al., 2004; Nielson, 2005). It is necessary that the seismic

vulnerability of bridges for various damage states be evaluated while carrying out a transportation

system seismic risk analysis. The generation of vulnerability information in the form of fragility

curves is a common approach when assessing bridge seismic vulnerability (Shinozuka et al.,

2000; Choi et al., 2004; Nielson, 2005; Padgett, 2007). A fragility curve provides a conditional

probability that gives the likelihood that a structure will meet or exceed a certain level of damage

for a given ground motion intensity. Information provided from a fragility curve can be used for

prioritizing bridge retrofit, pre-earthquake planning and post-earthquake response and evaluation.

2

These curves usually account for a multitude of uncertainty sources related to estimating seismic

hazards, including bridge characteristics, type, configuration and others.

Presently, the bridge type mainly considered for fragility curve development is a straight

structure. However, in many instances bridges at major highway interchanges and in urban

environments have a horizontally curved (e.g., curved in plan) superstructure. Horizontally

curved steel bridges make up a significant portion of the approximately 597,500 bridges in United

States road network (FHWA, 2008). In fact, nationwide, over one third of all steel bridges

constructed are curved (Davidson et al., 2002). Therefore, the effect of radius of curvature on the

fragility of this bridge type should be investigated.

In addition, most research to date on estimating bridge fragility has focused on statistical

extrapolation of results for an individual bridge. To adequately assess seismic vulnerability of a

family of bridges (e.g., curved bridge) across various geographic regions, it is necessary that

fragilities be generated and estimated based on direct individual dynamic analysis or statistical

interpolation. In particular, when a large number of dynamic analyses are required to compute

seismic response of a population of bridges, it is vital to employ approximation methods to

efficiently and adequately predict seismic response. Therefore, statistical methodologies that can

assist with expanding the population of bridges that are studied without greatly increasing

computational time, such as via the use of response surface metamodels (RSMs), are desired.

1.2 Problem Statement

Horizontally curved, steel, I-girder bridges continue to be built with increasing frequency

in all seismic zones in the United States. Even though such bridges are more vulnerable than

straight bridges during an earthquake, currently there is no seismic vulnerability criterion

incorporating fragility curves in the United States for these types of bridges.

3

1.3 Objective, Scope and Organization

Seismic fragility curves that solely consider straight bridges have been developed using

time consuming nonlinear time-history analyses. To efficiently produce fragility curves for a

population of horizontally curved steel I-girder bridges, the use of response surface metamodels

(RSMs), one of a number of rigorously generated approximate analysis methods based on

statistical methodologies, is proposed. Therefore, the ultimate objective of this research was to

generate fragility curves for a family of horizontally curved steel I-girder bridges in a specific

geographic region (i.e., Pennsylvania, Maryland, New York) using RSMs. Effects of various

curved bridge parameters, including radius of curvature, on the fragility curves developed for the

studied bridge family were also investigated. Secondary objectives for the research included: (1)

application of 3-D nonlinear analytical models to horizontally curved steel bridge dynamics, (2)

development of the spherical bearing model, (3) key parameters on seismic response for

horizontally curved steel bridges.

To accomplish these objectives, the research was organized as follows:

• Literature Search that reviewed dynamic studies on curved steel bridges, recent studies

on bridge seismic fragility curves, and development of seismic fragility curves using a

metamodeling techniques.

• RSM Development that reviewed RSMs and experimental designs, determined

appropriate experimental design to generate seismic bridge fragility analyses, and

developed the RSM methodology for bridge fragilities.

4

• 3-D Modeling that developed 3-D nonlinear analytical models for horizontally curved

steel I-girder bridges, validated 3-D nonlinear analytical model based on experimental

data, and investigated seismic response of horizontally curved steel I-girder bridges.

• Synthetic Ground Motion Development that determined a target region for the

horizontally curved steel I-girder bridge inventory, performed inventory statistical

analyses for horizontally curved steel I-girder bridges across the target region, and

developed synthetic acceleration time histories.

• Parameter Screening that established screening experiments, performed nonlinear time

history analysis using the 3-D analytical models, and identified key horizontally curved

steel I-girder bridge parameters influencing seismic response using statistical screening

• Seismic Fragility Curve Development that constructed RSMs for the target region,

identified the appropriate seismic capacities and performance levels and produced

seismic fragility curves using RSMs in conjunction with Monte Carlo simulation.

Chapter 2

LITERATURE SEARCH

In recent decades, an increased awareness of seismic susceptibility for various structural

systems has occurred as a result of major earthquakes. Subsequently, a number of studies related

to potential economic loss estimation after prescribed levels of earthquakes damage have

occurred (Shinozuka et al., 2000; Basoz et al., 1997; Padgett, 2007). Fragility studies, which can

help quantify potential economic losses, have been widely used for these seismic vulnerability

assessments. Bridge fragility curves are a critical component of these assessments and, as a result,

the number of developed fragility curves has grown as a result of this research. In addition to an

increase in the development of bridge fragility curves, techniques to efficiently assess the seismic

risk of other essential infrastructure components, such as building systems, using metamodeling

techniques have also seen increasing research emphasis.

This chapter begins with a historical review of analytical and empirical modeling

advances as well as dynamic studies for horizontally curved steel bridges. It continues with a

discussion of seismic fragility curves derived using empirical data from past earthquakes, expert

opinions or by the use of analytical methods. Lastly, it presents recent contributions from seismic

vulnerability assessment utilizing metamodeling techniques for other structures. A critical

appraisal of the state-of-the-art in the field of seismic fragility curves is also included.

2.1 Historical Review

During the last 50 years, studies on horizontally curved steel bridges have largely focused

on modeling, design, and field tests under static and pseudo-static loads (Chang et al., 2006;

6

Huang, 1996; Nevling et al., 2006; Zureick et al., 1999, 2000; White et al., 2001, Kim, 2004;

Jung, 2006), while there have been a relatively few studies on seismic analysis and behavior of

horizontally curved steel bridges (Abdel et al., 1988; Al-Baijat, 1999; Hosoda et al. 1992; Wu and

Najjar, 2007). Relevant previous studies on horizontally curved steel bridges are summarized in

the following section.

2.1.1 Horizontally Curved Steel Bridges

Rigorous research on horizontally curved steel bridges in the United States started in

approximately 1970 when the Consortium of University Research Teams (CURT) project, a

large-scale research project funded by 25 states and managed by the Federal Highway

Administration (FHWA), was initiated. The consortium conducted experimental and analytical

research to expand on existing information regarding horizontally curved steel girders.

Simplified analysis and design methods with accompanying aids and computer programs were

also developed. This project formed the basis for development of the initial Guide Specifications

for Horizontally Curved Highway Bridges (AASHTO, 1980). The specifications were

periodically updated (AASHTO, 1980; AASHTO, 1993; AASHTO, 2004) and generally

contained limited provisions on seismic design and analysis.

One of the earliest studies of the dynamic response of such bridges was conducted by

Shore et al. (1968). The study derived differential equations for out-of-plane vibrational motion

of a horizontally curved bridge due to a constant moving force and investigated the effect of

torsional inertia on free vibration. Chaudhuri and Shore (1977) developed a dynamic analysis

method for a horizontally curved, steel, I-girder bridge under smoothly moving mass loadings

similar to seismic loadings. A parametric study was also completed to examine the effects of

horizontal curvature, rigidity ratio, and weight and frequency ratios on response of the bridge to

flexural and torsional vibrations. That same year Rabizadeh and Shore (1975) published research

7

that used the finite element technique to examine forced vibration response of horizontally curved

box girder bridges.

These early studies largely focused on fundamental curved bridge forced vibration

behavior. Later studies looked at actual seismic response. Abdel et al. (1988) conducted seismic

analyses of a curved steel box girder bridge modeled using a space frame configuration with

beam elements to represent the superstructure. Seismic response (e.g., displacements, shear

forces, etc.) at the abutments, mid-span and an interior pier were investigated. A broad seismic

response analysis of a curved bridge accounting for material and geometric nonlinearities at the

bearings was completed by Hosoda et al. (1992). It was found that seismic response at a

substructure unit and at the bearing in the tangential direction was larger than corresponding

longitudinal responses of a similar straight bridge while seismic response in the radial direction

was smaller than transverse response in a similar straight bridge. Al-Baijat (1999) analyzed the

seismic behavior of single span curved steel I-girder bridges subjected to static and earthquake

loadings using 3-D finite element analysis. Each bridge was composed of steel girders and

reinforced concrete deck slabs. The seismic analysis consisted of an earthquake loading applied

in the radial direction based on the 1940 El Centro Earthquake. This analysis indicated that the

bending moment would be compatible with that of a straight bridge if closely spaced lateral

bracing was used. It was also found that the amount of torsion in the girders from the seismic

loading could be as high as 27% of the bending moment in a similar, straight bridge. It was also

stated that span lengths of less than approximately 30m (98.4ft) are preferred from a seismic

perspective for single span horizontally curved bridges based on the Guide Specifications for

Horizontally Curved Highway Bridges (AASHTO, 1993).

As is evident from this review of curved bridge literature, certain studies have

investigated the seismic response horizontally curved steel bridges. However, to date, none of the

studies have attempted to generate fragility curves.

8

2.2 Fragility Curves

The fragility curve probabilistic methodology was initially developed for the assessment

of nuclear facility vulnerabilities to blast and dynamic loadings in the late 1970s and early 1980s.

Since then, the methodology has expanded to other structural fields, including bridge engineering

(Mander et al., 1999; Shinozuka et al., 2000; Karim et al., 2003; Mackie and Stojadinovic, 2004;

Choi, 2004; Nielson, 2006; Padgett, 2007). Seismic bridge fragility curves, also called seismic

bridge vulnerability functions, describe the probability that a bridge will exceed a specific

damage state for a given ground motion parameter. Fragility curves for bridge performance play

an important role in their overall seismic assessment (Murachi, 2004; Choi, 2004; Nielson, 2006;

Padgett, 2007). They are particularly useful in regions of slight or moderate seismicity, such as

the Eastern, Central, and Southeastern United States, where bridge officials are beginning to

develop retrofit programs and to conduct pre-earthquake planning (Choi, 2004).

The conditional probability that a structure will meet or exceed a specified level of

damage for a given ground motion intensity measure is expressed as follows (Murachi, 2003;

Choi, 2004; Nielson, 2006; Padgett, 2007):

][ yIMLSPFragility == (2.1)

where LS is a certain limit state or damage level of the bridge or bridge component, IM is the

ground motion intensity measure, and y is the realized condition of the ground motion intensity

measure, often taken as peak ground acceleration or spectral acceleration at the fundamental

period of bridge. One can see from this function that, given a ground motion intensity measure,

prediction of the damage level may be made for each bridge for which a fragility curve is defined.

Figure 2-1 gives the continuous form of a set of fragility curves and their interpretation for a

specific ground motion intensity.

9

Figure 2-1: Sample Set of Fragility Curves.

There are a number of different methodologies that have been used to develop structure

seismic fragility curves and the following sections give the current state-of-the-art for these

methodologies. Advantages or shortcomings for each methodology are also addressed.

2.2.1 Expert Based Fragility Curves

Expert based fragility curves are generated using expert-opinion earthquake damage and

loss estimates for industrial, commercial, residential, utility and transportation facilities. As a

result, this methodology naturally involves subjectivity, resulting in a high level of uncertainty.

For instance, the Applied Technology Council (ATC) developed the Earthquake Damage

Evaluation Data for California (ATC-13) that includes tools for use in the generation of damage

probability matrices (ATC, 1985). Due to the limited amount of data available, expert-opinions

were solicited to evaluate the level of damage to industrial and transportation facilities subjected

to earthquakes. A survey was performed based on the following the Delphi method, in which

several rounds, or iterations, of questionnaires were distributed (ATC, 1985). Survey participants

10

were queried to the probability of a facility being in a particular damage state for given different

levels of ground motion using the Modified Mercalli Intensity scale (ATC, 1985). Seventy-one

experts participated at some point of the questionnaire process, but only five were bridge experts.

In addition, this document had little correlation to actual earthquake damage reports and was

based only on the experience and number of experts queried. Damage probability matrices were

created for only two classes of bridges; major spans over 152.4m (500 ft) and conventional spans

less than 152.4m (500 ft).

2.2.2 Empirical Fragility Curves

Empirical fragility curves are generated from actual earthquake data and give a general

idea about the relationship between structure damage levels and ground motion indices. After the

1989 Loma Prieta, 1994 Northridge and 1995 Kobe earthquakes, empirical bridge fragility curves

became more common as a direct result of actual ground motion and bridge damage data.

Empirically based fragility generation for highway bridges has been presented by several

researchers for the 1989 Loma Prieta and 1994 Northridge earthquakes (Basoz and Kiremidjian,

1997; Der Kiureghian, 2002; Shinozuka et al., 2000; and Elnashai et al., 2004) as well as for the

1995 Kobe earthquake by Yamazaki et al. (1999).

Basoz and Kiremidjian (1997) initially developed empirical fragility curves for bridges

using Peak Ground Accelerations (PGAs) derived from damage data from the Loma Prieta and

Northridge earthquakes. In particular, they used logistic regression analysis to generate the

fragility curves based on a damage probability matrix for multiple span bridges as shown in Table

2-1 and Figure 2-2, respectively. Shinozuka et al. (2000) used the maximum likelihood method

to develop lognormal distribution function parameters for fragility curves. A Monte Carlo

technique (Shinozuka et al., 2000) was also presented to simulate bridge condition states and

highway network damage. With the aid of damage data, collected after the 1994 Northridge

11

earthquake, fragility curves were developed for each bridge on a limited access expressway

network in Los Angeles County and Orange County in California. Der Kiureghian (2002)

presented a Bayesian framework (Geysekns et al., 1998) for estimating the fragility of civil

infrastructure systems based on field observations. The approach accounted for aleatoric

uncertainties arising from inherent variation and random errors in the values of the parameters

and their estimates, as well as epistemic uncertainties arising from model error, measurement

error and small sample size.

Table 2-1: Damage Probability Matrix for Multi-Span Bridges (%) (Basoz and Kiremidjian, 1997).

USGS Peak Ground Acceleration (g)

Observed Damage

0.15-0.2 0.2-0.3 0.3-0.4 0.4-0.5 0.5-0.6 0.6-0.7 0.7-0.8

None 98.5 92.3 86.2 66 37 55.6 81 Minor 1 2.8 9.2 4.3 22.2 14.8 9.5 Moderate 0.5 2.8 4.6 19.2 22.2 18.5 4.8 Major 0 2.1 0 10.6 18.5 11.1 4.8 Collapse 0 0 0 0 0 0 0

USGS Peak Ground Acceleration (g)

Observed Damage

0.8-0.9 0.9-1.0 1.0-1.1 1.1-1.2 1.2-1.3 1.3-1.4

None 72 24.1 64.3 50 0 100 Minor 0 17.2 7.1 50 0 0 Moderate 16 24.1 21.4 0 0 0 Major 4 31 0 0 0 0 Collapse 8 3.5 7.1 0 0 0

Based on actual damage data obtained from the 1995 Kobe earthquake, Yamazaki et al.

(2000) proposed empirical fragility curves for highway bridges in Japan and showed the

relationship between damage that occurred and ground motion indices. The ground motion

indices consisted of the PGA, the peak ground velocity (PGV) and the Japan Meteorological

Agency (JMA) seismic intensity, which was also estimated using the Kriging technique (Cressie,

1993). Damage data and ground motion indices were related to each damage rank determined by

12

PGA, PGV, and JMA. The damage ranks were mainly comprised of “As”, “A”, “B”, “C” and

“D”. “As” was the highest damage rank which described bridge collapse. “A” represented

extensive damage of bridge and “B” stood for moderate damage. “C” represented small damage

and “D” stood for minor/no damage. Figure 2-3 shows the empirical fragility curves developed

for highway bridges in Japan with respect to PGA.

Multiple Span Bridges(USGS pga values - unconditional on damage)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4PGA (g)

Prob

abili

ty o

f Exc

eedi

ng a

Dam

age

Stat

e

minormoderatemajorcollapse

Figure 2-2: Empirical Fragility Curves of Multi-Span Bridges for PGA (Basoz and Kiremidjian, 1997).

Even though the empirical method to create fragility curves is straightforward, it has

some shortcomings and limitations. One limitation has been the lack of sufficient damage data

for different bridge types and levels of damage, which has resulted in several significant groups

of bridges, such as horizontally curved steel bridges, being ignored (Basoz and Kiremidjian,

1997; Der Kiureghian, 2002). Another is related to subjectivity: when post-earthquake

assessments of bridges are made and damage levels are assigned, there is often a discrepancy

between damage levels that two different inspectors would assign (Basoz and Kiremidjian, 1997;

Yamazaki et al., 1999).

13

Figure 2-3: Empirical Fragility Curves of Bridge Structures for PGA (Yamazaki, 2000).

2.2.3 Analytical Fragility Curves

Analytical fragility curves are generated based on numerical simulation that considers

different levels and types of ground motions. Analytical fragility curves can be commonly used

to evaluate the seismic performance of highway bridges when actual bridge damage and ground

motion data does not exist. Many studies related to bridge analytical fragility curves have been

conducted using a variety of different methodologies. Because damage states are related to

structural capacity (C) and the ground motion intensity parameter is related to structural demand

(D), the analytical fragility is often described as shown in Equation 2.2 (Murachi, 2003; Choi,

2004; Nielson, 2006; Padgett, 2007):

⎥⎦⎤

⎢⎣⎡ ≥= 1

CDPPf (2.2)

14

where C is the structural capacity, D is the structural demand, Pf is the analytical fragility or

probability of failure.

An analytical fragility curve is often modeled using a lognormal cumulative distribution

function where structural demand and capacity are assumed to be lognormally or normally

distributed (Hwang and Jaw, 1990). This approach is chosen because is has shown to be a good

fit for actual performance in the past and is convenient for manipulation using conventional

probability theory (Wen et al., 2003). Therefore, a closed form solution for the fragility can be

represented by a lognormal cumulative distribution function as shown in Equation 2.3 (Melchers,

1999).

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

+=

22

/

CD

CDf

ββ

SInSP Φ (2.3)

where ][⋅Φ is the standard normal probability integral, CS is the median value of the structural

capacity defined for the damage state, Cβ is the dispersion or lognormal standard deviation of the

structural capacity, DS is the seismic demand in terms of a chosen ground motion intensity

parameter and Dβ is the logarithmic standard deviation for the demand.

A number of methodologies have been used to analytically assess structural capacity and

demand, which, as shown above, are necessary quantities for analytical fragility curve

development. The methodologies range from simplistic to comparatively rigorous. One of the

simplest and least time consuming approaches may be performing an elastic spectral response

analysis of a group of bridges to investigate structural capacity and demand. This has been used

by Yu et al. (1991) to develop fragility curves for highway bridges in Kentucky. They modeled

each of the bridge piers as single degree of freedom (SDOF) structures and then estimated their

response using an elastic response spectrum. This methodology was upgraded by Jernigan and

15

Hwang (2002) and by Hwang et al. (2000) with the intent of providing a simplified way for

practicing engineers to assess seismic vulnerability of bridges. Seismic demand determined

through elastic spectral analysis component capacities were identified according to the 1995

edition of the FHWA Seismic Retrofitting Manual for Highway Bridges (FHWA, 1995) and

damage states were established by evaluating the Capacity/Demand (C/D) ratio. An alternate

approach has been to use non-linear static methods, commonly called capacity spectrum methods,

to help develop analytical fragility curves. This approach uses a converted non-linear static

pushover curve in conjunction with a reduced response spectrum to generate fragility curves by

Karim and Yamazaki (2001), Mander and Basoz (1999), Dutta and Mander (1998), Shinozuka et

al. (2000), and Mander (1999). This was the methodology adopted for generation of seismic

bridge fragility curves for Hazards U.S, often referred to as HAZUS (FEMA, 2003).

To date the most rigorous methodology for developing analytical fragility curves is

through the utilization of nonlinear time history analysis (NLTHA) for determining seismic

demand. Although this approach tends to be the most computationally expensive, it is one of the

most reliable methodologies available (Shinozuka et al., 2000). The NLTHA methodology has

been developed and adopted in various forms for bridge fragility assessment by Hwang and Huo

(1994), Shinozuka et al. (2000), Hwang et al. (2000), Mackie and Stojadinovic (2004), Karim and

Yamazaki (2003), Elnashai et al. (2004), Choi et al. (2004), and Nielson (2006). While there are

some subtle differences in approach between researchers, the general NLTHA procedure follows

the same, basic steps. NLTHA for the seismic response of bridges has been recently performed

using the OpenSees program (Choi et al., 2004, Nielson, 2006, and Padgett, 2007). The initial

step for developing analytical seismic fragility curves is to select a suite of ground motions

representative of the region of interest. The ability of this suite to capture such inherent

uncertainties as earthquake source, wave propagation, and soil conditions dictates the

effectiveness of the fragility curve development procedure to propagate these aleatoric

uncertainties. In regions with recorded strong ground motion, such as Japan, Greece, or

16

California, earthquake ground motion records from past events have been used to input ground

motions and in some cases scaled to various levels of excitation when employed in an

incremental dynamic analysis (Elnashai et al., 2004; Karim and Yamazaki, 2003; Mackie and

Stojadinovic, 2003). In regions of moderate seismicity, such as the central and southeast United

States (e.g., Georgia, South Carolina, Tennessee, etc.), there may likely be little or no available

strong ground motion records, so synthetic ground motions are often adopted. Shinozuka et al.

(2000) utilized these time histories to develop fragility curves for Memphis, Tennessee and Choi

et al. (2004) used them for developing fragility curves for the central and southeastern United

States. Wen and Wu (2001) used this approach to develop simulated ground motions for three

cities (e.g., Memphis, etc.) in mid-America such that the median of the response spectra matched

the uniform hazard response spectra for a 10% and a 2% exceedance probability in 50 years. The

developed synthetic ground motions were then used as inputs in the NLTHAs to establish

analytical bridge fragility curves for the central and southeastern United States (Nielson, 2006).

More recently, Rix and Fernandez (2004) developed synthetic earthquake ground motions for

Memphis for various earthquake magnitudes and hypocentral distances in which nonlinear site

response was considered and a NLTHA for each earthquake-bridge sample was performed. For

each simulation, peak structural responses for key elements (e.g. column ductilities, bearing, and

abutment displacement, etc.) were produced.

Even though the analytical method to create fragility curves through the utilization of

NLTHA is reasonable, it has one shortcoming. It involves the time-consuming analysis processes

needed to construct fragility curves for a family of bridges. One way to overcome this

shortcoming is by employing metamodels.

17

2.2.4 Analytical Fragility Curve Development Using Metamodels

A metamodel is a less complex and more efficient surrogate for a higher fidelity model.

Metamodels are efficient tools for engineering analyses of complex model groups since fewer

calculations are required and desired accuracy in maintained. The most popular technique to

build a metamodel function to date is the response surface approach that typically employs

second-order polynomial models to fit system responses based on least squares regression

techniques. Recently, response surface metamodels (RSMs) have been efficiently used to

generate analytical fragility curves for building structures (Franchin et al., 2003; Rossetto and

Elanshai, 2004; Towashiraporn, 2004; Dueñas-Osorio, 2004). The RSM is simply described as a

mathematical polynomial regression (PR) function (Wu and Hamada, 2000) and an explanation

of RSMs is discussed in Chapter 3 in more detail.

In particular, RSMs in connection with probability approaches (e.g., First Order

Reliability Method, Monte Carlo simulation, etc.) have been efficiently used for generating

seismic fragility curves for populations for steel and reinforced concrete (RC) civil engineering

structures, but not bridges. For instance, seismic fragility analyses of RC structures have been

carried out using RSMs in conjunction with First Order Reliability Method (FORM) analysis to

construct empirical limit-state functions (Franchin et al. 2003). This method has been also used

to develop fragility curves for 3-D steel moment resisting frame structures with random geometry

to capture regional variability of 3-D frame structures in the central United States (Dueñas-Osorio,

2004). Fragility curves for a population of low-rise, in-filled RC structures having inadequate

seismic performance were developed in connection with mathematical polynomial regression

(PR) functions using analytical damage statistics (Rossetto and Elanshai, 2004). RSM

metamodels in conjunction with Monte Carlo simulation based on experimental designs have also

been successfully applied to the seismic response of 2-D steel moment resisting frame structures

18

(Towashiraporn, 2004) and to 3-D steel moment resisting frame structures with random geometry

(Seo et al., 2005).

2.3 Conclusion

This review presented research related to seismic analysis of horizontally curved steel

bridges and the state-of-the-art seismic fragility curve creation techniques for bridges and other

structures. Review of the literature has demonstrated a lack of research related to the seismic

analysis and generation of fragility curves for horizontally curved steel bridges.

As a result of the literature search, the research proposed herein applies a RSM

methodology based on existing nonlinear time-history analysis approaches to generate fragility

curves for a family of horizontally curved steel bridge. Fragility curves that are generated will be

for a representative family of horizontally curved steel bridges. The following Chapter will

address how the RSMs methodology, in conjunction with Monte Carlo simulation, is integrated

into a procedure to determine horizontally curved steel bridge seismic fragilities.

Chapter 3

RESPONSE SURFACE METAMODEL METHODOLOGY FOR GENERATION OF BRIDGE FRAGILITY CURVE

To determine the seismic response of a family of horizontally curved steel bridges using

an analytical approach, simulations, typically based on finite element techniques, are required.

The use of such models can require a number of time consuming computations for a large

population of structures to be analyzed. To efficiently complete the work, sampling techniques of

a reasonable, but often small, number of complex seismic response analyses of bridges have been

used to replace brute-force simulation. However, the probability distribution derived by fitting a

small number of data points may not be representative of the actual population and even the

statistical simplification can be difficult and time consuming when the probability density

function is nonlinear. To increase the population and efficiently and accurately compute seismic

responses of large population of bridges, RSMs in connection with Monte Carlo simulation can

be performed on a representative model with little computational cost to obtain an accurate

seismic response probability distribution.

3.1 Metamodels

A metamodel is a statistical approximation of a complex and implicit phenomenon.

Response is estimated in a closed-form function of input variables that is computationally simpler

to run (Simpson et al., 2001). If the true but unknown relationship between response (y) and a

vector of input variables (ξ) in nature is represented as:

( )ξ= fy (3.1)

20

then a metamodel g (ξ) is sought to approximate the true relationship f (ξ). The relationship

between y and ξ becomes:

( ) εgy +ξ= (3.2)

where ε represents a total error term. This error term is the sum of a lack-of-fit or bias error

(εbias) resulting from approximation of f (ξ) with g (ξ) or an approximation of the reality and a

random error (ε random) due solely to experimental and observational error (i.e., repeating

experiments at a specific set of ξ produces different values of y). The error term is assumed to be

a zero-mean random variable. However, the random error term (ε random) does not exist in the

case of computer analysis where repeated analytical evaluations of ξ always yield the same value

of y. Expectation of the response function is in the form:

[ ] ( )ξ= gyE (3.3)

which can be derived by running computer analysis codes at predefined levels of ξ (i.e.,

experimental designs), observing responses, and fitting data to an appropriate model.

As shown in Figure 3-1, metamodels involve three general steps: (1) selecting an

experimental design for generating data, (2) choosing a metamodels to represent the data, and (3)

training and validation of the selected metamodels using observed data.

21

Figure 3-1: Steps for Constructing Metamodels (Simpson et al., 2001).

3.2 Experimental Designs

The first step out of the three general metamodeling steps shown in Figure 3-1 is to

construct an adequate experimental design. The experimental design represents a sequence of

experiments to be performed, expressed in terms of factors set at specified levels. An

experimental design is represented by a matrix X where the rows denote experiment runs, and the

columns denote particular factor settings. The most common designs are a Full Factorial design

(FFD) and Central Composite Design (CCD). Figure 3-2 gives a graphical representation of FFD

and CCD.

22

Figure 3-2: Schematic Representation of (a) Full Factorial Design (FFD) and (b) Central Composite Design (CCD) for Three Variables (Simpson et al., 2001).

3.2.1 Full Factorial Design

Basic experimental design is a FFD. The number of design points dictated by a FFD is

the product of the number of levels for each factor. The most common are 2k for evaluating main

effects and interactions and 3k for evaluating main and quadratic effects and interactions for k

factors at 2 and 3 levels, respectively. The size of a full factorial experiment exponentially

increases with the number of factors, leading to an unmanageable number of experiments for

problems with several input variables (Simpson et al., 2001; Wu and Hamada, 2000).

The most common are 2(k-f) designs, in which f indicates the number of fractionations of

the full factorial design. For example, in a 26 (64) runs case, a half-fraction design will require

only 32 runs; while a quarter-fraction will require 16 runs. The number of fractionations is limited

by the resolution required to achieve significant statistical inferences. The resolution of a two-

level fractional factorial design, denoted by R, is the number of factors involved in the lowest-

order effect in the defining relation or acceptable confounding scheme.

To solely classify significant factors, screening processes are required. One specific

family of fractional factorial designs frequently used for this identification is the two-level

Plackett-Burman design (PBD) (Wu and Hamada, 2000). When screening, interactions are

X2

X1

X3

X2

X1

X3

X2

X1

X3

X2

X1

X3

X2

X1

X3

X2

X1

X3

X2

X1

X3(a) (b)

23

considered negligible, hence PBD allows unbiased estimation of the main effects and only

requires one more design point than the number of factors. It also provides the smallest possible

variance so that it contains the smallest possible combinations. A two-level PBD can be

efficiently used to perform a screening study with a suitable combination size.

3.2.2 Central Composite Design

Central Composite Design (CCD) requires fewer design points than FFD (Simpson et al.,

2001; Wu and Hamada, 2000) and is typically the most popular class of designs used in a RSMs

methodology in connection with Monte Carlo simulation (Wu and Hamada, 2000;

Towashiraporn, 2004; Dueñas-Osorio, 2004). This consists of a complete 2k factorial design,

where the variable levels are coded to -1 and +1 values to represent minimums and maximums

from the data set. The experimental use of CCD consists of three parts for a three-dimensional

data set. The first eight points (blue dots) in Figure 3-3 denote the data points that are

investigated and form a 23 design. Because they are on the corners of the 23 data “cube”, they are

called cube points or corner points. The next six points (cross) form three pairs of points along

the three coordinate axes and are therefore called axial points or star points. The last point (red

dot) is at the center of the design region and is called a center point.

Selection of the location of the axial or star points prompts further discussion since it

affects the rotatability property of the CCD. Rotatability in the design ensures that the variance

of the estimated response is constant and small at a fixed distance from the center point. The

CCD is considered rotatable if:

4/1)2( kα = (3.4)

24

α is always greater than unity and this locates the star points outside the original range. As a

result, each input variable has to be evaluated at 5 levels (-α, -1, 0, +1, and +α) so that the

efficiency of the input variable estimates is considerably increased. This may not be practical in

some instances where it is physically difficult or impossible to extend the experiment beyond the

region defined by the upper and lower limits of each input variable (e.g. -1, 0, +1). The research

completed herein seeks an efficient method with the least computational effort to efficiently

generate RSMs. The rotatability requirement can be dropped in such cases and the distance α is

set at 1. For α = 1, the star points are placed at the center of the faces of the cube. For example,

for k = 3 the six star points (Figure 3-3) are at the center of the six faces of the 23 cube and the

design is therefore called a face center cube. Approaching RSM development using this approach

keeps the design space manageable at three levels (Wu and Hamada, 2000). In addition, it has

been shown that these are effective designs if the design region is a cube (Towashiraporn, 2004)

and, for this study, the selected variables and their ranges were independent of each other (e.g.,

number of span, radius of curvature, etc.), which rendered the region naturally cuboidal. Results

from such an approach prove that the RSM provides good prediction even without the rotatability

property (Wu and Hamada, 2000; Towashiraporn, 2004).

Replications of the center points provide a means for estimating pure experimental error.

However, this type of error does not exist in computer analysis. Hence, only one replicate of the

center point is required. As a result, the total number of distinct design points for RSM

development is N = 2k + 2k + 1.

25

Figure 3-3: Central Composite Design (CCD) for Three Dimensions.

3.3 Response Surface Metamodels

RSMs are one of the most widely-used metamodel tools to generate analytical fragility

curves for a population of structures (Towashiraporn, 2004; Dueñas-Osorio, 2004). RSMs have

been applied by a number of researchers (Simpson et al. 1997; Cundy 2003; Dueñas-Osorio,

2004) for designing complex engineering systems and dynamic structural systems (Cundy 2003;

Towashiraporn 2004; Dueñas-Osorio 2004; Rossetto and Elanshai 2004). Typically a RSM

function consisting of polynomial models limits the order of the polynomials to two (Wu and

Hamada, 2000) since lower-degree models contain fewer terms and require fewer experiments to

be performed (Cundy 2003; Towashiraporn 2004; Dueñas-Osorio 2004; Rossetto and Elanshai

2004). A second-order polynomial model can be expressed as follows:

εxxβxβxββyk

i

k

i

k

i

k

ijjiijiiiii ++++= ∑ ∑ ∑∑

= =

−

= >1 1

1

1

20 (3.5)

x

x

x

x

cube points

center point

star points

26

where y is the dependent variable such as seismic response, xi , xj are independent variables such

as radius of curvature, β0 , βi , βii , βij are coefficients to be estimated, and k is the number of input

variables. β parameters are calculated using least-squares regression to fit the response surface

approximations to empirical data or to data generated from NLTHAs. Through this RS function,

seismic response can be computed.

Alternatively, the model can be written in the form of general linear model as follows:

εzββyp

iii ++= ∑

−

=

1

10 (3.6)

where p is the number of parameters to be estimated. A vector of dummy first-order variables z

replaces the original input variable x that includes quadratic terms (Wu and Hamada, 2000)

A general matrix form of the linear model can be written as:

εZY +β= (3.7)

where:

1

2

1

.

.

.

×⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

NNy

yy

Y is a vector of actual responses,

27

pNpNN

p

zz

zz

×−

−

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

1,1

1,111

...1...

..

.

.

.

.

.

.

....1

Z is a matrix of constants,

11

1

0

.

.

.

×− ⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=β

ppβ

ββ

is a vector of unknown parameters, and

1

2

1

.

.

.

×⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

NNε

εε

ε is a vector of error terms with expectation E[ε] = 0.

Consequently, a random vector Y has form of:

[ ] ZβYE = (3.8)

The parameters of the polynomials are usually determined by a least squares regression

analysis by fitting values to existing experimental data points. The method of least squares

selects the values (b0, b1,…, bp-1) for unknown parameters (β0, β1,…, βp-1) such that they minimize

the sum of squares of the differences between the actual output (y) and the approximated or fitted

outputs (ŷ). Mathematically, the least squares method minimizes:

( )∑=

−=N

uuu yyS

1

2)(ˆ)( bb (3.9)

28

where S is defined as the sum of squares function, N is the number of experimental points (N > p),

and b is a vector of least squares estimates of parameters β.

The estimates of the polynomial parameters can be obtained by solving the following

matrix equation:

( ) ( )= − YZ'ZZ'b 1 . (3.10)

The resulting fitted response surface function becomes

∑ ∑ ∑∑= =

−

= >

+++=k

i

k

i

k

i

k

ijjiijiiiii xxbxbxbby

1 1

1

1

20ˆ . (3.11)

Least-square regression analysis gives parameter estimates for the response surface

function. The next step is to evaluate adequacy of fit of the model. There are a number of

statistical measures that can be used to verify linear regression models. However, statistical

testing is inappropriate in the cases where outputs are computed by deterministic computer runs

and random error (εrandom) does not exist (Welch et al. 1990, Simpson et al. 2001). The simplest

measure for verifying model adequacy in deterministic computer experiments is the coefficient of

determination (R2):

SSTSSRR 2 = (3.12)

29

where: ( )NY1YXbSSR

2′−′′= is the Error Sum of Squares,

( )NY1YYSST

2′−′= is the Total Sum of Squares, and

1′ is a unity vector.

The value of R2 characterizes the fraction of total variation of the data points explained

by the fitted model. It has a value between 0 and 1 (with 1 being a perfect fit). However, R2 can

be misleading in some cases since it always increases as more input variables are added. An

adjusted 2R ( 2AR ), which takes into account the number of parameters in the model, is

introduced for evaluating the goodness-of-fit. It can be computed as follows:

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

⋅−−=pN1NR11R 22

A . (3.13)

A value of 2AR close to unity indicates a good fit of the response surface model to the

experimental data points. Papila and Haftka (2000) suggested the value of 2R (or 2AR ) of at

least 0.9 to ensure adequate approximation of the model.

Even though the 2AR value explains how well the model fits to the experimental points,

the value does not reflect its prediction potential for points not used to generate it. To verify the

overall accuracy of the RSM, statistical tests at additional random data points in the design space

must be performed. Those tests contain the Average Absolute Error (%), the Maximum Absolute

Error (%), and the Root Mean Square Error (%) (Venter et al., 1997). These measures are

defined as follows:

30

∑

∑

=

=

⋅

−⋅⋅= N

1ii

N

1iii

yN1

yyN1

100AvgErr% ; (3.14)

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⋅

−⋅=

∑=

N

1jj

ii

iy

N1

yy100MaxMaxErr% ; (3.15)

∑=

⋅

⋅⋅= N

1iiy

N1

PRESSN1

100RMSE% ; and (3.16)

( )∑=

−=N

1i

2ii yyPRESS .

where: N is the number of experimental points,

y is the actual output, and

ŷ is the approximated or fitted outputs.

The Average Absolute Error (%) is summation of the absolute value of each deviation

from the original data set. It is useful to measure the spread of data while considering the effect

of the total data set. The Maximum Absolute Error (%) is the sum of a maximum relative error

and a maximum absolute error and is represented by the same linear function for each of the

integration variables. It is used to measure the error of the worst-case scenario. The Root Mean

Square Error (%) is the expected value of the square of the error and is used to measure average

magnitude of the error.

31

3.4 Response Surface Metamodels for Seismic Fragility Assessment

The previously described RSMs and experimental designs have been used to rapidly

construct seismic bridge fragility curves across geographic regions. The procedure for

calculating seismic fragility based on the use of RSMs is described in Figure 3-4.

The first step is to define the input and output parameters for the RSMs. The input

parameters for target horizontally curved steel I-girder bridges are identified based on experience

and bridge inventory data, and consist of macro- (e.g. geometric and structural) and micro-(e.g.

material) parameters (Towashiraporn, 2004; Dueñas-Osorio, 2004). The range of each input

parameter applicable to the region of interest is defined based on inventory data. Outputs are

based on defined damage states and can also be based on damage indices or peak structural

response, such as curvature ductility of a column or bearing deformation.

The second step of the process starts with screening of the selected structural parameters

to identify the pareto-optimal parameters (most significant parameters) to define the RSM. This

identifies which of the selected predictor parameters are statistically significant for affecting the

response of the target bridges and ensures that the RS function will be efficient. Next, design of

experiments for RSMs is used to generate the surrogate model. The CCD method is used because,

as mentioned previously, it is straightforward and provides an efficient set of input parameter

combinations (also known as experimental sampling). An ensemble of representative ground

motion records are randomly selected to apply to each combination for which the nonlinear time

history response is computed. As mentioned previously, the NLTHA is carried out using the

OpenSees program (Mazzoni et al., 2008) because of its ease of implementation for earthquake

analyses. Open System for Earthquake Engineering Simulation (OpenSees) is a software

framework for simulating the seismic response of structural and geotechnical systems (Mazzoni

et al., 2008).

32

Peak deformation of critical components (bearings and abutments) and pier column

curvature ductility of the horizontally curved steel bridges obtained from the NLTHA are used as

the output variables corresponding to each set of input parameter combinations. These quantities

are selected as main evaluation parameters since they are identified as key elements for seismic

bridge evaluation based on FEMA HAZUZ-MH (FEMA, 2003) loss assessment criteria. Finally,

a polynomial RS function is computed from the selected samples using least-squares regression.

The third step involves estimating the seismic response (e.g., peak deformation, column

curvature ductility, etc.) of horizontally curved steel I-girder bridges using the developed RSMs

instead of performing a large number of additional nonlinear dynamic time-history analyses. The

probability of the chosen response to exceed a certain damage limit state can be extracted from

the distribution of the simulation results. This probability value is conditioned on a specific

earthquake intensity level (e.g., peak ground acceleration, etc.) and represents one point on a

fragility curve. Repetition of the process over the different level of earthquake intensity provides

probability values of exceedance at other intensity levels and the fragility curves can be created

for the target portfolio of horizontally curved steel bridges in regions of interest.

33

Figure 3-4: Fragility Construction Flowchart using RSMs. 33

34

3.5 Conclusion

Chapter 3 dealt with the description of Response Surface Metamodels and experimental

designs. Both RSMs and experimental designs are used to rapidly construct seismic bridge

fragility curves across geographic regions. The methodology for calculating seismic fragility

based on the use of RSMs was described as well.

Chapter 4

3-D ANALYTICAL MODELING APPROACH OF HORIZONTALLY CURVED STEEL BRIDGE

Prior to developing fragility curves following the methodology proposed in Chapter 3, an

appropriate analytical modeling approach for the horizontally curved steel I-girder bridge must be

established. The selected procedure is represented as Step 1 in the fragility curves construction

methodology shown in Figure 3-4. The chosen analytical modeling approach should be validated

based on existing experimental data, if possible.

In general, analysis approaches for such bridges have included line girder methods, the

V-load method and various 3-D modeling techniques. Studies have been completed that applied

validated 3-D modeling techniques, using what are termed “3-D analytical” models that employ

line elements for all of the idealized bridge components, including the concrete deck, girders,

bearings, piers, abutments, and foundations to predict the seismic response of straight steel girder

bridges and to generate fragility curves (Choi, 2004; Nielson, 2005; Nielson, 2006; Padgett,

2007). Similar 3-D analytical modeling techniques have been proposed for curved steel bridges,

but were applied to static response (Chang et al. 2006). These studies have shown that modeling

using this method provides many advantages over more sophisticated finite element models when

studying the effects of multiple parameters on behavior (Choi, 2004; Nielson, 2005; Nielson,

2006; Chang et al. 2006; Padgett, 2007). These advantages include clearer explanation of

structural behavior for bridge engineers, efficient treatment of bridge components when

constructing the model (e.g., girder, cross-frame, boundary conditions, bearing, abutment, and

pier column etc.), and reduced computer running time. Therefore, the 3-D analytical method is

proposed and examined herein as the tool to explore the seismic response of representative curved

steel I-girder bridges having skewed supports. The analytical method is validated statically using

36

comparisons of predicted girder vertical and lateral bending moments to results obtained from

field testing of an in-service, three-span continuous, curved, steel I-girder bridge.

4.1 Modeling Approach

3-D analytical modeling is performed using the OpenSees program (Mazzoni et al.,

2008). OpenSees has been shown to accurately reproduce seismic response for various structures

in the past, including bridges (Choi, 2004; Nielson, 2005; Nielson, 2006; Padgett, 2007). It is an

object-oriented software framework for simulation applications in earthquake engineering using

finite element methods. Because it is open-source software, it also has the potential for a

community code for earthquake engineering.

Curved bridge framing is represented using 3-D conventional 6-DOF beam elements,

referred to as BeamColumn elements in OpenSees (Mazzoni et al., 2008), with lumped masses,

calculated using tributary dimensions, being placed at each node to represent self-weight effects.

The 6-DOF beam elements have been shown to be suitable for modeling of curved bridge

framing (Chang et al. 2006). Model construction initiates with calculation of section properties

for the: girders; cross-frames; concrete deck; concrete column; concrete pier caps; foundation

components and rigid link elements between flanges to flanges and girder to concrete deck. The

cross-frames are modeled with OpenSees truss elements and the concrete deck is modeled with

BeamColumn elements (Mazzoni et al., 2008). The concrete pier columns were modeled with

displacement-based geometrically and materially nonlinear beam column elements, termed

DispBeamColumn elements in OpenSees (Mazzoni et al., 2008). In order to represent a

distributed plasticity of pier columns, and abutments resulting in nonlinear structural behavior

under seismic loadings, DispBeamColumn elements are used. The components are illustrated and

labeled in Figure 4-1. Figure 4-2 shows the 3-D analytical model of a typical horizontally curved

steel bridge.

37

Girder Bearing

AbutmentPier Column

Deck

Pier Cap

Piles

Foundation

Piles

Figure 4-1: Illustration of Typical Curved Steel Bridge Components.

Concrete centroid

Girder centroid

Rigidlink elements

Beam-Column elements

Transverse elements

Beam-Column elements

Lumped mass for girders and cross-frames

Top-flange centroid

Lumped mass for reinforced concrete deck

Concrete deck zone

Steel girder zone

Concrete bent beam zone

DispBeamColumnElement

Concrete columnzone

Foundation zone

Rigidlink elements

Bottom-flange centroid

Zerolength elements


Figure 4-2: 3-D Analytical Model of Typical Curved Steel Bridge.

38

4.1.1 Superstructure

As mentioned previously, certain superstructure elements, including the concrete deck

and steel girders, are represented using an elastic BeamColumn element in OpenSees because the

superstructure is expected to remain linearly elastic (Choi, 2004; Nielson, 2005; Nielson, 2006;

Padgett, 2007) under earthquake loading. Previous research has indicated that most steel bridge

superstructure components are typically expected to behave elastically during an earthquake

(Imbsen, 2006). The section properties of deck and girder in any curved bridge that is modeled

were separately calculated using the y- and z-axes shown in Figure 4-3. The composite action

between the concrete deck and steel girders is assumed (Choi, 2004; Nielson, 2005; Nielson,

2006; Padgett, 2007). Rigid link elements are used to mimic composite action by coupling girder

and concrete deck nodes in OpenSees. Rigid link rods constrain only translational degrees-of-

freedom (Chang et al., 2006). Cross-frames are idealized using truss elements connected to each

top-flange and bottom-flange centroidal axis. Nominal material properties are used for both the

concrete and steel, with the concrete having a modulus of elasticity of 27.8GPa (4030ksi) while

the steel has a modulus equal to 200GPa (29000ksi).

Concrete deck Roadway centerline

Steel girders

Shear transfer interface

Parapet

Steel cross-framesY

Z

CL

Figure 4-3: Typical Curved Steel Bridge Superstructure Cross Section.

39

4.1.2 Substructure

To represent nonlinear behavior of the substructure during an earthquake, a more detailed

discussion of modeling of the substructure is provided than superstructure. Included in the

substructure computational models are the columns, pier caps, abutments and footings. The 3-D

analytical modeling approach for the substructure is based on work by Neilson (2005).

Following Neilson’s approach, the columns are generated in OpenSees using

displacement-based beam-column elements (DispBeamColumn Element), which include

reinforcement effects and can represent geometrical and material nonlinearities. These elements

were selected because, as discussed earlier, it is expected that the columns will respond to seismic

loads in a nonlinear fashion. Integration of the DispBeamColumn element along its length is

based on Gauss-Legendre quadrature rule (Mazzoni et al., 2008). The fiber section that

represents the reinforcement bars in the concrete has a general geometric configuration formed by

sub regions of regular shapes (e.g. quadrilateral, and circular regions) called patches. In addition,

layers of reinforcement bars can be specified for each column. Section properties for a typical

pier are established by assigning appropriate geometric dimensions to the fiber elements and

adequate constitutive models for both the concrete and steel reinforcement, respectively.

Reinforced concrete sections being included in a concrete pier are represented with three

constitutive models: (1) the unconfined (cover) concrete; (2) the confined concrete; and (3) the

reinforcing steel as shown in Figure 4-4. Unconfined concrete is modeled using the OpenSees

Concrete01 material model. This material uses the Kent-Scott-Park (Karsan, 1969) model which

utilizes a degraded linear uploading/reloading stiffness and a residual stress to represent nonlinear

dynamic behavior (Mazzoni et al., 2008). The model for the confined concrete, which is inside

the transverse reinforcing steel cage, is slightly different from that of the unconfined concrete.

The maximum stress and associated strain for the confined concrete is given as cKf ' and

40

=0ε K.0020 , respectively. Each value of cKf ' and =0ε K.0020 can be calculated using K

Equation 4.1:

c

yhs

ffρ

K'

1+= (4.1)

where cf ' is the unconfined compressive cylinder strength, sρ is the ratio of the volume of steel

hoops to the volume of the concrete core measured to the outside of the peripheral hoop and yhf

is the yield strength of the steel hoops (Park et al., 1982).

Reinforcing steel(DispBeamColumn element)

Unconfined (cover) concrete(DispBeamColumn element)

Confined (uncover) concrete(DispBeamColumn element)

Y

Z

Figure 4-4: Circular Reinforced Concrete Column.

The concrete pier caps are modeled in OpenSees using a combination of

DispBeamColumn elements and rigid links. The combination of these two elements includes

reinforcement effects and can represent geometrical and material nonlinearities. The section for

the concrete pier caps is created the same way as that for the columns. A representative

discretization of a typical pier and concrete cap is presented in Figure 4-5.

41

Rigid link

DispBeamColumn Element

Rigid link


Figure 4-5: Discretization of Typical Pier.

Typical curved steel bridge abutments primarily resist vertical loads and horizontal loads

that consist of tangential and radial loads during earthquakes. Horizontal seismic loads can place

great demands on bridge abutments via varying earth pressures and, consequently, increase the

seismic demands placed onto the superstructure. Therefore, to adequately transfer ground motion

to the abutments and, subsequently, the bridge, it is important to consider those effects on

modeling of the curved steel bridge substructure. The abutment model presented by Nielson

(2005a) and adopted herein assimilates the findings from a number of past studies (Caltrans,

1990, 1999; Maroney et al., 1994). To represent the tangential stiffness, the active

(positive/tension) action of the abutment is dictated by the pile stiffness, while in passive action

the contribution of the piles and passive pressure of the soil against the abutment backwall are

considered. The passive action reflecting the soil property contribution uses a quadralinear

model, and the pile stiffness degrades from its initial stiffness before reaching an ultimate

strength of 119kN/pile (26.7 k/pile) as dictated by Caltrans (Caltrans, 1990). The pile

contribution is the component considered in both radial and tangential abutment response. The

42

possible values of initial stiffness and ultimate deformation for the soil property contribution are

11.5 - 28.8 kN/mm/m (20.0 - 50.1 k/in/ft) and 6% - 10%, respectively (Maroney et al., 1994;

Martin and Yan, 1995; Caltrans, 1999). However, the ultimate passive soil pressure is assumed

to be 0.37 MPa (0.05 ksi). The specific modeling for soil property contribution is detailed in

Nielson (2005).

Similar to the abutment, the foundations are a vital part of the substructure of a bridge

system since all inertial forces from the structures must be transferred to them. The foundation

model for the pile presented by Nielson (2005) and adopted herein also assimilates the findings of

a number of past studies (Ma and Deng, 2000). Pile foundations are modeled with simplified

linear translational and rotational springs. The vertical and horizontal stiffness, and pile

grouping, are considered in deriving the aggregate horizontal and rotational pile group stiffnesses.

The specific modeling of pile foundations are detailed in Nielson (2005).

4.2 Model Validation

3-D analytical model verification is a critical step to ensure that a given model is

producing accurate results. All analytical models should be scrutinized, and to the best extent

possible, tested against experimental results. However, it is not possible to validate the 3-D

analytical model of existing curved steel bridges against seismic experimental data because data

on curved steel I-girder bridges under actual seismic loadings is not readily available. Therefore,

the 3-D analytical modeling approach is validated via comparison to experimental data from a

curved bridge under static loads.

43

4.2.1 Examined Bridge Description

As shown in Figure 4-6, the structure used for static model validation is a curved steel I-

girder bridge located in central Pennsylvania. It has been examined previously by Nevling et al.

(2006) and McElwain et al (2000). The three-span continuous bridge is composed of five ASTM

A572 Grade 50 steel plate girders and the abutment skew varies between 29o and 52 o (south to

north) relative to the traffic direction as shown in Figure 4-7. Bridge support conditions are also

shown in Figure 4-7.

Figure 4-6: Picture of Examined Curved Steel I-Girder Bridge.

Elevations, sections and details for the bridge are shown in Figure 4-8. This bridge has

three spans that are 23.5, 30.6 and 23.5 m (77.1, 100.4 and 77.1 ft) long. The superstructure is

supported using multi-column piers consisting of 914.4 mm (36.0 in) wide by 1066.8 mm (42.0

in) deep reinforced concrete pier caps. Each cap is supported by three 914.4 mm (36.0 in) φ by

6400 mm (251.9 in) tall circular reinforced concrete columns which, in turn, are tied to the

footings.

44

2952

LL1L2

L3

: Restrained from longitudinal movement

o

o

: Restrained from transverse movement

Note : All other bearings are free to move in longitudinal and transverse direction

South abutment

Pier #1 Pier #2North abutment

NG1

G2G3

G4G5

Z

X

Figure 4-7: Examined curved steel I-girder bridge (Nevling, 2003).

Pier columns are spaced horizontally at 4.0 m (13.1 ft) on center. The cap at Section A-A

in Figure 4-8 (c) uses 16-#9 reinforcing bars across the section while transverse steel is provided

by #5 stirrups spaced 203.2 mm (8 in) on average and #6 stirrups spaced 228.6 mm (9 in) on

average. The cap at Section B-B in Figure 4-8 (c) uses 10-#9 reinforcing bars across the section

while transverse reinforcement is provided by #5 stirrups spaced 203.2 mm (8 in) on average and

#6 stirrups spaced 381.0 mm (15 in) on average. Columns use 9-#11 bars for longitudinal

reinforcement and are contained by #4 transverse bars spaced at 304.8 mm (12 in) as shown in

Figure 4-8(c). The footing uses 74-#6 longitudinal bars and #5 transverse bars spaced at 541.9

mm (21.3 in) as shown in Figure 4-8(c). The foundation wall is 11.9 m (39.0 ft) long, 3.4 m (11.2

ft) wide and 0.7 m (2.3 ft) thick with the reinforcement being placed on its bottom face. The

abutments consist of spread footings on piles as shown in Figure 4-8(a) with a 1.6 m (5.2 ft) tall

backwall. Ten driven piles are used to support both abutments. No piles are used to support

interior piers. The design strength for the concrete is 20.7 MPa (3 ksi) while the reinforcing steel

has yield strength of 414 MPa (60 ksi).

45

23.5m 30.6m 23.5m77.6m

South abutment Pier #1 Pier #2 North Abutment

(a) General Elevation

C C

D

A B

E

(b) Pier #1

46

9 #11 bars

#4 bars @ 304.8mm

10 #9 bars16 #9 bars#5 bars @ 203.2mm

#6 bars @ 228.6mm #6 bars @ 381.0mm

#5 bars @ 203.2mm

Section A-A Section B-B

Section D-DSection C-C

Section E-E

74 #6 bars

#5 bars @ 541.9mm

#5 bars @ 541.9mm

(c) Concrete Member Reinforcing Layout

11.46m

Type A diaphragm

Type A diaphragm

Type B diaphragm

Type B diaphragm

0.95m4 Spaces @ 2.39m = 9.56m 0.95m

PedestrianRailing

0.33m 1.53m 0.53m 4.27m 4.27m 0.53m

Roadway centerline

G1 G2 G3 G4 G5

(d) Superstructure Section (Nevling, 2003)

Figure 4-8: Curved Steel I-Girder Bridge Configuration.

47

Girders are spaced 2.39 m (7.84 ft) center-to-center and cross-frames are placed between

them as shown in Figure 4-8(d). All girders have 1219 mm (48 in) x 13 mm (0.5 in) webs with

356 mm (14 in) wide top and bottom flanges of varying thickness as shown in Table 4-1. Two

different K-shaped cross-frame types are used in the bridge. Type A frame top and bottom

chords are composed of 88.9 mm (3.5 in) x 88.9 mm (3.5 in) x 9.5 mm (0.4 in) double angles and

diagonals are composed of 88.9 mm (3.5 in) x 88.9 mm (3.5 in) x 9.5 mm (0.4 in) angles. Type B

top chords are composed of WT14 x 49.5s with bottom chords composed of 88.9 mm (3.5 in) x

88.9 mm (3.5 in) x 9.5 mm (0.4 in) double angles and diagonals composed of 88.9 mm (3.5 in) x

88.9 mm (3.5 in) x 9.5 mm (0.4 in) angles. As shown in Table 4-2, the radius of curvature is

178.49 m (585.6 ft) to the exterior girder (G5) and maximum span length (L2) is 31.56 m (103.5

ft) to the interior girder (G1).

Table 4-1: Girder Dimensions.

Girder Top flange, mm (in) Web, mm (in) Bottom flange, mm (in)

G1, G2 356x16 (14x0.6) 1219x13 (48x0.5) 356x25 (14x0.9) G3, G4, G5 356x16 (14x0.6) 1219x13 (48x0.5) 356x32 (14x1.3)

Table 4-2: Radius and Span Lengths.

Radius, m (ft) L, m (ft) L1, m (ft) L2, m (ft) L3, m (ft) G1 168.94 (554.3) 80.24 (263.3) 23.83 (78.2) 31.56 (103.5) 24.84 (81.5) G2 171.32 (562.1) 79.05 (259.4) 23.67 (77.7) 31.15 (102.2) 24.23 (79.5) G3 173.71 (569.9) 77.97 (255.8) 23.52 (77.2) 30.77 (100.9) 23.67 (77.7) G4 176.10 (577.8) 76.98 (252.6) 23.38 (76.7) 30.42 (99.8) 23.17 (76.0) G5 178.49 (585.6) 76.07 (249.6) 23.25 (76.3) 30.10 (98.8) 22.72 (74.5)

48

4.2.2 3-D Analytical Model

Spherical bearings are used for all girders. A spherical bearing fixed in the tangential

direction is placed under G3 at Pier #2 and spherical bearings fixed in the radial direction are

placed at abutments for G3. All other locations had bearings that are free to translate in both the

tangential and radial directions. Figure 4-9 details representative bearings.

(a) General view

(b) Detailed view

Figure 4-9: Photos of Spherical Bearing System.

Spherical Bearing Diameter 0.3m

Spherical bearing fixed in longitudinal direction

49

Since spherical bearing seismic response may be nonlinear (Roeder et al., 1995), their

moment-rotational behavior is modeled in OpenSees using the program’s Steel01 and Hysteretic

material models as shown in Figure 4-10(a). In order to represent the nonlinear behavior of

spherical bearings, the hysteretic material available in OpenSees program (Mazzoni et al., 2008)

is coupled with Steel01 in parallel. Spherical bearings used in the curved steel bridge were made

of A36 steel and the Steel01 material used to model those bearings reflected a nominal initial

stiffness Ke of 200 GPa (29000 ksi) and a nominal strain-hardening ratio b of 0.014. The

hysteretic material model, which utilizes four different linear zones to approximate nonlinear

hysteretic behavior, contained an initial stiffness K1 of 312.5 GPa (45355 ksi), a second stiffness

K2 of 3 GPa (435 ksi), a third stiffness K3 1.25 GPa (181 ksi), and a final stiffness K4 of -312.5

GPa (-45355 ksi). All numerical values are obtained using a trial and error process with data

supplied from work by Roeder et al. (1995) that examined spherical bearing under cyclic loads.

Figure 4-10(b) presents experimental moment-rotational behavior results by Roeder et al. (1995)

with analytical results superimposed. As shown in Figure 4-10(b), the analytical model gives

reasonable approximation of real bearing behavior at 10,000 cycles.

(a) OpenSees spherical bearing model

50

(b) Moment-rotation hysteresis (Roeder 1995)

Figure 4-10: Analytical Modeling of Spherical Bearing.

The superstructure model is developed in OpenSees following the method outlined in

Section 4.1.1. Small straight sections are used to depict the curvature of the girders and concrete

deck. Nodes are placed at cross-frame locations. Longitudinal and transverse elements that

represent the behavior of the slab are used since the slab is expected to remain linearly elastic

under seismic loading (Choi, 2004; Nielson, 2005; Nielson, 2006; Padgett, 2007). Substructure

modeling is also developed following the methods outlined in Section 4.1.2. As stated earlier,

more concern of the substructure model is required to represent nonlinear behavior of

substructure components than the superstructure model. Figure 4-11 shows the 3-D OpenSees

analytical model for the curved steel bridge.

51

Figure 4-11: 3-D Analytical Model of Horizontally Curved Steel Bridge.

4.2.3 Validation Procedure

Validation occurred by comparing experimental data from a series of static tests to

predictions from the 3-D analytical model. Comparisons occurred for vertical and lateral bending

moments. Field tests that were compared are summarized in the section that follows.

4.2.3.1 Field Testing

Static testing used for validation was performed by Nevling (2003). The curved steel

bridge was tested using two trucks of known weight as listed in Table 4-3. Truck longitudinal

positions shown in Table 4-4 were selected to induce extreme live-load effects, with positions

measured along the arc relative to the lead axle for the three-axle truck. The two-axle truck was

positioned transversely at a clear distance of 0.79 m (2.6 ft) from the first truck as shown in

52

Figure 4-12. The sidewalk located on the west side of the bridge precluded placing the truck

nearer to the interior girder. The static testing positions that were selected for model validation

developed maximum vertical and lateral bending effects on the interior and exterior girders at

instrumented sections shown in Figure 4-13. In this study, all girders in the positive bending

section were solely considered to validate 3-D analytical model of curved steel bridge because the

significant discrepancy between maximum vertical and lateral bending moments obtained from

the previous analytical model and the test (Nevling, 2003) occurred at the selected section. The

four locations for model validation were locations that induced critical bending effects to the

interior and exterior girders at the instrumented sections.

Table 4-3: Test Truck Parameters (Nevling, 2003).

Parameter Three-Axle Truck Two-Axle Truck

Total Weight 245 kN (55.1 kips) 169 kN (37.9 kips) Front 62 kN (13.9 kips) 53 kN (11.9 kips)

Front Axle Weight (center-to-center) 4.1m (13.5 ft) 4.1m (13.5 ft) Middle Axle to Rear Axle (center-to-center) 1.3m (4.3 ft) N/A

Tire Width 0.3m (1 ft) 0.3m (1 ft) Width between Tires (out-to-out) 2.4m (8 ft) 2.3m (7.5ft)

Table 4-4: Summary of Single Truck Load Cases (Nevling, 2003).

Test number Truck Transverse Position, m (ft) Truck Longitudinal Position from South End of Bridge, m (ft)

Static 1 0.6 (2) from east parapet 65.2 (214) Static 2 0.6 (2) from east parapet 42.7 (140) Static 3 0.6 (2) from west parapet 46.3 (152) Static 4 0.6 (2) from west parapet 34.7 (114)

53

Figure 4-12: Static Test Truck Positions (Nevling, 2003).

: Strain Transducer

South abutment

Pier #1 Pier #2North abutment

NG1

G2G3

G4G5

C

CEF

Figure 4-13: Girder Instrument Locations Over Pier – Selected for Validation (Nevling,

2003).

4.2.3.2 Static 1

The 3-D analytical model for the curved bridge predicts girder vertical bending moments

obtained from Static Test 1 within an average 9.9% and predicts girder lateral bending moments

within average 12.8% for this test. As shown in Figure 4-14(a), the 3-D analytical model tends to

predict smaller values at all girder section C-C and section F-F, and larger values at section E-E

moments than those observed in the field. On the other hand, as shown in Figure 4-14(b), the 3-D

analytical model tends to predict larger values for girder lateral bending moments at all girders

54

than those observed in the field. At G2 C-C the vertical bending moment obtained from the

simulation of the 3-D analytical model was -39.1 kN·m (-28.8 k·ft) while this testing moment was

-33.0 kN·m (-24.3 k·ft) so that the maximum percent difference of 18.5% for vertical moments

occurs at the section as shown in Figure 4-15. At G4 C-C the lateral bending moment obtained

from the 3-D analytical model was 1.9 kN·m (1.4 k·ft) and the moment observed in the field was

2.4 kN·m (1.8 k·ft) so that the maximum percent difference of 18.5% for lateral moments occurs

at this section as shown in Figure 4-15. Even though the maximum percent differences of both

moments are quite high, it is apparent that the graph generated from the 3-D analytical model are

similar to the graph for static test 1 because of the small magnitudes between the results.

-60

-50

-40

-30

-20

-10

0

G5 F-F G5 E-E G5 C-C G4 C-C G3 C-C G2 C-C G1 C-C

Instrumental Locations

Gird

er V

ertic

al B

endi

ng M

omen

t, kN

-m 3-D Analytical ModelTest

(a) Girder Vertical Bending Moments

55

0

1

2

3

4

5



Gird

er L

ater

al B

endi

ng M

omen

t, kN

-m

3-D Analytical ModelTest

(b) Girder Lateral Bending Moments

Figure 4-14: Comparison Graph – Bending Moments; Static Testing 1.

0%

20%

40%

60%

80%

100%



Perc

ent D

iffer

ence

Vertical Bending MomentLateral Bending Moment

Figure 4-15: Percent Difference Histogram – Bending Moments; Static Testing 1.

56

4.2.3.3 Static 2

The 3-D analytical model for the curved bridge predicts the girder vertical bending

moments obtained from Static Test 2 within an average 8.2% and predicts girder lateral bending

moments within average 14.3% for this test. As shown in Figure 4-16(a), the 3-D analytical

model tends to predict larger values at G5 E-E and G5 C-C and smaller values at G5 F-F and G1

to G4 for girder vertical bending moments than those observed in the field. As shown in Figure

4-16(b), the 3-D analytical model tends to predict larger values at G2 and G4 and smaller values

at G1, G3 and G5 for girder lateral bending moments than those observed in the field. At G1 C-C

the vertical bending moment obtained from the simulation of the 3-D analytical model was 23.0

kN·m (16.9 k·ft) and this moment observed in the field was 30.0 kN·m (22.1 k·ft) so that the

maximum percent difference of 23.3% for vertical moments occurs at the section as shown in

Figure 4-17. At G3 C-C the lateral bending moment obtained from the 3-D analytical model was

3.0 kN·m (2.2 k·ft) and this moment observed in the field was 3.7 kN·m (2.7 k·ft) so that the

maximum percent difference of 19.5% for lateral moments occurs at this section as shown in

Figure 4-17. Even though the maximum percent differences of both moments are quite high, it

appears the graph generated from the 3-D analytical model are similar to the graph for static test 2

because of the small magnitudes between the results.

57

0

40

80

120

160

200



Gird

er V

ertic

al B

endi

ng M

omen

t, kN

-m



0

3

6

9

12

15



Gird

er L

ater

al B

endi

ng M

omen

t, kN

-m




58

0%

20%

40%

60%

80%

100%

G5 F-F G5 E-E G5 C-C G4 C-C G3 C-C G2 C-C G1 C-CInstrumental Locations

Per

cent

Diff

eren

ce



4.2.3.4 Static 3




model tends to predict slightly larger values at G5 F-F and G4 and smaller values at G1 to G3 for

girder vertical bending moments than those observed in the field. As shown in Figure 4-18(b),

the 3-D analytical model tends to predict smaller values at G1, G4, G5 F-F and C-C and slightly

larger values at G2 for girder lateral bending moments than those observed in the field. At G1 C-

C the vertical bending moment obtained from the simulation of the 3-D analytical model was 27.0

kN·m (19.9 k·ft) and this moment observed in this field was 36.0 kN·m (26.5 k·ft) so that the

maximum percent difference of 25% for the moments occurs at this section as shown in Figure 4-

19. At G1 C-C the lateral bending moment obtained from the model was 2.24 kN·m (1.6 k·ft) and

the moment observed in the field was 2.8 kN·m (2.1 k·ft) so that the maximum percent difference

59

of 20.0% for the lateral moments occurs at this section as shown in Figure 4-19. Even though the

maximum percent differences of both moments are quite high, it is clear that the graphs generated

from the 3-D analytical model are similar to the graphs for static test 3 because of the small

magnitudes between the results.

0

10

20

30

40



Gir

der V

ertic

al B

endi

ng M

omen

t, kN

-m 3-D Analytical Model

Test


0

1

2

3



Gird

er L

ater

al B

endi

ng M

omen

t, kN

-m




60

0%

20%

40%

60%

80%

100%


Perc

ent D

iffer

ence



4.2.3.5 Static 4




model tends to predict larger values at only G1 and smaller values at G2 to G5 for girder vertical

bending moments than those observed in the field. As shown in Figure 4-20(b), the 3-D

analytical model tends to predict smaller values at all girders for girder lateral bending moments

than those observed in the field. At G1 C-C the vertical bending moment obtained from the

simulation of the 3-D analytical model was 145.1 kN·m (106.7 k·ft) and this moment observed in

this field was 133.0 kN·m (97.8 k·ft) so that the maximum percent difference of 9.2% for the

moments occurs at the section as shown in Figure 4-21. At G2 C-C the lateral bending moment

obtained from the 3-D analytical model was 9.9 kN·m (7.28 k·ft) and the moment observed in the

field was 10.6 kN·m (7.8 k·ft) so that the maximum percent difference of 21.5% for the lateral

61

moments occurs at this section as shown in Figure 4-21. Though the maximum percent

differences of both moments are high, it appears that the graphs generated from the 3-D analytical

model are similar to the graph for static test 4 because of the small discrepancy of the magnitudes

between the results.

0

50

100

150

200

250



Gir

der V

ertic

al B

endi

ng M

omen

t, kN

-m



0

3

6

9

12

15



Gird

er L

ater

al B

endi

ng M

omen

t, kN

-m




62

0%

20%

40%

60%

80%

100%


Per

cent

Diff

eren

ce



4.2.3.6 Discussion

The 3-D analytical modeling approach for curved bridge under static loadings can predict

girder vertical bending moments measured from the tests within an average of 9.1% and girder

lateral bending moments measured from the tests within an average 13.4%. According to

previous work examining curved steel bridge behavior (Chang et al., 2006), 3-D analytical

models predicted experimental vertical and lateral bending response of curved steel bridges tested

in a controlled, laboratory setting under static loads within an average of 11.4%. As mentioned

previously, having error levels with an average of 11.2% with standard deviation of 5.5% for 3-D

analytical model predictions of the response of an actual structure under live loads appears

acceptable. Though the maximum percent differences of both moments for all static tests are a

little high, the gaps between the 3-D analytical model and static tests were small so that the trends

for the model were very similar to those for static tests.

63

4.3 Seismic Response Methodology

A sample deterministic earthquake analysis using the previously discussed 3-D analytical

modeling approach is performed on the three-span curved steel I-girder bridge to examine its

seismic response. This section presents the deterministic seismic response procedure of how to

efficiently investigate mode shapes and seismic response for the existing curved steel I-girder

bridge. The seismic response methodology is based on the results obtained from analytical

seismic analysis for the curved steel bridges because the experimental data for curved steel

bridges under actual seismic loadings is not readily available.

GLD

(0,0)

GTD

South abutmentPier #1 Pier #2

North abutmentG1

G2G3

G4G5

Z

X

Figure 4-22: Seismic Loading Direction.

As mentioned in Section 4.1, material nonlinearities are considered for the substructure

units and geometric nonlinearities for reinforced concrete columns and caps. For the purpose of

this procedural investigation, five percent Rayleigh damping was used when extracting seismic

responses from a NLTHA completed using the Newmark method. To elicit seismic response, El

Centro ground motions obtained from the Pacific Earthquake Engineering Research Center, were

64

simultaneously applied to the bridge in the global longitudinal direction (Figure 4-22: GLD)

initially and then the global transverse direction (Figure 4-22: GTD). It is highly probable that

though the global longitudinal direction controls in a deterministic setting (Rashidi and Ala

Saadeghvaziri, 1997; Shinozuka, 1998), again to investigate all critical seismic responses the

global transverse direction is considered. However, seismic response will be captured in

tangential and radial directions relative to the curved bridge superstructure because results are

typically reported in this fashion for horizontally curved structures.

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 10 20 30 40 50

Time (sec)

Acc

eler

atio

n (g

)

(a) El Centro ground motion

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5

Period (sec)

Spec

tral

acc

eler

atio

n (g

)

(b) Response spectrum

Figure 4-23: Ground Motion Used for Illustration of Seismic Responses (Chopra, 2000): (a) Time History (b) Response Spectrum (5% Damping).

65

The El Centro ground motion has a peak ground acceleration (PGA) of 0.313 g and a

response spectrum with five percent damping as presented in Figure 4-23. The interval for the

time history is 0.01sec and the duration is 40sec. The ground motions are applied to the pier

footings and the abutments simultaneously.

4.3.1 Mode shapes

The first four mode shapes for the curved bridge are presented in Figure 4-24. It appears

that the first mode, which has a period of 0.872 s, is more sensitive to bending effects rather than

pure torsional effects. The second mode, which is dominated by coupled bending-torsion effects,

has a shorter period of 0.402 s. The third mode, which has a period of 0.383 s, is similar to the

second mode. The fourth mode, which has a period of 0.382 s, is dominated by torsion effects.

Effects of curvature on the fundamental modes are clearly demonstrated due to pure torsion and

coupled bending-torsion. After appropriately investigating the mode shapes of the existing

curved steel I-girder bridge, the seismic responses for the bridge are efficiently found using the 3-

D analytical model in the following section.

(a) First Mode Shape – 0.872s

66

(b) Second Mode Shape – 0.402s

(c) Third Mode Shape – 0.383s

(d) Fourth Mode Shape – 0.382s

Figure 4-24: Mode Shapes for Curved Bridge; First to Fourth.

67

4.3.2 Seismic Response

To efficiently explore the seismic response of existing curved steel I-girder bridges, one

curved steel I-girder bridge located in Lewistown, Pennsylvania, was previously selected and this

bridge was subjected a PGA of 0.313 g as stated earlier. This section is to illustrate the nature of

its response to seismic loadings (i.e., PGA of 0.313g). Figure 4-25 presents tangential and radial

displacement time histories for the concrete deck at the middle of span (L2) (Figure 4-7) under

the different loading scenarios. For this seismic event the maximum tangential displacement was

approximately 100 mm (3.9 in) while maximum radial displacement was approximately 75mm

(2.9 in).

Pier column seismic response was also monitored and presented in terms of moment-

curvature hysteresis. Figure 4-26 shows this response for the two right-most columns for Pier 1

and 2, as seen in Figure 4-8(a). These are referred to as Column 1 and Column 2, respectively.

The maximum moment obtained from the columns under longitudinal loading is on the order of

1500 kN·m (1102 k·ft) resulting in a curvature of about 0.012 1−m (0.00365 1−ft ), while the

maximum moment obtained from the columns under transverse loading is on the order of 1500

kN·m (1102 k·ft) at a curvature of about 0.0251−m (0.0076 1−ft ). Also, the maximum curvature

obtained from the columns under global longitudinal loading is 0.0311−m (0.0094 1−ft ), while the

maximum curvature obtained from the columns under global transverse loading is

0.0291−m (0.0088 1−ft ). It helps estimate the column capacity consisting of moment and

curvature, consequently it can be used for assessing the damage states for the columns.

68

-200

-150

-100

-50

0

50

100

150

200

0 5 10 15 20 25 30 35 40 45

Time, sec

Dis

plac

emen

t, m

m

Tangential displacementRadial displacement

(a) Global Longitudinal Loading

-200

-150

-100

-50

0

50

100

150

200

0 5 10 15 20 25 30 35 40 45

Time, sec

Dis

plac

emen

t, m

m

Tangential displacementRadial displacement

(b) Global Transverse Loading

Figure 4-25: Deck Displacement Time Histories for the Curved Steel I-Girder Bridge under (a) Global Longitudinal Loading (b) Global Transverse Loading.

69

Column 1: Longitudinal Loading & Response

-2000

-1500

-1000

-500

0

500

1000

1500

2000

-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04

Curvature, 1/m

Mom

ent,

kN-m

Column 2: Longitudinal Loading & Response

-2000

-1500

-1000

-500

0

500

1000

1500

2000

-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03

Curvature, 1/m

Mom

ent,

kN-m

(a) Global Longitudinal Loading

70

Column 1: Transverse Loading & Response

-2000

-1500

-1000

-500

0

500

1000

1500

2000

-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04

Curvature, 1/m

Mom

ent,

kN-m

Column 2: Transverse Loading & Response

-1500

-1000

-500

0

500

1000

1500

2000

-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03

Curvature, 1/m

Mom

ent,

kN-m

(b) Global Transverse Loading

Figure 4-26: Column 1 & 2 of the Curved Steel I-Girder Bridge Under (a) Global Longitudinal Loading (b) Global Transverse Loading.

As shown in Figure 4-26, a distinguishing characteristic of the seismic hysteresis for both

Column 1 and Column 2 is the open hysteresis loops that are indicative of inelastic response.

Another way of looking at the deformation or curvature of the columns is through a

curvature ductility demand ratio, cμ , which is given in Equation 4.3 (Nielson, 2005).

71

yieldc κ

κμ max= (4.3)

Where yieldκ is the curvature in the column which causes first yield of the outer most

reinforcing bar (MacGregor, 1997) and maxκ is the maximum curvature demand for the column

throughout the loading event. Again, the curvature ductility ( cμ ) for this study is defined in

Equation 4.3 as the maximum realized curvature divided by the yield curvature or curvature at

yield of the outer most steel reinforcing bar. Under global longitudinal loading, the ductility

demand is around 6.1 while global transverse loading produces a ductility demand of 5.7 for

Column 1. The same trend is observed for the other column line. It is clear from these results

that the global longitudinal direction tends to be the slightly more critical direction for this

structure as far as column response is concerned.

Another set of components which are of interest are the spherical bearings since the

bearings are critical components that can be severely damaged during earthquakes. Their

response is given in terms of force-displacement hysteresis from which maximum deformations

can be attained. Maximum response of the bearing (see Figure 4-7: nearest at most-outside

girder, G5) under global longitudinal and global transverse earthquake loadings is presented in

Table 4-5. The maximum deformation of the spherical bearing under longitudinal loading is

approximately 130 mm (5.1 in) while the maximum deformation under the transverse loading is

approximately 35 mm (1.4 in), both of which are excessive.

The abutments’ response in passive action (soil) and active action (piles) in the global

longitudinal direction and transverse direction were obtained from the analysis. The passive and

active maximum response of the abutments in the global longitudinal direction appears as

presented for the south abutment in Table 4-5. In passive action, the response became nonlinear

resulting in a deformation which exceeded 129 mm (5.07 in) and in active action, the response

72

became nonlinear resulting in a deformation of 85 mm (3.35 in). Deformation of the abutments

under global transverse loading never exceeded the linear range with maximum deformations not

much greater than 10 mm (0.4 in) as shown in Table 4-5.

Table 4-5: Maximum Displacements at bearing and abutment.

Maximum displacements, mm(in) Global longitudinal loading Global transverse loading

Spherical bearing 130(5.11) 35(1.37) South abutment (Passive) 129(5.07) 8(0.31) South abutment (Active) 85(3.35) 7(0.28)

4.4 Conclusions

In Chapter 4, 3-D analytical modeling approach for horizontally curved steel bridges was

presented. The proposed analytical approach was validated statically using comparisons of

predicted girder vertical and lateral bending moments to results obtained from field testing of an

in-service, three-span continuous, curved, steel I-girder bridge in Pennsylvania. The 3-D

analytical modeling approach was able to predict girder vertical bending moments measured from

the static tests within an average of 9.1% and girder lateral bending moments measured from the

static tests within an average 13.4%. Having error levels with an average of 11.2% with standard

deviation of 5.5% for 3-D analytical model predictions of the response of an actual structure

under live loads appears acceptable. This is because the gaps between the 3-D analytical model

and static tests were small and the trends for the model were very similar to those for static tests.

Based on the validation for the 3-D analytical model, a sample deterministic earthquake

analysis using the modeling approach was performed on the bridge to explore its seismic response

using the proposed seismic response methodology was presented. The methodology to efficiently

examine the bridge subjected to seismic loading was based on analytical results obtained from the

3-D analytical model because the experimental data on such bridges under actual seismic

73

loadings is not readily available. From the seismic analyses conducted in this Chapter, it appears

the seismic behavior of columns and of the abutments under global longitudinal earthquake

loading were the slightly more vulnerable components. The global longitudinal responses of the

bearings also displayed vulnerabilities which are cause for concern. As mentioned previously,

the seismic analysis utilizing the analytical modeling approach is a basic tool for finding critical

seismic responses which will be used as an output for seismic vulnerability assessment for a

family of horizontally curved steel bridges.

Chapter 5

HORIZONTALLY CURVED STEEL BRIDGE INVENTORY AND GROUND MOTION DEVELOPMENT

To complete a study that develops fragility curves for a family of horizontally curved

steel I-girder bridges for a specified region in the United States, it is essential to have an

understanding of the bridge inventory in the region of interest and to establish appropriate

parameters for that family that may affect their seismic response. This procedure can be included

as Step 1 of the proposed fragility curve construction methodology shown in Figure 3-4. For the

present study, which focuses on a region that includes Pennsylvania, a partial curved bridge

inventory from three states (Maryland, New York, Pennsylvania) was used to develop a

statistically significant family of bridges. Figure 5-1 indicates the region is in a low to moderate

seismic zone, having a 10% PGA with a 2% probability of exceedence for a 50 year recurrence

interval.

To establish fragility curves using RSMs for this geographical region, an inventory

statistical analysis is carried out based on available construction plans. These construction plans

are collected from the Maryland Department of Transportation (MDOT), the New York

Department of Transportation (NYDOT) and the Pennsylvania Department of Transportation for

a region that focused on Philadelphia (PennDOT).

Presented in this Chapter are typical horizontally curved steel I-girder bridge classes from

Maryland, New York, and Pennsylvania obtained from these construction plans. Class statistics

used to examine the resulting horizontally curved steel I-girder bridge family are presented and

characteristics used to elicit key parameters that affect seismic response are also shown. Finally,

appropriate ground motions applied to the family to elicit seismic response are presented.

75

Figure 5-1: Maryland, New York, Pennsylvania Considered in the Inventory Study with Hazard Map (USGS, 2002).

5.1 Horizontally Curved Steel Bridge Inventory Analysis

To adequately perform a preliminary seismic fragility assessment for a given bridge

family, appropriate classification is needed with respect to many relevant geometric parameters.

This was performed for the inventory of horizontally curved steel I-girder bridge construction

plans obtained for this study. 355 horizontally curved steel I-girder bridges, with and without

skew, from the state of Maryland, New York and Pennsylvania (focusing on Philadelphia) were

included in the inventory study. Of these 355 bridges, 129 were horizontally curved steel I-girder

bridges without skew (36% of the total) and 226 were a combination of skewed and curved steel

I-girder bridges (64% of the total). As mentioned previously, this study focuses purely on curved

76

bridges without skew so the 226 structures having skew were not included in any future statistical

analyses.

Curved steel bridges that remained in the inventory were divided into horizontally single,

double, and multiple curved steel I-girder bridges. The horizontally single, double, and multiple

curved steel I-girder bridges have single-curvature, two-curvatures, and more than two-

curvatures, respectively. 99 of the remaining inventory were horizontally, single curved bridges

(77%), 18 were horizontally double curved bridges (14%) and 12 had more than two-curvatures

(9%). Figure 5-2 shows horizontally curved steel I-girder bridge classification with respect to the

number of curvatures. This inventory work focused on the 99 horizontally curved steel I-girder

bridges with a single-curvature.

0

20

40

60

80

100

120

Single Curved Bridge Double Curved Bridge Multiple Curved Bridge

Number of Curvatures

Coun

ts

Figure 5-2: Horizontally Curved Steel I-Girder Bridge Counts with respect to the Number of Curvatures.

77

5.2 Potential Key Parameters for Horizontally Curved Steel Bridge

To determine potential key parameters affecting horizontally curved, steel, I- girder

bridge seismic response for the fragility assessment utilizing RSMs, it is necessary to examine the

characteristics of the curved steel I-girder bridges selected in the previous section. These

parameters can be divided into 2 groups: global geometric parameters that are used to

characterize individual bridges, termed macro-level parameters; and random parameters for the

bridge material properties and the damping ratio of the bridge system, called micro-level

parameters.

Important macro-level parameters were identified by looking at National Bridge

Inventory (NBI) data (FHWA, 2008) and investigating more detailed characteristics from the

bridge plans. NBI data provides the following information related to bridge geometric

parameters:

a) Number of spans;

b) Maximum span length;

c) Deck width; and

d) Vertical under clearance (column height)

From the construction plans other key parameters affecting seismic response, including:

radius of curvature; girder spacing; and cross-frame spacing were obtained (Brockenbrough,

1986; Kim, 2004). These parameters were selected because it has been shown in previous studies

that they have significant influence on seismic responses. Micro-level material property

parameters, including: concrete compressive and tensile strength; concrete Young’s Modulus;

reinforcement yield strength and Young’s modulus are also obtained from the plans. Note that

these micro-level parameters focused on the bridge substructure, rather than the superstructure.

78

As previously mentioned, substructure deficiencies have been used to assess bridge seismic

susceptibility regardless of structure type and geometry and are adopted as the primary

assessment tool here, as well (FEMA, 2003; Murachi, 2003; Choi, 2004; Nielson, 2006; Padgett,

2007). All parameters are considered as random variables.

Ranges for each of the macro- and micro-level parameters can be determined by

examining plans for the MDOT, NYDOT, and PennDOT curved bridges that remain in the

family. Statistical examination of the data allows for development of probability distributions for

key parameters influencing seismic response (i.e. probability density functions and cumulative

distribution functions) and these variations can be directly implemented into the seismic

vulnerability assessment methodology. The following subsections deal with each parameter’s

range and distribution for the family that is examined.

5.2.1 Macro-Level Parameters

As mentioned previously, macro-level parameters consisted of number of spans,

maximum span length, deck width, column height, radius of curvature, girder spacing, and cross-

frame spacing. Each macro-level parameter’s range and distribution from examination of the

family are described as follows.

5.2.1.1 Number of Spans

A frequency histogram for the number for spans in the 99 horizontally curved, steel, I-

girder bridges that remain in the family is shown in Figure 5-3. From this figure it is clear that

the number of spans take on discrete values so a discrete probability distribution is appropriate.

Because of the nature of discrete probability distributions, it is unnecessary to fit the data to a

known distribution but rather to examine the data frequency for each span number. Therefore, a

79

probability mass function (PMF) for the number of spans was generated as shown in Figure 5-4

by counting the number of curved steel bridges with a particular number of spans and dividing by

the total number of bridges. Table 5-1 gives some of the statistics for this parameter and, for this

inventory, the average number of spans is 2.53 with a standard deviation of 1.73. The median

value was 1.5 and the corresponding mode was 2. This reveals that the majority of horizontally

curved steel I-girder bridges in the family have two spans.

Table 5-1: Number of Span Statistics for Horizontally Curved Steel Bridges.

Class Mean Std. Dev. Median Mode Curved Steel I-Girder Bridge 2.53 1.73 1.5 2

0

5

10

15

20

25

30

35

40

45

50

1 2 3 4 5 6 7 8 9 10 11 12 13

Number of spans

Cou

nts

Figure 5-3: Frequency Histograms for Number of Spans.

80

0.0

0.3

0.5

0.8

1.0

1 2 3 4 5 6 7 8 9 10 11 12 13

Number of spans

Prob

abili

ty

Figure 5-4: Probability Density Function for Number of Spans.

5.2.1.2 Maximum Span Length

Maximum span length has also been used as a parameter that assesses bridge seismic

susceptibility using vulnerability functions (Nielson, 2005). Plans from the family were reviewed

to elicit span length statistics (e.g., mean, median, standard deviation, etc.) and results are

presented in Table 5-2. From this data the mode of for the maximum span length for the region is

43.8m (143.9ft). The median value is 43.1m (141.5ft) and the mean is 43.1m (141.5ft).

Table 5-2: Span Length Statistics for Curved Steel I-Girder Bridges.

Class Mean, m(ft) Std. Dev., m(ft) Median, m(ft) Mode, m(ft) Curved Steel I-Girder Bridge 43.8 (143.9) 13.7(45.1) 43.1(141.5) Not Unique

Even though maximum span length statistics are useful for showing data tendencies and

dispersion as presented in Table 5-2, they do not sufficiently describe the data for the purposes of

this study Therefore, development of an empirical cumulative distribute function (CDF) for the

81

maximum span length is required, as this gives a complete description of the data distribution and

allows for implementation of Monte Carlo simulation techniques necessary for seismic

vulnerability assessment using RSMs. The CDF for maximum span length for the inventory is

given in Figure 5-5. As shown in Figure 5-5, the CDF has a 0.01 probability when the maximum

span length approaches 15.2m (50ft) and a probability of 1 when the maximum span length

approaches 91.4m (300ft). This implies that there are no existing curved steel bridges in the

selected inventory with a maximum span length smaller than 15.2 m (50 ft) or larger than 91.4 m

(300 ft). Therefore this study included the span lengths between 15.2 m (50 ft) and 91.4 m (300

ft).

0.0

0.3

0.5

0.8

1.0

0 6 12 19 25 31 37 43 49 55 61 67 73 80 86 92 98

Maximum span length, m

Prob

abili

ty

Figure 5-5: Cumulative Distribution Function for Maximum Span Length.

5.2.1.3 Deck Width

Similar to span length, the mean, median, standard deviation and mode for the deck width

were established from the selected inventory and are presented in Table 5-3. The mean out-to-out

width is 12.9m (42.3ft) with a standard deviation of 4.4m (14.5ft). The median value is 12.2m

(40ft) and mode is 12.8m (42ft). So that these deck width statistics could be effectively applied

82

to future Monte Carlo simulations, an empirical CDF was calculated from the data set and is

presented in Figure 5-6. As shown in Figure 5-6, the CDF has 0.01 of probability when the width

approaches 6.7m (22ft), which indicates that deck widths less than 6.7m (22ft) are not in the

selected inventory. The CDF has a probability of 1 when it approaches 33.8m (111ft), which

shows that deck widths larger than 33.8m (111ft) are not in the selected inventory. Therefore this

study included deck widths varying between 8.5 m (27.9 ft) and 17.3 m (56.8 ft) based on mean

value plus one standard deviation, not the maximum and minimum values since they were

deemed impractical.

Table 5-3: Deck Width Statistics.

Class Mean, m(ft) Std. Dev., m(ft) Median, m(ft) Mode, m(ft) Curved Steel I-Girder Bridge 12.9(42.3) 4.4(14.5) 12.2(40) 12.8(42)

0.0

0.3

0.5

0.8

1.0

0 3 6 9 12 16 19 22 25 28 31 34 37 40

Deck width, m

Prob

abili

ty

Figure 5-6: Cumulative Distribution Function for Deck Width.

5.2.1.4 Column Height

It is widely understood that the seismic response of a bridge structure is sensitive to

column height because relatively slender columns can support a superstructure dynamically with

83

a lot of mass (Nielson, 2006). This parameter has traditionally been included as a major influence

for bridge seismic fragility assessments (Choi, 2004; Nielson, 2006; Padgett, 2007). Typically,

the column heights used for seismic studies are measured from the centerline of column/pier cap

to the centerline of the bridge column foundation and that methodology is employed here

(Nielson, 2006). For the selected bridge inventory some basic column height statistics are

presented in Table 5-4. The mean column height is 6.5m (21.3ft) with a standard deviation of

4.3m (14.2ft). The median value is 5.3m (17.5ft) and mode is 4.6m (15ft).

Table 5-4: Column Height Statistics.

Class Mean, m(ft) Std. Dev., m(ft) Median, m(ft) Mode, m(ft) Curved Steel I-Girder Bridge 6.5(21.3) 4.3(14.2) 5.3(17.5) 4.6(15)

Again, it was established that an empirical CDF needs to be employed for column height

because it provides efficient graphical representation of the distribution used for RSMs in

conjunction with Monte Carlo simulation. The CDF is shown in Figure 5-7. This CDF has initial

probability (around 3%) when the column height is 1.5 m (around 5 ft) which, again, is indicative

that column heights less than 1.5 m (around 5 ft) are not in the selected bridge inventory. In

addition, the CDF has a probability of 1 when the column height approaches 32 m (105 ft). This

indicates that column heights larger than 32.3 m (106 ft) are not in the selected bridge inventory.

Therefore, this study included the column heights varying between 10 m (32.8 ft) and 32.3 m

(106 ft), not the minimum value since it was deemed impractical.

84

0.0

0.3

0.5

0.8

1.0

0 4 8 12 16 20 24 28 32 36 40

Column Height, m

Prob

abili

ty

Figure 5-7: Cumulative Distribution Function for Column Height.

5.2.1.5 Radius of Curvature

Radius of curvature is known to be one of the key parameters influencing both curved

bridge static and dynamic response (Brockenbrough, 1986; Senthilvasan, 2002) and, as a result,

was included in the current study. Radius of curvature as reported herein is measured to the

exterior girder and some basic statistics for the selected inventory are presented in Table 5-5.

Note that the inventory data set, which included eight straight steel I-girder bridges as

benchmarking samples, did not include these bridges in radius of curvature statistics. The mean

exterior girder radius of curvature is 513.5 m (1684.8 ft) with a standard deviation of 542.9 m

(1781.3 ft) and the median is 304.6 m (999.5 ft), respectively. The mode is 250 m (820 ft), so the

mode value is considerably different from the mean which reveals that the majority of curved

steel bridges in the family had a radius of curvature of 250 m (820 ft) to the exterior girder.

Table 5-5: Radius of Curvature Statistics.

Class Mean, m(ft) Std. Dev., m(ft) Median, m(ft) Mode, m(ft) Curved Steel I-Girder Bridge 513.5(1684.8) 542.9(1781.3) 304.6(999.5) 250 (820)

85

Again, empirical CDFs were employed and the plot for radius of curvature is shown in

Figure 5-8. The CDF has an initial probability (around 1%) when the radius of curvature is 240m

(787.4ft). The CDF has probability of 1 when the radius approaches 3492m (11456.7ft) which

indicates that bridges with an exterior girder radius greater than 3492m (11456.7ft) do not exist in

the selected bridge inventory. Therefore this study included radius of curvature varying from 240

m (787.4 ft) to 3492 m (11456.7 ft), not the minimum value since it was deemed impractical.

0.0

0.3

0.5

0.8

1.0

0 457 914 1372 1829 2286 2743 3200

Radius of Curvature, m

Prob

abili

ty

Figure 5-8: Cumulative Distribution Function for Radius of Curvature.

5.2.1.6 Girder Spacing

Studies conducted by Brockenbrough (1986) and Kim (2004) have determined that girder

spacing is a key factor influencing both static and dynamic response of curved steel bridges and

that parameter was included in this study. Girder spacing is measured between adjacent girder

centerlines from the construction plans and some basic statistics are presented in Table 5-6. The

mean for the girder spacing is 2.6 m (8.5 ft) with a standard deviation of 0.4 m (1.2 ft) and the

median is 2.5 m (8.3 ft), respectively. The mode is not unique because the twelve bridges among

86

99 bridges, which have either same 2.4 m (8 ft) or 2.7 m (9 ft) girder spacing, exist in the selected

inventory.

Table 5-6: Girder Spacing Statistics.

Class Mean, m(ft) Std. Dev., m(ft) Median, m(ft) Mode, m(ft) Curved Steel I-Girder Bridge 2.6(8.5) 0.4(1.2) 2.5(8.3) Not Unique

The generated empirical CDF is shown in Figure 5-9. The CDF has a probability of 1 at

3.2 m (10.6 ft) with the range of spacing between 1.5 m (4.8 ft) and 3.2 m (10.6 ft). Therefore,

this study included a range of girder spacing between 1.5 m (4.8 ft) and 3.2 m (10.6 ft).

0.0

0.3

0.5

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.4 2.9 3.4 3.9 4.4

Girder Spacing, m

Pro

babi

lity

Figure 5-9: Cumulative Distribution Function for Girder Spacing.

5.2.1.7 Cross-Frame Spacing

Similar to girder spacing, studies conducted by Brockenbrough (1986) and Kim (2004)

determined that cross-frame spacing is another key factor influencing both static and dynamic

response of curved steel bridges. Basic statistics for cross-frame spacing for the selected

inventory are presented in Table 5-7. The mean for the girder spacing is 5.4 m (17.6 ft) with a

standard deviation of 1.2 m (3.9 ft) and a median of 5.2 m (17.2 ft). The mode is 7.3 m (24 ft).

87

Table 5-7: Cross-Frame Spacing Statistics.

Class Mean, m(ft) Std. Dev., m(ft) Median, m(ft) Mode, m(ft) Curved Steel I-Girder Bridge 5.4(17.6) 1.2(3.9) 5.2(17.2) 7.3(24)

Again, an empirical CDF for the cross-frame spacing is shown in Figure 5-10. The CDF

has a probability of 1 at 7.3 m (24 ft) and the range for cross-frame spacing is between 2.7 m (8.8

ft) and 7.3 m (24 ft). Therefore this study included cross-frame spacing varying between 2.7 (8.8

ft) and 7.3 m (24 ft).

0.0

0.3

0.5

0.8

1.0

0.0 0.9 1.8 2.7 3.7 4.6 5.5 6.4 7.3 8.2 9.1

Cross-Frame Spacing, m

Prob

abili

ty

Figure 5-10: Cumulative Distribution Function for Cross-Frame Spacing.

5.2.2 Micro-Level Parameters

As mentioned previously, material property parameters and the damping ratio of the

overall bridge system were deemed micro-level parameters because properties material properties

also possess uncertainties due to manufacturing processes or inherent unpredictability within

materials themselves (Galambos et al., 1982). Other micro-level parameters in terms of local

superstructure geometric variability (e.g., web thickness, etc.) were not deemed this micro-level

88

parameter setup. The local geometric superstructure parameters have little variability based on

the family statistics. The local parameters would have little influence on the overall behavior of

seismic response for the bridge system. Instead of this micro-level parameter setup, the relevant

superstructure parameters were previously included in macro-level parameter setup. Again, this

is because the macro-parameters (e.g., radius of curvature, etc.) related to the superstructure have

more influence on the static and dynamic responses of horizontally curved steel bridges

(Brockenbrough, 1986; Kim, 2004) than local geometric superstructure parameters.

The micro-level parameters considered consist of concrete compressive and tensile

strength, concrete and steel reinforcing bar Young’s Modulus, steel reinforcing bar yield strength

and damping ratio. Table 5.8 shows the lower and upper bounds of these parameters for the

inventory data set. Similar to the previous parameters, lower and upper bounds for each micro

parameter are identified from selected inventory construction plans. The resulting range of

parameters can be included in the Monte Carlo simulations.

Table 5-8: Potential Micro-Level Parameters for Response Surface Model Generation.

Micro-level parameters Minima Maxima

Damping ratio 2 6 Concrete compressive strength, MPa (ksi) 20.7(3) 34.1(5) Concrete tensile strength, MPa (ksi) 1.7(0.24) 2.7(0.40) Concrete Young's Modulus, MPa (ksi) 17(2500) 31(4500) Steel reinforced bar Young's Modulus, MPa (ksi) 192920(28000) 206700(30000) Steel reinforced bar yield strength, MPa (ksi) 289.4 (42) 345 (50)

5.3 Synthetic Ground Motions

When assessing the vulnerability of a horizontally curved steel bridge class to the

associated seismic hazard of a particular region, it is helpful to have ground motion time histories

89

that are representative of the area. However, ground motions records for the specific region of

interest, in and around Pennsylvania, do not exist. Therefore, synthetic acceleration time histories

can be generated and used instead. Synthetic ground motion records have been developed for the

Central and Eastern United States by Rix and Fernandez-Leon (2004) and used in Charleston area

by Padgett (2007). Therefore, these synthetic ground motions were used for the present study.

Rix and Fernandez-Leon (2004) developed synthetic ground motions using stochastic

ground motion models. Source models were constructed by Rix and Frankel et al. (1996) to help

capture the impact of modeling uncertainty. Synthetic ground motion sets were developed for

three body wave magnitudes (5.5, 6.5 and 7.5), which are related to seismic waves that move

through the interior of the earth, as opposed to surface waves that travel near the earth's surface.

In addition, four hypocentral distances (10, 20, 50 and 100 km), which is the distance between

hypocenter (earthquake starting point) and observer, were used with each scenario event for a

given magnitude and distance and twenty ground motion records were developed for each

distance. Ground motions for a 10 km hypocentral distance at a body wave magnitude of 7.5

were not available. The 220 ground motions were developed using Frankel’s model as presented

in Table 5-9 and allowed the inclusion of soil nonlinearity and uncertainties in the site response

parameters.

Corresponding PGA distributions for a classification of input earthquake loadings are

shown in Figures 5-11. PGA values range from 0.008 g to 0.646 g. The 30 ground motions used

for this study were randomly selected from the 220 ground motions discussed in earlier literature

(Choi, 2004; Nielson, 2006; Padgett, 2007). For these 30 ground motions the PGA ranges from

0.06 g to 0.50g while the spectral accelerations at 0.87 seconds range from 0.0003 g to 1.702 g.

Figure 5-12 shows each representative ground motion at PGA values of 0.01g, 0.23g, and 0.44g.

Each PGA zone was a function of the PGA distributions and contains ten synthetic ground

motions. Ten ground motions are deemed appropriate to generate fragility curves at each level

when considering uncertainty of earthquakes based on past literature (Choi, 2004; Nielson, 2006;

90

Padgett, 2007). Figure 5-13 shows the mean response spectrum for the 30 selected ground

motion sets. These response spectra were calculated based on 5% critical damping by Rix and

Fernandez (2004). The frequency content of this ground motion suite is more clearly presented

by examining the mean response spectrum shown in Figure 5-13. Therefore this study included

10 synthetic ground motions at each PGA level.

Table 5-9: Sets of Ground Motion Records (Rix and Fernandez-Leon 2004); The presence of A. in a field means “Available”, while N.A. means “Not Available”.

Frankel et al. (1996) Model

Magnitude Distance (km)

5.5 6.5 7.5

10 A. A. N.A. 20 A. A. A. 50 A. A. A. 100 A. A. A.

0

50

100

150

0-0.1 0.1-0.2 0.2-0.3 0.3-0.4 0.4-0.5 0.5-0.6 0.6-0.7

PGA, g

Freq

uenc

y

Figure 5-11: Histogram of PGA Values of Rix and Fernandez Ground Motion Suite. (Rix

and Fernandez-Leon 2004).

91

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0 5 10 15 20 25 30

Time, sec

Acc

eler

atio

n, g

PGA of 0.07gPGA of 0.23gPGA of 0.44g

Figure 5-12: Each Representative Rix and Fernandez Ground Motion at PGA Zone of 0.01g, 0.24g, and 0.44g.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2

Period, sec

Spe

ctra

l Acc

eler

atio

n, g

MeanMean+Std. Dev.Mean-Std. Dev.

Figure 5-13: Mean and Mean ± One Standard Deviation of Response Spectra - Rix and Fernandez (Rix and Fernandez-Leon 2004).

92

5.4 Conclusions

In this Chapter, potential key parameters which affect seismic response of horizontally

curved steel I-girder bridges are determined by examination of statistical analysis from an

inventory of 99 horizontally curved steel I-girder bridges in Maryland, New York, and

Pennsylvania (around Philadelphia). Synthetic ground motions to be used as input earthquake

loadings are also described. The information from this inventory analysis is not only used to

identify typical curved steel I-girder bridge types for RSM generation but to also identify basic

structural and geometric characteristics as well as appropriate probability density functions

corresponding to key parameters. Results from the statistical inventory study are used for

screening analysis that assisted with identifying significant seismic response parameters.

93

Chapter 6

SCREENING OF HORIZONTALLY CURVED STEEL BRIDGE PARAMETERS

A screening or sensitivity study of the potentially significant parameters which influence

horizontally curved steel I-girder bridge seismic response is included in Step 2 in the fragility

curve construction methodology presented in Figure 3-4. The sensitivity study is necessary to

efficiently perform seismic vulnerability assessment of a family of horizontally curved steel

bridges using RSMs. The potential parameters for assessment of the seismic structural response

of horizontally curved steel structures were defined as mentioned in Chapter 5 and are included in

the sensitivity study. Macro-parameter predictor variables included geometric items (e.g.,

number of span, radius of curvature, column height, etc.), while micro-parameters included

dynamic structural parameters (e.g., damping) and material properties (e.g., concrete compressive

strength, reinforcing bar yield strength, etc.). A screening analysis to be used to identify most

significant parameters will be accomplished using a combination of Design of Experiments

(DOE) approaches and statistical tools employed using commercially available software to run a

least-squares regression analysis. Representative synthetic ground motions selected based on

PGA distributions were used as the earthquake loadings and, via nonlinear time-history analysis

(NLTHA) for each combination selected from the DOE, seismic response information of interest

is obtained.

Figure 6-1 gives an overview of the parameter screening procedure. Again, the first step

of screening is to determine potential macro-and micro-level parameters which may influence

seismic response of horizontally curved steel I-girder bridges. The second step of screening is to

establish an ensemble of synthetic ground motions in the region of interest. These were

completed in Chapter 5. The third screening step is to conduct NLTHA to yield maximum

94

seismic responses for each combination of parameters. The last step is to generate Pareto optimal

plots of potential parameters via a least-squares regression analysis using JMP program which is

reliability analysis platforms integrated with RSMs. Finally, the most significant curved steel

bridge parameters that impact seismic response are identified. The following sections include

more detailed descriptions of the parameter screening setup and data analysis procedures.

Figure 6-1: Screening Procedure.

95

6.1 Screening Experiments for Inputs

Screening of potential macro- and micro-level parameters for their relative significance is

performed using an experimental design (Figure 6-1). In particular, this step is intended to

generate the screening experimental design using JMP program. The initial screening

experiments consist of only inputs, while the complete screening experiments contain the inputs

and the corresponding outputs for the potential parameters.

A conventional full factorial experiment is initially selected in conjunction with a two-

level design, with each parameter considered at two values (upper and lower bounds) traditionally

noted as (+) and (-), respectively. For each combination of parameter levels, the experiment, or in

this case the bridge, is analyzed and responses of interest are monitored. Setups such as these

require the running of 2k experiments, where k is the number of parameters being considered.

Table 6-1 details this concept using two representative parameters, A and B.

Table 6-1: Sample Full Factorial Experimental Design.

Parameters Run A B

1 - - 2 - + 3 + - 4 + +

Investigating all possible combinations of factor levels for each model allows for a

detailed exploration of the effects of each factor. It also allows for study of interaction effects

that may or may not exist between parameters. Although the use of 2k parameters is an ideal

setup, it becomes computationally expensive when a large number of parameters must be

considered. As presented in Table 6-2, the 13 potential curved steel bridge parameters require the

running of 132 , or 8192 analyses. This number of analyses is not feasible when a suite of ground

motions needs to be applied to each combination.

96

Table 6-2: Selected Horizontally Curved Steel I-Girder Bridge Parameters.

Parameter category Parameters

X1 Number of span X2 Maximum span length, m (ft) X3 Deck width, m (ft) X4 Maximum column height, m (ft) X5 Radius of curvature, m (ft) X6 Girder spacing, m (ft)

Macro parameters

X7 Cross-frame spacing, m (ft) X8 Damping Ratio X9 Concrete Compressive Strength, Mpa (ksi) X10 Concrete Tensile Strength, Mpa (ksi) X11 Concrete Young's Modulus, Mpa (ksi) X12 Steel Young's Molulus, Mpa (ksi)

Micro parameters

X13 Steel Yield Strength, Mpa (ksi)

To efficiently reduce computation time, Plackett-Burman design (SAS, 2008) is

employed in this screening study. As mentioned in Chapter 3, for Plackett-Burman design (PBD)

one specific family of fractional factorial designs is frequently used for capturing optimal

parameters (Wu and Hamada, 2000). When PBD is used for screening processes, interactions are

considered negligible to avoid unnecessary design spaces requiring time-consuming analyses and

only main predictor effects are considered. In addition, a two-level design uses minima and

maxima of the 13 input variables to form the Plackett-Burman space table that can be obtained

from Experiments Planning, Analysis and Parameter Design Optimization (Wu and Hamada,

2000). As a result of the consideration for the main predictor effects and the use of PBD space

table, the twenty PBD are generated for 13 potential parameters (Appendix A). The twenty

combinations represent the representative bridges that can be selected from the 99 bridge

inventory in accordance with PBD pattern. Table 6-3 displays the selected input parameters from

Chapter 5 and their corresponding upper and lower level values. Again, there are a total of

twenty experimental cases by using the generated PBD matrix implemented to JMP program

(SAS, 2008) for the 13 input parameters as shown in Appendix A. Values of -1 and +1 denote

97

the minima and maxima of the input variables, respectively. Twenty detailed bridge analysis

models are generated corresponding to the combinations of input parameters outlined in the PBD

table (Appendix A). For example, bridge pattern 1 is constructed by using maximum values for

radius of curvature (X5), damping ratio (X8), concrete Young’s Modulus (X11), steel reinforcing

bar Young’s Modulus (X12) and steel reinforcing bar yield strength (X13) while using minimum

values for span number (X1), maximum span length (X2), deck width (X3), maximum column

height (X4), girder spacing (X6), cross-frame spacing (X7), concrete compressive strength (X9)

and concrete tensile strength (X10).

For the RSM construction case in Chapter 7, however, RSM construction will consider

both interactions and main effects for DOE using CCD (Chapter 3) because both effects would

significantly impact the final fragility curves. Seismic responses corresponding to each bridge

pattern outlined in Appendix A is investigated by performing twenty analyses using the 3-D

analytical models and synthetic ground motions discussed in Section 5.3.

Table 6-3: Two Level Predictor Parameters for Plackett-Burman Experimental Design.

Lower level Upper level Parameter

category Parameters -1 1

X1 Number of Span 1 3 X2 Maximum Span Length, m (ft) 15.2 (50) 57.6 (300) X3 Deck Width, m (ft) 8.5 (27.9) 17.3 (56.8) X4 Maximum Column Height, m (ft) 10 (32.8) 32.3 (106) X5 Radius of Curvature, m (ft) 240 (787.4) 3492(11456.7) X6 Girder Spacing, m (ft) 1.46 (4.8) 3.23 (10.6)

Macro parameters

X7 Cross-Frame Spacing, m (ft) 2.7 (8.8) 7.31 (24) X8 Damping Ratio 0.2 0.8 X9 Concrete Compressive Strength, MPa (ksi) 20.7(3) 34.1(5) X10 Concrete Tensile Strength, MPa (ksi) 1.7(0.24) 2.7(0.4) X11 Concrete Young's Modulus, MPa (ksi) 17(2500) 31(4500) X12 Steel reinforced bar Young's Modulus, MPa (ksi) 192920(28000) 206700(30000)

Micro parameters

X13 Steel reinforced bar Yield Strength, MPa (ksi) 289.4 (42) 345 (50)

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6.2 Screen Experiments for Outputs: Seismic Response

Earthquake analyses using NLTHA are discussed herein for the twenty bridge patterns

containing combinations of the previously discussed 13 curved bridge parameters. The seismic

response for each of the twenty patterns shown in Table A-1 in Appendix A is obtained from the

NLTHA and these outputs are used to populate the PBD table shown in Table A-2 to A-4 in

Appendix A.

OpenSees was used to run NLTHA for each of the bridge patterns following the

procedure discussed in Chapter 4. The twenty selected bridges were subjected to a suite of 0.2g

synthetic ground motions, the mean PGA reported for the region from past studies (Rix and

Fernandez-Leon, 2004). These motions were applied to the pier footings and the abutments,

simultaneously. Once these analyses were carried out critical seismic response magnitudes were

extracted.

As stated in Chapter 4, the synthetic ground motions were simultaneously applied to each

bridge model in OpenSees in the global longitudinal direction initially and then the global

transverse direction. However, seismic responses (i.e., steel bearings, abutments, and column

curvatures) were captured in tangential and radial directions relative to the curved bridge

superstructure. All nodes representing steel bearings, abutments, and columns in each bridge

model in OpenSees were monitored and the maximum seismic responses at each component were

captured. In particular, the bearing and abutment were separately modeled with ZeroLength

available in OpenSees as shown in Figure 6-2. The bearing OpenSees model represented actual

bearing using bearing force-displacement relationship in presented Chapter 4, while the abutment

OpenSees model mimics the actual behavior of abutments using the 3-D analytical modeling

approach presented in Chapter 4. Therefore, the maximum seismic response for each bearing and

abutment were simultaneously monitored, but the values were separately extracted since those

seismic behaviors were different. In the past, the maximum deformation at the bearings and

99

abutments, as well as the curvature ductility of columns have been important indicators of

earthquake damage to bridges and have been used as vital output for fragility curve development

(Mander et al., 1996; Hwang et al., 2000; Choi, 2000; Nielson, 2006; Padgett, 2007). Also,

qualitative damage levels related to these variables have been developed by FEMA (FEMA,

2003) and by Nielson (2006).

Girder Node

Abutment & Pile Node

Deck

Bearing ZeroLength Element Model

Abutment ZeroLength Element Model Bearing Node

Girder Element

Pile ZeroLength Element Model

Figure 6-2: Specified Analytical Model between Bearing and Abutment.

By capturing the seismic responses for twenty horizontally curved steel I-girder bridges,

those values (outputs) corresponding to the 13 input variables are presented in Table A-2 and A-4

in Appendix A. Using the complete PBD tables, the most significant horizontally curved steel I-

girder parameters are identified in the following section.

6.3 Parameter Screening

In this section, the most significant parameters which importantly influence seismic

responses of the bridge family are determined by performing the statistical analysis using the

complete PBDs presented in Table A-2 to A-4 in Appendix A. This procedure is included in the

final stages in Figure 6-1.

100

After establishing the complete PBDs (Section 6.1 and Appendix A), a parameter

screening process is employed to identify the contribution of each parameter to curved steel

bridge system response. One of the popular screening methods is to systematically increment

each input variable and compute the seismic responses (e.g., maximum deformation at bearing,

etc) for each case. This approach was used for the current study with the assistance of a statistical

program, JMP (SAS, 2008). The rank-ordered output yields what is often called a Pareto optimal

solution (Montgomery, 1997). The Pareto optimal solution is a means to visually determine the

most significant contributors to a response. The Pareto optimal plot provides the individual

influence of each PBD variable on the response using horizontal bars and the cumulative effect of

the variable group is plotted using a line and this line is used to determine the most significant

parameters for seismic response.

To complete PBDs in Table A-2 and A-4 in Appendix A, again, a selected seismic

response output variable (y) is used to represent each seismic response of importance, such as

tangential maximum deformation at the abutment, radial maximum deformation at abutment, and

so on. The output variables (y) were obtained from NLTHA (Chapter 6.2) for each horizontally

curved steel I-girder bridge model from the selected inventory in Chapter 5. By performing the

statistical analysis using JMP program based on the complete PBDs and a first order regression

model, Pareto plots for each seismic response are generated. Note that the first order regression

model is implemented into JMP program. The first order regression model (Equation 6.1) is

generated for identifying the influence each input parameter has on the selected output

calculation:

131322110 xβxβxββy +⋅⋅⋅+++= (6.1)

101

where y = output variable, or maximum radial and tangential deformations at

bearing and abutments (mm), and column curvature;

xi = potential horizontally curved steel bridge parameters; and

βi = coefficient estimates representing the main effect of xi

The resulting response values are scaled by multiplying results by the standard deviation

of each input parameter so that the estimates can be reasonably compared with each other. The

Pareto plot then details the composition of an absolute value of each scaled estimate normalized

to the sum, which allows for generation of a cumulative distribution of the highest to the lowest

scaled estimate. Parameters with the highest scaled estimate are the most influential.

Figures 6-3 to 6-7 display the statistical results for the twenty bridges subjected to the 0.2

g synthetic ground motions. Figure 6-3 (a) to 6-7 (a) are the Pareto plot of the scaled estimates

for each input parameter. The black solid curve in the figures indicates the cumulative

contribution to the overall response while the individual contribution or scaled estimate is

indicated by the horizontal bar. The blue dotted line is the 80% contribution line for seismic

response, while red four vertical dotted lines in the figures indicate the cumulative probability

corresponding to each input parameter. In general, when statistically determining significant

seismic parameters using the Pareto plot those that contribute to more than 80% of the seismic

response are deemed significant (Towashiraporn, 2004; SAS, 2008).

Another way to obtain a qualitative impression of the significance of each input variable

is through the prediction profiler plots shown in Figure 6-3 (b) to 6-7 (b). They are plots between

an output variable (i.e., maximum deformation at bearings and abutments, or column curvature

ductility) and each of the input variables (x) while other inputs are held at their mean values. The

influence of each input variable can be inferred from the slope of the resulting line. A steeper

102

line over the variable range means that the input variable is more influential on an output than the

others.

As discussed in Chapter 6.1, the input parameters in the figures are: number of spans

(X1); maximum span length (X2); deck width (X3); maximum column height (X4); radius of

curvature (X5); girder spacing (X6); cross-frame spacing (X7); damping ratio (X8); concrete

compressive strength (X9); concrete tensile strength (X10); concrete Young’s Modulus (X11);

steel reinforcing bar Young’s Modulus (X12) and steel reinforcing bar yield strength (X13). It is

apparent from Figure 6-3 (a) and (b) that, for maximum tangential deformation at the abutments,

the first seven parameters (X2, X7, X1, X6, X5, and X11) contribute to almost 80% of the overall

response. For maximum radial deformation at the abutments, Figure 6-4 (a) and (b) show that the

first six parameters (X5, X1, X2, X6, X8, and X13) influence almost 80% of the overall response.

Figure 6-5 (a) and (b) show that, for column curvature ductility, the first six parameters

(X1, X2, X3, X8, X7, and X6) contribute to almost 80% of the overall response. Figure 6-6 (a)

and (b) show that the first six parameters (X1, X2, X5, X8, X7, and X13) contribute to nearly

80% of the overall tangential deformation at the bearings. Similarly, from Figure 6-7 (a) and (b)

the results of maximum radial deformation at the bearings are dominated by the first four

parameters (X5, X1, X2, and X3).

The results show that, although the relative significance of the various parameters may

change for a given output variable; in general the same parameters remain the most significant.

Therefore, the consistently significant parameters (i.e. the number of spans, radius of curvature,

maximum span length, girder spacing, and cross-frame spacing) were selected as the most

significant input parameters for the RSMs.

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(a) Pareto Plot of Input Parameters

(b) Prediction Profiler

Figure 6-3: Results of Maximum Tangential Deformation at Abutments.

103

104



Figure 6-4: Results of Maximum Radial Deformation at Abutments.

104

105



Figure 6-5: Results of Maximum Column Curvature Ductility.

105

106



Figure 6-6: Results of Maximum Tangential Deformation at Bearings. 106

107



Figure 6-7: Results of Maximum Radial Deformation at Bearings.

107

108

6.4 Conclusions

In this chapter, a screening or sensitivity study of the potentially significant parameters

which influence horizontally curved steel I-girder bridge seismic response is described in detail.

The screening procedure helped identify the most significant seismic parameters.

Table 6-4 lists the most significant horizontally curved steel I-girder bridge parameters

found from the screening. The parameters are listed in order of significance, where ranking is

determined from the number of response measures for which it was shown to be significant.

Table 6-4: Summary of Most Significant Horizontally Curved Steel I-Girder Bridge Parameters.

Monitored Bridge Component Rank 1 Rank 2 Rank 3 Rank 4 Rank 5 Rank 6 Tangential Deformation (Abutment) X2 X7 X1 X6 X5 X11 Radial Deformation (Abutment) X5 X1 X2 X6 X8 X13 Column Curvature Ductility X1 X2 X3 X8 X7 X6 Tangential Deformation (Bearing) X1 X2 X5 X8 X7 X13 Radial Deformation (Bearing) X5 X1 X2 X3 N/A N/A

The five most significant parameters have been determined to be the number of spans

(X1), radius of curvature (X5), maximum span length (X2), cross-frame spacing (X7), and girder

spacing (X6). These parameters can be used as the optimal input parameters in the RSMs.

Chapter 7

SEISMIC FRAGILITY CURVES FOR HORIZONTALLY CURVED STEEL BRIDGES

For construction of seismic fragility curves, RSMs are generated (see Figure 3-4). The

RSM function is created by regression analysis and the function consists of parameters that have

significant influence on seismic response of horizontally curved steel I-girder bridges. The most

significant parameters (e.g., number of span, radius of curvature, etc.) were determined by

performing the sensitivity study in Chapter 6. For construction of the seismic fragility curves, the

RSMs need to be integrated into a Monte Carlo simulation to not only assist with accounting for

significant parameter uncertainty, but also to assist with evaluating seismic performance at a

given performance level (e.g., radial and tangential maximum deformations at abutments and

bearing, and column curvature, etc.) for a family of horizontally curved steel I-girder bridges.

The performance level used to establish the relationship between seismic quantitative

performance levels obtained from previous research (Maroney et al., 1994; Mander, 1996; Hwang

et al., 2000; Choi, 2004; Nielson, 2006; Padgett, 2007) and existing definitions of qualitative

seismic performance levels obtained from FEMA HAZUZ-MH (FEMA, 2003). Each simulation

can be carried out using inputs selected from significant parameter probability distribution

functions from Chapter 5 and 6. The probability of the chosen response exceeding a certain

performance level can be extracted from the distribution of the simulation results. This

probability value is conditioned on a specific earthquake intensity level and represents one point

on a fragility curve. Repetition of the process over the different levels of earthquake intensity

provides exceedance probability values at other intensity levels, and the entire fragility curve can

be created. In addition, the fragility curves are generated at four different qualitative performance

110

levels from the FEMA Hazards U.S. Multi-Hazard (HAZUS-MH) loss assessment package (i.e.

slight, moderate, extensive, and complete).

7.1 RSMs Construction

This section is intended to develop the damage prediction RSM functions specific to the

examined horizontally curved steel bridge family at each prescribed synthetic earthquake

intensity level (i.e., peak ground acceleration). The RSMs are formulated from the most

significant horizontally curved steel I-girder bridge parameters determined using the screening

parameters in Chapter 6. The RSM construction process is shown in detail in Figure 7-1.

Similar to the screening procedure presented in Chapter 6, a combination of experimental

design and NLTHAs are used for generating RSMs. To consider not only single variable effects,

but also the effects of interaction of the most significant parameters, more experimental design

spaces are necessary. Therefore, the RSM construction considers both interactions and the main

effects for DOE using a three level CCD (Chapter 3) because both could significantly impact the

final fragility curves. Therefore, the three level CCD is selected for RSM construction with each

parameter considered at three values (upper, center and lower bounds) traditionally noted as (+1),

(0) and (-1), respectively. Table 7-1 displays the five most significant parameters identified

using the Pareto optimal and prediction profiler plots from Chapter 6 and their corresponding

upper, center and lower level values. These parameters are used in combination with an

earthquake intensity level parameter to formulate the RSMs. Peak ground accelerations (PGAs)

are used as the representative earthquake intensity indicator for fragility development (Choi,

2004; Nielson, 2006; Padgett, 2007).

It has also been shown that, when generating fragility curves using RSM functions, it is

important to consider the inherent uncertainties of earthquake ground motions (Choi, 2004;

Nielson, 2006; Padgett, 2007). As a result, it can be necessary to consider many potential ground

111

motion possibilities. For this study, an ensemble of synthetic ground motions used for a

generation of fragility curves in previous work (Choi, 2004; Nielson, 2006; Padgett, 2007) was

applied to the pier footings and the abutments simultaneously.

Figure 7-1: RSMs Construction for Horizontally Curved Steel Bridges.

112

Table 7-1: Screened Most Significant Parameters for RSMs models.

Lower Level Center Level Upper Level

Most Significant Parameters

-1 0 1

X1 Number of spans 1 2 3 X2 Maximum span length, m (ft) 15.2 (50.3) 53.3(174.9) 91.4 (300) X5 Radius of curvature, m (ft) 240 (787.4) 1866 (6122.0) 3492(11456.7) X6 Girder spacing, m (ft) 1.46 (4.8) 2.35(7.7) 3.23 (10.6) X7 Cross-frame spacing, m (ft) 2.7 (8.8) 5.005(16.4) 7.31 (24) Xeq Peak Ground Acceleration, g 0.1 0.55 1

As mentioned above, the randomly extracted suite of synthetic ground motions are

applied in OpenSees to each curved bridge containing the five most significant parameters. The

suite of synthetic ground motions was scaled to have an average PGA of 0.1g, 0.55g and 1.0g to

examine seismic response for a broader range of earthquake scenarios. The three level CCD

spaces for the five most significant parameters (X1, X2, X5, X6, and X7) and one earthquake

intensity level parameter (Xeq) are shown in Table B-1 in Appendix B. The 45 bridge models are

generated corresponding to the combinations of the most significant parameters and earthquake

intensity level parameters outlined in the CCD table by running CCD experimental design in JMP

program (SAS, 2008).

It is worth demonstrating how a response computation is achieved for each of the

experimental design combinations. A specific combination (Pattern 1 in Table B-1 in Appendix

B) contains the following input variables: X1=-1, X2=1, X5=-1, X6=-1, X7=-1, and Xeq=-1. The

variable set is interpreted as a simple span curved steel bridge with a maximum span length of

91.4m, a radius of curvature of 240m, a girder spacing of 1.46m, a cross-frame spacing of 2.7m,

and a suite of synthetic ground motions with an average PGA of 0.1g. All other curved steel

bridge parameters that are deemed less influential are fixed at their mean values. Based on the

experimental CCD table, maximum concrete column curvature and maximum radial and

tangential deformations at the bearings and abutments are computed from the NLTHAs. The

mean and standard deviation of these variables are then computed and recorded in the

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experimental CCD tables (Table B-2 to B-4 in Appendix B). Appendix B includes the mean and

standard deviation of the maximum tangential and radial deformations extracted from the

abutments of each bridge pattern, the maximum curvature ductility extracted from the pier

columns for each bridge pattern, the maximum tangential and radial deformations captured from

the bearings for each bridge pattern. Similar to obtaining those responses in the screening

analysis, all nodes representing bearings, abutments, and columns in each bridge model in

OpenSees were monitored and the maximum seismic responses at each component were captured.

These tables are used to generate the RSMs using statistical analyses.

By using a least-square regression analysis of each CCD table, RSM functions consisting

of the five most significant parameters in association with given PGAs is developed. The RSM

functions are generated for each seismic response (e.g., maximum tangential deformation at

bearing, etc.). They can be shown symbolically and mathematically in the following equations.

The equations are composed of the two response surface models, μy and σy . In these models,

the first term predicts an expected or a mean value of the critical seismic response due to a suite

of synthetic ground motions, while the second term represents earthquake-to-earthquake

dispersion in response computation and consequently incorporates randomness in earthquake

excitations. Assuming a normal distribution, the RSM can be mathematically expressed as:

[ ]iσiμi yNyy || ˆ,0ˆˆ += (7.1)

for i = 1 … 5

where 1y = maximum tangential deformation at an abutment;

2y = maximum radial deformation at abutment;

3y = maximum curvature ductility of column;

114

4y = maximum tangential deformation at bearing; and

5y = maximum radial deformation at bearing

1|ˆ μy = mean value of maximum tangential deformation at an abutment;

2|ˆ μy = mean value of maximum radial deformation at abutment;

3|ˆ μy = mean value of maximum curvature ductility of column;

4|ˆ μy = mean value of maximum tangential deformation at bearing; and

5|ˆ μy = mean value of maximum radial deformation at bearing

[ ]1|ˆ,0 σyN = normal distribution plus one standard deviation for maximum tangential

deformation at an abutment;

[ ]2|ˆ,0 σyN = normal distribution plus one standard deviation for maximum radial

deformation at abutment;

[ ]3|ˆ,0 σyN = normal distribution plus one standard deviation for maximum curvature

ductility of column;

[ ]4|ˆ,0 σyN = normal distribution plus one standard deviation for maximum tangential

deformation at bearing; and

[ ]5|ˆ,0 σyN = normal distribution plus one standard deviation for maximum radial

deformation at bearing

As a result of the least-square regression analysis of each CCD table, RSM polynomial

models for each seismic response were derived. The RSMs for the mean and standard deviation

115

of the maximum tangential deformation at an abutment are represented in Equation 7.2, and

Equation 7.3, respectively. Note that the RSM equation incorporates variables representing the

number of spans (X1), maximum span length (X2), radius of curvature (X5), cross-frame spacing

(X7), girder spacing (X6) and PGA (Xeq).

+++−++=− 76521| 307.0633.0324.0916.0555.2964.36ˆ xxxxxy abutmenttangentialμ

+−−++ 512221

21 422.0820.3163.0470.1881.27 xxxxxxxeq

−−−−+ 6562612552 560.0041.0509.0680.0412.0 xxxxxxxxx (7.2)

++−++ 7675727126 766.0464.0624.0091.0976.1 xxxxxxxxx

++−++ eqeqeqeq xxxxxxxxx 652127 396.0636.0607.0752.1689.0

21 783.0185.0 eqeq xxx −

and

+++−++=− 76521| 059.0111.0007.0166.0432.0780.8ˆ xxxxxy abutmenttangentialσ

+−−++ 512221

21 053.0865.0023.0397.0657.3 xxxxxxxeq

−−−−+ 6562612552 089.0012.0050.0208.0070.0 xxxxxxxxx (7.3)

++−++ 7675727126 121.0053.0092.0028.0425.0 xxxxxxxxx


21 289.3015.0 eqeq xxx −

The resulting seismic response prediction model for the maximum abutment tangential

deformation mean is derived by evaluating the above functions at each parameter level (i.e. -1, 0,

and +1). 2D prediction plots for the maximum tangential deformation mean at an abutment are

shown in Figure 7.2. In the prediction plots, the seismic response is plotted against each of the

116

input parameters, while other input variables are fixed at their center point values. The purpose

of this plot is to observe the trend of the response due to effects from varied input parameters and

fixed control parameters. The influence of each input parameter can be inferred from the

steepness of the red line (Figure 7-2) as a steeper line over the parameter range means that the

input parameter is more influential on maximum abutment tangential deformation. It appears that

Xeq (PGA) and X1 (number of spans) affected the seismic response more than the other

parameters.

Figure 7-2: Responses Surface Plots for Mean of Maximum Tangential Deformation at Abutment of Horizontally Curved Steel Bridges.

117

The RSMs for the mean and standard deviation of abutment maximum radial

deformations at are presented in Equation 7.4, and Equation 7.5, respectively. Note that the RSM

equation incorporates variables representing the number of spans (X1), maximum span length

(X2), radius of curvature (X5), cross-frame spacing (X7), girder spacing (X6) and PGA (Xeq).

+++−−+=− 76521| 263.0018.0361.1128.0574.0521.11ˆ xxxxxy abutmentradialμ

++++− 512221

21 139.0173.0233.0071.0896.14 xxxxxxxeq

+++++ 6562612552 250.0563.0068.0875.0036.0 xxxxxxxxx (7.4)

++++− 7675727126 628.0068.0544.0261.0222.0 xxxxxxxxx

+−−++ eqeqeqeq xxxxxxxxx 652127 149.0996.0161.0808.0756.0

21 684.4012.0 eqeq xxx +

and

+++−−+=− 76521| 079.0009.0349.0049.0117.0870.2ˆ xxxxxy abutmentradialσ

−+−++ 512221

21 011.0006.0059.0007.0416.3 xxxxxxxeq

+++++ 6562612552 073.0141.0031.0245.0002.0 xxxxxxxxx (7.5)

++−+− 7675727126 145.0010.0105.0094.0003.0 xxxxxxxxx

−−−++ eqeqeqeq xxxxxxxxx 652127 049.0208.0077.0206.0215.0

21 279.1027.0 eqeq xxx +

The seismic response prediction model for mean maximum abutment radial deformation

is derived by evaluating the above functions at each parameter level = -1, 0, and +1. 2D

prediction plots in terms of mean of the maximum radial deformation are shown in Figure 7.3.

118

Similar to the previous interpretation of the prediction plots, therefore, the plots (Figure 7.3)

indicate that X5 (radius of curvature) has more impact on the seismic responses than the other

parameters, except for Xeq (PGA).

Figure 7-3: Responses Surface Plots for Mean of Maximum Radial Deformation at Abutment of Horizontally Curved Steel Bridges.

The RSMs function for the mean and the standard deviation of curvature ductility at

column or abutment are represented in Equation 7.6 and Equation 7.7 below. Note that the RSM

119

equation incorporates variables representing the number of spans (X1), maximum span length

(X2), radius of curvature (X5), cross-frame spacing (X7), girder spacing (X6) and PGA (Xeq).

+++−++= 76521| 011.0099.0067.0094.0331.0060.4ˆ xxxxxy ductilityμ

+−−++ 512221

21 039.0412.0039.0169.0823.2 xxxxxxxeq

−−−−+ 6562612552 094.0020.0060.0083.0113.0 xxxxxxxxx (7.6)

++−++ 7675727126 121.0043.0087.0021.0209.0 xxxxxxxxx


21 219.0049.0 eqeq xxx +

and

+−−+++= 76521| 016.0002.0006.0015.0028.0589.0ˆ xxxxxy ductilityσ

++−++ 512221

21 011.0066.0006.0017.0205.0 xxxxxxxeq

−−−−+ 6562612552 022.0032.0018.0005.0011.0 xxxxxxxxx (7.7)

+++++ 7675727126 005.0005.0005.0014.0037.0 xxxxxxxxx

++−−+ eqeqeqeq xxxxxxxxx 652127 026.0002.0008.0017.0005.0

21 016.0010.0 eqeq xxx −

The seismic response prediction model for mean of curvature ductility at columns or

abutments is derived by evaluating the above functions at each parameter level = -1, 0, and +1.

2D prediction plots in terms of mean of the curvature ductility are shown in Figure 7.4. Similar

120

to the previous interpretation of Figures 7-2 and 7-3, X1 (number of span) appears to have more

effect than the other parameters, again disregarding Xeq (PGA) as shown in Figure 7-4..

Figure 7-4: Responses Surface Plots for Mean of Curvature Ductility at Column or Abutment of Horizontally Curved Steel Bridges.

The RSMs function for the mean and the standard deviation of maximum tangential

deformation at a bearing are represented in Equations 7.8 and 7.9, respectively. Note that the

RSM equation incorporates variables representing the number of spans (X1), maximum span

121

length (X2), radius of curvature (X5), cross-frame spacing (X7), girder spacing (X6) and PGA

(Xeq).

+++−++=− 76521| 869.0801.1955.0614.2293.7348.103ˆ xxxxxy bearingtangentialμ

+−−++ 512221

21 216.1713.10463.0071.4077.80 xxxxxxxeq

−−−−+ 6562612552 607.1116.0462.1864.1177.1 xxxxxxxxx (7.8)

++−++ 7675727126 198.2334.1790.1260.0561.5 xxxxxxxxx


21 449.0529.0 eqeq xxx +

and

+++−++=− 76521| 154.0329.0209.0531.0406.1899.14ˆ xxxxxy bearingtangentialσ

+−−++ 512221

21 247.0601.1067.0522.0003.15 xxxxxxxeq

−−−−+ 6562612552 327.0023.0231.0205.0228.0 xxxxxxxxx (7.9)

++−++ 7675727126 444.0238.0354.0071.0861.0 xxxxxxxxx


21 948.5070.0 eqeq xxx +

The seismic response prediction model for mean of maximum tangential deformation at

bearing is derived by evaluating the above functions at each parameter level = -1, 0, and +1. 2D

prediction plots in terms of mean of the seismic response are shown in Figure 7.5. Similarly to

the curvature ductility, it appears that X1 (number of spans) has a larger effect on the seismic

responses than the other parameters, except for Xeq (Figure 7-5).

122

Figure 7-5: Responses Surface Plots for Mean of Maximum Tangential Deformation at Bearing of Horizontally Curved Steel Bridges.

The RSMs function for the mean and the standard deviation of maximum radial

deformation at bearing are represented in Equation 7.10, and Equation 7.11, respectively. Note

that the RSM equation incorporates variables representing the number of spans (X1), maximum

span length (X2), radius of curvature (X5), cross-frame spacing (X7), girder spacing (X6) and

PGA (Xeq).

123

+++−−+=− 76521| 748.0040.0889.3355.0638.1340.32ˆ xxxxxy bearingradialμ

++++− 512221

21 379.0424.0648.0178.0851.42 xxxxxxxeq

+++++ 6562612552 705.0607.1211.0480.2113.0 xxxxxxxxx (7.10)

++++− 7675727126 817.1178.0549.1737.0560.0 xxxxxxxxx


21 384.14019.0 eqeq xxx +

and

+++−−+=− 76521| 235.0011.0156.1075.0408.0192.6ˆ xxxxxy bearingradialσ

+−−++ 512221

21 012.0503.0128.0357.0910.12 xxxxxxxeq

−++++ 6562612552 194.0467.0142.0911.0037.0 xxxxxxxxx (7.11)

++−+− 7675727126 577.0062.0369.0253.0514.0 xxxxxxxxx

−−−++ eqeqeqeq xxxxxxxxx 652127 199.0758.0279.0682.0841.0

21 116.9114.0 eqeq xxx +

The seismic response prediction model for mean of maximum radial deformation at

bearing is derived by evaluating the above functions at each parameter level = -1, 0, and +1. 2D

prediction plots in terms of mean of the seismic response are shown in Figure 7.6. Similar to

mean of maximum radial deformation at abutment, it appears that X5 (radius of curvature) more

considerably affected the seismic responses than the others, except for Xeq (PGA) in Figure 7.6.

124

Figure 7-6: Responses Surface Plots for Mean of Maximum Radial Deformation at Bearing of Horizontally Curved Steel Bridges.

It is necessary to statistically evaluate overall RSM accuracy. Accuracy performance

measures relate to the ability of the RSM functions to reproduce the behavior for a family of

horizontally curved steel I-girder bridges over the considered range of parameter values.

To statistically evaluate RSM performance, predictions from the original 3-D

horizontally curved steel I-girder bridge model are compared with the derived RSM functions

(Cundy 2003; Towashiraporn 2004; Venter et al, 1997). We consider three measures of

125

metamodel performance, the mean absolute error (MAE), the maximum absolute error (MAX)

and the root mean square error (RMSE) (Venter et al, 1997) as discussed in Chapter 3. For the

purpose of these statistical tests, 45 combinations of input parameters are randomly generated.

Seismic response from the computer simulation and the continuous RSM function are calculated

for each combination and those statistical measures are computed as presented in Table 7-2. The

measures quantify the percentage error between the RSM prediction and the 3-D numerical

simulation in OpenSees. It can be seen that the level of MAE and RMSE in the RSM model is

reasonably low while the MAX percentages are relatively high to the MAE and RMSE

percentages. This is deemed acceptable because reasonably low MAE and RMSE indicate a

strong association between the actual and the predicted responses. Even though MAX

percentages are relatively high, it means that some extreme errors occur. Because the mean

RSMs with one standard deviation can cover extreme error, this approach is deemed agreeable,

again.

Table 7-2: Statistical Performance Measures for RSMs.

Statistical Error Measure Curved Bridge Components MAE(%) MAX(%) RMSE(%)

Tangential Deformation (Abutment) 15.2 31.7 11.3

Radial Deformation (Abutment) 14.5 34.5 5.4

Curvature Ductility 12.7 23.5 6.5

Tangential Deformation (Bearing) 15.7 33.2 5.9

Radial Deformation (Bearing) 25.8 56.5 5.4

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7.2 Seismic Performance Levels

This section outlines the procedure used to establish the relationship between seismic

quantitative performance levels obtained from previous research (Maroney et al., 1994; Mander,

1996; Hwang et al., 2000; Choi, 2004; Nielson, 2006; Padgett, 2007) and existing definitions of

qualitative seismic performance levels. Vulnerability is preliminarily assessed qualitatively by

applying FEMA HAZUZ-MH (FEMA, 2003) loss assessment criteria, developed for straight

bridges but applied here as an initial assessment tool. The qualitative description of the four

damage states (slight, moderate, extensive and complete) is given in Table 7-3. The next task

was to assign a quantitative measure to each of the performance levels for each of the bridge

components.

Table 7-3: HAZUS Qualitative Performance Levels (FEMA, 2003).

Performance

Level Description

Slight

Minor cracking and spalling to the abutment, cracks in shear keys at abutments, minor spalling and cracks at hinges, minor spalling at the column (damage requires no more than cosmetic repair) or minor cracking to the deck.

Moderate

Any column experiencing moderate (shear cracks) cracking and spalling (column structurally still sound), moderate movement of the abutment (<2inch), extensive cracking and spalling of shear keys, any connection having cracked shear keys or bent bolts, keeper bar failure without unseating, rocker bearing failure or moderate settlement of the approach.

Extensive

Any column degrading without collapse - shear failure - (column structurally unsafe), significant residual movement at connections, or major settlement approach, vertical offset of the abutment, differential settlement at connections, shear key failure at abutments.

Complete Any column collapsing and connection losing all bearing support, which may lead to imminent deck collapse, tilting of substructure due to foundation failure.

Quantitative performance levels are related to selected response aspects, such as

deformations at the bearing and abutment and column curvature. The quantitative performance

127

levels were developed using an approach from previous research (Maroney et al., 1994; Mander,

1996; Hwang et al., 2000; Choi, 2004; Nielson, 2006; Padgett, 2007). This work focused on

experimental results and the interpretation of bridge component behavior to develop bridge

performance level quantities corresponding to qualitative information presented in Table 7-3.

The resulting quantitative limit states from Nielson (2005) for the bearings, abutments and

column curvature ductility are presented in Table 7-4. The table gives the median ( cS ) and

dispersion ( cβ ) for these components. In Table 7-4, Abutment-Passive has only two quantitative

performance levels (i.e., slight and moderate), while Abutment-Active has three quantitative

performance levels (i.e., slight, moderate and extensive).

Table 7-4: Performance Levels for Bridge Components (Nielson, 2005).

Slight Moderate Extensive Complete Component

cS cβ cS cβ cS cβ cS cβ

Column Curvature Ductility ( φμ ) 1.29 0.59 2.10 0.51 3.52 0.64 5.24 0.65

Steel Bearing Rocker-Long, mm (in)

37.4 (1.47)

0.6 (0.02)

104.2 (4.1)

0.55 (0.02)

136.1 (5.36)

0.59 (0.02)

186.6 (7.35)

0.65 (0.03)

Steel Bearing Rocker-Tran, mm (in)

6.0 (0.24)

0.25 (0.01)

20.0 (0.8)

0.25 (0.01)

40.0 (1.57)

0.47 (0.02)

186.6 (7.35)

0.65 (0.03)

Abutment-Passive, mm (in) 37.0 (1.47)

0.46 (0.02)

146.0 (5.75)

0.46 (0.02) N/A N/A N/A N/A

Abutment-Active, mm (in) 9.8 (0.39)

0.70 (0.03)

37.9 (1.5)

0.90 (0.04)

77.2 (3.04)

0.85 (0.03) N/A N/A

Abutment-Tran, mm (in) 9.8 (0.39)

0.70 (0.03)

37.9 (1.5)

0.90 (0.04)

77.2 (3.04)

0.85 (0.03) N/A N/A

Quantities were determined for horizontally curved steel I-girder bridges for bearing

movements, for abutment movements and for bending of the pier columns. For the bearings and

abutments, they focused on radial and tangential deformations. For the columns, they focused on

ductility demand based on a curvature ratio calculated using Equation 4.3 (Nielson, 2005).

128

Quantities given in the table were applied herein by correcting them for horizontal curvature by

performing inventory analysis. Sample resulting corrected values for a bridge with a mean

subtended angle of 10.5o as an example are given in Table 7-5. For quantitative performance

levels for the selected family of horizontally curved steel bridges each different value for all

considered bridge curvatures were used to evaluate the studied horizontally curved steel bridges.

Similar to Table 7-4, Abutment-Passive has only two quantitative performance levels, while

Abutment-Active has three quantitative performance levels.

Table 7-5: Performance Levels for Horizontally Curved Bridge.

Slight Moderate Extensive Complete Component Median Median Median Median

Column Curvature Ductility 1.31 2.14 3.58 5.33 Steel Bearing Tangential Deformation, mm (in) 38.0 (1.5) 106.0 (4.2) 138.4 (5.5) 189.8 (7.5) Steel Bearing Radial Deformation, mm (in) 6.8 (0.3) 19.1 (0.8) 24.9 (0.1) 34.1 (1.3) Abutment-Passive Tangential Deformation, mm (in) 37.6 (1.5) 148.5 (5.8) N/A N/A Abutment-Passive Radial Deformation, mm (in) 6.8 (0.3) 26.7 (1.1) N/A N/A Abutment-Active Tangential Deformation, mm (in) 10.0 (0.4) 38.5 (1.5) 78.5 (3.1) N/A Abutment-Active Radial Deformation, mm (in) 1.8 (0.07) 6.9 (0.3) 14.1 (0.6) N/A

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7.3 Seismic Fragility Curve Generation

This section presents the procedure for using the RSMs as a tool for seismic vulnerability

assessment and fragility curve construction. As discussed in Chapter 5, the bridge family used

for fragility curve construction is composed of 99 existing horizontally curved steel I-girder

bridges located in Pennsylvania, New York, and Maryland. All bridges differ with respect to the

five most significant parameters identified in Chapter 6 (e.g., number of spans, radius of

curvature, etc.) in some fashion. Using this information, Figure 7-7 details the seismic fragility

curve construction process.

As discussed before, RSMs employ Monte Carlo simulation to establish the probability

of seismic response parameter exceeding a prescribed value at a given earthquake intensity level.

As stated before, PGA will be used to establish the intensity levels. Monte Carlo simulations

using 10,000 trial-runs, deemed an efficient number to accurately estimate an exceedence

probability by Towashiraporn (2004), are carried out on the RSMs. The Monte Carlo simulations

employ the probability density functions (PDFs) determined from statistical inventory analysis in

Chapter 5. The PDFs are associated with the most significant parameters. The exceedance

probability at a certain performance level (e.g., curvature ductility, radial and tangential

deformation at bearing and abutment) can be computed from a suitably large number of outputs

obtained from these simulations. This yields a fragility value for a specific level of PGA. The

process is repeated for other synthetic ground acceleration record sets so that vulnerability for

various earthquake intensity levels can be plotted in the form of a fragility curve.

130

Figure 7-7: Development of Seismic Fragility Curves.

131

7.3.1 Seismic Fragility Curves of Bridge Component

Seismic fragility curves of bridge components are valuable to the extent that they help to

highlight their sensitivity to certain seismic demands. As a result, they assist with decision-

making when determining suitable retrofit/rehabilitation strategies.

To develop fragility curves, the seismic demands of each component are computed using

its RSMs. Each demand is evaluated against a corresponding performance level using cS ,

defined as the median value of an intensity measure for the chosen performance level from the

FEMA HAZUS-MH (FEMA, 2003) loss assessment package.

Monte Carlo simulations using 10,000 trial-runs, deemed an efficient number to

accurately estimate an exceedence probability by Towashiraporn (2004), were performed on the

RSMs associated with the most significant parameters. To appropriately compute seismic

demands using the RSM simulations, each PDF, having a mean and standard deviation

determined by a statistical inventory analysis, was applied to the corresponding RSMs. The PDF

plots for each of the most significant parameters identified in Chapter 5 are shown in Figure 7-8.

Based on these PDFs, a discrete distribution is used for the number of spans (X1), a normal

distribution is used for cross-frame spacing (X7) and lognormal distributions were used for the

remaining parameters.

Figure 7-8: Probability Density Functions for Significant Curved Bridge Parameters.

132

Seismic fragility curves are shown in Figures 7-9 to 15 for each bridge component using

the specific described procedure. To generate the curves, the Monte Carlo Simulation associated

with RSMs was again applied using Crystal Ball program. Figure 7-9 shows generated fragility

curves for curvature ductility of column. Fragility curves were generated for the slight, moderate,

extensive, and complete damage levels as outlined by FEMA (FEMA, 2003). It appears that the

fragility curve representing slight damage has an exceedence probability of 1 over all PGA levels.

This means that slight column damage could occur during very low earthquakes for PGAs of less

than 0.1 g. The fragility curve for moderate damage level has an exceedence probability of 1 for

PGAs between 0.3 g and 1 g, while the curve for extensive damage level has an exceedence

probability of 1 for PGAs between 0.55 g and 1 g. The fragility curve for complete damage level

has an exceedence probability of 1 for PGAs between 0.8 g and 1 g. Based on the evaluation of

these seismic fragility curves, it appears that slight to severe column damage could occur during

even low to moderate earthquakes for PGAs between 0.1 g and 0.3 g in the target regions (e.g.,

Pennsylvania, Maryland, and New York).

Fragility Curves for Column

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Figure 7-9: Seismic Fragility Curves of Column Curvature Ductility at Horizontally Curved Steel Bridges.

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Figure 7-10 shows generated fragility curves for tangential deformation at a bearing. The

fragility curves were generated using computed maximum tangential deformations at the bearing.

Figure 7-11 shows the fragility curves for radial deformation at a bearing. The fragility curves

are created using maximum radial deformations. Both fragility curves consist of the four damage

states detailed in Section 7.2. For tangential deformation at a bearing, it appears that the fragility

curve for slight damage has an exceedence probability of 1 for PGAs between 0.24 g and 1 g.

The fragility curve for moderate damage has an exceedence probability of 1 for PGAs between

0.6 g through 1 g, while the curve for extensive damage has an exceedence probability of 1 for

PGAs between 0.74 g and 1 g. The fragility curve for complete damage has an exceedence

probability of 1 for PGAs between 0.96 g and 1 g. For radial deformation at a bearing, the

fragility curve for slight damage has an exceedence probability of 1 for PGAs between 0.1 g and

1 g. The fragility curve for moderate damage level has an exceedence probability of 1 ranging for

PGAs between 0.25 g through 1 g, while the fragility curve for extensive damage level has an

exccedence probability of 1 for PGAs between 0.46 g and 1 g. The fragility curve for complete

damage level has an exceedence probability of 1 for PGAs between 0.7 g and 1 g. By comparing

two fragility curves, the radial deformation bearing appears to be the more vulnerable bearing

component at all damage levels.

Fragility curves for the abutment passive response in the tangential direction are shown in

Figure 7-12. Note that the passive response is partially provided by the soil and partially

provided by the piles. The fragility curves were generated using the computed maximum

tangential deformation at abutment and the two performance levels listed in Table 7-5. Figure 7-

13 shows fragility curves for the abutment passive response in the radial direction. The fragility

curves were created using maximum radial deformation at the abutments and the two

performance levels presented in Table 7-5. Since for the abutment only two performance levels

exist as shown in Table 7-5, resulting fragility curves in Figure 7-12 and 13 consist of two

performance levels. For the passive tangential abutment deformations, Figure 7-12 shows that

134

the fragility curve for slight damage has an exceedence probability of 1 for PGAs between 0.6 g

and 1 g. The fragility curve for moderate damage level has an exceedence probability of 0 over

all PGA levels, which means that moderate damage may not occur in the examined region. For

passive radial abutment deformations, the fragility curve for slight damage has an exceedence

probability of 1 for PGAs between 0.4 g and 1 g as shown in Figure 7-13. The fragility curve for

moderate damage level has an exceedence probability of 1 for PGA over 1 g. By comparing the

two fragility curves, the radial passive response appears be more vulnerable over the slight and

moderate damage levels.

Fragility Curves for Bearing (Tangential)

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Figure 7-10: Seismic Fragility Curves for Bearing (Tangential Deformation) at Horizontally Curved Steel Bridges.

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Fragility Curves for Bearing (Radial)

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Figure 7-11: Seismic Fragility Curves for Bearing (Radial Deformation) at Horizontally Curved Steel Bridges.

Fragility Curves for Abutment (Passive-Tangential)

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Figure 7-12: Seismic Fragility Curves for Abutment (Passive-Tangential Deformation) at Horizontally Curved Steel Bridges.

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Fragility Curves for Abutment (Passive-Radial)

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Figure 7-13: Seismic Fragility Curves for Abutment (Passive-Radial Deformation) at Horizontally Curved Steel Bridges.

Figure 7-14 shows generated fragility curves for the abutment active response in

tangential direction. Note that the active response occurs as the abutment is pulled away from the

backfill. The fragility curves were generated using the computed maximum tangential

deformation at abutment and the three performance levels listed in Table 7-5. Figure 7-15 shows

generated fragility curves for the abutment active response in radial direction. The fragility

curves were created using maximum radial deformation at abutment and the three performance

levels states presented in Table 7-5. Since for the abutment only three performance levels exist as

shown in Table 7-5, resulting fragility curves in Figure 7-14 and 15 consist of three performance

levels. For the active tangential abutment deformations, Figure 7-14 shows that the fragility

curve for slight damage has an exceedence probability of 1 for PGAs between 0.2 g and 1 g. The

fragility curve for moderate damage level has an exceedence probability of 1 for PGAs between

0.65 g and 1 g. The fragility curve for the extensive damage level has an exceedence probability

of 0.25 at PGA of 1 g. For active radial abutment deformation, Figure 7-15 shows that the

fragility curve for slight damage has an exceedence probability of 1 for PGAs between 0.97 g and

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1 g. The fragility curve for moderate damage level has an exceedence probability of 1 for PGAs

between 0.45 g and 1 g, while the fragility curve for extensive level has an exceedence

probability of 1 for PGAs between 0.9 g and 1 g. By comparing the two fragility curves, the

radial active response appears be more vulnerable over the slight, moderate and extensive damage

levels.

In summary, the radial deformations at the bearings appear to be the most vulnerable

components for the studied family of curved steel bridges at all damage states. At the moderate

damage state the columns appear to be the most vulnerable component. The abutments appear to

be the least vulnerable components at all damage states.

Fragility Curves for Abutment (Active-Tangential)

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Figure 7-14: Seismic Fragility Curves for Abutment (Active-Tangential Deformation) at Horizontally Curved Steel Bridges.

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Fragility Curves for Abutment (Active-Radial)

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Figure 7-15: Seismic Fragility Curves for Abutment (Active-Radial Deformation) at Horizontally Curved Steel Bridges.

7.3.2 Holistic Seismic Fragility Curves

It is of interest to generate fragility curves for a complete bridge system (representing all

99 horizontally curved steel I-girder bridges that were examined) by combining the previously

generated fragility curves for use in seismic loss estimation packages (e.g., FEMA HAZUS-MH).

In addition, a holistic bridge fragility curve can depict a bridge’s vulnerability across regions. A

method for generation of this type of fragility using joint probability density functions (Joint

PDFs) is briefly presented and then holistic fragility curves are generated.

Holistic fragility curves are generated by convolving the Joint PDF computed from each

RSM model, which describes the failure domain for each damage state (Ang and Tang, 1975;

Nielson, 2005). This is again carried out numerically through Monte Carlo simulation to estimate

the probability of exceedance across a range of PGAs. Figure 7-16 demonstrates the intersection

of a bi-variate Joint PDF with its failure domain.

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Figure 7-16: Bi-variate Joint PDF Integrated Over System Failure Domains (Ang and Tang, 1975).

Monte Carlo simulation in association with RSMs is a process in which N random

samples are generated from the PDFs. This is done for both the seismic demand side and the

seismic capacity side. The paired realizations of these distributions are compared and evaluated

for failure. Tracking of the system failure is accomplished through the use of an indicator

function. The indicator function for the two dimensional case shown in Figure 7-16 is given in

Equation 7.12.

⎩⎨⎧

∉∈

=ijji

ijjiF Fxxif

FxxifI

),(0),(1

(7.12)

where xi and xj are realizations of the ith and jth distributions and Fij is defined by the ith and jth

performance levels.

The probability of being in the selected level at a given value of the intensity measure

][ PGAIMLSP = , is estimated by Equation 7.13.

140

N

IPGAIMLSP

N

iFi∑

=== 1][ (7.13)

where LS is a certain limit state, damage level or performance level of the bridge or bridge

component, IM is the ground motion intensity measure, and PGA is the realized condition of the

ground motion intensity measure.

This numerical integration scheme is carried out for a reasonable range of the selected IM.

The resulting probabilities are recognized as approximations to the CDF of the underlying

distribution. Therefore, a simple linear regression is carried out to estimate the two parameters of

the lognormal distribution.

By performing the above procedure with five parameters (e.g., column curvature ductility,

etc.) the exceedance probabilities can be computed and representative of estimates of the CDF for

the curved bridge family system fragility curves. Values for the lognormal fragility distribution

for each damage state are estimated through regression analysis. As a result of the application of

this method, holistic bridge fragility curves for the studied family are produced as shown in

Figure 7-17.

Similar to the fragility curves for each bridge component, the holistic fragility curves

were also generated for slight, moderate, extensive, and complete damage levels. It appears that

the holistic fragility curve representing slight damage for the complete bridge system has an

exceedence probability of 1 for PGAs between 0.2 g and 1 g. This means that slight damage for

the bridge system occurs during low earthquake for a PGA between 0.1 g and 0.2 g. The fragility

curve for moderate damage has an exceedence probability of 1 for PGAs between 0.5 through 1,

while the curve for extensive damage has an exceedence probability of 1 for PGAs between 0.8 g

and 1 g. The fragility curve for complete damage level has an exceedence probability of 1 at over

PGAs of 1 g. Based on the evaluation of the holistic seismic fragility curves, it appears that slight

141

to extensive damage for the complete bridge system (99 horizontally curved steel I-girder

bridges) could occur during even low to moderate earthquakes in the target regions (i.e.,

Pennsylvania, Maryland, New York) for PGAs between 0.1 g and 0.3 g.

Holistic Fragility Curves

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Figure 7-17: Holistic Seismic Fragility Curves of A Family of Horizontally Curved Steel I-

Girder Bridges.

7.3.3 Fragility Curve Case Studies

It is beneficial that the seismic vulnerability assessment methodology developed herein

be applied to specific horizontally curved steel bridges to assess its effectiveness and resulting

damage levels. The application of RSMs allows for this to be accomplished rapidly. To

demonstrate how holistic fragility curves could be generated for specific bridges, a group of ten

bridges is selected from the inventory used for the current study. Selected bridges are different in

basic properties and geometric configuration. Important bridge parameters are listed in Table 7-6.

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Table 7-6: Characteristics of Ten Existing Curved Steel Bridges.

Name Location Class Number of Span

Radius of Curvature,

m (ft)

Girder Spacing,

m (ft)

Cross-Frame

Spacing, m (ft)

Deck Width, m(ft)

Concrete Compressive

Strength, MPa (ksi)

B-1 MD Horizontally

Curved Steel Bridge

3 240.2 (787.9) 2.6 (8.5) 3.8

(12.4) 9.8

(32.0) 23.9(3.5)

B-2 MD Horizontally

Curved Steel Bridge

3 381.0 (1250.0) 2.6 (8.5) 4.6

(15.0) 12.8

(42.0) 23.9(3.5)

B-3 MD Horizontally

Curved Steel Bridge

3 1164.3 (3819.7) 2.7 (8.9) 7.4

(24.3) 12.8

(42.0) 23.9(3.5)

B-4 NY Horizontally

Curved Steel Bridge

2 349.3 (1145.9) 2.7 (8.9) 7.3

(24.0) 10.4

(34.0) 27.3(4.0)

B-5 NY Horizontally

Curved Steel Bridge

1 871.1 (2857.8) 3.0 (9.8) 6.8

(22.3) 13.4

(44.0) 27.3(4.0)

B-6 NY Horizontally

Curved Steel Bridge

2 1746.2 (5729.0) 2.7 (8.9) 6.2

(20.4) 12.8

(42.0) 27.3(4.0)

B-7 NY Horizontally

Curved Steel Bridge

3 630.3 (2068.0) 3.0 (9.8) 5.7

(18.6) 13.4

(44.0) 27.3(4.0)

B-8 PA Horizontally

Curved Steel Bridge

3 304.8 (1000) 2.4 (8.0) 7.0

(23.0) 16.2

(53.0) 23.9(3.5)

B-9 PA Horizontally

Curved Steel Bridge

1 3036.1 (9960.8) 2.6 (8.6) 7.6

(25.0) 33.7

(110.7) 23.9(3.5)

B-10 PA Horizontally

Curved Steel Bridge

3 3492.0 (11456.7) 1.5 (4.8) 7.3

(24.0) 6.9

(22.5) 23.9(3.5)

The holistic fragility curves for each specific bridge are generated based on the previous

procedures discussed in Section 7.3.2. Figure 7-18 shows fragility curves constructed for four

performance levels corresponding to slight, moderate, extensive and complete damage for Bridge

B-1.

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Fragility Curves for B-1

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Figure 7-18: Seismic Fragility Curves for B-1.

Fragility curves that are specific to the other nine curved steel bridges are depicted in

Figure 7-19 to Figure 7-27. In order to look at effects of each curved bridge parameter on the

holistic seismic fragility curves, a fragility-to-fragility comparison is carried out. For the effect of

cross-frame spacing, the fragility curves for Bridge B-1 (Figure 7-18) are compared to those

curves for Bridge B-8 (Figure 7-25) at each damage level. These bridges are compared because,

as shown in Table 7-6, there is a considerable difference for cross-frame spacing between Bridge

B-1 and B-8 while little difference exists for the other parameters. It appears that the cross-frame

spacing has some influence on the holistic fragilities for curved steel bridges at slight damage

level, but little influence on the fragilities at moderate, extensive and complete damage levels.

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For the effect of number or spans, fragility curves for B-3 (Figure 7-20) are compared to

those for B-5 (Figure 7-22) at all damage levels. It appears that number of spans has a significant

impact on the holistic fragilities for curved steel bridges at all damage levels.

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For the effect of radius of curvature, the fragility curves for B-5 are compared to those for

B-9. It is apparent that radius of curvature has a significant impact on holistic fragilities for

horizontally curved steel bridges at all damage levels. B-5, that is tightly curved, appears to be

more vulnerable to than B-9, which is slightly curved. When the fragility curves for B-4, which

is tightly curved, are compared to those for B-6, which is moderately curved, it shows similar

results. Meanwhile, through these comparisons it appears that the girder spacing has little

influence on holistic fragilities.


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In summary, it can be concluded that number of span and radius of curvature have

influence on fragilities for horizontally curved steel bridge at all damage levels based on the

above comparison evaluation between specific bridge fragilities. Also, the cross-frame spacing

have somewhat influence on fragilities for horizontally curved steel bridges at only slight damage

level, and the girder spacing has little influence on holistic fragilities.

149

7.4 Conclusions

In this Chapter, the methodology for seismic bridge fragility curve construction in

associated with RSMs is described. The RSMs consisting of the five most significant parameters

(e.g., number of span, radius of curvature, etc.) are identified for each bridge component (e.g.,

bearing deformation, etc.). The RSMs models are determined based on 45 combinations using

CCD and by running a least-square regression analysis. Seismic fragility curves are generated

utilizing the RSMs in conjunction with Monte Carlo simulations. The seismic fragility curves

address four performance levels (i.e., slight, moderate, extensive, and complete) provided by the

FEMA HAZUZ-MH (FEMA, 2003) seismic loss estimation program. The fragilities are

generated for bridge components, for complete curved steel bridge systems by combining the

bridge component fragilities, and for specific curved steel bridges obtained from the horizontally

curved steel bridge family.

Chapter 8

CONCLUSIONS, IMPACT AND FUTURE RESEARCH

This study developed fragility curves for a family of horizontally curved steel I-girder

bridges utilizing statistical analysis, RSMs and Monte Carlo simulation in conjunction with

numerical models. This Chapter summarizes important findings learned from the study,

contributions along with outlining future research required to continue advancing this topic.

8.1 Summary and Conclusions

The main propose of this study is to provide reliable fragility curves for a family of

horizontally curved steel I-girder bridges across the specific regions (i.e., Pennsylvania,

Maryland, New York) using RSMs methodology in conjunction with Monte Carlo simulation.

Prior to the generation of fragility curves, 3-D modeling for finding seismic demands of

the bridges was proposed in Chapter 4. The 3-D modeling was applied to a horizontally curved

steel I-girder bridge located in central Pennsylvania. The approach was developed using the

OpenSees program. One unique aspect of the modeling component was representing the

spherical bearings. The spherical bearing used for all curved girders was modeled to represent

their moment-rotational behavior in OpenSess using the program’s steel01 and hysteretic material

models. All numerical values used for the steel01 and hysteretic material models, were obtained

using a trial and error process with data supplied from work by Roeder et al. (1995) that

examined the bearings under cyclic loads. The analytical model provided reasonable

approximation of real nonlinear bearing behavior at 10,000 cycles. The models were also

validated with static experimental data from field testing of an in-service horizontally curved steel

bridge. Seismic demands for the bridge were also investigated analytically.

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To increase efficiency of the RSMs methodology, screening of the potential key

parameters that can influence seismic responses of horizontally curved steel bridges was carried

out in Chapter 6. The key predictors for assessment of the seismic structural response of

horizontally curved steel bridge were defined as mentioned in Chapter 5. The inventory study of

the horizontally curved bridge inventory for three states (i.e., Pennsylvania, Maryland, New

York) was used as Eastern and Northern United State region representatives for developing a

statistically significant family of bridges. The macro-parameter predictor variables included

geometric items (i.e., number of span, maximum span length, deck width, maximum column

height, radius of curvature, girder spacing, and cross-frame spacing), while micro-parameters

included dynamic structural parameters (i.e., damping) and material properties (i.e., concrete

compressive and tensile strength, concrete and steel reinforcement Young’s modulus, steel

reinforced bar yield strength). By performing the NLTHAs, seismic responses were obtained and

the screening analysis setup was accomplished using the results incorporated into the

experimental design. The five key parameters obtained from development off the screening

analysis included:

• Number of span

• Maximum span length

• Radius of curvature

• Girder spacing

• Cross-frame spacing

In Chapter 7, the seismic fragility curves were constructed using RSMs consisting of the

five key parameters in similar fashion to the screening analysis. The RSMs were generated by a

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regression analysis. For the construction of the seismic fragility curves, the RSMs was integrated

into Monte Carlo simulations to deal with key parameter uncertainty and to assist with evaluating

seismic performance at a given performance level (e.g., column ductility curvature, etc.) for

bridge components (bearing, column, and abutment), a family of horizontally curved steel bridges

(system), and specific bridge obtained from the bridge inventory. The fragility curves were

generated at four different performance levels (i.e., slight, moderate, extensive, and complete

damage classification levels) and with different levels of earthquake intensity.

Notable findings obtained from development of the fragility curves for the selected

bridge components (bearing, column, and abutment) included:

• Bearing radial deformations appeared to be the most susceptible component for

seismic loads at most damage states for the bridges that were studied. This means the

bearing radial deformations in curved bridges under PGAs between 0.1 g and 1 g were

more critical than other bridge components. Seismic rehabilitation that adequately

controls radial large deformations in horizontally curved steel bridges under low to high

earthquake zones (PGAs between 0.1g and 1 g) should be studied. For example,

reducing bearing vulnerability can be expected by replacing the existing spherical

bearings with less vulnerable bearings.

• Pier column curvature ductility appears to be the most vulnerable components at

the moderate damage state. This means that the fragility curve for the moderate damage

level has an exceedence probability of 1 for PGAs between 0.3 g and 1 g and the

probability was the highest among other components. To reduce column vulnerability in

moderate and high seismic zones (PGAs between 0.3 g and 1 g), increasing column

ductility by using column rehabilitation, such as steel jacketing, may be needed.

153

• Abutment passive and active tangential deformations appear to be the least

vulnerable components at the slight and moderate damage states. Although abutment

vulnerability were the least among the studied components, it is still possible that minor

cracking and spalling to abutments under passive and active tangential deformations

could occur.

The main finding from the holistic fragility curves that were developed was:

• Slight that means minor cracking and spalling to bearing, abutment and column

to extensive damage that means column degrading without collapse, and vertical offset of

the abutment could occur during low to moderate earthquakes in the target regions. In

order to reduce seismic vulnerability for slight and extensive damage levels, seismic

rehabilitation plans for controlling minor to major cracks for bearings, abutment and

column in the existing horizontally curved steel bridges may be required.

When the developed fragility curves were applied to ten randomly selected curved

bridges from the inventory that was studied, findings included:

• Span number and radius of curvature having the most influence on fragilities at

all damage levels.

• Cross-frame spacing having the most influence on fragilities at the slight damage

level.

• Girder spacing having little influence on fragilities.

154

Based on the above investigation, it could be recommended that attempts should be made

to avoid constructing curved bridges with many spans and tightly curved bridges if

seismic vulnerability in the considered regions becomes a concern. If those bridge types

are needed, a seismic vulnerability assessment should be carried out.

8.2 Impact

The key impact of this research is the capability to generate 3-D analytical simulation-

based seismic fragility curves of a family of horizontally curved steel bridges using the RSMs

methodology. This is because it provides primary decision-making information for seismic

retrofits/rehabilitations for such bridges. This framework also results in a number of key benefits

and contributions which include:

• Development of RSMs that are practical to compute seismic demands for

horizontally curved steel bridges due to quick computing time.

• Determination of key seismic parameters for a select horizontally curved steel

bridge family.

• Generation of seismic fragility curves for a horizontally curved steel bridge

family using RSMs methodology.

• Application of a preexisting infrastructure damage level approach to seismic risk

mitigation for the fragility curves for horizontally curved steel bridges.

155

8.3 Areas for Future Research

Work from the present study could be extended through additional research in the

following areas:

• Development of seismic fragility curves for a broader family of skewed and

curved, steel, I-girder, bridges.

• Examining other types of metamodels, such as a Kriging model that includes a

discontinuity slope to efficiently and accurately estimate seismic demands.

• Studying the effectiveness of seismic retrofits via the comparison of retrofitted

and original bridge structures.

• Examine the effects of soil liquefaction on seismic response and fragility curves

for horizontally curved steel bridges.

• Develop a relationship between bridge curved, steel, I-girder bridge damage and

economic loss and a means to aggregate individual bridge losses into a network.

• Extension of the bridge vulnerability that were examined to other hazards.

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Appendix A

PLACKETT-BURMAN DESIGN

Table A-1: Plackett-Burman Experimental Design Table for 13 Input Parameters.

Table A-2: PBD Table for Maximum Tangential and Radial Deformations at the Abutment.

Table A-3: PBD Table for Maximum Curvature Ductility. Table A-4: PBD Table for Maximum Tangential and Radial Deformations at the

Bearing.

167

Table A-1: Plackett-Burman Experimental Design Table for 13 Input Parameters.

Pattern X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13

1 -1 -1 -1 -1 1 -1 -1 1 -1 -1 1 1 1

2 -1 1 1 -1 1 -1 -1 1 1 1 1 -1 1

3 -1 1 -1 1 -1 -1 -1 1 1 -1 -1 1 -1

4 -1 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 -1

5 1 -1 1 -1 -1 -1 -1 -1 1 1 -1 -1 1

6 1 -1 -1 -1 1 1 -1 -1 1 -1 -1 1 1

7 1 1 1 1 1 1 1 1 1 1 1 1 1

8 -1 -1 1 -1 1 1 1 1 -1 1 -1 1 -1

9 1 1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1

10 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 -1

11 -1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1

12 -1 1 -1 -1 1 1 1 -1 1 -1 1 -1 -1

13 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 -1 1

14 -1 -1 -1 1 -1 -1 1 -1 -1 1 1 1 1

15 1 1 -1 -1 -1 -1 1 1 1 1 -1 1 -1

16 1 -1 -1 1 -1 1 1 1 1 -1 1 -1 1

17 1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 -1

18 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 1

19 -1 -1 1 1 1 -1 1 -1 1 -1 -1 -1 -1

20 -1 -1 1 1 -1 1 -1 -1 1 1 1 1 -1

-1 = Lower Bound, +1 = Upper Bound

168

Table A-2: PBD Table for Maximum Tangential and Radial Deformations at the Abutment.

Parameters Abutment

Pattern X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13

Maximum Tangential Deformations,

mm (in)

Maximum Radial Deformations,

mm (in) 1 -1 -1 -1 -1 1 -1 -1 1 -1 -1 1 1 1 7.8 (0.31) 2.5 (0.10) 2 -1 1 1 -1 1 -1 -1 1 1 1 1 -1 1 7.9 (0.31) 2.8 (0.11) 3 -1 1 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 10.9 (0.43) 6.0 (0.24) 4 1 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 -1 10.8 (0.43) 5.9 (0.23) 5 1 -1 1 -1 -1 -1 -1 -1 1 1 -1 -1 1 12.8 (0.50) 7.1 (0.28) 6 1 -1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 9.9 (0.39) 6.7 (0.26) 7 1 1 1 1 1 1 1 1 1 1 1 1 1 19.4 (0.76) 8.3 (0.33) 8 -1 -1 1 -1 1 1 1 1 -1 1 -1 1 -1 7.9 (0.31) 2.4 (0.09) 9 1 1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 10.9 (0.43) 4.8 (0.19)

10 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 -1 14.9 (0.59) 8.6 (0.34) 11 -1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 12.3 (0.48) 7.6 (0.30) 12 -1 1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 16.8 (0.66) 6.5 (0.26) 13 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 -1 1 13.2 (0.52) 7.2 (0.28) 14 -1 -1 -1 1 -1 -1 1 -1 -1 1 1 1 1 13.5 (0.53) 6.1 (0.24) 15 1 1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 15.7 (0.62) 7.3 (0.29) 16 1 -1 -1 1 -1 1 1 1 1 -1 1 -1 1 12.5 (0.49) 5.9 (0.23) 17 1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 -1 10.7 (0.42) 4.8 (0.19) 18 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 1 13.2 (0.52) 7.4 (0.29) 19 -1 -1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 8.3 (0.33) 2.7 (0.11) 20 -1 -1 1 1 -1 1 -1 -1 1 1 1 1 -1 9.1 (0.36) 5.6 (0.22)

168

169

Table A-3: PBD Table for Maximum Curvature Ductility.

Parameters Columns Pattern

X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 Maximum Curvature Ductility

1 -1 -1 -1 -1 1 -1 -1 1 -1 -1 1 1 1 0.9 2 -1 1 1 -1 1 -1 -1 1 1 1 1 -1 1 1.0 3 -1 1 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 2.3 4 1 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 -1 1.7 5 1 -1 1 -1 -1 -1 -1 -1 1 1 -1 -1 1 2.6 6 1 -1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 2.5 7 1 1 1 1 1 1 1 1 1 1 1 1 1 2.1 8 -1 -1 1 -1 1 1 1 1 -1 1 -1 1 -1 1.5 9 1 1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 2.6

10 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 -1 3.7 11 -1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 2.2 12 -1 1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 3.9 13 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 -1 1 2.6 14 -1 -1 -1 1 -1 -1 1 -1 -1 1 1 1 1 2.4 15 1 1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 3.8 16 1 -1 -1 1 -1 1 1 1 1 -1 1 -1 1 3.1 17 1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 -1 2.9 18 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 1 3.9 19 -1 -1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1.0 20 -1 -1 1 1 -1 1 -1 -1 1 1 1 1 -1 1.4

169

170

Table A-4: PBD Table for Maximum Tangential and Radial Deformations at the Bearing.

Parameters Bearings Pattern

X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 Maximum Tangential

Deformations, mm (in)

Maximum Radial Deformations,

mm (in) 1 -1 -1 -1 -1 1 -1 -1 1 -1 -1 1 1 1 10.8 (0.43) 2.4 (0.09) 2 -1 1 1 -1 1 -1 -1 1 1 1 1 -1 1 10.9 (0.43) 2.5 (0.10) 3 -1 1 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 21.3 (0.84) 13.1 (0.52) 4 1 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 -1 21.7 (0.85) 13.0 (0.51) 5 1 -1 1 -1 -1 -1 -1 -1 1 1 -1 -1 1 21.7 (0.85) 20.0 (0.79) 6 1 -1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 20.5 (0.81) 4.5 (0.18) 7 1 1 1 1 1 1 1 1 1 1 1 1 1 21.7 (0.85) 14.1 (0.56) 8 -1 -1 1 -1 1 1 1 1 -1 1 -1 1 -1 14.2 (0.56) 3.2 (0.13) 9 1 1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 23.8 (0.94) 5.1 (0.20)

10 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 -1 25.2 (0.99) 14.8 (0.58) 11 -1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 20.4 (0.80) 13.4 (0.53) 12 -1 1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 24.0 (0.94) 6.4 (0.25) 13 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 -1 1 19.8 (0.78) 9.5 (0.37) 14 -1 -1 -1 1 -1 -1 1 -1 -1 1 1 1 1 19.8 (0.78) 8.7 (0.34) 15 1 1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 24.7 (0.97) 13.3 (0.52) 16 1 -1 -1 1 -1 1 1 1 1 -1 1 -1 1 20.1 (0.79) 11.6 (0.46) 17 1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 -1 17.8 (0.70) 9.6 (0.38) 18 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 1 24.3 (0.96) 13.2 (0.52) 19 -1 -1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 13.5 (0.53) 3.2 (0.13) 20 -1 -1 1 1 -1 1 -1 -1 1 1 1 1 -1 17.3 (0.68) 7.6 (0.30)

170

Appendix B

CENTRAL COMPOSITE DESIGN

Table B-1: Central Composite Design Spaces for Five Most Significant Parameters and Earthquake Intensity Level Parameter.

Table B-2: CCD Table for Maximum Tangential and Radial Deformations at the

Abutment. Table B-3: CCD Table for Maximum Curvature Ductility. Table B-4: CCD Table for Maximum Tangential and Radial Deformations at the

Bearing.

172

Table B-1: Central Composite Design Spaces for Five Most Significant Parameters and Earthquake Intensity Level Parameter.

Pattern X1 X2 X5 X6 X7 Xeq

1 -1 1 -1 -1 -1 -1 2 -1 1 1 -1 1 -1 3 -1 1 1 -1 -1 1 4 0 0 0 0 0 1 5 1 1 1 -1 -1 -1 6 1 1 1 1 -1 1 7 -1 -1 -1 1 -1 -1 8 0 0 -1 0 0 0 9 0 0 0 0 -1 0 10 0 0 0 0 0 -1 11 0 1 0 0 0 0 12 -1 -1 1 1 -1 1 13 1 1 1 -1 1 1 14 1 -1 1 -1 1 -1 15 1 1 1 1 1 -1 16 0 0 0 1 0 0 17 1 1 -1 -1 1 -1 18 -1 -1 -1 -1 -1 1 19 -1 1 1 1 -1 -1 20 1 -1 1 -1 -1 1 21 1 -1 -1 1 1 -1 22 -1 1 1 1 1 1 23 0 0 0 0 1 0 24 1 -1 -1 -1 1 1 25 1 -1 -1 -1 -1 -1 26 -1 -1 1 -1 -1 -1 27 0 0 1 0 0 0 28 1 0 0 0 0 0 29 1 -1 -1 1 -1 1 30 1 -1 1 1 -1 -1 31 1 1 -1 -1 -1 1 32 -1 -1 -1 1 1 1 33 -1 0 0 0 0 0 34 1 1 -1 1 1 1 35 1 1 -1 1 -1 -1 36 1 -1 1 1 1 1 37 -1 1 -1 1 1 -1 38 -1 1 -1 -1 1 1 39 -1 -1 1 -1 1 1 40 -1 1 -1 1 -1 1 41 -1 -1 1 1 1 -1 42 -1 -1 -1 -1 1 -1 43 0 0 0 0 0 0 44 0 -1 0 0 0 0 45 0 0 0 -1 0 0

-1 = Lower Bound, 0 = Center Bound, and +1 = Upper Bound

173

Table B-2: CCD Table for Maximum Tangential and Radial Deformations at the Abutment.

Significant Curved Bridge Parameters Abutment Mean Standard Deviation

Pattern X1 X2 X5 X6 X7 Xeq Maximum tangential

deformations, mm Maximum radial

deformations, mm Maximum tangential


deformations, mm

1 -1 1 -1 -1 -1 -1 5.6 (0.22) 2.6 (0.10) 1.4 (0.06) 0.9 (0.04) 2 -1 1 1 -1 1 -1 5.8 (0.23) 2.4 (0.09) 1.4 (0.06) 0.9 (0.04) 3 -1 1 1 -1 -1 1 58.8 (2.31) 27.3 (1.07) 8.5 (0.33) 6.8 (0.27) 4 0 0 0 0 0 1 67.6 (2.66) 30.0 (1.18) 9.8 (0.39) 7.5 (0.30) 5 1 1 1 -1 -1 -1 6.2 (0.24) 2.2 (0.09) 1.5 (0.06) 0.7 (0.03) 6 1 1 1 1 -1 1 62.0 (2.44) 30.7 (1.21) 8.8 (0.35) 7.4 (0.29) 7 -1 -1 -1 1 -1 -1 5.9 (0.23) 2.8 (0.11) 1.4 (0.06) 1.0 (0.04) 8 0 0 -1 0 0 0 33.2 (1.31) 14.2 (0.56) 7.9 (0.31) 3.6 (0.14) 9 0 0 0 0 -1 0 34.9 (1.37) 12.7 (0.50) 8.3 (0.33) 3.2 (0.13)

10 0 0 0 0 0 -1 5.4 (0.21) 2.3 (0.09) 1.3 (0.05) 0.8 (0.03) 11 0 1 0 0 0 0 35.9 (1.41) 12.5 (0.49) 8.6 (0.34) 3.2 (0.13) 12 -1 -1 1 1 -1 1 56.4 (2.22) 27.4 (1.08) 8.0 (0.31) 6.6 (0.26) 13 1 1 1 -1 1 1 65.1 (2.56) 32.3 (1.27) 9.2 (0.36) 7.8 (0.31) 14 1 -1 1 -1 1 -1 5.9 (0.23) 2.4 (0.09) 1.5 (0.06) 0.9 (0.04) 15 1 1 1 1 1 -1 6.6 (0.26) 2.5 (0.10) 1.8 (0.07) 0.8 (0.03) 16 0 0 0 1 0 0 38.4 (1.51) 12.1 (0.48) 9.2 (0.36) 3.1 (0.12) 17 1 1 -1 -1 1 -1 5.6 (0.22) 3.2 (0.13) 1.4 (0.06) 1.2 (0.05) 18 -1 -1 -1 -1 -1 1 55.3 (2.18) 38.7 (1.52) 7.8 (0.31) 9.4 (0.37) 19 -1 1 1 1 -1 -1 4.5 (0.18) 2.6 (0.10) 1.1 (0.04) 1.0 (0.04) 20 1 -1 1 -1 -1 1 64.7 (2.55) 35.2 (1.39) 9.2 (0.36) 8.5 (0.33) 21 1 -1 -1 1 1 -1 5.8 (0.23) 3.4 (0.13) 1.5 (0.06) 1.3 (0.05) 22 -1 1 1 1 1 1 61.5 (2.42) 33.5 (1.32) 8.7 (0.34) 8.1 (0.32) 23 0 0 0 0 1 0 41.1 (1.62) 11.8 (0.46) 9.8 (0.39) 3.0 (0.12) 24 1 -1 -1 -1 1 1 64.6 (2.54) 35.2 (1.39) 9.1 (0.36) 8.5 (0.33) 25 1 -1 -1 -1 -1 -1 5.4 (0.21) 3.1 (0.12) 1.3 (0.05) 1.2 (0.05) 173

174

26 -1 -1 1 -1 -1 -1 4.8 (0.19) 2.8 (0.11) 1.2 (0.05) 1.1 (0.04) 27 0 0 1 0 0 0 42.7 (1.68) 10.6 (0.42) 10.2 (0.40) 2.7 (0.11) 28 1 0 0 0 0 0 47.0 (1.85) 11.5 (0.45) 11.2 (0.44) 2.9 (0.11) 29 1 -1 -1 1 -1 1 64.8 (2.55) 35.3 (1.39) 9.2 (0.36) 8.5 (0.33) 30 1 -1 1 1 -1 -1 5.9 (0.23) 3.4 (0.13) 1.5 (0.06) 1.3 (0.05) 31 1 1 -1 -1 -1 1 66.1 (2.60) 36.0 (1.42) 9.4 (0.37) 8.7 (0.34) 32 -1 -1 -1 1 1 1 61.5 (2.42) 33.5 (1.32) 8.7 (0.34) 8.1 (0.32) 33 -1 0 0 0 0 0 30.5 (1.20) 11.3 (0.44) 7.3 (0.29) 2.9 (0.11) 34 1 1 -1 1 1 1 73.1 (2.88) 39.8 (1.57) 10.3 (0.41) 9.6 (0.38) 35 1 1 -1 1 -1 -1 6.5 (0.26) 3.8 (0.15) 1.6 (0.06) 1.4 (0.06) 36 1 -1 1 1 1 1 58.4 (2.30) 31.8 (1.25) 8.3 (0.33) 7.7 (0.30) 37 -1 1 -1 1 1 -1 5.1 (0.20) 3.6 (0.14) 1.3 (0.05) 1.4 (0.06) 38 -1 1 -1 -1 1 1 55.5 (2.19) 33.4 (1.31) 7.9 (0.31) 8.1 (0.32) 39 -1 -1 1 -1 1 1 50.0 (1.97) 28.1 (1.11) 7.1 (0.28) 6.8 (0.27) 40 -1 1 -1 1 -1 1 56.5 (2.22) 31.8 (1.25) 8.0 (0.31) 7.7 (0.30) 41 -1 -1 1 1 1 -1 4.6 (0.18) 4.7 (0.19) 1.2 (0.05) 1.8 (0.07) 42 -1 -1 -1 -1 1 -1 4.3 (0.17) 5.8 (0.23) 1.1 (0.04) 2.2 (0.09) 43 0 0 0 0 0 0 33.6 (1.32) 11.8 (0.46) 8.0 (0.31) 2.8 (0.11) 44 0 -1 0 0 0 0 31.0 (1.22) 10.9 (0.43) 7.4 (0.29) 2.6 (0.10) 45 0 0 0 -1 0 0 32.3 (1.27) 11.3 (0.44) 7.7 (0.30) 2.7 (0.11)

174

175

Table B-3: CCD Table for Maximum Curvature Ductility.

Significant Curved Bridge Parameters Column Mean Standard Deviation Pattern

X1 X2 X5 X6 X7 Xeq Maximum Curvature Ductility Maximum Curvature Ductility

1 -1 1 -1 -1 -1 -1 1.0 0.5 2 -1 1 1 -1 1 -1 1.0 0.4 3 -1 1 1 -1 -1 1 6.8 0.7 4 0 0 0 0 0 1 7.5 0.8 5 1 1 1 -1 -1 -1 1.2 0.5 6 1 1 1 1 -1 1 7.0 0.7 7 -1 -1 -1 1 -1 -1 1.8 0.5 8 0 0 -1 0 0 0 3.7 0.5 9 0 0 0 0 -1 0 3.8 0.6

10 0 0 0 0 0 -1 1.1 0.4 11 0 1 0 0 0 0 4.0 0.6 12 -1 -1 1 1 -1 1 5.9 0.8 13 1 1 1 -1 1 1 7.3 0.7 14 1 -1 1 -1 1 -1 1.1 0.4 15 1 1 1 1 1 -1 1.3 0.2 16 0 0 0 1 0 0 4.2 0.6 17 1 1 -1 -1 1 -1 1.2 0.2 18 -1 -1 -1 -1 -1 1 6.2 0.6 19 -1 1 1 1 -1 -1 1.0 0.2 20 1 -1 1 -1 -1 1 7.3 0.7 21 1 -1 -1 1 1 -1 1.3 0.2 22 -1 1 1 1 1 1 6.9 0.7 23 0 0 0 0 1 0 4.5 0.6 24 1 -1 -1 -1 1 1 7.3 0.7 25 1 -1 -1 -1 -1 -1 1.2 0.2 26 -1 -1 1 -1 -1 -1 1.1 0.2 175

176

27 0 0 1 0 0 0 4.7 0.7 28 1 0 0 0 0 0 5.2 0.7 29 1 -1 -1 1 -1 1 7.3 0.7 30 1 -1 1 1 -1 -1 1.3 0.2 31 1 1 -1 -1 -1 1 7.5 0.8 32 -1 -1 -1 1 1 1 6.9 0.7 33 -1 0 0 0 0 0 3.4 0.5 34 1 1 -1 1 1 1 8.2 0.8 35 1 1 -1 1 -1 -1 1.4 0.3 36 1 -1 1 1 1 1 6.6 0.7 37 -1 1 -1 1 1 -1 1.1 0.2 38 -1 1 -1 -1 1 1 5.9 0.6 39 -1 -1 1 -1 1 1 5.3 0.5 40 -1 1 -1 1 -1 1 6.0 0.6 41 -1 -1 1 1 1 -1 1.0 0.2 42 -1 -1 -1 -1 1 -1 1.0 0.2 43 0 0 0 0 0 0 3.7 0.5 44 0 -1 0 0 0 0 3.4 0.5 45 0 0 0 -1 0 0 3.5 0.5

176

177

Table B-4: CCD Table for Maximum Tangential and Radial Deformations at the Bearing.

Significant Curved Bridge Parameters Bearing

Mean Standard Deviation Pattern X1 X2 X5 X6 X7 Xeq Maximum tangential


deformations, mm Maximum tangential


deformations, mm 1 -1 1 -1 -1 -1 -1 16.1 (0.63) 7.3 (0.29) 4.5 (0.18) 2.6 (0.10) 2 -1 1 1 -1 1 -1 16.5 (0.65) 6.7 (0.26) 4.7 (0.19) 2.7 (0.11) 3 -1 1 1 -1 -1 1 168.8 (6.65) 78.8 (3.10) 33.3 (1.31) 26.1 (1.03) 4 0 0 0 0 0 1 194.1 (7.64) 86.7 (3.41) 38.3 (1.51) 28.7 (1.13) 5 1 1 1 -1 -1 -1 17.6 (0.69) 6.3 (0.25) 4.7 (0.19) 2.0 (0.08) 6 1 1 1 1 -1 1 177.9 (7.00) 88.3 (3.48) 34.8 (1.37) 27.0 (1.06) 7 -1 -1 -1 1 -1 -1 16.7 (0.66) 8.0 (0.31) 4.4 (0.17) 2.6 (0.10) 8 0 0 -1 0 0 0 92.9 (3.66) 39.8 (1.57) 13.3 (0.52) 8.3 (0.33) 9 0 0 0 0 -1 0 97.5 (3.84) 35.8 (1.41) 14.0 (0.55) 7.5 (0.30)

10 0 0 0 0 0 -1 15.4 (0.61) 6.6 (0.26) 3.6 (0.14) 2.2 (0.09) 11 0 1 0 0 0 0 100.5 (3.96) 35.1 (1.38) 14.4 (0.57) 7.3 (0.29) 12 -1 -1 1 1 -1 1 161.9 (6.37) 78.6 (3.09) 31.6 (1.24) 24.1 (0.95) 13 1 1 1 -1 1 1 186.8 (7.35) 92.7 (3.65) 36.5 (1.44) 28.4 (1.12) 14 1 -1 1 -1 1 -1 16.8 (0.66) 6.9 (0.27) 4.8 (0.19) 2.7 (0.11) 15 1 1 1 1 1 -1 18.8 (0.74) 7.0 (0.28) 5.5 (0.22) 2.1 (0.08) 16 0 0 0 1 0 0 107.3 (4.22) 34.1 (1.34) 15.4 (0.61) 7.1 (0.28) 17 1 1 -1 -1 1 -1 16.0 (0.63) 9.3 (0.37) 4.6 (0.18) 3.7 (0.15) 18 -1 -1 -1 -1 -1 1 158.8 (6.25) 111.3 (4.38) 31.0 (1.22) 34.1 (1.34) 19 -1 1 1 1 -1 -1 12.8 (0.50) 7.4 (0.29) 3.6 (0.14) 2.9 (0.11) 20 1 -1 1 -1 -1 1 185.8 (7.31) 101.2 (3.98) 36.3 (1.43) 31.0 (1.22) 21 1 -1 -1 1 1 -1 16.6 (0.65) 9.6 (0.38) 4.7 (0.19) 3.8 (0.15) 22 -1 1 1 1 1 1 176.5 (6.95) 96.2 (3.79) 34.5 (1.36) 29.5 (1.16) 23 0 0 0 0 1 0 114.8 (4.52) 33.1 (1.30) 16.5 (0.65) 6.9 (0.27) 24 1 -1 -1 -1 1 1 185.3 (7.30) 101.0 (3.98) 36.2 (1.43) 30.9 (1.22) 25 1 -1 -1 -1 -1 -1 15.3 (0.60) 8.9 (0.35) 4.4 (0.17) 3.5 (0.14) 177

178

26 -1 -1 1 -1 -1 -1 13.6 (0.54) 7.9 (0.31) 3.9 (0.15) 3.1 (0.12) 27 0 0 1 0 0 0 119.4 (4.70) 29.8 (1.17) 17.1 (0.67) 6.2 (0.24) 28 1 0 0 0 0 0 131.3 (5.17) 32.4 (1.28) 18.9 (0.74) 6.8 (0.27) 29 1 -1 -1 1 -1 1 185.9 (7.32) 101.3 (3.99) 36.3 (1.43) 31.0 (1.22) 30 1 -1 1 1 -1 -1 16.8 (0.66) 9.7 (0.38) 4.8 (0.19) 3.9 (0.15) 31 1 1 -1 -1 -1 1 189.8 (7.47) 103.4 (4.07) 37.1 (1.46) 31.7 (1.25) 32 -1 -1 -1 1 1 1 176.6 (6.95) 96.2 (3.79) 34.5 (1.36) 29.5 (1.16) 33 -1 0 0 0 0 0 85.4 (3.36) 31.8 (1.25) 12.3 (0.48) 6.6 (0.26) 34 1 1 -1 1 1 1 209.7 (8.26) 114.3 (4.50) 41.0 (1.61) 35.0 (1.38) 35 1 1 -1 1 -1 -1 18.6 (0.73) 10.8 (0.43) 5.3 (0.21) 4.3 (0.17) 36 1 -1 1 1 1 1 167.8 (6.61) 91.4 (3.60) 32.8 (1.29) 28.0 (1.10) 37 -1 1 -1 1 1 -1 14.7 (0.58) 10.3 (0.41) 4.2 (0.17) 4.1 (0.16) 38 -1 1 -1 -1 1 1 159.4 (6.28) 96.0 (3.78) 31.1 (1.22) 29.4 (1.16) 39 -1 -1 1 -1 1 1 143.4 (5.65) 80.8 (3.18) 28.0 (1.10) 24.8 (0.98) 40 -1 1 -1 1 -1 1 162.1 (6.38) 91.3 (3.59) 31.7 (1.25) 28.0 (1.10) 41 -1 -1 1 1 1 -1 13.2 (0.52) 13.4 (0.53) 3.8 (0.15) 5.3 (0.21) 42 -1 -1 -1 -1 1 -1 12.4 (0.49) 16.7 (0.66) 3.5 (0.14) 6.6 (0.26) 43 0 0 0 0 0 0 93.9 (3.70) 32.9 (1.30) 13.5 (0.53) 4.7 (0.19) 44 0 -1 0 0 0 0 86.7 (3.41) 30.3 (1.19) 12.4 (0.49) 4.4 (0.17) 45 0 0 0 -1 0 0 90.2 (3.55) 31.6 (1.24) 12.9 (0.51) 4.5 (0.18)

178

VITA

Junwon Seo

Junwon Seo was born on January 12, 1980 in PyungTaek, South Korea. He graduated

with a Bachelor of Sciences in Civil Engineering in the Summer of 2001 from Konyang

University (Graduate Summa Cum Laude). He then entered Yonsei University in the Fall of

2001 and the Georgia Institute of Technology in the Fall of 2004 to pursue a Master of Science in

Civil Engineering with a focus on Structural Engineering. After receiving his two Master of

Science degrees in Civil Engineering, Junwon continue his graduate work by pursuing a Ph.D. at

the Pennsylvania State University.

seismic vulnerability assessment of a family of

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