evaluating the performance-based seismic design of …

EVALUATING THE PERFORMANCE-BASED SEISMIC DESIGN OF RC BRIDGES

ACCORDING TO THE 2014 CANADIAN HIGHWAY BRIDGE DESIGN CODE

by

Sepideh Ashtari

B.Sc., Sharif University of Technology, 2009

M.A.Sc., the University of British Columbia, 2012

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

in

THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES

(Civil Engineering)

THE UNIVERSITY OF BRITISH COLUMBIA

(Vancouver)

December 2018

© Sepideh Ashtari, 2018

ii

The following individuals certify that they have read, and recommend to the Faculty of Graduate

and Postdoctoral Studies for acceptance, the dissertation entitled:

Evaluating the performance-based seismic design of RC bridges according to the 2014 Canadian

Highway bridge design code

submitted by Sepideh Ashtari in partial fulfillment of the requirements for

the degree of Doctor of Philosophy

in Civil Engineering

Examining Committee:

Carlos E. Ventura, Department of Civil Engineering

Supervisor

W.D. Liam Finn, Department of Civil Engineering

Supervisory Committee Member

Ricardo Foschi, Department of Civil Engineering


Donald Anderson, Department of Civil Engineering

University Examiner

Annalisa Meyboom, School of Architecture and Landscape Architecture

University Examiner

Additional Supervisory Committee Members:

Don Kennedy, Associated Engineering


Sharlie Huffman, BC Ministry of Transportation


iii

Abstract

The 2014 edition of the Canadian Highway Bridge Design Code, CSA S6-14, has adopted a

performance-based design approach for the seismic design of lifeline and major-route bridges in

highly seismic zones. This addition offers many opportunities as well as some challenges with

regards to implementing the CSA S6-14 performance-based design provisions in practice. This

thesis aims to identify these challenges through a critical review of the CSA S6-14 performance-

based design provisions and to address a number of them within the scope of the thesis. The

motivation behind conducting the present study is to prepare a reference document for engineers

to better comprehend and implement the new provisions in practice. The focus of the thesis is on

the performance-based design of new reinforced concrete bridges with ductile substructures.

The addressed challenges are related to CSA S6-14 performance verification framework,

calibration of performance criteria, and appropriate numerical models to evaluate the established

performance criteria. A deterministic and a probabilistic framework are recommended to be used

with the CSA S6-14 performance-based design approach. The applications of each of the

frameworks are demonstrated through two detailed case studies and the advantages and

disadvantages of each framework are discussed. The performance criteria of the code are

compared against the recommended criteria in the literature and other design guidelines.

Moreover, the strain limits of the code are examined to predict the damage to a number of tested

reinforced concrete bridge columns. A thorough comparison of the CSA S6-14 and the updated

strain limits of the BC MoTI Supplement to CSA S6-14 is presented. Finally, common modelling

techniques for reinforced concrete structures including distributed and concentrated plasticity

models are employed to predict the response of a number of tested bridge columns. Mesh-

sensitivity issues due to the localization of plastic strains at critical sections or elements of

distributed plasticity models are discussed and the methods to rectify the issue are presented and

compared. A simple solution is proposed to eliminate the post-processing effort that is required

to verify the strain limits of the code in distributed plasticity models, for which material model

regularization is used to deal with the mesh-sensitivity issue.

iv

Lay Summary

The 2014 edition of the Canadian Highway Bridge Design Code (CSA S6-14) introduced a

major shift in the seismic design of bridges by adopting a performance-based design approach.

The performance-based design approach is based on meeting specified performance criteria at

multiple seismic hazard levels.

While the new design approach provides many opportunities for achieving better seismic

designs with more reliable structural performances during earthquake events, it faces a number

of challenges too. The present study provides a critical review of the CSA S6-14 performance-

based design approach and identifies the challenges of implementing the code in practice. A

number of the identified challenges with regards to analysis, design framework, performance

criteria, and performance verification are addressed subsequently.

The thesis helps practicing engineers better understand and implement the performance-

based design provisions in their projects and in that sense, it serves as a reference for practical

purposes.

v

Preface

This dissertation is an original intellectual product of Sepideh Ashtari under the primary

supervision of Prof. Carlos Ventura. Identification and design of the research program, analysis,

result presentations, and writings were done by Sepideh Ashtari. Some parts of this thesis has

been already published, as specified below. The rest of the research material in the thesis is

unpublished to this date.

- Portions of Chapter 5 have been published in Ashtari and Ventura (2018) “Verifying the

seismic performance of concrete bridge columns according to the 2014 Canadian

Highway Bridge Design Code,” proceedings of the 11th National Conference on

Earthquake Engineering, Los Angeles, California, US. I was the principal author of the

paper and conducted all the numerical analyses and writing of the paper. Professor

Ventura provided assistance in revising the manuscript.

- A portion of Chapter 7 has been published in the EGBC Professional Practice Guidelines

on the Performance-Based Seismic Design of Bridges in BC as Ashtari et al. (2018)

“Appendix D1: Reinforced Concrete Bridge.” This paper contains only the results for the

response spectrum analysis of the case study. Mr. Khan provided the proportioning of the

bridge model. I conducted all the numerical analyses and wrote the manuscript. Professor

Ventura and Mr. Khan reviewed the manuscript. Dr. Atukorala provided technical

assistance with the geotechnical inputs for the model. A version of the above material

with the addition of nonlinear time-history analysis has been published in Ashtari and

Ventura (2017b) “A case study on implementing the performance-based seismic

provisions of the 2014 Canadian Highway Bridge Design Code,” proceedings of

International Workshop on Performance-Based Seismic Design of Structures, Shanghai,

China.

- Portions of Chapter 8 has been published in Ashtari et al. (2017) “A case study on

evaluating the performance criteria of the 2014 Canadian Highway Bridge Design

Code,” Proceedings of 39th IABSE Symposium, Vancouver, Canada. I formulated the

concept, performed all the numerical analyses, and wrote the first draft of the paper. Mr.

vi

Kennedy provided the drawings of the Trent River Bridge and helped with the technical

and practical aspects of the paper. Professors Finn and Ventura provided technical and

academic inputs and revised the paper.

- Section D.1 of Appendix D have been published in Ashtari and Ventura (2017a)

“Correlation of damage with seismic intensity measures for ductile concrete bridge

columns in British Columbia,” Proceedings of 16th World Conference on Earthquake

Engineering, Santiago, Chile. I conducted all the analysis and wrote the paper. Professor

Ventura provided technical input and revised the paper.

vii

Table of Contents

Abstract ......................................................................................................................................... iii

Lay Summary ............................................................................................................................... iv

Preface .............................................................................................................................................v

Table of Contents ........................................................................................................................ vii

List of Tables ............................................................................................................................... xii

List of Figures ............................................................................................................................ xvii

Acknowledgements ................................................................................................................... xxii

Chapter 1: Introduction ................................................................................................................1

1.1 Motivation and Objectives .............................................................................................. 1

1.2 Scope ............................................................................................................................... 2

1.3 Overview of the Thesis ................................................................................................... 3

Chapter 2: Background on Performance-Based Seismic Design ..............................................5

2.1 A Shift in Seismic Design Philosophy ............................................................................ 5

2.2 Progress of PBD Procedures in Building Industry ......................................................... 7

2.3 Progress of PBD Procedures in Bridge Industry........................................................... 11

2.4 Performance-Based Design vs Assessment .................................................................. 14

2.5 Review of a Number of Existing Design/Assessment Frameworks ............................. 16

2.5.1 Load and Resistance Factor Design (LRFD) ............................................................ 16

2.5.2 PEER Framework ..................................................................................................... 17

2.5.3 ASCE/SEI 7-16 Performance Assessment Framework ............................................ 19

2.6 Overview of the CSA S6-14 PBD Provisions............................................................... 21

Chapter 3: Merits and Challenges of the CSA S6-14 PBD Approach ....................................24

viii

3.1 General Challenges of Performance-Based Procedures ............................................... 24

3.2 Particular Challenges of the CSA S6-14 PBD Approach ............................................. 27

3.3 Merits of CSA S6-14 PBD approach ............................................................................ 28

3.4 What Is Required to Implement the CSA S6-14 PBD Approach in Practice ............... 29

Chapter 4: Assessment Frameworks for CSA S6-14 PBD .......................................................31

4.1 Assessment Framework Requirements for CSA S6-14 PBD ....................................... 31

4.2 Deterministic Assessment Framework for CSA S6-14 PBD ........................................ 31

4.2.1 Implementation Steps................................................................................................ 32

4.3 Probabilistic Assessment Framework for CSA S6-14 PBD ......................................... 36

4.3.1 Demand and Capacity Factored Design (DCFD) ..................................................... 36

4.3.2 Implementation Steps................................................................................................ 41

Chapter 5: Examining Modelling Alternatives of RC Bridges for PBD .................................46

5.1 Modelling Alternatives for RC Bridges ........................................................................ 46

5.1.1 Linear Elastic Models with Cracked Section Properties ........................................... 47

5.1.2 Concentrated Plasticity Models ................................................................................ 49

5.1.3 Distributed Plasticity Models .................................................................................... 50

5.2 Localization in Distributed Plasticity Models of RC Components ............................... 53

5.2.1 Background in Fracture Mechanics .......................................................................... 53

5.2.2 Localization in Distributed Plasticity Beam-Column Elements ............................... 54

5.2.3 The Importance of Addressing Localization Issues for PBD of RC Bridges ........... 55

5.2.4 Options for Dealing with Localization Issues in DBE and FBE .............................. 57

5.3 Validating Distributed and Concentrated Plasticity Models of RC Bridge Columns ... 63

5.3.1 Selected Bridge Column Tests .................................................................................. 64

5.3.2 Bridge Column Models ............................................................................................. 68

ix

5.3.3 Validation of the Distributed Plasticity Models for Test 328 ................................... 72

5.3.4 Sensitivity of Monotonic Response to GFcc Value ................................................... 81

5.3.5 Adjusting Local Strain Response .............................................................................. 84

5.3.6 Comparison of the Options for Dealing with Localization Issues ............................ 90

5.3.7 Validation of the Concentrated Plasticity Model for Test 328 ................................. 91

5.3.8 Comparison of the Distributed and Concentrated Plasticity Models of Column 32891

5.3.9 Comparison of the Models for the Other Tests ......................................................... 94

Chapter 6: Evaluating the Strain Limits of CSA S6-14 for RC Bridges ................................97

6.1 Strain Limits in CSA S6-14 and BC MoTI Supplement............................................... 97

6.2 Performance Criteria for RC Bridges in Literature and Other Design Guidelines ....... 99

6.2.1 Strain Limits............................................................................................................ 100

6.2.2 Curvature Limits ..................................................................................................... 102

6.2.3 Deformation Limits ................................................................................................. 103

6.3 Examining the Strain Limits of CSA S6-14 and BC MoTI Supplement .................... 104

6.3.1 Damage Observed in the Tests ............................................................................... 104

6.3.2 Damage Predicted by the Models ........................................................................... 105

6.3.3 Comparison and Discussion .................................................................................... 106

Chapter 7: Implementation of the Frameworks for CSA S6-14 PBD...................................110

7.1 Description of the Bridge ............................................................................................ 110

7.2 Bridge Site Properties ................................................................................................. 111

7.3 Probabilistic Seismic Hazard Analysis ....................................................................... 112

7.4 Performance Criteria and Analysis Requirements ...................................................... 113

7.5 Structural Analysis Models ......................................................................................... 115

7.6 Fundamental Period of the Bridge .............................................................................. 117

x

7.7 Ground Motion Selection and Scaling ........................................................................ 117

7.8 Response Spectrum Analysis ...................................................................................... 123

7.9 Nonlinear Time-History Analysis ............................................................................... 123

7.10 Pushover Analysis ....................................................................................................... 123

7.11 PBD Assessment Using the Deterministic Framework .............................................. 124

7.12 PBD Assessment Using the Probabilistic Framework ................................................ 128

7.13 Discussion ................................................................................................................... 138

7.13.1 Comparison of the Deterministic vs Probabilistic Frameworks ......................... 138

7.13.2 Question of Uniform Reliability (The Value of Loss Analysis) ......................... 140

Chapter 8: Case Study-Seismic Evaluation of the Trent River Bridge ................................142

8.1 Description of the Bridge ............................................................................................ 142

8.2 Soil Properties ............................................................................................................. 145

8.3 Bridge Model .............................................................................................................. 146

8.3.1 Choice of Analysis Tool ......................................................................................... 146

8.3.2 Material Models ...................................................................................................... 146

8.3.3 Structure .................................................................................................................. 147

8.3.4 Foundations ............................................................................................................. 147

8.3.5 Abutments ............................................................................................................... 148

8.4 Performance Criteria ................................................................................................... 149

8.5 Structural Analysis ...................................................................................................... 149

8.6 PBD Assessment ......................................................................................................... 151

8.6.1 Assuming Major-route Importance Category ......................................................... 153

8.6.2 Assuming Lifeline Importance Category ................................................................ 153

8.6.3 Assuming Other Importance Category ................................................................... 156

xi

8.7 FBD Assessment ......................................................................................................... 157

8.8 Conclusions ................................................................................................................. 157

Chapter 9: Summary and Future Research ............................................................................159

9.1 Summary of Thesis ..................................................................................................... 159

9.2 Main Contributions ..................................................................................................... 162

9.3 Future Research .......................................................................................................... 163

References ...................................................................................................................................165

Appendices ..................................................................................................................................174

Appendix A CSA S6-14 Tables for PBD ............................................................................... 174

Appendix B Deriving Expressions for Post-Processing Local Strain Response .................... 178

Appendix C Acceleration and Displacement Response Spectra of the Selected Records for

NTHA in Chapter 7 and Chapter 8 ......................................................................................... 180

C.1 Crustal Suite (Chapter 7) ........................................................................................ 180

C.2 Subcrustal Suite (Chapter 7) ................................................................................... 183

C.3 Subduction Suite (Chapter 7) .................................................................................. 187

C.4 Suite of Motions for Chapter 8 ............................................................................... 191

Appendix D PBD Aid Using the Correlation of Damage with Seismic Intensity Measures .. 195

D.1 Study1: Developing the PBD Aid ........................................................................... 195

D.2 Study2: Examining the PBD Aid for the Implementation Example of Chapter 5 .. 207

xii

List of Tables

Table 2.1 Rehabilitation objectives in ASCE 41-06 (2006). ........................................................ 10

Table 2.2 Seismic performance criteria in ATC-32 (1996a) and ATC-18 (1997) ........................ 13

Table 2.3 Bridge damage assessment stage of the five-level performance evaluation procedure

developed by the University of California, San Diego (Hose and Seible 1999) .......................... 13

Table 2.4 Minimum performance levels for retrofitted bridges in FHWA Seismic Retrofitting

Manual (2006) ............................................................................................................................... 13

Table 2.5 Performance goals in ASCE/SEI 7-16 .......................................................................... 19

Table 4.1 ASCE/SEI 7-16 assumed uncertainty values for component deformation demand ..... 42

Table 4.2 ASCE/SEI 7-16 assumed uncertainty values for component deformation capacity .... 42

Table 5.1 Specimen properties of the test columns (Lehman et al. 2004) .................................... 66

Table 5.2 Material properties of the test columns (Lehman et al. 2004) ...................................... 66

Table 5.3 Imposed displacement histories in mm for columns with different aspect ratios

(Lehman et al. 2004) ..................................................................................................................... 66

Table 5.4 Empirical plastic hinge length for the selected test columns ........................................ 71

Table 5.5 Calculated regularized strain of the unconfined and confined concrete at 80% strength

loss, and the ultimate rupture strain and post-peak slope of the reinforcement steel for Column

328 models .................................................................................................................................... 75

Table 5.6 Failure displacements of the DBE and FBE models of Column 328 with and without

material regularization .................................................................................................................. 78

Table 5.7 Calculated values of ε20c-PH for the Column 328 model with LIP=LP=376 mm ............ 84

Table 5.8 Column 328 DBE models: (a) displacements corresponding to the specified strain

limits, (b) mapped strain limits based on the regularized material models, (c) displacements

corresponding to the modified strain limits .................................................................................. 88

xiii

Table 5.9 Column 328 FBE models: (a) displacements corresponding to the specified strain

limits, (b) mapped strain limits based on the regularized material models, (c) displacements

corresponding to the modified strain limits .................................................................................. 89

Table 5.10 Column 328 fibre hinge model, displacements corresponding to the specified strain

limits ............................................................................................................................................. 93

Table 5.11 Average displacements of Table 5.8 and Table 5.9 for the FBE and DBE models of

Column 328 ................................................................................................................................... 93

Table 6.1 CSA S6-14 concrete and reinforcing steel strain limits ................................................ 99

Table 6.2 BC MoTI Supplement to CSA S6-14 concrete and reinforcing steel strain limits ....... 99

Table 6.3 Strain limits associated to the flexural damage states of reinforced concrete columns 99

Table 6.4 ATC-32 (1996) strain limits (AC1 2016) ................................................................... 101

Table 6.5 Strain limits by Kowalsky (2000) ............................................................................... 101

Table 6.6 Bridge performance/design parameters SRPH-1 by Hose and Seible (Hose and Seible

1999; Transportation Research Board 2013) .............................................................................. 101

Table 6.7 Mean tensile strains in longitudinal reinforcement corresponding to visual damage

states as reported by Vosooghi and Saiidi (2010; ACI 2016) (f’c is in ksi) ................................ 101

Table 6.8 Displacements cycle in meter reported for the first occurrence of the damage states in

the tested bridge columns (Lehman et al. 2004) ......................................................................... 105

Table 6.9 Recorded average strain values corresponding to the first occurrence of the damage

states in the tested bridge columns (Lehman et al. 2004) ........................................................... 105

Table 6.10 Predictions of the regularized FBE and DBE models for the cyclic displacements in

meter corresponding to the first occurrence of the specified strain limits .................................. 106

Table 6.11 Comparison between the measured and predicted displacements by the regularized

FBE and DBE models, corresponding to the first occurrence of damage states (all values are in

meter) .......................................................................................................................................... 109

xiv

Table 7.1 CSA S6-14 and BC MoTI strain limits associated to the performance levels of a major-

route bridge ................................................................................................................................. 114

Table 7.2 CSA S6-14 and BC MoTI strain limits associated to the performance levels of a major-

route bridge ................................................................................................................................. 114

Table 7.3 Composite section properties of the deck at different sections .................................. 116

Table 7.4 Deaggregation of 2%/50 UHS of Victoria at period of 1.61 s. Results of the

deaggregation at the 0.56 were very similar to the 1.61 s and are not reported separately. ....... 119

Table 7.5 Source contribution to the UHS of Victoria at different hazard levels and at different

periods (the sum of the contributions in each row is equal to 1). ............................................... 121

Table 7.6 Selected ground motion records for time history analysis .......................................... 121

Table 7.7 Column drift ratio capacities associated to the first occurrence of the damage states 124

Table 7.8 Mean column drift demands from NTHA in the longitudinal (x) and transverse (y)

directions for crustal, subcrustal, subduction suites, and all 33 records ..................................... 126

Table 7.9 Column drift demands from RSA and NTHA in the longitudinal (x) and transverse (y)

directions, along with the predicted damage (M: minimal, Y: yielding of longitudinal

reinforcements, SL1 & SL2: serviceability limit 1 & 2, SP1 & SP2: cover spalling 1& 2, B:

reduced buckling)........................................................................................................................ 126

Table 7.10 Ratio of the drift demand to drift capacity of the columns in the longitudinal (x) and

lateral (y) directions, and the reserve drift capacity for each hazard level ................................. 127

Table 7.11 Maximum longitudinal and lateral displacement of the deck ................................... 127

Table 7.12 Median drift ratio demands of the bridge columns from NTHA in the longitudinal (x)

and lateral (y) directions, and the calculated record-to-record variabilities ............................... 130

Table 7.13 Demand and capacity aleatory and epistemic uncertainty values ............................. 130

Table 7.14 Obtained values for the demand and hazard curve parameters ................................ 130

Table 7.15 Demand and capacity factors calculated for the longitudinal and lateral directions 130

Table 7.16 Confidence factor values calculated for various confidence intervals ..................... 130

xv

Table 7.17 Implied factor of safety in evaluating the lateral and longitudinal response of the

bridge columns in terms of drift ratio with/without considering the confidence interval .......... 130

Table 7.18 CSA S6-14 performance criteria checks using factored demand and capacity drift

ratios of the bridge columns in the longitudinal and lateral direction (P: pass; F: fail; FD: factored

drift ratio demand in %, FC: factored drift ratio capacity in %; λ90%: confidence factor for 90%

confidence interval; D: median demand; C: median capacity) ................................................... 132

Table 7.19 BC MoTI performance criteria checks using factored demand and capacity drift ratios

of the bridge columns in the longitudinal and lateral direction For CSA S6-14 performance

criteria, the use of the DCFD framework indicates that the columns do not meet any of the

specified criteria in the longitudinal direction, while the deterministic framework suggest that

they meet the ones at 2%/50. ...................................................................................................... 132

Table 7.20 Recalculated confidence factors of Table 7.16calculated for lower confidence

intervals ....................................................................................................................................... 140

Table 8.1 Calculated SSI spring constants for the Trent River Bridge site Class D model........ 148

Table 8.2 CSA S6-14 and BC MoTI strain limits associated to the performance levels of lifeline

and other bridges. ........................................................................................................................ 150

Table 8.3 Selected motions for NTHA of the Trent River Bridge on site class D ..................... 151

Table 8.4 Drift ratio capacities of the columns associated to the first occurrence of the damage

states ............................................................................................................................................ 152

Table 8.5 Column maximum drift demands from RSA in the longitudinal (x) and lateral (y)

directions, along with the predicted damage (N: none, M: minimal damage, Y: yielding of

longitudinal reinforcements) ....................................................................................................... 154

Table 8.6 Column mean drift demands from NTHA in the longitudinal (x) and lateral (y)

directions, along with the predicted damage............................................................................... 154

Table 8.7 Ratios of the drift demand to drift capacity of the columns the reserve drift capacities

assuming Major-route importance category for the bridge ......................................................... 155

xvi


assuming Lifeline importance category for the bridge ............................................................... 155


assuming Other importance category for the bridge ................................................................... 156

Table A.1 CSA S6-14 site classification for seismic site response ..…………………………. 174

Table A.2 CSA S6-14 Seismic performance category based on 2475-year return period spectral

values .......................................................................................................................................... 174

Table A.3 CSA S6-14 Requirements for PBD and FBD ............................................................ 174

Table A.4 CSA S6-14 minimum analysis requirements for 2%/50 and 5%/50 hazard levels.... 175

Table A.5 CSA S6-14 minimum analysis requirements for 10%/50 hazard level ..................... 175

Table A.6 CSA S6-14 Minimum performance levels for PBD approach................................... 175

Table A.7 CSA S6-14 Performance criteria (continued) ............................................................ 176

Table D.1 Strain-based damage Criteria ..................................................................................... 197

Table D.2 Selected ground motion records for time history analysis ......................................... 199

Table D.3 Prediction of the damage to the 1.0s period bridge column at different hazard levels

using smaller suites of records. PGD was utilized as the intensity measure; μn is the average of

the maximum drift ratios for n records with the highest PGD values ........................................ 205

Table D.4 Correlation coefficients of longitudinal and lateral drift ratios at 2%/50 hazard level

with PGD, PGV, and PGA of H1 and H2 components ................................................................ 209

Table D.5 Mean longitudinal drift ratios (%) of the smaller suites sampled based on H1-PGV

correlation ................................................................................................................................... 209

Table D.6 Mean lateral drift ratios (%) of the smaller suites sampled based on H2-PGA

correlation ................................................................................................................................... 209

xvii

List of Figures

Figure 2.1 SEAOC Vision 2000 recommended seismic performance objectives for buildings

(SEAOC 1995) ................................................................................................................................ 9

Figure 2.2 Visualization of the first generation performance-based earthquake engineering

(Moehle and Deierlein 2004) .......................................................................................................... 9

Figure 4.1 The deterministic PBD assessment framework flowchart.......................................... 35

Figure 4.2 The probabilistic PBD assessment framework flowchart .......................................... 45

Figure 5.1 Types of nonlinear component models: (a) and (b) are concentrated plasticity models;

(c), (d), and (e) are distributed plasticity models (NEHRP 2013) ................................................ 47

Figure 5.2 Priestley et al. chart for finding the effective stiffness of cracked RC circular sections

(1996) ............................................................................................................................................ 49

Figure 5.3 Element level and sectional level response of a cantilever beam modelled using a

single FBE with (a) elastic strain-hardening (b) elastic-perfectly plastic, and (c) strain-softening

section responses from Coleman and Spacone (Coleman and Spacone 2001) ............................. 56

Figure 5.4 Schematic post-peak crushing energy of (a) unconfined concrete and (b) confined

concrete (reproduced from Pugh, 2012) ....................................................................................... 60

Figure 5.5 Schematic post-peak energy of reinforcing steel (reproduced from Pugh, 2012) ....... 63

Figure 5.6 (left) geometry and reinforcement of the tested specimens, (right) test configuration

and instrumentation (Calderone et al. 2001; Lehman et al. 2004) ................................................ 65

Figure 5.7 Imposed lateral displacement time history (Lehman et al. 2004) ............................... 66

Figure 5.8 Cyclic force-displacement responses of the selected test columns (PEER 2003) ....... 67

Figure 5.9 Schematic distributed and concentrated plasticity models of the columns in OpenSees

(left) and SAP2000 (right) ............................................................................................................ 68

Figure 5.10 Concret02 constitutive model in OpenSees (PEER 2017) ........................................ 70

xviii

Figure 5.11 Comparison of the simulated cyclic force-displacement response of Column 328

using DBE with (a) 0.076 m, (b) 0.152 m, (c) 0.305 m, (d) 0.610 m element size and 2

integration points, and FBE with (a`) 0.610 m element size and 2 integration points, (b`) 0.914 m

element size and 3 integration points with the experimental results ............................................ 73

Figure 5.12 Regularized cyclic force-displacement response of Column 328 using DBE with (a)

0.076 m, (b) 0.152 m, (c) 0.305 m, (d) 0.610 m element size and 2 integration points, and FBE

with (a`) 0.610 m element size and 2 integration points, (b`) 0.914 m element size and 3

integration points with the experimental results ........................................................................... 76

Figure 5.13 Monotonic response of the DBE and FBE models of Column 328 before material

regularization ................................................................................................................................ 78

Figure 5.14 Monotonic response of the DBE and FBE models of Column 328 after material

regularization ................................................................................................................................ 78

Figure 5.15 Comparison of the cyclic and monotonic responses of the (a) DBE model and (b)

FBE model with LIP=LP against the test results of Column 328 ................................................... 80

Figure 5.16 Comparison of the cyclic and monotonic responses of the (a) DBE model and (b)

FBE model with LIP=0.5 LP against the test results of Column 328 ............................................. 81

Figure 5.17 Sensitivity of the monotonic force-displacement response of the Column 328 (a)

DBE and (b) FBE models to GFcc value ....................................................................................... 82

Figure 5.18 Mapping strain limits in the post-peak region of (a) confined concrete and (b)

reinforcing steel materials ............................................................................................................. 86

Figure 5.19 Comparison of the monotonic response of the regularized (a) DBE and (b) FBE

models of Column 328 with the model satisfying LP=LIP ............................................................ 91

Figure 5.20 Monotonic force-displacement response of the SAP2000 Fibre Hinge model of

Column 328 ................................................................................................................................... 92

Figure 5.21 Comparison of the regularized (a) DBE and (b) FBE models with the concentrated

plasticity models of Column 328 .................................................................................................. 93

xix

Figure 5.22 Comparison of the predictions of the regularized DBE and FBE models with the

fibre hinge models for Column (a) 415, (b) 430, (c) 828, and (d) 1028 ...................................... 96

Figure 7.1 Schematic elevation view of the bridge .................................................................... 111

Figure 7.2 Schematic cross-section of the bridge at the pier ..................................................... 111

Figure 7.3 UHS (left) and design spectra (right) for site class D at 2%/50, 5%/50, and 10%/50

hazard levels................................................................................................................................ 112

Figure 7.4 Fibre cross section of the columns in SAP2000 (left) and the moment-curvature

response of the section in the plastic hinge region (right) (the response is calculated under the

dead load). ................................................................................................................................... 118

Figure 7.5 View of the bridge model in SAP2000..................................................................... 118

Figure 7.6 Mean spectra for the crustal, subcrustal, subduction suites, and all 33 records vs the

target spectrum (the range over which the mean spectra are matched to the target spectrum, are

shown with the vertical lines). .................................................................................................... 122

Figure 7.7 Longitudinal and lateral drift ratio demands versus the spectral acceleration at periods

of 1.61 s and 0.56 s, respectively, using all 33 records. .............................................................. 129

Figure 7.8 Mean annual frequency versus the spectral acceleration at periods of 1.61 s and 0.56 s

for Victoria .................................................................................................................................. 129

Figure 7.9 MAF of exceeding limit states for longitudinal deck displacement ......................... 133

Figure 7.10 Column fragility curves for the CSA S6-14 performance criteria in the (a)

longitudinal and (b) lateral directions. ........................................................................................ 135

Figure 7.11 Column fragility curves for the BC MoTI Supplement performance criteria in the (a)

longitudinal and (b) lateral directions. ........................................................................................ 135

Figure 7.12 Factored demand to factored capacity ratio in the lateral and longitudinal directions

versus the mean annual frequency of exceeding the performance limits states of CSA S6-14. . 136

Figure 7.13 Factored demand to factored capacity ratio in the lateral and longitudinal directions

versus the mean annual frequency of exceeding the performance limits states of BC MoTI

Supplement. ................................................................................................................................ 136

xx

Figure 7.14 Factored demand to factored capacity ratio with confidence interval of 90% in the

lateral and longitudinal directions versus the mean annual frequency of exceeding the

performance limits states of CSA S6-14. .................................................................................... 137

Figure 7.15 Factored demand to factored capacity ratio with confidence interval of 90% in the

lateral and longitudinal directions versus the mean annual frequency of exceeding the

performance limits states of BC MoTI Supplement. .................................................................. 137

Figure 8.1 Relative location of the Trent River Bridge with respect to Victoria and Vancouver

..................................................................................................................................................... 143

Figure 8.2 Elevation view of the Trent River Bridge ................................................................ 144

Figure 8.3 Plan view of the Trent River Bridge ......................................................................... 144

Figure 8.4 Deck, pier diaphragm, and cap-beam cross-sections (dimensions in mm) .............. 144

Figure 8.5 Connection of the deck to the end diaphragm .......................................................... 145

Figure 8.6 Pier 1 and Pier 2 elevation views (looking North) ................................................... 145

Figure 8.7 Spine 3D model of the Trent River Bridge in SeismoStruct .................................... 149

Figure 8.8 Bridge design spectra for site class C and D at 2%/50 hazard level ........................ 150

Figure 8.9 Mean acceleration and displacement spectra for the 11 motions vs the target spectrum

(the period range over which the mean spectra are matched to the target spectrum, are shown

with the vertical lines). ................................................................................................................ 151

Figure 8.10 Pushover curves for the Trent River Bridge on site class C in the lateral direction152

Figure 8.11 Drift ratios of the bridge columns from NTHA at 2%/50 ...................................... 154

Figure 8.11 Comparison of the axial force-bi-axial bending moment demand at 2%/50 hazard

level with the column interaction diagram.................................................................................. 158

Figure B.1 Mapping strain limits in the post-peak region of (a) confined concrete and (b)

reinforcing steel ………………………………………………………………………………. 179

Figure D.1 Maximum drift ratios versus PGV and Vmax/Amax for the three bridge columns ... 201

xxi

Figure D.2 (left column) Maximum drift ratios versus PGD for the three bridge columns and

(right column) maximum drift ratios versus PGD of the individual earthquake sources for the

bridge column with fundamental periods of 1.0 s....................................................................... 202

Figure D.3 Maximum longitudinal and lateral drift ratios versus PGD, PGV, and PGA of H1 and

H2 components of the crustal suite.............................................................................................. 210


H2 components of the subcrustal suite ........................................................................................ 211


H2 components of the subduction suite ....................................................................................... 212


H2 components of all records ...................................................................................................... 213

xxii

Acknowledgements

The creation of this research work would not have been possible without the constant support

and encouragement of my supervisor Dr. Carlos Ventura. He motivates his students not only to

advance in their research but also to develop skills that would further help them succeed in their

future paths. It has been a pleasure and privilege to work under his supervision.

I would also like to express my deepest gratitude to Dr. Liam Finn, who was a true mentor

to me. He patiently devoted hours guiding me to move forward in my research, while keeping

my ideas and efforts focused on the main theme of the thesis. His continuous support and kind

feedbacks inspired me to excel my work every step of the way.

I owe particular thanks to Mr. Don Kennedy from the Associated Engineering, who never

hesitated to respond to my questions. His many years of practical seismic design experience

helped me to better understand the industry and connect my work to the needs of current

practice. He was very kind to provide the Trent River Bridge drawings that were used in one of

the case studies of this thesis.

I offer my enduring gratitude to other members of my supervisory committee, Dr. Ricardo

Foschi, Dr. Anoosh Shamsabadi form Caltrans, and Ms. Sharlie Huffman. Their feedbacks were

essential to the success of this work.

I also wish to gratefully acknowledge the funding for this research project, which was

partly supported through the Four-Year Fellowship award from the University of British

Columbia and partly provided by the Natural Sciences and Engineering Research Council of

Canada (NSERC).

Many thanks go to the wonderful faculty and staff members of UBC Civil Engineering

Department, who were always ready to assist me throughout my graduate studies. I also would

like to thank all my friends and colleagues at UBC Earthquake Engineering Research Facility.

Special thanks are owed to my dear friend, Dr. Armin Bebamzadeh, who inspires every single

person around him by his passion for research.

I would like to dedicate this work to my parents, whose love and kindness gave me

strength and will all the way. I could not make it this far without their endless love and support.

xxiii

To my parents, who inspired me every step of my life,

and to whom I shall always remain grateful.

1

Chapter 1: Introduction

1.1 Motivation and Objectives

The release of the new seismic performance-based design (PBD) provisions in the 2014

Canadian Highway Bridge Design Code (2014) marked one of the boldest and most progressive

changes ever made to Canadian seismic design provisions. The new provisions stemmed from a

radically different seismic design philosophy, compared to the force-based design (FBD)

philosophy incorporated in the previous versions of the code. The use of the PBD design

approach became mandatory for major-route and lifeline bridges, as well as irregular bridges, in

highly seismic zones. The new provisions also aimed to facilitate the seismic design of

innovative new systems, for which no elastic seismic force reduction factors were available to be

used with the FBD approach.

As promising as all the advantages of the new PBD approach were anticipated to be,

concerns were expressed about the followings:

- Whether the recommended performance criteria were suitable or adequate to guarantee

meeting the expected performance levels;

- And, whether the practicing engineers were able to fully comprehend and adopt the new

provisions, as envisioned by the code committee.

The motivation behind conducting the present study was to address the above-mentioned

concerns, within the scope established for the thesis. Based on this motivation, the main

objectives of this thesis are:

- First, to provide a critical review of the recommended provisions and to identify and

highlight the challenges of implementing the PBD provisions in practice.

- Second, to address a number of the identified challenges, within the established scope of

the thesis as specified in Section 1.2.

- Third, to provide detailed case studies on implementing the PBD provisions and the

recommended solutions to address the identified challenges.

- Finally, to serve as a reference for practicing engineers to better understand and

implement the PBD provisions of CSA S6-14.

2

The present study is the first of its kind to focus on the PBD provisions of CSA S6-14, with an

aim to achieve the above objectives. As such, this thesis serves a great purpose to the progress

and improving of PBD practice in the Canadian bridge industry.

1.2 Scope

The thesis is focused on the seismic performance-based design of new bridges with ductile

reinforced concrete substructures. This class of bridges is one of the most prevailing classes of

bridge structures in British Columbia, Canada, and therefore, it was selected for this study. Only

the performance criteria relevant to ductile reinforced concrete bridges were studied. Evaluating

the seismic performance of existing or retrofitted bridges is beyond the scope and interest of the

thesis.

In preparing the case studies, multi-span reinforced concrete bridges with steel girders or

pre-stressed concrete girders were utilized. The case studies included two types of abutment

conditions, namely seat-type abutment with expansions joints and semi-integral abutment. Other

types of abutment conditions such as integral abutments were not covered in the examples. It is

recognized that there are numerous configurations possible for a bridge within this class and

further studies may include archetypes of common ductile reinforced concrete bridge

configurations in British Columbia, instead of individual examples. In addition, the limited scope

of the case studies does not compromise the applicability of the outcomes to other bridges within

this class of bridges. One-of-a-kind bridges or bridges with special seismic resistant systems

were not addressed herein. Although, the identified challenges of implementing the PBD

provisions and some of the provided solutions are still valid and applicable to these bridges as

well.

The modelling of soil-structure interaction effects was limited to foundation flexibility

effects, using compliance springs. Kinematic interaction and foundation damping were not

considered in the case studies, as including their effects were not indispensable to the focus of

the thesis. The study of the remaining capacity of bridges in aftershock events is out of the scope

of the thesis, and it is not covered. In selecting ground motions for nonlinear-time history

analysis, only far-field ground motions were considered and near-fault motions were excluded to

3

narrow the scope of the study. The effects of directionality on the response of the studied bridges

were ignored, as the case study bridges were relatively short with regular configurations.

1.3 Overview of the Thesis

The thesis opens with a background literature review on the performance-based seismic design

of structures with a focus on bridge structures in Chapter 2. The chapter reviews the progress of

PBD procedures in both building and bridge industries and summarizes the essence of a number

of existing design and assessment frameworks. An overview of the CSA S6-14 PBD provisions

is then included to serve as a reference throughout the thesis for those readers who are not

familiar with the provisions.

Chapter 2 is followed by a critical review of the CSA S6-14 PBD provisions in Chapter 3.

The chapter identifies the merits and challenges of implementing the new PBD provisions in

practice. Once these challenges are recognized, the chapter lists the actions required to address

them and to facilitate the implementation of the PBD provisions. This chapter sets the flow of the

thesis, as the following chapters each try to address a part of the identified challenges, within the

established scope of the thesis in Section 1.2.

The first of these challenges is addressed in Chapter 4, which is dedicated to searching for

suitable frameworks for CSA S6-14 PBD approach. The chapter recommends both a

deterministic and a probabilistic framework to be utilized with the CSA S6-14 PBD approach to

verify the performance objectives. The details and steps of each framework are presented.

Chapter 5 encompasses suitable modelling alternatives for verifying the performance of

reinforced concrete bridges. The chapter covers the strain localization issues associated with the

use of strain-softening materials in distributed plasticity models of RC components. A series of

previously tested reinforced concrete bridge columns are then employed to validate the models.

The predictions of different modelling alternatives are compared against one another to give a

perspective on the capability of each model to accurately predict the test results.

Chapter 6 is a follow-up chapter to Chapter 5 and aims to evaluate the strain limits of CSA

S6-14 and BC MoTI Supplement to CSA S6-14 (2016). A brief background review on strain and

deformation limits recommended in various references for predicting the damage states of ductile

reinforced concrete components is presented. The last part of the chapter uses the validated and

4

verified models of Chapter 5 and the strain limits of CSA S6-14 and BC MoTI Supplement to

predict the damage to the test columns and compares the predictions with the actual test results to

evaluate the strain limits.

Chapter 7 provides a detailed application of both of the deterministic and probabilistic

assessment frameworks introduced in Chapter 4 to a reinforced concrete bridge with steel girder.

This chapter takes all the elements introduced and examined in the previous chapters into

account, and combines them together within a case study. A full comparison of the two

frameworks and their advantages and disadvantages are discussed at the end of the chapter.

Chapter 8 constitutes the second case study of the thesis. The Trent River Bridge, a major-

route reinforced concrete bridge with pre-stressed concrete girders, is selected for this study.

However, the bridge is treated as a newly designed bridge so that it can be employed for the

purpose of this thesis. Moreover, the performance of the bridge is evaluated assuming three

different scenarios, where the bridge is assumed to be a lifeline, major-route, and other bridge.

The focus of the chapter is to identify the governing performance criteria in each scenario. This

is especially important for practicing engineers, since it helps them to reduce the design trial and

error process by initiating the design verification at the governing performance level.

Chapter 9 concludes the thesis by summarizing the research outcomes and main

contributions and discussing the visions of the author for future research in this area. The thesis

also includes four appendices. Appendix A simply provides copies of the CSA S6-14 tables used

with the PBD approach. Appendix B presents the step-by-step derivation of the expressions in

Chapter 4 for adjusting the local monotonic response of RC components modelled with

distributed plasticity models. Appendix C contains the acceleration and displacement response

spectra of the motions selected for nonlinear time-history analysis in Chapter 7. Finally,

Appendix D demonstrates a design aid as a solution to reduce the trial and error effort when

nonlinear time-history analysis is employed to verify PBD. The application of the design aid is

demonstrated on a number of reinforced concrete bridge columns designed to CSA S6-14, as

well as the case study bridge of Chapter 7. This was presented as an appendix, to keep the main

body of the thesis coherent.

5

Chapter 2: Background on Performance-Based Seismic Design

This chapter presents a brief review of the evolution of performance-based design procedures

and their implementation in bridge and building codes and design guidelines. The review

summarizes some of the well-recognized challenges that performance-based design procedures

have faced and overcome through the last two decades. This is followed by a selective review of

the three widely accepted frameworks for design/assessment of structures, including load and

resistance factor design, PEER performance-based earthquake engineering framework, and the

more recent ASCE/SEI 7-16 (2016) framework. The purpose of the review is to provide a

background and a point of comparison for Chapter 4, where a deterministic and a probabilistic

assessment framework are selected for the CSA S6-14 PBD approach. Finally, an overview of

the CSA S6-14 PBD provisions is presented to conclude the chapter.

2.1 A Shift in Seismic Design Philosophy

The eleventh edition of the Canadian Highway Bridge Design Code (CSA S6-14) has introduced

performance-based design approach for the seismic design of lifeline and major-route bridges in

highly seismic zones, and considers it as an option for the design of other bridges. The force-

based design (FBD) approach in the previous edition of the code, CSA S6-06 (2006), was quite

similar to AASHTO LRFD Bridge Design Specifications (2004). In both cases, seismic design

forces acting on ductile substructure elements were obtained by dividing the resulting forces

from elastic analysis by a reduction factor, R, which accounted for the amount of ductility in the

elements. The values of R for different ductile substructure elements ranged from 2 to 5 and were

in fact, lower than the expected displacement ductility capacities of the elements. This

conservatism was due to the fact that the procedure was intended to be applied to designing of a

wide range of bridge geometries (Tehrani 2012). The seismic design forces were also affected by

the importance of bridge through the importance factor, and by the type of soil profile through

the site coefficients.

While FBD has long been the standard seismic design approach recommended by various

guidelines and codes, and has been widely adopted in practice, several problems have been

associated with the use of this design approach. Priestley et al. (Priestley, M. J. N. and Calvi

6

2007; Calvi et al. 2013) has summarized some of the main issues with the FBD approach,

including the following:

1. The equal displacement rule, which relates inelastic displacement demands to elastic

demands, does not hold in many cases.

2. The implicit assumption of equal displacement ductility demand for different ductile

substructure elements when using the same force reduction factor may not hold. For

instance, columns with similar cross-section but different heights have different

displacement ductility capacities. Nevertheless, the same force reduction factor applies to

all of them.

3. In FBD, it is assumed that stiffness is independent of strength, and thus yield curvature is

proportional to flexural strength. However, it has been shown that this assumption is

invalid, and for a given section the yield curvature is independent of strength, and

stiffness is proportional to strength.

One major consequence of the aforementioned issues is that the structures designed with similar

R values may experience different levels of damage at a given hazard level; i.e. the FBD

reduction factors are not fully representatives of the expected level of damage to the structures.

The goal of seismic design changed after the occurrence of major earthquakes in the 1990s,

such as the 1994 California Northridge earthquake and the 1995 Japan Kobe earthquake1

(Ghobarah 2001; Jalayer 2003). The extent of the damage to buildings and infrastructure due to

these earthquakes was unexpectedly significant and enormous direct and indirect financial losses

were incurred. It became clear that there was a need for new seismic design approaches that

would enable designing structures with more predictable performances. Providing collapse

prevention was neither the primary, nor the sole goal of seismic design any more. Instead,

controlling and limiting the level of damage to structures and the subsequent losses became the

primary design objective. PBD emerged and evolved from this shift in seismic design

1 Although earlier examples of this shift in seismic design practice can be found-such as the two-level

seismic design in British Columbia in mid-‘80s, or the displacement-based seismic retrofit of bridges in British

Columbia commencing in 1990-the occurrence of the aforementioned events are typically considered as the

motivation of sudden movement towards adopting performance-based seismic design philosophy.

7

perspective, and ever since its birth, it has been implemented gradually in various forms in many

seismic design and retrofit guidelines and standards around the world.

2.2 Progress of PBD Procedures in Building Industry

The progress of seismic PBD procedures can be roughly broken down into three stages:

1. Displacement-based seismic design

2. First generation performance-based earthquake engineering

3. Next generation performance-based earthquake engineering

The above classification is based on the evolution of performance-based procedures both

in terms of concept and framework. The performance-based design label can be applied to any

design approach that ensures that structures meet certain performance objectives. In view of this

definition, all limit state design procedures can be considered as performance-based procedures.

Conventionally, in seismic design the performance was synonymous to strength, and only

strength-based criteria were considered. However, as mentioned earlier, the seismic design

philosophy has gradually evolved towards designing for improved seismic performance in terms

of enhanced safety and reduced damage. Attention was paid to quantifying the inelastic

deformation and ductility capacities of structural components, which were more indicative of the

structural damage. In the early stages, the pure seismic FBD was improved by the addition of

displacement checks and related modification of the design strength. This improved design

approach was referred to as displacement-based design or performance-based design (Priestley,

MJN 2000).

The early efforts towards improving the seismic design approach evolved into the first

generation of performance-based design and assessment procedures. Several conceptual

frameworks were proposed such as SEAOC Vision 2000 (1995), ATC-40 (1996b), FEMA 273

(1997) and its later version FEMA 356 (2000a), and FEMA 350 (2000b). The first generation

performance-based procedures were different from earlier procedures in the definition of

performance objectives and their relation to the expected intensity of seismic ground motions.

The performance objective in Vision 2000 was defined as “an expression of the desired

performance level for each earthquake design level” (Krawinkler 1999). The coupling of seismic

ground motion levels with performance levels as described in Vision 2000 is shown in Figure

8

2.1. As illustrated, the performance levels were labeled with names that were representative of

the expected level of overall damage (FEMA 2006). To verify the level of damage from analysis,

specific values of structural response parameters were associated to each performance level and

the overall level of damage. Examples of such response parameters included ductility, story drift

indices, damage indices, floor acceleration, and velocity (Bertero and Bertero 2002). A

schematic visualization of the first generation performance based procedures is illustrated in

Figure 2.2.

While the first generation performance-based earthquake engineering took some major

steps towards designing for enhanced seismic performance, they suffered from three main

shortcomings as listed by Moehle and Deierlein (2004):

1. Inaccuracy of simplified analysis techniques in predicting engineering demands,

including static and linear methods. In case of using more refined analysis methods such

as non-linear analysis, a lack of calibration between calculated demands and component

performance is a major issue.

2. The relations between engineering demand and component performance are defined

based on laboratory test results, analytical models, or engineering judgment, which

results in inconsistency in the established relations. Consistent approaches for defining

these relations are necessary.

3. The assumption of equality of the overall system performance to the worst performance

calculated for any component in the structure. This assumption does not hold for many

structural systems, and does not give a realistic prediction of the system capacity.

One of the key documents showcasing the implementation of performance-based

procedures was ASCE 41-06 (2006), Seismic Rehabilitation of Existing Buildings. The

document was the continuation of FEMA 356 and it can be considered as a link between the first

and next generation performance-based procedures. The performance assessment approach

adopted in this document was quite comprehensive and considered many aspects of the

performance of structural and non-structural components. In this approach, the rehabilitation

objective selected from Table 2.1 determined how many hazard levels were considered in

performance assessment and what performance level was expected at each hazard level. For

9

Figure 2.1 SEAOC Vision 2000 recommended seismic performance objectives for buildings (SEAOC 1995)

Figure 2.2 Visualization of the first generation performance-based earthquake engineering (Moehle and

Deierlein 2004)

instance, if the building was rehabilitated for the basic safety objectives (BSO) of “k” and “p”,

then it should have been at life safety performance level under rare earthquakes and at collapse

prevention performance level under MCE. The document linked target building performance

levels to structural and non-structural performance levels, and indicated which combinations of

structural and non-structural performances were acceptable for a target building performance

level. Each of the structural and non-structural performance levels were defined qualitatively and

quantitatively based on the type of structural element or non-structural system/component. ASCE

10

41 also provided separate acceptance criteria for deformation-controlled and force-controlled

actions on an element-by- element basis employing either linear or nonlinear methods.

Several shortcomings were identified with the ASCE 41-06 performance assessment

approach. The approach did not directly assess the economic losses, which is of great concern

when design decisions are made. The performance of the system was assessed in terms of

individual structural and non-structural components rather than a system. Also, the reliability of

the approach in delivering the expected performances was not characterized.

Target Building Performance Levels

Operational

(1-A)

Immediate

Occupancy

(1-B)

Life Safety

(3-C)

Collapse

Prevention

(5-E)

Des

ign H

azar

d L

evel

Frequent

(50%/50-72 years) a b c d

Occasional

(20%/50 -225 years) e f g h

Rare

(10%/50-474 years) i j k l

MCE

(2%/50-2475 years) m n o p

Notes: The rehabilitation objectives in the above table may be used to represent the

following three specific rehabilitation objectives:

- Basic Safety Objectives (BSO): k and p

- Enhanced Objectives: k and m, n, or o; p and i or j, k and p and a, b, e, or f;

m, n, or o alone

- Limited Objectives: k alone; p alone; c, d, g, h, or l alone

Table 2.1 Rehabilitation objectives in ASCE 41-06 (2006).

With an aim of overcoming the shortcomings of the first generation performance-based

procedures, and to better communicate the performance objectives to stakeholders, the next

generation performance-based procedures were developed. The hallmark of the next generation

procedures was expressing the performance objectives in terms of the expected value of some

decision variable such as amount of loss within a probabilistic framework, which would take into

account the sources of uncertainty effectively. The loss in this context, primarily referred to

direct economic loss (repair cost), indirect economic loss (downtime and business interruption),

and casualties (injuries and death) (Whittaker et al. 2007). Two major projects, namely the ATC-

11

58 and later ATC 58-1, were devoted specifically to develop the next generation of performance-

based seismic design and assessment guidelines for new and existing buildings. The outcomes of

the two projects were documented under FEMA 445 (2006) and FEMA P-58-1 (2012). These

two documents contained a generic probabilistic framework and methodology for the next

generation performance-based procedures, which were developed based on breaking down the

process into the following four distinctive elements:

1. Hazard Analysis

2. Structural Analysis

3. Damage Analysis

4. Loss Analysis

The above framework for performance-based earthquake engineering was developed

originally at the Pacific Earthquake Engineering Research Center (PEER), and is also referred to

as the PEER PBEE framework. There are numerous ongoing studies to develop and enhance

fundamental elements of the PEER PBEE framework. The main goal of these studies is to lay

out the necessary foundation for implementing the next generation performance-based

procedures in design guidelines and ultimately in practice.

2.3 Progress of PBD Procedures in Bridge Industry

The PBD procedures in bridge industry progressed in parallel with the building industry,

although it had a different manifestation in design guidelines. The basic seismic design

philosophy for ordinary bridges used to be a no-collapse-based design, i.e. to prevent collapse

during major earthquakes and to withstand more frequent earthquakes with minimal damage.

Since 1989, there has been a shift in the California Department of Transportation (Caltrans)

seismic design guidelines from FBD towards displacement-based design with an emphasis on

capacity design (Duan and Li 2003). For instance, Caltrans Seismic Design Criteria 1.7 (Caltrans

2013) requires that three displacement-based criteria must be met, including global displacement,

demand ductility, and capacity ductility. Another example is AASHTO guide specifications for

seismic bridge design (AASHTO 2011), which also uses a displacement-based design approach.

However, the first distinctive appearance of PBD concept for the seismic design of bridges can

be found in ATC-32 (1996a) and ATC-18 (1997) documents. In these documents, performance

12

levels in terms of overall structural damage and serviceability were described for two levels of

functional evaluation and safety evaluation ground motions, for ordinary and important bridges

(Table 2.2). The structural damage for each performance level was described as minimal,

repairable, or significant. The repairable damage referred to a level of damage that could be

repaired with minimum risk of functionality. The significant damage on the other hand referred

to a level of damage that would require closure for repairs, although exposing minimum risk of

collapse. With regards to the descriptions of performance levels, these documents could be

considered as the first generation seismic performance-based procedures for bridges.

Another appearance of PBD concept was the use of a five-level performance evaluation

approach by the University of California, San Diego (UCSD) for the development of a bridge

performance database. The approach was developed based on the results of the tests conducted at

UCSD on bridge components and systems. The classification of performance in five levels

followed the previous work in buildings, while the trend for bridge industry was moving towards

a two or three-level approach (Hose and Seible 1999). The performance evaluation consisted of

three stages. In the first stage called bridge damage assessment, the classification of structural

damage was related to socio-economic descriptions as shown in Table 2.3 . The second stage

called bridge performance assessment employed the same five levels in the first stage to describe

performance qualitatively and quantitatively. This stage involved field investigations of

structural damage after an earthquake, detailed assessment of laboratory, and/or detailed

analysis. The final stage of the assessment procedure, called bridge performance/design

parameters, involved investigating the correlations of a series of quantitative parameters,

including strain limits, curvature ductility, drift ratio, etc., with the qualitative performance

levels. The aim of this stage was to identify the parameters that would correlate best with the

specified performance levels, and to use these parameters later for design.

One other document that specifically employed performance-based concepts was the

FHWA Seismic Retrofitting Manual for Highway Structures (2006). The guideline categorized

bridges as either standard or essential with three anticipated service life periods of 0-15 (ASL1),

16-50 (ASL2), and >50 years (ASL3). Two seismic hazard levels with 100-year and 1000-year

return periods were considered. The minimum performance levels were then defined for a certain

type of bridge with an anticipated service life period and a specified hazard level (Table 2.4).

13

Level of Damage and Post-Earthquake Service Levels

Important Bridge Ordinary Bridge

Ground Motion at Site Service Damage Service Damage

Functional Evaluation Ground Motion Immediate Minimal Immediate Repairable

Safety Evaluation Ground Motion Immediate Repairable Limited Significant

Table 2.2 Seismic performance criteria in ATC-32 (1996a) and ATC-18 (1997)

Level

Damage

Classification Damage Description

Repair

Description

Socio-economic

Description

I No Barely visible cracking No repair Fully Operational

II Minor Cracking Possible repair Operational

III Moderate Open cracks, Onset of spalling Minimum

repair

Life Safety

IV Major Very wide cracks, Extended concrete

spalling

repair Near Collapse

V Local

Failure/Collapse

Visible permanent deformation,

buckling/rupture of reinforcement

replacement Collapse

Table 2.3 Bridge damage assessment stage of the five-level performance evaluation procedure developed by

the University of California, San Diego (Hose and Seible 1999)

Bridge Importance and Service Life Category

Standard Essential

Earthquake Ground Motion ASL1 ASL2 ASL3 ASL1 ASL2 ASL3

50%/75-100 years PL0 PL3 PL3 PL0 PL3 PL3

7%/75-1000 years PL0 PL1 PL1 PL0 PL1 PL2

Notes: The tolerable damages in each performance level is as follows:

- PL0 (No Minimum): no minimum level of performance is recommended.

- PL1 (Life Safety): significant damage is sustained and service is significantly disrupted, but life

safety is assured. The bridge may be replaced after a large earthquake.

- PL2 (Operational): sustained damage is minimal and after inspection and clearance of debris, full

service for emergency vehicles should be available. Bridge should be repairable disregarding the

traffic flow restriction.

- PL3 (Fully Operational): sustained damage is negligible, and after inspection and clearance of

debris, full service for all vehicles is available. Damage to the bridge is repairable without

interrupting traffic.

Table 2.4 Minimum performance levels for retrofitted bridges in FHWA Seismic Retrofitting Manual (2006)

14

Four performance levels were specified in terms of overall structural damage and serviceability,

including no minimum, life safety, operational, and fully operational levels. A description of the

structural damage levels termed as negligible, minimal, and significant was provided. Although

the retrofitting manual defined performance objectives for different importance, service life, and

hazard levels, it did not clarify how the performance levels should be linked to structural damage

limit states. It would have been the task of engineers to establish this link and set criteria that

would deliver the anticipated performance (Transportation Research Board 2013).

In line with the above changes in design and retrofitting guidelines, PBD was adopted in

the latest edition of the Canadian Highway Bridge Design Code as a standard design approach.

The basic premise of the CSA S6-14 PBD approach is to meet multiple performance criteria

defined in terms of tolerable structural damage, and serviceability objectives at multiple hazard

levels. CSA S6-14 has indeed taken a significant step towards enhancing seismic design practice

of bridges in Canada. It is also fair to say that such explicit adoption of PBD is quite unique

among the current bridge design guidelines. If held to its premises, the performance-based design

would deliver bridges with more predictable and reliable seismic performances.

2.4 Performance-Based Design vs Assessment

The structural design problem under any loading condition such as seismic loading, entails

finding a valid structural system that satisfies the design limit state(s) (objective function) subject

to a number of limitations (constraints). Two major considerations in structural design process

are uncertainties and optimization (Royset et al. 2001). Uncertainties must be properly taken into

account to ensure that the safety of design is not compromised due to randomness in materials or

input loading, etc., or due to the lack of knowledge or bias. Optimization of design is also

desirable since it maximises benefits (minimizes cost) of satisfactorily meeting the design limit

states. Therefore, the structural design problem is essentially an optimization problem under

uncertainties.

As mentioned in Section 2.2, conventionally the seismic design limit state was formulated

based on strength criteria. With the progress of PBD procedures, the seismic design problem

included more limit states to meet multiple performance criteria at multiple hazard levels.

Moreover, the design goal in PBD is often to meet the specified limit states with an acceptable

15

level of confidence over the service life of the structure (Zhang and Foschi 2004; Moller et al.

2015). To evaluate the level of confidence for each limit state, reliability analysis techniques are

required to estimate the probability of failure. Consequently, finding the best solution to a PBD

problem involves conducting optimization of structural design under reliability-based

constraints. Achieving such optimal solution is not straight forward. A lot of research effort has

been dedicated to facilitate the solution of these complex optimization problems and a number of

techniques have been developed (Madsen et al. 1986; Wen 2001; Zhang and Foschi 2004;

Ellingwood, Bruce R. and Wen 2005; Haukaas 2008). Even so, the solution process is very

computationally intensive and most often requires using reliability analysis software. This is a

downside, since most often the reliability analysis tools are not commercially available or

compatible with the common structural analysis tools.

All of the above-mentioned hurdles are the reasons for engineers to take a practical

approach to achieve an acceptable but not necessarily optimal design solution. Such solution is

obtained through trial and error process. For the initial trial, the structure is proportioned based

on the best practice and designer’s experience, which are the key factors to reduce the number of

trial and error efforts before an acceptable solution is reached. The better the initial design, the

less the number of trials. For each design trial, performance limit states are assessed and the

probabilities of failures are evaluated. If the trial meets all of the specified performance limit

states with the specified minimum levels of confidence, then the design is deemed acceptable.

Otherwise, modifications are made to the design trial and the process is repeated. While such

approach does not necessarily find an optimal solution, it provides a good enough feasible

solution. The initial proportioning of the structure based on experience, guarantees the feasibility

of solution in many cases.

Throughout the rest of this thesis, the second approach is assumed to solve PBD problems.

While the first approach has an evident upside in scientific applications, it is too complex for

design code applications with the current state of practice. Therefore, the second approach is

more suitable for the purpose of this thesis. Within the context of the second approach, design

and assessment are inter-connected, as the design process is comprised a series of assessments on

design trials. As a result, in many instances throughout the thesis, the two terms of performance-

16

based design framework and performance-based assessment framework are used

interchangeably.

2.5 Review of a Number of Existing Design/Assessment Frameworks

2.5.1 Load and Resistance Factor Design (LRFD)

Load and resistance factor design or LRFD is a design approach that incorporates both load and

resistance factors to achieve an implicit desired level of reliability for the applicable range of

structures. LRFD is the term used in the United States to refer to multiple-factor design formats.

Other names have been used to refer to such formats in their earlier implementations in design

guidelines, such as partial safety factor format in 1940 reinforced concrete and geotechnical

standards of Denmark, ultimate strength design in 1950 reinforced concrete standard of the

American Concrete Institute, and limit state design in the National Building Code of Canada

since 1977 (Madsen et al. 1986). Using LRFD approach, the design is deemed acceptable for the

specified limit states if the following inequality is met:

ϕRn≥ ∑ γ

iQ

ni

i

1

(2.1)

where Rn is the normal strength calculated from the given equations in design guidelines and Qni

is the load effect, ϕ is the resistance factor, and γi is the load factor for load effect i, respectively.

The LRFD was an improvement to the earlier allowable stress design approach, which utilized a

single factor of safety in the design equation as follows:

Rn/F.S.≥ ∑ Qni

i

1

(2.2)

By using more than one factor, LRFD provided more consistency in the provided level of safety

for complex loading conditions and various structural members. The calibration of the load and

resistance factors in LRFD is done considering that resistance, R, and load effect, Q, are random

variables and by finding optimum ϕ and γi factors, which would satisfy P[R<Q]<Pf (the

probability of resistance being exceeded by the load effect is less than Pf ). Pf is the allowable

probability of exceeding a limit state. For code calibration purposes, Pf is preferably expressed in

terms of a reliability index, β, as follows:

17

β= -Φ-1(Pf) (2.3)

In which, Φ is the Gaussian (or Normal) cumulative distribution function2. Typical values of β

range from 2 to 6, for which increase of one unit corresponds roughly to the decrease of one

order of magnitude in Pf (Galambos 1981). It is possible to calibrate ϕ and γi factors using

various reliability analysis techniques. A common practice in code calibration in earlier days was

to utilize first-order second-moment reliability analysis, which considered only a central value

such as mean and a measure of dispersion such as coefficient of variation for random parameters.

This method however, did not consider the actual distribution of random variables. More refined

methods such as first-order and second order reliability analysis methods may be utilized instead.

A thorough application of such methods to develop the load factors for the American National

Standard A58 can be found in Ellingwood et al. (1980).

2.5.2 PEER Framework

The Pacific Earthquake Engineering Research Center (PEER) formulated a probabilistic

framework for the seismic assessment of structures, which was based on acceptable probability

(or the associated frequency) of exceeding specific performance levels (Cornell and Krawinkler

2000). To obtain such probability, PEER broke down the process of performance-based

assessment into four elements of hazard analysis, structural analysis, damage analysis, and loss

analysis. Subsequently, the annual frequency of exceeding a performance level is obtained

through a triple integral, using the total probability theorem as follows:

λ(DV)= ∭ G⟨DV | DM⟩ |dG⟨DM | EDP⟩| |dG⟨EDP | IM⟩| |dλ(IM)| (2.4)

In the above formula, the IM, EDP, DM, and DV denote intensity measure, engineering demand

parameter, damage measure, and decision variable; G(A|B) denotes conditional probability of

exceeding B given A; and 𝜆(DV) is the annual frequency of exceeding. The four elements of the

PEER framework can be briefly described as below:

2 It should be noted however that in general, R-Q does not necessarily have a normal distribution, but for

most practical cases an approximate value of the reliability index β can be obtained this way (see Cornell 1969;

Madsen et al. 1986).

18

- Hazard Analysis:

In this step, one or more ground motion intensity measures are evaluated and appropriate

input ground motions for response history analysis are selected. Standard IMs are peak

ground acceleration and spectral acceleration, which can be evaluated through conventional

probabilistic seismic hazard analysis (PSHA). IMs are expressed as the mean annual

probability of exceedance for the particular location and design characteristics of a facility.

- Structural Analysis:

Having IM and input ground motions determined, structural analysis is performed next to

obtain EDP values. The most common EDPs for structural components in buildings are inter-

storey drift ratios, inelastic component deformations and associated forces; and for non-

structural components are inter-storey drift ratios, floor accelerations, and floor velocities. To

determine the relation between the EDPs and IM, inelastic simulations are typically carried

out, in which aspects of structural and geotechnical engineering and soil-structure-

foundation-interaction are considered. One of the procedures developed by PEER to

systematically calculate the conditional probability of p(EDP|IM) is the incremental dynamic

analysis (IDA) (Vamvatsikos and Cornell 2002).

- Damage Analysis:

In this step the EDPs are related to the physical damage to a facility through damage

measures. DMs provide the description of damage to structural/non-structural components

and contents, along with the necessary repairs and functionality condition of the facility. The

conditional probability of p(DM|EDP) is typically referred to as fragility relation or fragility

curve.

- Loss Analysis:

The last step in the framework is to aggregate the loss due to foreseen damage and calculate

decision variables most suitable for decision making. This is achieved by integrating

p(DV|DM) with the mean annual DM probability of exceedance p(DM), obtained in the

previous step.

19

2.5.3 ASCE/SEI 7-16 Performance Assessment Framework

ASCE7/SEI 7-16 (2016) defines the performance goals in terms of the tolerable probability of

collapse under the maximum considered earthquake (MCER) ground motions. This probability

depends on the risk category of the building, as shown in Table 2.5. The stated collapse

performance goals are evaluated implicitly through prescribed set of analysis rules and

acceptance criteria, as proposed in Chapter 16 of the standard.

Risk Category Tolerable Probability of Collapse Ground Motion Level

I or II 10% MCER

III 6% MCER

IV 3% MCER

Table 2.5 Performance goals in ASCE/SEI 7-16

The acceptance criteria ensure that the building meets the performance-goals of Table 2.5,

and are categorized as follows:

- Global acceptance criteria defined for the average story drifts, maximum story drifts, and

residual story drifts

- Element-level acceptance criteria including the ones defined for force-controlled (brittle)

components and the ones defined for deformation controlled (ductile) components

For force-controlled components, the acceptance criteria follow the framework employed by the

PEER TBI guidelines (2008) with the following expression (Haselton et al. 2014):

λ FmeanDemand ≤ ϕ Fn,e

(2.5)

In which, 𝜆 is a calibration parameter, ϕ is the strength reduction factor, and Fn,e is the nominal

strength computed considering expected material properties. Assuming that both of component

demand and capacity follow a lognormal distribution, the ratio of 𝜆/ϕ is calibrated to meet the

tolerable probabilities of failure depending on how critical the component is to the collapse of

the building. For instance, for critical components the calibration of 𝜆/ϕ is done for 10%

probability of failure, and for ordinary components for 25% probability of failure. An overview

of the calibration process is presented in the commentary of Chapter 16 of ASCE7/SEI 7-16. The

calibrated acceptance criteria for force-controlled components (with the exception of capacity-

controlled components) are as follows (Haselton et al. 2015):

20

- For critical components, where the failure of the component would likely lead to a

progressive global collapse of the building, it is required that:

2.0 IeFmeanDemand ≤ FmeanStrength (2.6)

- For ordinary components, where the failure of the component would lead to a local collapse,

then:


- For non-critical components, where the failure of the component would not lead to any

structural instability, then:


In the above expressions ϕ=1 is used to calculate the mean strength, and Ie is the importance

factor, which is equal to 1.00 for seismic risk category I and II, 1.25 for III, and 1.5 for IV. The

reason for using the Ie factor is to account for the lower tolerable probabilities of collapse for

seismic risk categories of III and IV.

For critical deformation-controlled components, the acceptance criterion is proposed as

(Haselton et al. 2015):

DriftmeanDemand

≤ 0.3 to 0.5/Ie Driftcapacity

(2.9)

For ordinary deformation-controlled components, the criterion is modified as:

DriftmeanDemand

≤ 0.5 to 0.7/Ie Driftcapacity

(2.10)

The lower bound capacities in the above two expressions, correspond to the case where no

redistribution path for the gravity loads exist in the structural system, and upper bound values

correspond to the otherwise. The deformations of non-critical deformation controlled

components are limited by the global acceptance criteria and no element-level capacities are

defined for these components. Similar to force-controlled components, drift capacity factors are

obtained to ensure tolerable probabilities of collapse. For instance, in Equation (2.9), the factor

of 0.5 implies a 40% probability of building collapse if the drift capacity is exceeded in a single

component, and the factor of 0.3 implies a 100% probability of collapse.

21

2.6 Overview of the CSA S6-14 PBD Provisions

Section 4.4 of CSA S6-14 named as “earthquake effects”, contains the seismic design provisions

of both of the PBD and FBD approaches. The following steps encapsulate the essence of the

CSA S6-14 PBD approach:

1. Determining importance category:

Bridges are categorized in terms of importance as lifeline, major-route, and other bridges.

Major-routes are used for emergency response and movement of people after an earthquake,

and thus the bridges on these routes have higher priorities. Lifeline and major-route bridges

are both located on major-routes, but compared to major-route bridges, life-line bridges are

typically more complex. They require more resources, money, and time to restore following

an earthquake and their downtime have serious and extensive impacts on the economy of the

region of lives of the people. Examples of such bridges are Lions’ Gate Bridge in Vancouver,

British Columbia, and Golden Gate Bridge in San Francisco, California. Other bridges refer

to those bridges located on minor and local routes.

2. Performing seismic hazard analysis and constructing bridge design spectrum:

CSA S6-14 performs PBD at three hazard levels with 475-year, 975-year, and 2475-year

return periods. These will correspond to 10%, 5%, and 2% probabilities of exceedance in 50

years, respectively. For brevity, we will refer to these hazard levels with 10%/50, 5%/50,

and 2%/50 notations from here on through the rest of this thesis. The 5% damped spectral

values of the uniform hazard spectrum (UHS) at the specified hazard levels can be obtained

by conducting probabilistic seismic hazard analysis (PSHA) of the bridge site, using

programs such as EZ_FRISK (The Fugro Consultants 2015). Alternatively, the values of the

uniform hazard spectrum for select locations can be obtained from the 2015 National

Building Code of Canada seismic hazard calculator available online at the Natural Resource

Canada Website (ref). The 2015 hazard maps of Canada (ref) gives values for the peak

ground acceleration (PGA), peak ground velocity (PGV), and peak ground displacement

(PGD), and the 5% damped spectral acceleration at periods of 0.2 s, 0.5 s, 1.0 s, 2.0 s, and

10.0 s. The UHS values are calculated for the reference ground condition, i.e. site class C of

the code (see Table A.1 for site classifications in CSA S6-14). For constructing bridge

22

design spectrum, the UHS values should be multiplied by the recommended period-

dependent site coefficients, to reflect the bridge site ground condition.

The 5% damped spectral values of the UHS should also be modified for damping ratios. The

damping modification factor is calculated from the following expression:

RD= (

0.05

ξ)

0.4

(2.11)

In which ξ is the damping ratio of the bridge, but should not be taken greater than 0.1. If ξ <

0.05, then the 5% damped spectral values are increased. On the other hand, if the bridge

abutment is designed to mobilize the backfill soil and satisfies the conditions of Clause

4.4.3.5 of CSA S6-14, then the 5% damped spectral values can be reduced. For cases where

the abutment is not specifically designed to mobilize the backfill soil, then no modification is

necessary.

3. Determining seismic performance category:

The seismic performance category indicates the level of seismic design requirements for the

bridge. The higher the category number, the higher the requirements of seismic design, either

using PBD or FBD. The seismic performance category is determined based on the principal

period of the bridge in the considered direction, site specific spectral acceleration values at

2%/50 hazard level (the site specific spectral accelerations are the UHS spectral values for

reference site C modified by the site coefficients for the site class at the bridge site), and the

importance of the bridge. Table A.2 copies the CSA S6-14 Table 4.10 for determining the

seismic performance category of a bridge. In case the seismic performance categories of a

bridge for the two principal directions are different, the higher performance category

governs.

4. Determining minimum analysis requirements and design approach:

CSA S6-14 does not mandate PBD for design of all bridges. Table A.3 copies the design

approach requirements of the code (for regular and irregular bridge specification see Clause

4.4.5.3.2 of CSA S6-14). For lifeline bridges regardless of their seismic performance

category and regularity, PBD is the only acceptable design approach, while major-route and

other bridges can in some cases be designed using FBD approach. It is evident that all

bridges may be designed according to PBD requirements.

23

The code also recommends minimum analysis requirements for each of the specified hazard

levels, as copied in Table A.4 and Table A.5. In these tables, ESA is elastic static analysis,

EDA is elastic dynamic analysis (response spectrum analysis, elastic time-history analysis),

ISPA is inelastic static pushover analysis, and NTHA is nonlinear time-history analysis.

5. Determining minimum performance levels and performance criteria:

CSA S6-14 defines minimum performance levels in terms of serviceability and tolerable

structural damage criteria at each of the three hazard levels, as shown in Table A.6. The

minimum performance levels are determined based on the importance of the bridge. For

instance, the minimum performance level of a major-route bridge at 10%/50 hazard level is

minimal damage and immediate service.

Once the minimum performance levels are determined, the PBD criteria can be specified

from Table 4.16 of the code. For instance, to ensure minimal damage in RC bridges, the

compressive strains of concrete should not exceed 0.004 and the reinforcing steel strains

should not exceed yield. The criteria in Table 4.16 are not defined specifically for a particular

type of bridge or lateral force resisting system. Additional performance criteria may be

established for a bridge depending on the type of the bridge structure and its importance.

6. Verifying that the design meets the performance objectives:

In order for the bridge to meet the PBD requirements, it must satisfy all the specified

performance criteria at each hazard level. However, the code does not provide specific

recommendations regarding how engineers should verify that the design meets the

performance objectives.

24

Chapter 3: Merits and Challenges of the CSA S6-14 PBD Approach

This chapter highlights the merits of the provisions of seismic performance-based design (PBD)

in CSA S6-14 (2014) and identifies the challenges that need to be addressed to successfully

implement these provisions in practice. The challenges are categorized as either “general” or

“particular”. General challenges are the main challenges that still hinder the implementation of

PBD procedures in design guidelines, including CSA S6-14. Particular challenges however,

concern only the CSA S6-14 PBD approach and require improvement to the current approach in

future editions of the code. Finally, based on the discussion presented in this chapter, a number

of the identified challenges are selected to be addressed later in the following chapters of the

thesis.

3.1 General Challenges of Performance-Based Procedures

Some of the challenges of the first generation PBD procedures were outlined in Section 2.2. The

next generation performance-based earthquake engineering also faces a number of challenges

ahead of successful implementation in practice. It should also be noted that the necessary basis

for implementation of PBD procedures is significantly less developed for bridge structures

compared to buildings. With that in mind, some of the major challenges of performance-based

procedures with respect to design and assessment of bridges in general are as follows:

1. There is not a unified agreed-upon framework for performance-based design/assessment. The

PEER PBEE framework has been the most referenced framework for research applications.

However, as explained in Section 2.4, seismic design guidelines and codes tend to simplify

the design problem and incorporate simpler deterministic frameworks.

2. The next-generation PBD procedures heavily rely on the use of fragility curves within

probabilistic frameworks. There is still a resistance within the engineering community to use

probabilistic frameworks that utilize fragility curves. Probabilistic frameworks have gained

acceptance for only some specific design problems such as the design of bridge piers against

vessel impacts (see Foschi 2007), or in corrosion studies. This lack of interest is partly due to

the fact that practicing engineers may not be fully familiar with the probability concepts and

reliability analysis and how they can be applied to engineering problems. But primarily, the

lack of interest is due to disbelief in the usefulness of fragility curves for design problems.

25

3. Another obstacle in utilizing a probabilistic framework for the next-generation PBD

procedures in many instances is the lack of reliable data for defining the intervening random

variables. This would result in incorporating inaccurate or subjective statistical inferences to

obtain the properties of the corresponding probability distributions of the variables.

4. Frequently, the tools required for implementation of a probabilistic framework for the next-

generation PBD procedures are not readily accessible outside of academia or available in

commercial software packages. One example is reliability analysis. Despite the many

software packages that have been developed to perform reliability analysis, none have been

integrated seamlessly into commercial structural analysis programs. In addition, in many

cases the developed tools are intended for design and assessment of buildings only.

5. There are very few guidelines and documents dedicated exclusively to the implementation of

next generation PBD procedures. The most comprehensive guidelines on the subject are

FEMA P-58-1 (2012). The second volume provides a step-by-step guide on the

implementation of the methodology introduced in the first volume, and the third volume is

dedicated to the supporting electronic material and documentations. No equivalent guidelines

have been published so far that specifically addresses performance-based seismic design of

bridges.

6. The complexity of the next generation performance-based assessment procedures, demands

enormous computational effort even for the design of simple facilities. Designing by trial and

error utilizing the next generation PBD procedures could become unfeasible considering the

current limitations.

7. To accurately predict the extent of structural damage, damage models are needed that

quantify the structural damage in terms of measurable and meaningful engineering

parameters. A damage model is essentially a capacity model that relates the structural

damage to engineering parameters. Examples of damage models are fragility curves, damage

indices, and other predictive expressions. The values of engineering parameters should be

obtainable via structural analysis, so that calculated demands can be checked against the

capacity obtained from damage models. While there has been a great deal of effort invested

in developing more accurate and reliable damage models of structural components, it seems

26

that there is still a huge gap in knowledge for quantifying physical damage. Some

components such as concrete columns have been the subject of many studies. For these

components, more accurate damage models exist that quantify the level of damage at

different damage states of the components. Example of such models can be found in the work

of Berry and Eberhard (2003) on the flexural damage of concrete columns. They have

developed models that predict the strains corresponding to concrete cover spalling and bar

buckling damage states for concrete columns. Employing these models, one is able to

calculate the associated drift of a column corresponding to the first occurrence of the two

aforementioned damage states using the Priestley et al. (1996) procedure. However, for many

other components, predictive damage models have not yet been developed and the

quantification of damage relies on subjective interpretations of physical damage in terms of

engineering demand parameters. In addition, the developed damage models may not

necessarily provide accurate predictions of damage. For instance, all fragility curves, and

many of the other damage models take only one engineering parameter as an input. However,

it is argued that the state of damage in a component cannot be described accurately utilizing

only a single input parameter. Example of such arguments can be found in Gardoni et al.

(2002).

The aforementioned general challenges of implementing PBD can be summarized as the follows:

1. Need for a unified agreed-upon framework (probabilistic vs deterministic, academic vs

codified)

2. Disbelief in usefulness or lack of familiarity with probabilistic performance-based

frameworks in practice

3. Need for more reliable data to use with probabilistic performance-based frameworks

4. Need for available commercial tools for enhancing the use of probabilistic procedures

5. Need for guidelines for implementing the PBD procedures for other structures than

buildings

6. Need for a systematic performance-based design procedure to avoid high computational

costs

7. Need for more accurate predictive damage models for all structural components

27

3.2 Particular Challenges of the CSA S6-14 PBD Approach

Aside from the general challenges with the next-generation performance-based design, there are

particular challenges in the way of implementing the CSA S6-14 PBD provisions:

1. The code does not provide a clear framework for verifying that a design meets the specified

performance criteria of the PBD approach. To be more specific, it is not clear how the

performance criteria of the code should be formulated in terms of performance limit states

and be evaluated for a structure. From the review of literature presented in Chapter 2:, the

lack of such framework becomes evident.

2. There is inconsistency in how the deformation-controlled and force-controlled performance

criteria of the code should be verified. To verify force-controlled limit states such as the

shear of ductile components, the code recommends using FBD approach, which employs

load and resistance factors. In contrast, for verifying the deformation-controlled performance

criteria such as the strain limits of RC bridges, no load and resistance factors are specified.

This will result in inconsistent reliability achieved for the deformation-controlled versus the

force-controlled limit states.

3. The performance criteria of the code are primarily defined in terms of the performance of

components. It is not clear if the criteria are met, how the performance of components would

relate to the performance of system. It was mentioned earlier in Section 2.2, that ASCE 41

links target building performance levels to structural and non-structural performance levels.

Similar concept could also be used in relating the performance of components or sub-

assemblies to the performance of bridge structural system.

4. It is not specified whether a bridge designed to meet the code performance levels, will

achieve a uniform reliability at all hazard levels or this is of any concern at all.

5. The recommended minimum analysis requirements of the code include elastic dynamic

analysis and linear time history analysis (Table A.4 and Table A.5). These methods may

provide reasonable estimation of inelastic global displacement demand if the conditions of

equal displacement rule hold. Unlike ASCE-41, the code does not recommend modifications

to elastic displacements to incorporate the effects of hysteric behaviour. Moreover, elastic

28

analysis procedures cannot be directly used to check local strain response, the amount of

dissipated energy, or other engineering demand parameters that are indicators of structural

damage. There is no clear recommendation in the code and its commentary on how these

methods should be utilized for such purpose.

6. Other than the above challenges, there are some ambiguities in some of the phrases in the

code. An important one is in the use of expected versus nominal material properties for

performance evaluations. It is stated that for the prediction of the extensive damage and

probable replacement performance levels, expected material properties may be used.

Following that, two different sets of material properties will be required for evaluating the

different performance levels of a bridge. This may lead to inconsistent interpretations of

bridge performance at different hazard levels.

The aforementioned particular challenges of implementing the CSA S6-14 PBD approach can be

summarized as follows:

1. Need for a clear design/assessment framework for the PBD approach

2. Inconsistency in verifying deformation-controlled versus force-controlled performance

criteria

3. Vaguely defined relation of component performance to system performance

4. Vaguely definition of design goals and minimum target reliabilities

5. Vague guidelines on using elastic analysis methods to evaluate the structural damage

parameters

6. Ambiguity in the use of expected and nominal material properties for performance

evaluation

3.3 Merits of CSA S6-14 PBD approach

All the limitations of the FBD approach mentioned in Section 2.1, which are addressed by the

PBD approach, are only part of the merits of the PBD approach. The most important motivation

behind introducing the PBD approach in CSA S6-14 was to encourage and require the discussion

of expected post-earthquake performance of bridges between owners and engineers. PBD

facilitates a wide range of opportunities in conceptual and detailed design of innovative and low-

29

damage structural systems. Most often the seismic performance of such systems cannot be

adequately demonstrated using the FBD approach. Kennedy et al. (2017) provide examples of

such systems, including system with:

Base isolated bearings (lead-rubber bearings, laminated elastomeric bearings, friction-

pendulum bearings, sliding systems, or combination of these items)

Dampers (shock absorbers) and lock-up devices (shock transmitters)

Ductile fuses yielding in flexure or shear

Ductile intermediate or end diaphragms

Bucking restrained braces or ductile or semi-ductile braces

Rocking foundations

FRP (for improved ductility or strength)

3.4 What Is Required to Implement the CSA S6-14 PBD Approach in Practice

The general and particular challenges of the CSA S6-14 PBD approach should not undermine its

merits over the traditional FBD approach and the value that it brings to the seismic design

practice. Rather these challenges should be viewed as opportunities to take the next step towards

implementing the PBD approach in practice.

Some of the general challenges mentioned in Section 3.1 need more resources and time to

be addressed, and in some cases, are subjects of many ongoing research studies. Producing more

test data and developing more accurate models and predictive equations for damage are under

this category. However, the successful implementation of CSA S6-14 PBD approach is not

conditioned on addressing all of the challenges at once. In fact, the particular challenges listed in

the previous section are the ones, which require immediate attention. In line with these particular

challenges, the following actions items are required:

1) Providing an assessment framework for the CSA S6-14 PBD approach; such framework

should be simple enough to implement and should incorporate the elements of the code

PBD approach.

2) Defining acceptance criteria for verifying the code deformation-controlled limit states.

This would entail defining and calibrating load and resistance factors for deformation

performance criteria.

30

3) Investigating whether the CSA S6-14 performance levels result in a uniform reliability

across hazard levels or certain performance levels will control the design.

4) Investigating appropriate structural models and analysis techniques, which would allow

predicting the performance criteria of the code with acceptable accuracy.

5) Providing case studies on implementing the CSA S6-14 approach including the

suggested changes for engineers in practice.

The subsequent chapters of this manuscript aim to tackle the above action items within the scope

of the thesis.

31

Chapter 4: Assessment Frameworks for CSA S6-14 PBD

This chapter recommends a deterministic and a probabilistic assessment framework for verifying

the PBD of bridges according to CSA S6-14. The chapter starts by introducing assessment

framework options and listing the requirements of such a framework for the CSA S6-14 PBD

approach. Subsequently, the details of the two frameworks are described. The deterministic

assessment framework is formulated based on the reviewed literature. For the probabilistic

framework, demand and capacity factor design (DCFD) developed by Jalayer and Cornell

(Jalayer 2003; Jalayer and Cornell 2003) is considered suitable for further examination.

4.1 Assessment Framework Requirements for CSA S6-14 PBD

Ideally, the assessment framework for CSA S6-14 PBD should have the following features:

1. Should enable engineers to verify that their design meets the specified performance limit

states with acceptable probability of exceedance and satisfies minimum target reliabilities

as design objectives.

2. Should account for the uncertainties involved in the design problem.

3. Should be simple enough so that it can be easily adopted and implemented in practice.

4. The necessary tools for implementing the framework should be available to practicing

engineers.

5. Should be robust enough to be applicable to design of all types of bridges, and

accommodate future improvements.

4.2 Deterministic Assessment Framework for CSA S6-14 PBD

Deterministic frameworks are referred to as deterministic, since they do not incorporate random

variables as inputs, and the outputs of the assessment have deterministic values. The

deterministic framework here follows the CSA S6-14 PBD approach step-by-step (see Section

2.6 and Appendix A). However, it introduces additional elements for performance verification.

The additional elements that were considered in formulating the framework are as follows:

i. Reserve Capacity of a Structural Component or System:

Reserve capacity is defined as the ratio of the difference between capacity and demand to

capacity, as follows:

32

𝑅𝐶𝑖𝑗=1-𝐷𝑖𝑗/𝐶𝑖 (4.1)

where, RCij is the reserve capacity of a structural component or system for action i at hazard

level j, Dij is the demand for action i at hazard level j, and Ci is the capacity of the component

or system for action i.

ii. Target Reserve Capacity of a Structural Component or System:

The target reserve capacity represents the design objective, which should be met for design to

be deemed acceptable. The target reserve capacity is denoted here as RCij*, such that:

0≤ 𝑅𝐶𝑖𝑗*≤1 (4.2)

iii. Acceptance Criteria for Performance Limit States:

The performance of a structural component or system with respect to a performance criterion

is deemed acceptable if the reserve capacity of that component or system for the action

associated to the performance criterion is greater than or equal to the target reserve capacity

of the component for the same action.

𝑅𝐶𝑖𝑗≥𝑅𝐶𝑖𝑗* (4.3)

In the above definitions, component refers to the individual members of a system and in this case

a bridge structural system and actions are deformations and forces associated to independent

degrees of freedom of a component.

4.2.1 Implementation Steps

The steps of the deterministic framework for CSA S6-14 PBD are summarized in the flowchart

of Figure 4.1, which are as follows:

1. Determine importance, regularity, and seismic performance category of the bridge.

2. Perform preliminary design and service load design. Initial proportioning of the members

may be carried out based on experience or force-based design equations.

3. Identify possible ductile and non-ductile local and global failure mechanisms of the

bridge. For instance, flexural failure of concrete columns is a local ductile failure

mechanism, shear failure of concrete columns is a local non-ductile failure mechanism,

and unseating of deck is a global failure mechanism.

33

4. Determine appropriate performance criteria at the three hazard levels of 2%/50, 5%/50,

10%/50, based on the minimum performance requirements of the code and the identified

failure mechanism. The performance criteria should be expressed in terms of the limit

states of measurable displacement quantities such as strains, curvatures, etc., or forces

such as shear strength.

5. Obtain the UHS for the bridge site with the 2%, 5%, and 10% probabilities of exceedance

in 50 years and calculate the bridge design spectra at those hazard levels.

6. Depending on minimum analysis requirements, obtain seismic demands using either

response spectrum analysis (RSA) or non-linear time history analysis (NTHA). If

performing nonlinear time history analysis, select and scale a appropriate suite of ground

motions records following the recommendations of CSA S6-14 commentary, or more

sophisticated approaches as outlined in NIST GCR 11-917-15 (2013) or similar

references.

7. Perform structural analysis and determine the seismic demands from response spectrum

analysis or the mean demands from nonlinear time history analysis at each of the

specified hazard levels.

8. Determine capacities of the structural components for each of the related performance

limit states. The drift capacities of ductile components and the structural system can be

determined using nonlinear static pushover analysis. To do so, conduct pushover analysis

of the entire bridge in each of the two principal axes of the structure. Determine drift

ratios in each of the two principal directions that correspond to the first occurrence of the

limit states defined in Step 4 that are related to the ductile failure mechanisms. These

would be the drift capacities for those limit states. For force-controlled actions such as

shear, CSA S6-14 FBD or BC MoTI Supplement equations may be utilized to calculate

capacities.

9. Calculate reserve capacities at each hazard level for each of the limit states defined in

Step 4.

10. Determine desired target reserve capacities for the considered limit states. It should be

noted that the capacities calculated for force-controlled actions using FBD equations, are

factored capacities, unlike the drift capacities obtained as explained in Step 8.

34

11. Check the reserve capacities from Step 9 against the target reserve capacities from Step

10. If the reserve capacities are greater than or equal to the target reserve capacities, the

design is meeting the specified PBD requirements successfully; if not redesign the

components that do not meet the considered criteria and repeat steps 7 to 11. If the period

of the modified design changes from the initial design, or if the bridge system changes,

return to Step3.

If NTHA is conducted to obtain seismic demands, the performance limit states should be also

checked for each of the selected ground motion records. Per recommendation of the code

commentary, the number of ground motions, for which the bridge does not meet the specified

performance criteria, should be limited to one.

35

Figure 4.1 The deterministic PBD assessment framework flowchart

36

4.3 Probabilistic Assessment Framework for CSA S6-14 PBD

Upon reviewing the literature on performance-based design procedures and with regards to the

requirements of Section 4.1 for a suitable assessment framework, the demand capacity factored

design developed by Jalayer and Cornell (Jalayer 2003; Jalayer and Cornell 2003) appears to be

a suitable probabilistic framework to be adopted for the CSA S6-14 PBD approach.

The selected framework offers several advantages. Both Equation (4.19) and (4.20)

provide simplified closed-form solutions for verifying a performance limit state for a desired

MAF of exceedance with measurable confidence. The simplicity of these equations makes them

advantageous for using in engineering practice. The terms in both equations can be readily

obtained with the current available tools in practice and do not demand extra computational

beyond what is already required by the code. A summary of the framework formulation and main

expressions is given in the following section.

4.3.1 Demand and Capacity Factored Design (DCFD)

Demand and capacity factor design is a probability-based framework for seismic design and

assessment. Developed by Jalayer and Cornell, the framework is in fact a special case of the

PEER framework, where a scalar binary decision variable is employed that takes value of 1 when

the capacity for a limit states is exceeded and takes 0, otherwise. The main purpose of the

methodology is to ensure that the structural seismic design meets the specified performance

objectives with a desired guaranteed degree of confidence. The degree of confidence is measured

by setting an upper confidence bound on the probability of exceeding a performance limit state.

The work of Jalayer and Cornell has later been improved upon by other researchers, an example

of which can be found in Mackie et al. (2008). Here, only the main expressions employed in the

framework are reproduced for the convenience of the readers. A full derivation of the framework

can be found in Jalayer (2003) and Jalayer and Cornell (2003).

4.3.1.1 Mean Annual Frequency of Exceeding a Limit State Considering Aleatory

Uncertainty

The mean probability of exceeding a performance limit state is formulated with the following

two approaches (Jalayer 2003):

37

1. Displacement-based approach:

PLS=P[D>C]= ∑ ∑ P[

all x

D>C|D=d] P[D=d|Sa=x] P[Sa=x]

all d

(4.4)

2. IM-based approach:

PLS=P[D>C]= ∑ P[

all x

D>C|Sa=x] P[Sa=x] (4.5)

Both of the above formulations, use total probability theorem to decompose the probability of

demand exceeding capacity into a number of probabilities, which are easier to calculate. P[Sa=x]

is the likelihood that the intensity measure (in this case spectral acceleration) will be equal to a

specific value. This term can be obtained from probabilistic seismic hazard analysis (PSHA) of

the site. The other intermediate terms are conditional probabilities, which can be calculated from

structural analysis and damage analysis.

To facilitate the calculation of the intervening terms in Equations (4.4) and (4.5), some

simplifying assumptions are typically made. The first is to assume that the hazard curve in the

vicinity of Sa=x to follow a power-law relation as below (Jalayer 2003):

𝜈𝑆𝑎(Sa)=P[Sa>x]≅kox-k (4.6)

in which 𝜈𝑆𝑎 is the mean annual frequency (MAF) of exceeding the intensity level Sa. Parameters

of the power-law relation, ko and k, can be found by fitting a line to the hazard curve in the log-

log scale at Sa=x. It is also assumed that the median of the demand at each hazard level has a

power-law relation with the intensity measure, and its aleatory uncertainty can be quantified by a

lognormal random variable (Jalayer 2003):

𝐷 = ��𝐷|𝑆𝑎(𝑥)𝜀 = 𝑎𝑥𝑏𝜀 (4.7)

For the lognormal random variable ε, the median is ��ε=exp(mean(ln(ε))=1, and the

lognormal standard deviation is σln(ε)=βD|Sa also denoted by βRD. The parameters for both the

median power-law relation and the lognormal random variable ε are extracted from the results of

structural analysis performed at the specified hazard levels. Another simplifying assumption is

made for limit state threshold or capacity. It is assumed that the random variable C can be also

38

characterized as a lognormal random variable with median ηc, and lognormal standard deviation

βRC.

With the aforementioned assumptions and using the displacement approach, the MAF of

demand exceeding capacity is formulated once only considering the aleatory uncertainty, as

follows (Jalayer 2003):

��LS=νP[D>C]=��Sa(Sa

��C) exp( k2

2b2β

RD

2⁄ ) exp( k2

2b2βRC

2⁄ )

=ko(��c

a)

-k/b

exp( k2

2b2β

RD

2⁄ ) exp( k2

2b2β

RC

2⁄ )

(4.8)

In the above formula, the hazard level corresponding to the median capacity is multiplied by two

terms that account for the aleatory uncertainties in demand conditioned on intensity measure, and

capacity.

4.3.1.2 Mean Annual Frequency of Exceeding a Limit State Considering Aleatory and

Epistemic Uncertainty

While Equation (4.8) considers sources of aleatory uncertainty in calculating the MAF of

demand exceeding capacity, it does not address epistemic uncertainty3. To improve the equation

with this regard, it is assumed that the intervening random variables can be defined in the

following form (Jalayer 2003):

𝑋 = ��𝑥 𝜀𝜂 𝜀𝑥 (4.9)

where ��𝑥 is the current point estimate of median X, εη and εx are both lognormal random

variables with unit median to represent epistemic and aleatory uncertainties respectively. Using

the above form for the intervening random variables, the MAF of demand exceeding capacity is

obtained as (Jalayer 2003):

νLS=��Sa(Sa

ηC) exp(1/2 βUH

2) exp( k

22b

2(β

RD

2+β

UD

2)⁄ ) exp( k

22b

2(β

RC

2+β

UC

2)⁄ ) (4.10)

In the above expression, βUH, βUD, and βUC, refer to lognormal standard deviation for epistemic

uncertainty in hazard, demand, and capacity, respectively. A comparison between Equations

3 Aleatory uncertainty is due to inherent randomness, while epistemic uncertainty is due to uncertainty in

knowledge.

39

(4.8) and (4.10) reveals that the latter equation is essentially the former equation times the terms

that account for the epistemic uncertainty in hazard, demand, and capacity.

It is then assumed that the epistemic uncertainty in hazard can be accounted for by using

mean hazard instead of median, as follows (Jalayer 2003):

νLS=ν𝑆𝑎(Sa

ηC) exp( k2

2b2(β

RD

2+β

UD

2)⁄ ) exp( k

22b

2(β

RC

2+β

UC

2)⁄ ) (4.11)

Where:

ν𝑆𝑎(Sa

ηC)= ��Sa(Sa

ηC) exp(1/2 βUH

2) =ko(

��c

a)

-k/b

exp(1/2 βUH

2) (4.12)

4.3.1.3 Factored Demand and Capacity Format

Assuming a maximum allowable MAF of exceedance for a limit state denoted by Po, such that:

νLS≤Po (4.13)

It is possible to rearrange Equation (4.11) as follows (Jalayer 2003):

a(Po/ko)-b/k

exp( k 2b (βRD

2+β

UD

2)⁄ )≤ η

c exp( -k 2b (β

RC

2+β

UC

2)⁄ ) (4.14)

The LHS of the above inequality is the factored demand corresponding to MAF of exceedance of

Po, and the RHS is the factored capacity. To distinguish between the involved factors, they can

be denoted as the following (Jalayer 2003):

- Demand factor representing the aleatory uncertainty (in case of NTHA, record-to-record

variability) in demand

𝛾𝑅 = exp( k 2b βRD

2)⁄ (4.15)

- Demand factor representing epistemic uncertainty in demand

𝛾𝑈 = exp ( k 2b βUD

2)⁄ (4.16)

- Capacity factor representing aleatory uncertainty in capacity

ϕ𝑅

= exp( - k 2b βRC

2)⁄ (4.17)

- Capacity factor representing epistemic uncertainty in capacity

40

ϕ𝑈

= exp( - k 2b βUC

2)⁄ (4.18)

Using the above notations, Equation (4.14) can be written as:

γRγ

UD (Po) ≤ ϕ

Rϕ

UC

γD (Po) ≤ϕC (4.19)

The above inequality is the demand and capacity factored design (DCFD) format. This

representation is very similar to LRFD format and therefore is very convenient for use in

practice.

4.3.1.4 Factored Demand and Capacity Format with Confidence Level

The final addition to the DCFD format is to build a confidence interval around the limit state

frequency of exceedance. As such, it could be then stated that the structure would meet a

performance criterion for an allowable MAF of exceedance Po with a confidence level of x%.

The confidence interval is added to the DCFD format of Equation (4.19) through an additional

factor as follows (Jalayer 2003):

γD (Po) ≤𝜆𝑥ϕC (4.20)

where λx is the confidence factor corresponding to confidence level x, defined as (Jalayer 2003):

λx=exp(-βUT

(Kx - k 2b βUT

))⁄ (4.21)

In the above equation Kx is the standard Gaussian (Normal) variate associated with probability of

x not being exceeded, and βUT is the total epistemic uncertainty calculated as follows:

β

UT=√β

UD

2+β

UC

2

(4.22)

Using the DCFD format in Equation (4.20), the performance of a structure for a performance

criterion is deemed acceptable if the ratio of factored demand to capacity is less than λx. The

difference between Equation (4.20) and (4.19) is the addition of λx. It can be considered that

Equation (4.19) is a special case of Equation (4.20), where λx=1.

The DCFD format has been implemented in FEMA-351 (2000b), where Equation (4.20)

represents the acceptance criteria for evaluating the performance of existing welded steel

41

moment-frame building in terms of interstory drift, column axial load, and column splice tension.

In commentary of Section 3.6.1 of FEMA-351, it is explained that the confidence in the above

format is “calculated as a function of the number of standard deviations that factored-demand-

to-capacity-ratio λ lies above or below a mean value” and provides “a measure of the extent that

predicted behaviour is likely to represent reality.”

4.3.2 Implementation Steps

To implement the DCFD framework, the following parameters need to be determined:

1) Hazard parameters k and ko - see Equation (4.6)

2) Demand parameters a and b - see Equation (4.7)

3) Estimate of aleatory and epistemic uncertainties βRD, βUD, βRC, and βUC

- Calculating hazard parameters k and ko

The hazard curve parameters in Equation (4.6) can be obtained readily by fitting a power-law

curve in Microsoft Excel or similar tools to the segment of the hazard curve spanning between

2%/50, 5%/50, and 10%/50 hazard levels. To plot this segment, the values of the UHS spectral

acceleration at the fundamental period of the bridge should be plotted against the MAF of that

hazard level (hazard levels 2%/50, 5%/50, and 10%/50 have MAF of 0.000404, 0.00106, and

0.002105, respectively).

- Calculating demand parameters a and b

The demand curve parameters in Equation (4.7) can also be obtained by fitting a power-law

curve in Microsoft Excel or similar tools to the median of the demand parameter from NTHA at

2%/50, 5%/50, and 10%/50 hazard levels.

- Estimate of aleatory and epistemic uncertainties βRD, βUD, βRC, and βUC

The aleatory uncertainty in demand, i.e. βRD, is the only parameter that needs to be calculated.

For the other three uncertainty parameters, the values recommended in the literature can be

utilized. βRD is calculated by taking the standard deviation of the natural logarithm of the demand

parameter values in Microsoft Excel or similar tools. Since the NTHA is run at three hazard

levels of 2%, 5%, and 10% probability of exceedance in 50 years, three values of βRD will be

obtained. The assumption in developing the DCFD framework is that βRD does not change with

42

the change in the intensity measure (Jalayer 2003). As such, the arithmetic mean of the βRD

values at the three hazard levels can be used for the DCFD framework.

There are a few references in the literature, which provide estimate values for the other

three uncertainty parameters, βUD, βRC, and βUC, in case there is no option for more accurate

measure of these parameters4. Previously, FEMA-350 (2000b) and FEMA P-695 (FEMA 2009)

contained reference values for these parameters, but the most recent recommended values can be

found in Chapter 16 of ASCE/SEI 7-16 (2016). The document provides two different sets of

values for the uncertainty parameters to be used with component force and deformation. The

uncertainty values recommended for component deformation capacity are about twice as large as

those of the component force capacity, due to the lack of enough available test data on the

former. For later reference in the thesis, the uncertainty values recommended for deformation-

controlled actions are reproduced in Table 4.1 and Table 4.2.

Source of Uncertainty in the Deformation Value

Record-to-record variability for MCER ground motions (βRD) 0.4

Uncertainty from estimating deformation demands using structural model (βUD) 0.2

Variability from estimating deformation demands from mean of only 11 ground

motions (βRD)

0.13

Total 0.46

Table 4.1 ASCE/SEI 7-16 assumed uncertainty values for component deformation demand

Source of Uncertainty in the Deformation Value

Typical variability in prediction equation for deformation capacity from

available data (βRC)

0.6

Typical uncertainty in prediction equation for deformation capacity due to

extrapolation beyond data (βUC)

0.2

Uncertainty in as-built deformation capacity because of construction quality

and errors (βUC)

0.2

Total 0.66

Table 4.2 ASCE/SEI 7-16 assumed uncertainty values for component deformation capacity

The steps of implementing the DCFD framework for CSA S6-14 PBD are summarized in Figure

4.2, which are as follows:

4 The value of βRC or the aleatory uncertainty in capacity can be more accurately measured from test data.

The epistemic uncertainty in capacity, βUC, can also be estimated from the bias in test data. The source of βUD or the

epistemic uncertainty in demand according to the Appendix A of FEMA-350 is the inaccuracies in defining the

modelling parameters such as yield strength, viscous damping, foundation flexibility, etc. To calculate this

parameter, a series of structural models should be developed, in which the aforementioned parameters are varied and

time-history analysis is performed to calculate the demands.

43

1. Determine the bridge importance, regularity, and seismic performance category.

2. Perform preliminary design and service load design. Initial proportioning of the members

may be carried out based on experience or force-based design equations.

3. Identify possible ductile and non-ductile, local and global failure mechanisms of the

bridge. For instance, flexural failure of concrete columns is a local ductile failure

mechanism, shear failure of concrete columns is a local non-ductile failure mechanism,

and unseating of deck is a global failure mechanism.

4. Determine appropriate performance criteria at the three hazard levels of 2%/50, 5%/50,

10%/50, based on the minimum performance requirements of the code and the identified

failure mechanism. The performance criteria should be expressed in terms of the limit

states of measurable displacement quantities such as strains, curvatures, etc., or forces

such as shear strength.

5. Obtain the UHS for the bridge site with the 2%, 5%, and 10% probabilities of exceedance

in 50 years by performing probabilistic seismic hazard analysis, or alternatively, use the

tabulated values by the National Building Code of Canada (2015).

6. Calculate hazard parameters k and ko.

7. Select and scale an appropriate suite of ground motions records, following the

recommendations of CSA S6-14 commentary, or more sophisticated approaches as

outlined in NIST GCR 11-917-15 (2013) or similar references.

8. Perform NTHA and determine seismic demands at each of the specified hazard levels.

9. Calculate demand parameters a and b.

10. Determine capacities of the structural components for each of the related performance

limit states. The drift capacities of ductile components and the structural system can be

determined using nonlinear static pushover analysis. To do so, conduct pushover analysis

of the entire bridge in each of the two principal axes of the structure. Determine drift

ratios in each of the two principal directions that correspond to the first occurrence of the

limit states defined in Step 4 that are related to the ductile failure mechanisms. These

44

would be the drift capacities for those limit states. For force-controlled actions such as

shear, CSA S6-14 FBD equations may be utilized to calculate capacities.

11. Compute the demand and capacity factors using Equation (4.15) to (4.18).

12. Compute the confidence factor corresponding to the desired level of confidence (for

example 90%) using Equation (4.21).

13. Check factored drift demand against factored drift capacity for the desired confidence

interval in Step 12, using Equation (4.20). The factored demand at each hazard level is

checked against the factored capacity of those limit states, which are specified by the

code to be checked at that hazard level. Check force-controlled actions using the CSA S6-

14 FBD approach. If the factored capacities are greater than or equal to factored demand,

the design is meeting the specified PBD requirements successfully; if not redesign the

components that do not meet the considered criteria and repeat steps 8 to 13. If the period

of the modified design changes from the initial design then return to Step 6, or if the

bridge system changes return to Step3.

45

Figure 4.2 The probabilistic PBD assessment framework flowchart

46

Chapter 5: Examining Modelling Alternatives of RC Bridges for PBD

At the heart of performance-based design lies the ability to predict engineering demand

parameters that relate to structural performance from analysis. The key to achieve this is

generating appropriate models for structural analysis. An “appropriate model” in this manuscript

refers to any model, which has the following two main characteristics:

1. It enables evaluating the specified performance criteria of CSA S6-14 through predicting

the associated demand and/or capacity parameters.

2. It is computationally affordable and sufficiently accurate to use in practice.

The main objective of this chapter is to examine and compare the common structural

modelling options used in practice and academia for PBD of RC bridges. Special attention is

given to distributed plasticity models with inelastic beam-column elements and fibre sections, as

these models are very advantageous for evaluating local strain response of RC bridges. However,

the use of these models is challenged by mesh-dependency and localization issues, as will be

explained in this chapter. Some techniques such as material model regularization have been

proposed in the literature to deal with the mesh-dependency issue will be reviewed later. The

distributed plasticity models are subsequently utilized to predict the response of a tested RC

bridge column. The recommended methods for dealing with the mesh-dependency issue are

examined by comparing the test results with the model predictions and the effectiveness of each

method is assessed. Concentrated plasticity models are also utilized to predict the response of the

test column and comparison are made between the advantages and disadvantages of using each

of the distributed and concentrated plasticity models with respect to evaluating the strain limits

of CSA S6-14 PBD. The applicability of the final takeaways is examined on four more bridge

column tests.

5.1 Modelling Alternatives for RC Bridges

Structural models of bridges can be classified in several different ways with respect to how they

represent the geometry and behaviour of the actual structure. Each way of classification, allows

comparing models with respect to the feature that the classification is based on. For example,

bridge models are sometimes categorized in terms of the hierarchy of bridge structural system

into global, frame, and bent models. It is also possible to classify bridge models based on the

47

level of complexity in describing the geometry of structure, into three categories of lumped

parameter models, spine or grid models, and continuum models (Priestley, M. J. N. et al. 1996;

Caltrans 2015). The focus of this chapter however, is on another way of classification, which is

concerned with modelling the inelastic response of bridge components due to material

nonlinearity. This classification is especially meaningful for seismic analysis, since it identifies

the capabilities of different models in estimating the various inelastic demand and capacity

parameters that are used in seismic design evaluation.

Nonlinear models are often divided into two categories based on the degree of idealization in the

model (Figure 5.1). At one end, there are distributed plasticity models that are detailed in the

physical presentation of structures and their components. At the other end, there are concentrated

plasticity models, which are primarily phenomenological models of the nonlinear behaviour of

structural components (PEER and ATC 2010). It is also possible to estimate inelastic seismic

demands using modified elastic models. Each of the above three modelling alternatives, namely

concentrated plasticity, distributed plasticity, and linear elastic models with cracked sections

properties, are briefly reviewed and discussed in the following sections.

Figure 5.1 Types of nonlinear component models: (a) and (b) are concentrated plasticity models; (c), (d), and

(e) are distributed plasticity models (NEHRP 2013)

5.1.1 Linear Elastic Models with Cracked Section Properties

Linear elastic models are used when an elastic method of analysis such as EDA or ESA is

employed to estimate seismic demand parameters, or to obtain structural periods. To account for

the nonlinearity due to the initial cracking and yielding of concrete members, effective section

properties are assigned to these members. It is important to use reasonable values for effective

section properties, as the outcomes of elastic analysis are very sensitive to these values. There

48

are several recommendations regarding how the effective section properties should be

determined for various concrete members. Section 4.4.5.3.3 of CSA S6-14 defines the effective

flexural stiffness as the slope of the moment-curvature curve between the origin and the point

corresponding to the first yielding in the reinforcement steel, as follows:

EcIeff

= My

ϕy

(5.1)

In the above expression, Ec is the elastic modulus of concrete, Ieff is the effective moment of

inertia of the section, My is the moment at first yield of the section, and ϕy is the curvature of the

section at first yield. The effective shear stiffness of ductile RC elements is determined based on

the effective flexural stiffness:

(GA)eff

= Gc Acv

Ieff

Ig

(5.2)

Where (GA)eff is the effective shear stiffness of section, Gc is the shear modulus of concrete, Acv

is the effective shear area that may be taken equal to cross-sectional area, and Ig is the gross

moment of inertia of section.

Caltrans Seismic Design Criteria 1.7 (2013) uses a similar definition for the effective

flexural stiffness of concrete sections as CSA S6-14. Alternatively, it allows finding the flexural

stiffness of reinforced concrete columns from the charts developed by Priestley et al. (1996) for

circular and rectangular cross sections. In these charts, the ratio of the effective flexural stiffness

to the gross flexural stiffness is a function of axial load ratio and longitudinal reinforcement steel

ratio (Figure 5.2). Caltrans also recommends using 0.5Ig-0.75Ig to estimate the effective flexural

stiffness of box girder superstructures, noting that the lower bound value represents lightly

reinforced sections and the upper bound, heavily reinforced sections.

The shear and torsional stiffness of the section should also be modified according to the

recommendations of Caltrans SDC. The effective shear area of concrete members can be taken

as 80% of the gross area of the sections. Caltrans considers a significant reduction in the

torsional moment of area of columns after the onset of cracking and assumes that the effective

torsional moment of inertia of concrete columns can be reduced to 20% of the original value.

49

Figure 5.2 Priestley et al. chart for finding the effective stiffness of cracked RC circular sections (1996)

5.1.2 Concentrated Plasticity Models

In concentrated plasticity models, the nonlinearity and hysteretic behaviour of structural

components are lumped at discrete locations of the structure, while the rest of the structure is

elastic. The lumped plasticity is either in the form of a plastic hinge at a critical section of a

beam-column element or in the form of nonlinear spring elements (spring hinges). Some

commercial analysis tools such as CSI SAP2000 use concentrated plasticity models. In

SAP2000, plastic hinges are deformation-controlled hinges assigned to pre-determined locations

of beam or column elements, where nonlinear behaviour is expected to be concentrated. There

are several options in SAP2000 for defining a plastic hinge, depending on which actions of a

component the hinge is defined for. For instance, for the flexural response of frame elements,

uncoupled bending moment hinges in either of the principal axes of a section, may be utilized or

an axial force-biaxial moment interaction hinge may be used instead. Hinges can be defined in

terms of sectional response, i.e. the moment-curvature response of a section or element response,

i.e. the moment-rotation response of an element. If the first option is used, an approximate length

needs to be defined for the assigned plastic hinges. The program will then automatically integrate

the sectional response assigned to the hinges using the given length to produce element response.

The hinge properties and the back-bone curve can be defined by the user. Alternatively, one can

use predefined auto hinges in the program, which are based on FEMA 356 (2000a) and Caltrans

SDC 1.7 (2013) recommendations. Another way of deriving hinge properties is by defining

fibres for a hinge section. The sectional response is then calculated by the program from the

50

uniaxial response of individual fibres, which in turn depend on the material constitutive model

assigned to each fibre. In that case, the hinge is called a fibre hinge. Fibre hinges can capture the

axial force-biaxial moment interaction at the hinge section as well as the post-yield degradation

and softening, but are unable to model pinching and bond slip effects. Further details and a

thorough comparison of the SAP2000 plastic hinge options for columns can be found in the

PEER report by Aviram et al. (2008).

Another type of concentrated plasticity models are nonlinear spring hinge models. Spring

hinges are individual or a set of zero-length spring elements assigned to the ends of an elastic

beam-column element, which together captures the nonlinear response of that element. Just as

plastic hinges, spring hinges may be defined for different actions of a component. For instance,

three individual springs in series sometimes is employed to capture the interaction of axial, shear,

and flexural responses of a column (Elwood and Moehle 2005; Elwood and Moehle 2004).

Unlike plastic hinge option, which is not available in all structural analysis tools, spring hinges

are available in most of the commercial and academic analysis tools.

The strength of concentrated plasticity models is in their ability to capture the strength

degradation behaviour due to bar buckling, bond slip, and shear failure. They are also very

compatible with force and deformation limit state checks in codes and standards. However, hinge

behaviour in these models is primarily based on empirical models rather than theory, and

therefore it is dependent on the tests that the model was derived from. Moreover, the empirical

models may not be available for all types of components or the existing models may not be

applicable to configurations other than the tested ones. In that case, additional tests need to be

conducted to develop new empirical models. Alternatively, a continuum model may be used to

predict the component behaviour.

5.1.3 Distributed Plasticity Models

In distributed plasticity models the nonlinear behaviour is not limited to certain points in the

structure, and instead it is captured over the length, area, or volume of structural components.

Examples of distributed plasticity models include continuum finite element models, fibre section

inelastic beam-column elements, and finite-length hinge-zone inelastic beam-column elements.

Fibre section inelastic beam column elements include multiple fibre sections along the length of

51

the element. This allows capturing the inelastic action at several points along the elements as

opposed to concentrated plasticity models, where the plasticity is lumped at hinges. There are

two common formulations for inelastic beam-column elements, namely displacement-based

elements and force-based elements.

- Displacement-based elements (DBE):

Displacement-based elements are formulated based on the stiffness method. This means that for

these elements, compatibility is achieved in the strong form and equilibrium is achieved in the

weak form. The element deformations are interpolated from nodal displacements by using shape

functions that have the basic assumptions of linear curvature and constant average axial strain

along the length of the element (Pugh 2012; Pugh et al. 2015). In DBE formulation, the iterations

are performed at the structure level, and there are no internal iterations at the fibre, section, or

element levels.

- Force-based elements (FBE):

Force-based elements are formulated based on the flexibility method. Opposite to DBE, for these

elements the equilibrium is satisfied in the strong form and the compatibility is satisfied in the

weak form. The basic assumption of FBE formulation is constant axial load and linearly varying

moment along the length of the element (Pugh 2012; Pugh et al. 2015). To make the flexibility-

based formulation of FBE compatible with the displacement-based finite element programs, the

formulation of FBE has additional internal iterations at the element and section levels (Coleman

and Spacone 2001) on top of the iterations at the structure level. As a result, the formulation of

FBE is noticeably more complex than DBE. Models using FBE require greater computational

effort compared to models using DBE for the same number of elements, and may experience

convergence issues. However, when the inelastic flexural response of frame elements at plastic

hinges need to be calculated, FBE has an advantage over DBE. This is due to the fact that at

plastic hinges the curvature is highly nonlinear and FBE assumes linear varying curvature as

opposed to the constant curvature of DBE. Consequently, greater number of DBE is needed to

accurately estimate the curvature at the plastic hinge compared to FBE. This will compensate

considerably for the additional computational effort that individual FBE requires because of the

higher number of required iterations compared to DBE.

52

Continuum finite element models are another class of distributed plasticity models that use

continuum elements such as shells and solid elements. Continuum models are the most detailed

class of models and do not require the simplifying assumptions of the inelastic beam-column

elements about stress or strain field at the section level. They are inherently capable of capturing

the interaction of various actions such as shear, flexure, and torsion within a component by

detailed modelling of the geometry and the interacting mechanism within the component. The

material models assigned to continuum elements can range from a simple linear elastic isotropic

material model with just a few parameters to complex constitutive models with many parameters.

Due to their higher level of complexity, sometimes continuum models are not categorized under

the distributed plasticity models and are recognized as a separate class. The challenge with using

continuum models for seismic design evaluation purposes stems from three main reasons. Firstly,

continuum models are computationally way more expensive than other distributed plasticity

models. This is especially problematic when the entire structure is subjected to multiple ground-

motion records for NTHA and completing the analysis for a set of 11 or more records as required

by CSA S6-14 may take several days. The use of continuum models are further hampered by

convergence issues, when an implicit solution is used with NTHA. In that case an explicit

solution is often employed, which requires even more computational effort to achieve accurate

results, and besides explicit solution algorithms may not be available on all analysis platforms.

Secondly, the accuracy of continuum models depends greatly on the constitutive material

models, which accuracy in turn depends on the values assigned to the input parameters of the

models. As a result, to guarantee accuracy of continuum models, it is often required to calibrate

the constitutive model parameters to some test results. However, tests results are not always

available for all input parameters and performing additional tests can become expensive. If test

results are not available, then the input parameters need to be assigned subjectively. Finally, to

perform code-based checks on structural components, post-processing most often is needed to

obtain the relevant demand and capacity parameters from continuum models outputs. All the

aforementioned reasons, limits the use of continuum models for seismic design evaluation to

mostly academic purposes.

53

5.2 Localization in Distributed Plasticity Models of RC Components

5.2.1 Background in Fracture Mechanics

Localization in finite element analysis refers to the localization of plastic strain in critical

elements (or at sections) where strain softening is expected. The localization term has its roots in

the research of fracture mechanics and the use of smeared cracking model with strain softening

materials. The smeared cracking model, which was originally introduced by Rashid (1968),

replaces discrete crack lines by infinitely many parallel cracks with infinitely small opening that

are distributed continuously over the finite element (Bazant, Zdenek P. and Planas 1997). Strain

softening is the gradual decline of stress at increasing strain and it is incorporated in the smeared

cracking to reflect the growth of cracks until full fracture. As the use of smeared cracking gained

popularity in the finite element analysis of concrete structures, localization and mesh sensitivity

issues in these models were identified. It was recognized that a fracture cannot be consistently

and objectively described by a single softening stress-strain curve, and additional conditions,

called localization limiters, are necessary to prevent strain localization in finite element analysis.

Bazant and Planas (1997) provide a comprehensive theoretical background on why

localization occurs in finite element analysis of softening materials such as concrete. They first

examine the localization under static loading in a series of N equal strain softening elements and

then on a softening bar, which is the continuum model of the N discrete elements as N goes to

infinity. They conclude that the assumption of a simple stress-strain curve with strain softening

will result in the softening zone to have zero width and volume, the inelastic strain and fracture

energy to be zero, and the finite element computations to be mesh-dependent. They report that

the same conclusions apply to dynamic situations as well. Therefore, it is necessary to

complement continuum formulation of strain softening materials with some localization limiter.

Crack-band model proposed by Bazant and Colleagues (Bazant, Z. P. 1976; Bazant, Z. P. and

Cedolin 1979; 1980; 1983) uses the simplest localization limiter. In this model, the strain

softening stress-strain curve is associated with a certain width of the crack band as a reference

width that can be considered as a material property. Based on this reference width, the stress-

strain curve of the softening material for a finite element of any mesh size should be adjusted so

that the global response of the finite element model becomes objective (mesh-independent). This

54

adjustment is done based on preserving constant fracture energy in the post-peak region of the

softening stress-strain curve.

5.2.2 Localization in Distributed Plasticity Beam-Column Elements

Distributed plasticity beam-column elements exhibit similar localization and mesh-dependency

issues with strain softening material models. Both DBE and FBE exhibit these issues. However,

in FBE damage is localized at a critical section, while in DBE it is localized at a critical member

(Pugh 2012).

Coleman and Spacone (2001) demonstrated the localization and mesh-dependency in FBE

using a simple cantilever beam under imposed transverse tip displacements. They considered

three situations:

1. Cantilever beam with elastic-strain hardening section response

2. Cantilever beam with elastic-perfectly plastic section response

3. Cantilever beam with strain softening section response

They were interested to compare the force-displacement response at the element level, and the

moment curvature response at the section level of each of the above three models, when the

number of integration points was increased. Figure 5.3 copies the summary of their observations

for the three models. For a strain hardening section response, both of the element level (global)

and section level (local) response remained objective, with increasing the integration points from

3 to 8 points. The minor difference in the response of the element with 3 integration points was

due to lower accuracy in integrating the element integrals.

In the case of an elastic-perfectly plastic section response, the element level response

remained objective, while the section response was mesh-dependent. The post yield curvature

demand kept increasing with increasing the number of integration points. The loss of objectivity

in curvature prediction was argued to be due to the localization of the inelastic curvature at the

base integration point. Once the beam reaches its plastic moment capacity, with the addition of

the tip displacement, the inelastic curvature at the base integration point increases to correspond

to that level of global response, while the curvature in the other integration points remains

elastic. When the number of integration points increases, the length associated to each

integration point (we may call that the characteristic length) decreases. Therefore, the inelastic

55

curvature at the base point increases for elements with shorter characteristic lengths to result in

the same tip displacement.

For a softening section, both the element and section responses were mesh-dependent and

depended on the number of the integrations points and the associated characteristic length. It was

observed that the post-peak global response of the model became more brittle as the number of

integration points increased. RC bridge piers and columns with high axial load and subjected to

seismic loads exhibit this type of softening section response. In this case, as the characteristic

length decreases and the inelastic curvature increases, the compressive strain of concrete fibres

also increases. This will lead to more degradation in the material stiffness and consequently the

post-peak stiffness.

Coleman and Spacone did not demonstrate the localization in displacement-based

elements. Similar observations are reported for DBE as well (Pugh 2012), with the difference

that the localization occurs in a critical element and not at a section. When the number of DBE is

increased, the characteristic length associated to each element including the critical element at

the plastic hinge region reduces. This will again lead to mesh-dependent response for a strain

softening or elastic perfectly plastic section response.

5.2.3 The Importance of Addressing Localization Issues for PBD of RC Bridges

In Chapter 2, it was explained that the performance-based design of RC bridges according to

CSA S6-14 requires meeting several local strain limits and global displacement limits at different

hazard levels. This performance evaluation is only valid, if objective local and global responses

are obtained for the demand and capacity parameters. However, the distributed plasticity models

of RC components responding in the post-peak region will face the localization and mesh-

dependency issues. Therefore, the response of these models, whether local or global, is not

objective and cannot be used in performance evaluations5. Should distributed plasticity models

be employed for the analysis of RC bridges, it is inevitable to address the localization issues

prior to calculation of any response parameter. In the next section, some of the recommended

methods for dealing with the localization issues in DBE and FBE are briefly reviewed.

5 This is because the values of demand or capacity parameters predicted by these models vary depending on

the selected mesh size, while they are all checked against the same limits.

56

(a)

(b)

(c)

Figure 5.3 Element level and sectional level response of a cantilever beam modelled using a single FBE with

(a) elastic strain-hardening (b) elastic-perfectly plastic, and (c) strain-softening section responses from

Coleman and Spacone (Coleman and Spacone 2001)

57

5.2.4 Options for Dealing with Localization Issues in DBE and FBE

In the research related to structural analysis, only a few studies have addressed the localization

and mesh-dependency issues in concrete and other strain softening materials. There are few

analysis guidelines that have mentioned the problem of mesh sensitivity in distributed plasticity

models of RC components (NEHRP 2013), and even fewer that have recommended a solution

for that.

Two techniques have been recommended more frequently so far to deal with the

regularization issues: (1) adjusting the mesh size based on an empirical plastic hinge length, (2)

material model regularization.

Technique 1: Setting the mesh size based on an empirical plastic hinge length

This is a more common technique of dealing with mesh sensitivity in FBE and DBE. In this

technique, the length associated to the critical section (if FBE is employed) or the critical

element (if DBE is employed) is related to an empirical plastic hinge length. This way, the

physical aspect of localization is mirrored in the numerical model.

One commonly used expression to obtain empirical plastic hinge length can be found in

Priestley et al. (1996), which is also adopted in Caltrans SDC 1.7 (2013):

Lp= 0.08L+0.022fye

db > 0.044fye

db (mm, MPa) (5.3)

In which, L is the member length from the point of maximum moment to the point of contra-

flexure, fye is the expected yield strength of the longitudinal rebars, and db is the nominal

diameter of the longitudinal rebars.

PEER/ATC-72-1 (2010) is one of the few analysis guidelines, which recognizes the

localization issue in distributed plasticity models of RC components. In summary

recommendations for the modelling of planar and flanges concrete walls, it recommends using an

element size equal to the empirical plastic hinge length. Another version of this idea is

mentioned by Hachem et al (2003) for modelling circular RC bridge columns. Hachem and

colleagues suggest that for FBE to predict curvatures accurately, the following condition must be

satisfied:

58

w1= Lp

Lend𝐿 (L-Lp) ≈

Lp

Lend

(5.4)

Where w1 is the Gauss-Lobatto weight of the first integration point of the end element, L is the

total length of element, and Lend is the length of the end element at plastic hinge. If the element

has two integration points, the weight of each point is equal to 0.5. This will imply that the

length of end element should be 2Lp. Based on this reasoning, he suggests using a 2-point FBE in

the plastic hinge zones of the bridge column models. Similar technique is used by Lara (2011) to

model several tested RC bridge columns. Calabrese and colleagues (2010) have also looked at

the numerical issues in distributed plasticity models of RC frame elements for seismic analysis.

They report that if a regularization technique is not available, then for FB elements, it is common

to use the above technique. The downside of using this technique is that for short elements, it

may require using a small number of integrations points, or small number of elements. In both

cases accuracy is compromised and error will be introduced in response calculations. In these

cases, material regularization technique can be employed instead.

Technique 2: Material model regularization

Material regularization technique is based on preserving constant fracture energy of concrete in

compression, referred to as crushing energy. The concept of constant fracture energy of concrete

in tension is commonly used to address the localization and mesh-dependency issues in the

continuum finite element analysis of smeared crack models (Section 5.2.1). However, the results

of the experimental research on the compression failure of concrete cylinders suggested that this

is also a localized phenomenon and the amount of post-peak fracture energy does not depend on

the length of specimens (Jansen and Shah 1997; Lee and William 1997).

Using this concept, Coleman and Spacone (2001) suggested a simple regularization

technique for fibre section FBE. The crushing energy is defined as the area under the post-peak

stress-displacement of concrete, obtained from the following integral:

Gfc= ∫ σ dui (5.5)

59

In which, σ is the concrete stress and ui is the inelastic displacement. This energy can be related

to the material stress-strain curve through the characteristic length h. For FBE h is the length

associated to the critical integration point and is denoted by LIP, as follows:

Gfc=h ∫ σ dεi = LIP ∫ σ dεi (5.6)

Where, ε is the concrete strain. The above expression suggests that, if Gfc is preserved constant,

then the production of the characteristic length and the area under the stress-strain curve of

concrete in the post-peak region should remain constant. Consequently, changing the

characteristic length by varying either the mesh size or the number of integration points should

require adjusting the area under the stress-strain curve correspondingly. The shaded area in

Figure 5.4 Part (a) represents this area for a typical compressive stress-strain curve of unconfined

concrete and is equal to the ratio of Gfc to LIP. It should be noted that the post-peak softening

branch of the model is not necessarily linear. εo is the strain of unconfined concrete at the peak

stress of f’c and ε20u is the strain in the post-peak region of the curve corresponding to 80%

strength loss (i.e. stress is equal to 20% of f’c). ε20u is the parameter that is adjusted in the

concrete uniaxial stress-strain curve for regularization. Coleman and Spacone present an

expression for calculating ε20u, assuming the modified Kent and Park model (Pugh 2012) is used

for concrete:

ε20u=Gfc

0.6f'cLIP

- 0.8f

'

c

𝐸𝑐+εo (5.7)

In this expression, Ec is the elastic modulus of unconfined concrete in compression. The

modified Kent and Scott model assumes a linear softening branch. Given Gfc, the ε20u and

therefore the material model assigned to each critical section or element, should be changed

based on their characteristic length. Similar expressions can be utilized for confined concrete as

well. However, the confined concrete parameters should be used in place of the unconfined

concrete parameters. Figure 5.4 Part (b) illustrates the equivalent of Part (a) for confined

concrete. In this picture, the Gfcc and f’cc are the crushing energy and compressive strength of

confined concrete, respectively, εoc is the strain of confined concrete at f’cc, ε20c is the strain in the

post-peak region corresponding to 80% strength loss, and Ecc is the elastic modulus of confined

concrete in compression.

60

εo ε20u

f’c

0.2 f’c

Ec

σ

Gfc

LIP

εoc ε20c

f’cc

0.2 f’cc

Ecc

σ

Gfcc

LIP

(a) (b)

Figure 5.4 Schematic post-peak crushing energy of (a) unconfined concrete and (b) confined concrete

(reproduced from Pugh, 2012)

Preserving a constant crushing energy will ensure that the inelastic portion of the global

response becomes objective. However, the inelastic portion of the local response remains mesh-

dependent, since this response should vary with the size of the mesh so that it produces the same

global displacement. Coleman and Spacone suggested a very simple procedure to obtain an

objective prediction of curvature in FBE. The idea is simply to scale the inelastic part of the

curvature obtained from analysis to the inelastic curvature that is predicted using the empirical

plastic hinge length. The inelastic curvature from analysis can be approximated as:

ϕi

model=

δi

LIP (L2

-LIP

2) (5.8)

And the curvature is the sum of the elastic curvature and the scaled inelastic curvature:

ϕ=ϕe+SF ϕ

i

model (5.9)

The scale factors are obtained by taking the ratio of ϕi

predicted/ϕi

model, where the predicted

curvature is obtained by replacing LIP in Equation (5.8) with Lp obtained from Equation (5.3):

SF=wIPL2(1-wIP)

Lp(L-Lp) (single curvature) (5.10)

SF=wIPL2(2-wIP)

Lp(2L-Lp) (double curvature) (5.11)

61

Referring to the definition of the scale factors, Coleman and Spacone argued that if LIP = Lp then

no further post-processing is necessary to obtain objective curvatures. It becomes clear with this

explanation, that the first technique of dealing with location issues described in the previous

section can be considered as a special case of the material regularization technique. When it is

not possible or reasonable to set LIP equal to Lp, then material regularization technique must be

used.

The most recent extensive research on this topic was conducted by Pugh (2012; Pugh et al.

2015). He studied the localization issues in distributed plasticity models for numerical simulation

of reinforced concrete shearwalls and extended the work done by Coleman and Spacone to

include DBE as well as FBE. His main contribution was to recommend a more scientific

formulation for the crushing energy of both unconfined and confined concrete. Coleman and

Spacone recommended using a value of 20 N/mm for the crushing energy of unconfined concrete

based on the recommendation of Jansen and Shah (1997), up to a value of 30 N/mm. The

confined concrete crushing energy is approximated then as six times of the crushing energy for

the unconfined concrete (150-180 N/mm). Pugh conducted experimental tests on a number of

planar wall specimens and observed that the crushing energy can be formulated in terms of the

specified strength of unconfined and confined concrete. He recommended using the following

crushing energy values with FBE and DBE:

Gfc=2f'c ( N mm)⁄ (FBE) (5.12)

Gfc=0.56f'c ( N mm)⁄ (DBE) (5.13)

And the confined concrete crushing energy for both FBE and DBE can be estimated as:

Gfcc=1.7Gfc (5.14)

Then Equation (5.7) can be written for the confined concrete properties as:

ε20c=Gfcc

0.6f'ccLIP

- 0.8f

'

cc

𝐸𝑐𝑐+εoc (5.15)

The crushing energy recommended to be used with DBE is considerably lower than FBE. Pugh

argued that in DBE, the predicted axial load at the critical section is lower than the actual applied

62

load and as a result, the predicted curvature ductility values are greater than those predicted by

FBE. The reduction in the crushing energy values accounts for this difference.

Pugh made another improvement to the Coleman and Spacone’s work by suggesting

material regularization to be applied to reinforcing steel as well as concrete. Steel shows a strain-

hardening behaviour in the post-yield portion of the stress-strain curve. This will ensure

distribution of plasticity along the beam-column element instead of localization in a single

critical element or section. However, in a reinforced concrete element, for which at the critical

section concrete exhibits strain-softening, steel deformations also localize at the critical section

to conform to compatibility conditions. Consequently, steel strain demands at the critical section

become mesh-dependent and require material regularization.

The post-yield energy of steel material is referred to as hardening energy. Much like

concrete, this energy can be related to the stress-strain curve through a length measure. For steel

this length is the length, along which the inelastic deformation localizes and is taken equal to the

gage length used in the laboratory test. Figure 5.5 shows a simplified bi-linear steel stress-strain.

The shaded area under the post-yield portion of the curve represents the ratio of the hardening

energy, Gs, to the gage length, Lgage. Therefore, the hardening energy can be calculated as:

Gs=0.5 (εsu,exp-εy) (fu+f

y) Lgage (5.16)

Where εsu,exp is the expected rupture strain, εy is the yield strain, and fu and fy are the ultimate

tensile strength and yield strength, respectively. For regularizing the steel material, the ultimate

rupture strain assigned to the steel material model in the analysis should be modified based on

the length associated to the critical section or element. The underlying assumption again is that

the hardening energy is preserved constant. Using this assumption, Pugh derived the following

expression for the ultimate rupture strain in the regularized material model:

εsu=εy+(εsu,exp-εy)Lgage

LIP

(5.17)

63

εy εsu

fu

Es

ε

fybEs

Gs

Lgage

Figure 5.5 Schematic post-peak energy of reinforcing steel (reproduced from Pugh, 2012)

Pugh suggested using an 8 in (0.203 m) gage length required by ASTM A370 for numerical

modelling of wall specimen, if gage length was not reported in test results. Also, he argued that

Equation (5.17) can be used with Menegotto-Pinto steel model (1973) as well, and the

simplifications have insignificant impact on numerical results. Finally, it should be noted that the

post-yield hardening modulus will be changed based on the value calculated for the ultimate

rupture strain of the regularized material.

5.3 Validating Distributed and Concentrated Plasticity Models of RC Bridge Columns

The main purpose of this section is to apply the techniques of dealing with localization and

mesh-dependency issues discussed in the previous section, to distributed plasticity models of a

number of tested RC bridge columns. It is desirable to observe how these models would predict

the response of the columns before and after each of the two techniques have been applied to

them. The selected columns were tested under lateral quasi-static cyclic loading condition. In this

exercise the cyclic force-displacement response of the models is compared against the test results

and the monotonic response of the models is predicted using static pushover analysis. In addition

to the global force-displacement response, the local strains of concrete and reinforcing steel

fibres are checked for consistency. The aim is not to calibrate the models so that they would

reproduce the test results with the utmost accuracy; rather, it is an exercise to illustrate how well

with the current modelling techniques we are able to predict the performance of RC bridge

columns, when testing is not an option. This is in fact the likely case when a new structure is

designed for construction and testing of the main components is too expensive to be feasible or

simply is not possible. Once consistent local and global responses are achieved, the performance

64

criteria of CSA S6-14 and BC MoTI Supplement will be employed to predict the extent of

damage to the columns in the next chapter.

5.3.1 Selected Bridge Column Tests

Five reinforced concrete bridge column tests are selected from the PEER structural performance

database (2003). All of the selected columns have circular cross-section and are laterally

reinforced with spirals. This configuration of bridge columns is very common in the Canadian

bridge industry and therefore is relevant to the theme of this thesis. The selected tests are part of

the experimental program developed by Lehman and colleagues at PEER (Lehman and Moehle

2000a; Lehman and Moehle 2000b; Calderone et al. 2001; Lehman et al. 2004) to study the

cyclic performance of concrete bridge columns detailed for ductile flexural response in high

seismicity zones. The main focus of the program was on capturing the performance of the bridge

columns in terms of relatable engineering demand parameters at a range of damage states other

than failure of the component. The measured engineering demand parameters included global

lateral load and displacement and local concrete and steel strains values at different damage

states. The availability of these test results makes them appropriate for the purpose of this

section.

A schematic picture of the test configuration and instrumentation as well as the overall

geometry and reinforcement details of the tested specimens are shown in Figure 5.6. Table 5.1

lists the properties of test specimens including the test number, length, reinforcement details, and

axial load ratio, and Table 5.2 shows the material properties for the test specimens. The tests

specimens are one-third of the full-scale columns. All columns have similar diameter of 2 ft

(0.61 m). The spiral pitch is constant for Column 415 and 430 throughout the length of the

column, while it varies for Column 328, 828, and 1028. In these columns the pitch is smaller in

the plastic hinge zone (Lc in Figure 5.6) and is twice larger at the rest of the columns. Column

415 and 430 are different only in the amount of longitudinal reinforcement, whereas Column

328, 828, and 1028 vary in their length and thus their aspect ratio. The first digit in the test

number indicates the aspect ratio value, for instance 328 and 1028 has aspect ratios of 3 and 10,

respectively. The rest of the number indicates the longitudinal reinforcement ratio.

65

Figure 5.6 (left) geometry and reinforcement of the tested specimens, (right) test configuration and

instrumentation (Calderone et al. 2001; Lehman et al. 2004)

The axial load was applied to the columns through the high-strength rods at either side of the test

specimens and a spreader beam as shown in Figure 5.6. The lateral displacement history was

applied using an actuator attached to the top of the column. All specimens had similar

instrumentation and test procedure. First, the axial load was applied and was maintained

constant, while the lateral load was applied. Figure 5.7 shows the lateral load time-history in

terms of the target displacement ductility values, and Table 5.3 contains the amplitude values at

each displacement level that are imposed to the columns with varying aspect ratios. The loading

protocol for the post-yield cycles included three cycles at each amplitude and a following cycle

with one-third of the amplitude. The recorded cyclic force-displacement responses of the

columns are shown in Figure 5.8. Comparing the hysteretic response of the columns, it is notable

that Column 328, 415, and 430 demonstrate higher strength degradation at the last three loading

cycles, compared to Column 828 and 1028.

66

Reinforcement

Column

Length

(mm) Longitudinal

ρl

(%)

Spiral Spacing

(mm)

ρs

(%)

Confined Length

(mm)

Axial Load

Ratio

328 1829 28 No. 6 2.8 25/50 0.87 610 0.1

828 4877 28 No. 6 2.8 25/50 0.87 915 0.1

1028 6096 38 No. 6 2.8 25/50 0.87 1220 0.1

415 2438 22 No. 5 1.5 32 0.7 2438 0.1

430 2438 44 No. 5 3 32 0.7 2438 0.1

Table 5.1 Specimen properties of the test columns (Lehman et al. 2004)

Column f'c (MPa) fym (MPa) fum (Mpa) εsh εu fyhm (MPa)

328, 828, 1028 34 448 634 0.02 0.14 607

415, 430 31 497 662 0.02 0.13 607

Table 5.2 Material properties of the test columns (Lehman et al. 2004)

Figure 5.7 Imposed lateral displacement time history (Lehman et al. 2004)

Aspect Ratio

Displacement Level 3 4 8 10

Pre-cracking 1 2 4 5

Pre-yield1 3 3 15 20

Pre-yield2 5 8 45 64

Pre-yield3 10 19 89 127

μΔ≈1 15 25 133 191

μΔ≈1.5 20 38 178 254

μΔ≈2 30 51 267 381

μΔ≈3 51 76 445 635

μΔ≈5 71 127

μΔ≈7 102 178

μΔ≈10 132

Table 5.3 Imposed displacement histories in mm for columns with different aspect ratios (Lehman et al. 2004)

67

Figure 5.8 Cyclic force-displacement responses of the selected test columns (PEER 2003)

68

5.3.2 Bridge Column Models

Two separate analysis platforms are employed to model the tested bridge columns including

OpenSees and CSI SAP2000. OpenSees is an open source analysis platform, which is widely

used for academic purposes and is geared towards fulfilling current research demands, while

SAP2000 is a well-established commercial tool used by many engineers in practice. The reason

for using both platforms is that they have different strength and limitations, and therefore

employing both offers the chance to contrast the capabilities of the academic versus commercial

tools with respect to the modelling alternatives described earlier in this chapter. OpenSees is

used to generate distributed plasticity models of the test columns, whereas SAP2000 is used to

build concentrated plasticity models. A schematic illustration of the column models in both

platforms is shown in Figure 5.9.

Figure 5.9 Schematic distributed and concentrated plasticity models of the columns in OpenSees (left) and

SAP2000 (right)

- OpenSees Models:

The test columns are modelled using inelastic beam-column elements with fibre sections. Both

DBE and FBE are employed for the models. The base of the columns is restrained in all degrees-

of-freedom to mimic the fixed-base condition in the tests. A concentrated vertical load, equal to

the applied axial load in the tests and a unit lateral load is applied at the top of the columns. The

lateral load is imposed at the center of the loading zone at the top of the column, from where the

height of the column is measured (Figure 5.6, left). A displacement-controlled integrator is

employed in the analysis, for which the control node is the node where the lateral load is applied.

69

To create fibre sections, separate material models needs to be defined for the unconfined

and confined concrete, and reinforcement steel. Two sections are defined for the test columns

where the plastic hinge zone and the rest of the column have different confinement arrangement.

Concrete02 is used for both of the unconfined and confined concrete material models. This

model is developed based on the Kent-Scott-Park constitutive relationship and it has a bi-linear

tension response with linear tension softening. The compression response has an initial parabolic

segment up to the maximum compressive strength, followed by a linear softening segment, and a

final plateau, as demonstrated in Figure 5.10. The model takes seven calibration parameters

including the compressive strength of concrete (fpc), concrete strain at maximum strength

(epsc0), concrete crushing strength (fpcU), concrete strain at crushing strength (epsU), ratio

between unloading slope and initial slope (lambda), tensile strength of concrete (ft), and tension

softening stiffness (Ets). OpenSees does not automatically apply the confinement effects.

Therefore, the confined concrete material properties are manually obtained following the Mander

et al. confined concrete model (1988). The strain at the peak stress of the unconfined concrete,

the confined concrete in the plastic hinge zone, and outside of it, is assigned as 0.002, 0.0045,

and 0.0038, respectively. The ultimate strain capacity of the confined concrete can be calculated

using Priestley et al. expression (1996), as follows:

εcu=0.004+1.4ρ

s f

yh εfs

fcc

' (5.18)

In the above expression, ρs is the spiral reinforcement ratio, fyh is the spiral yield strength, εfs is

the spiral fracture strain, which can be taken equal to εsu,exp, and f’cc is the confined concrete

strength. It should be noted this expression has been formulated for confined concrete section

under compression. When used for member under bending or combination of bending and axial

compression, it tends to underestimate εcu by at least 50% (Priestley, M. J. N. et al. 1996;

Kowalsky 2000). Using the above formula, the ultimate strain capacity of Column 328 is

obtained as -0.0180 in the plastic hinge zone, and -0.011 outside of it.

For the reinforcement steel, Steel02 material model is employed. This is a Giuffre-

Menegotto-Pinto material model (1973) with isotropic strain hardening proposed by Filippou et

al. (1983). The model takes eleven calibration parameters. The strain hardening parameter,

70

Figure 5.10 Concret02 constitutive model in OpenSees (PEER 2017)

defined as the ratio of the post-yield stiffness to the initial elastic stiffness, is calculated as 0.01.

The ultimate strain capacity of the reinforcement steel is set to 0.09, based on the

recommendation of Caltrans SDC 1.7 (2013) for the reduced ultimate tensile strain of bar #10 or

smaller. To impose this limit on the response of the steel fibres, a MinMax uniaxial material

model is additionally assigned to the column sections with the limiting strain of 0.09 in tension

and infinity in compression. This material model will ensure that the steel fibres strength reaches

zero when the strain in those fibres is equal to 0.09, which will mimic the effect of fracture in the

reinforcement steel.

The beam-column elements are not capable of modelling the shear response of the

columns. To account for the shear stiffness loss of the columns, an elastic shear section was

integrated to the fibre section using section aggregator in models with FBE. This option does not

work with DBE in OpenSees, and instead a zero-length shear spring was added to the base of the

models with DBE, as suggested by Pugh (2012). The effective shear stiffness modulus can be

taken as follows (Oyen 2006):

Geff=0.1G≈0.04 E (5.19)

Where G is the elastic shear modulus and E is the Young’s modulus of elasticity. CSA S6-14

recommends obtaining effective shear stiffness using Equation (5.2). It should be noted however,

that accounting for the loss of shear stiffness is very important for the modelling of shear

71

dominated elements such as shearwalls. If the columns are dominated primarily by flexure

response, the effect of the shear stiffness loss is minimal and can be ignored.

- SAP2000 Models:

Unlike OpenSees, SAP2000 does not have the option of distributed plasticity models. Instead,

non-linear behaviour is modelled with concentrated plasticity models, assigning plastic hinges

with a specified length to elastic frame elements, as explained earlier in Section 5.1.2. Fibre

hinges are employed here to model the nonlinear response of the columns. The shear and torsion

behaviour of the cross section are elastic. So, the loss of shear stiffness should be captured by

applying shear area modification factors to the elastic frame elements. The plastic hinge length

assigned to fibre hinges can be calculated using Equation (5.3), as listed in Table 5.4. As shown

in Figure 5.9, plastic hinges should be assigned to the mid-point of the plastic hinge zone of the

columns. This will ensure that the length of the column that undergoes plastic rotation is correct.

Column L (mm) Lp (mm) Lp/L

415 2438 369 0.15

430 2438 369 0.15

328 1829 376 0.21

828 4877 578 0.12

1028 6096 676 0.11

Table 5.4 Empirical plastic hinge length for the selected test columns

SAP2000 has a built-in model to calculate confined concrete material properties from the

inputs for the unconfined concrete and confinement properties of a section, based on Mander et

al. confined concrete model (1988). The column sections were defined using the section designer

module and radial fibre arrangement was assigned to the sections. A nonlinear static load case

was defined for the vertical gravity load applied at the top of the columns. A nonlinear static

pushover analysis case was also defined using a displacement-controlled load application for the

top node of the columns.

While using fibre hinges, it should be noted that there was an elastic softening issue in

SAP2000 version 18.0.1 and prior, which was rectified in version 18.1.0 and later. The elastic

softening was due to double counting of the elastic flexibility in frame elements with fibre

72

hinges. This was the reason for the recommendations on using increased stiffness factors with

elements (see Aviram et al. 2008). To remove this issue, the elastic stiffness of frame elements is

now automatically set to zero throughout the tributary length of the fibre hinges.

5.3.3 Validation of the Distributed Plasticity Models for Test 328

The generated distributed plasticity models in OpenSees are used to predict the response of the

bridge column in Test 328. The localization and mesh sensitivity issues in distributed plasticity

models of the column are demonstrated for both DBE and FBE models and the two methods of

dealing with localization issues are applied to correct the predictions of the models. All the

investigations are performed on Column 328 model, as the exercise is similar for the rest of the

test columns. The final takeaways from the study on Column 328 are examined for Column 828,

1028, 415, and 430 in Section 5.3.8.

5.3.3.1 Cyclic Response without Regularization

The effect of changing the mesh size in DBE and FBE models on the cyclic response of the

models is demonstrated in Figure 5.11. The graphs show the displacement of the top node of the

column versus the base shear. Parts (a) to (d) compare the predicted cyclic response of the DBE

models with 0.076, 0.152, 0.305, and 0.610 m element size against the test results (the mesh size

is equal to 1/3, 1/6, 1/12, and 1/24 of the height of the column, respectively). All the DBE

models use two integration points per element. Therefore, the length associated to each

integration point is half of the element size (0.038, 0.076, 0.152, and 0.305 m, respectively).

Parts (a’) and (b’) show the FBE model responses with 0.610 and 0.914 m element sizes, having

two and three integration points per element, respectively (the mesh size is equal to 1/3 and 1/2

of the height of the column). The length associated to the integration point at the critical section

of the element at the base of the column is 0.305 and 0.152 m. The FBE and DBE models are

different only in the selected mesh size and otherwise are similar.

By examining the graphs, it is readily evident that changing the mesh size significantly

affects the force-displacement response of both the DBE and FBE models. Models with smaller

mesh size have smaller hysteresis loops, and they quickly degrade and loose strength. We will

refer to the displacement, at which there is a sudden drop in base shear at two consecutive cycles,

as the failure displacement.

73

(a) (b)

(c) (d)

(a’) (b’)

Figure 5.11 Comparison of the simulated cyclic force-displacement response of Column 328 using DBE with

(a) 0.076 m, (b) 0.152 m, (c) 0.305 m, (d) 0.610 m element size and 2 integration points, and FBE with (a`)

0.610 m element size and 2 integration points, (b`) 0.914 m element size and 3 integration points with the

experimental results

74

The DBE models in Parts (a), (b), and (c) predicts the failure displacement of the column to be at

0.05, 0.07, and 0.10 m, respectively, while the test results suggest that the failure displacement is

about 0.13 m. Therefore, all these models underestimate the displacement capacity of the column

at failure. Only the DBE model with 0.610 m mesh size provides a good estimation of the cyclic

response of the column. The reason for this, is that the length associated to the integration point

at the critical element at the base of the column (LIP = 0.305 m) is close to the empirical plastic

hinge length of Column 328 in Table 5.4 (Lp = 0.376 m). While for the other mesh sizes, LIP is

considerably smaller than Lp. This means that the length, over which the plastic strain and the

plastic curvature localize, is smaller than the physical plastic hinge length. Therefore, when the

fibres in the critical section reaches the strain value where the material model starts degrading

and eventually fails, the predicted failure displacement at that strain value is smaller than the

actual displacement of the column.

Comparing the two FBE models, the model with 0.914 m mesh size and three integration

points underestimate the displacement capacity of the column at 0.10 m, while the model with

0.610 m mesh size and two integration points provides an acceptable prediction of the response.

This again relates to how the length associated to the integration point at the critical section of

the column compares to the plastic hinge length of the column. For the FBE model in Part (b’),

LIP = 0.152 m, while for model in Part (a’), LIP = 0.305 m, which is closer to Lp = 0.376 m.

5.3.3.2 Cyclic Response with Regularization

Material regularization technique should be applied to each of the following four material

models individually: (1) confined concrete in the plastic hinge zone, (2) confined concrete

outside the plastic hinge zone, (3) unconfined concrete, and (4) reinforcement steel. The steps of

the regularization technique were explained in Section 5.2.4. Once the regularization is applied,

each of the above material models needs to be updated with the values for the post-peak

parameters, including the strain at the 80% loss of strength for unconfined and confined concrete,

and the ultimate rupture strain and the post-peak slope of reinforcement steel. The calculated

values of these parameters are listed in Table 5.5. In this table Lele is the mesh size, IPs is the

number of integration points used per element, LIP is the length associated to the integration point

at the critical section of a FBE or the critical DBE at the base of the column, ε20u, ε20c-PH, and ε20c-

75

R are the strains at the 80% loss of strength for the unconfined concrete, and the confined

concrete inside the plastic zone, and outside of it, respectively, and εsu and b are the ultimate

rupture strain and the post-peak slope of the reinforcement steel. It should be noted that the

integration scheme for the DBE models is Gauss-Legendre and for the FBE models Gauss-

Lobatto, and LIP is calculated based on Lele,, the number of integration points per element, and the

integration weights for each type of integration scheme.

Type Lele (m) IPs LIP (m) ε20u ε20c-PH ε20c-R εsu b

DBE 0.610 2 0.305 -0.0040 -0.0077 -0.0074 0.0607 0.0159

DBE 0.305 2 0.152 -0.0071 -0.0120 -0.0122 0.1193 0.0079

DBE 0.152 2 0.076 -0.0132 -0.0207 -0.0216 0.2363 0.0040

DBE 0.076 2 0.038 -0.0255 -0.0381 -0.0406 0.4703 0.0020

FBE 0.610 2 0.305 -0.0119 -0.0189 -0.0196 0.0607 0.0159

FBE 0.914 3 0.152 -0.0228 -0.0344 -0.0365 0.1193 0.0079

FBE 1.829 4 0.152 -0.0228 -0.0344 -0.0365 0.1193 0.0079

FBE 1.829 6 0.061 -0.0556 -0.0808 -0.0872 0.2948 0.0032

Table 5.5 Calculated regularized strain of the unconfined and confined concrete at 80% strength loss, and the

ultimate rupture strain and post-peak slope of the reinforcement steel for Column 328 models

The DBE and FBE models of Column 328 used in the previous section are updated with

the regularized material properties in Table 5.5 and their cyclic force-displacement response are

compared against the test results in Figure 5.12. Comparing Part (a) to (d) of Figure 5.12 with

Figure 5.11, reveals how regularizing the material models changes the cyclic response of the

DBE models. The following can be observed from this comparison:

Regularization of the DBE models allows the hysteretic behaviour to fully develop and

prevents the premature failure of the models with smaller mesh sizes in Parts (a) to (c).

The predicted failure displacements of the four DBE models are quite similar, unlike Figure

5.11.

The hysteretic loops of the four regularized material models have different shapes. The

models with smaller mesh sizes tend to develop larger hysteresis loops at similar

displacement levels.

The regularized FBE models have very similar hysteric response, both in terms of the shape

and the failure displacement. However, they both underestimate the failure displacement at

0.1 m (about 23%).

76

(a) (b)

(c) (d)

(a’) (b’)

Figure 5.12 Regularized cyclic force-displacement response of Column 328 using DBE with (a) 0.076 m, (b)

0.152 m, (c) 0.305 m, (d) 0.610 m element size and 2 integration points, and FBE with (a`) 0.610 m element

size and 2 integration points, (b`) 0.914 m element size and 3 integration points with the experimental results

77

5.3.3.3 Monotonic Response without and with Regularization

Once the cyclic response predictions of Column 328 are compared against the cyclic test results,

It is possible to use the same model to predict the monotonic response of the columns, although

there are no monotonic test results to compare to. It is of interest to see how material

regularization will affect the monotonic response of DBE and FBE models of Column 328 with

various mesh sizes.

Figure 5.13 shows the monotonic response of DBE models with 0.076, 0.152, 0.305, and

0.610 m mesh size, and FBE models with 0.610, 0.914, 1.828 m mesh size, and 2, 3, and 4 and 6

integration points per element, respectively. The cyclic test result is also shown in the figure so

that the monotonic response can be compared against the backbone of the cyclic response. We

will define the failure displacement for monotonic response as the displacement, at which the

base shear suddenly drops by 20% or more. Using this definition, the monotonic results also

suggest that the failure displacement of both of the DBE and FBE models is dependent on the

mesh size. These values are listed in Table 5.6 under the “without regularization” column. The

difference of the failure displacements between the models with the largest and the smallest

mesh sizes is about 300-350%, which is very significant. One cannot conclude, which of these

displacements are the failure displacement of the actual column, unless the mesh-sensitivity issue

is dealt with.

Figure 5.14 shows the monotonic response of the same models but after applying material

regularization. For both DBE and FBE models, the regularized models produce almost identical

force-displacement responses and have similar failure displacements. The failure displacements

of these models are also listed in Table 5.6 under the “with regularization” column. The

difference of the failure displacements between the models with the largest and the smallest

mesh sizes in this case is about 7-8%. The only outlier is the DBE model with 0.610 m mesh

size, which has a difference of about 10% in failure displacement with the 0.075 m model. Also

as evident in Part (a) of the figure, it predicts higher strength values as compared to the other

78

(a) (b)

Figure 5.13 Monotonic response of the DBE and FBE models of Column 328 before material regularization

(a) (b)

Figure 5.14 Monotonic response of the DBE and FBE models of Column 328 after material regularization

Failure Displacement (m)

Type Lele (m) IPs LIP (m) WO Reg. W Reg.

DBE 0.610 2 0.305 0.173 0.119

DBE 0.305 2 0.152 0.098 0.121

DBE 0.152 2 0.076 0.057 0.129

DBE 0.076 2 0.038 0.037 0.132

FBE 0.610 2 0.305 0.155 0.096

FBE 0.914 3 0.152 0.081 0.100

FBE 1.829 4 0.152 0.081 0.102

FBE 1.829 6 0.061 0.039 0.104

Table 5.6 Failure displacements of the DBE and FBE models of Column 328 with and without material

regularization

-700

-500

-300

-100

100

300

500

700

-0.15 -0.05 0.05 0.15

La

tera

l F

orc

e (k

N)

Displacement (m)

Test328DBE 0.076 mDBE 0.152 mDBE 0.305 mDBE 0.610 m

-700

-500

-300

-100

100

300

500

700

-0.15 -0.05 0.05 0.15

La

tera

l F

orc

e (k

N)

Displacement (m)

Test328FBE 1.828 m 6IPFBE 1.828 m 4IPFBE 0.914 m 3IPFBE 0.610 m 2IP

-700

-500

-300

-100

100

300

500

700

-0.15 -0.05 0.05 0.15

La

tera

l F

orc

e (k

N)

Displacement (m)

Test328DBE 0.076 mDBE 0.152 mDBE 0.305 mDBE 0.610 m

-700

-500

-300

-100

100

300

500

700

-0.15 -0.05 0.05 0.15

La

tera

l F

orc

e (k

N)

Displacement (m)

Test328FBE 1.828 m 6IPFBE 1.828 m 4IPFBE 0.914 m 3IPFBE 0.610 m 2IP

79

DBE models. This is because DBE models need more elements compared to FBE models to

achieve the same level of accuracy. Therefore, the minor discrepancy in the DBE model with

0.610 m mesh size with the other DBE models is due to using small number of elements (3

elements).

It is also possible to compare the monotonic failure displacements of the regularized

models with the cyclic failure displacement from the test result. Although it should be noted that

the two displacements may not necessarily be the same as in the cyclic test, the column

undergoes many cycles before reaching the failure displacement and meanwhile dissipates a lot

of energy through material nonlinearity. On the contrary, in the monotonic response, the column

is consistently pushed until reaching the failure displacement. As a result, the column dissipates

less energy through material nonlinearity and is more likely to endure less damage at the same

level of displacement when compared to the column under cyclic loading. The cyclic failure

displacement of Column 328 is about 0.13 m. Comparing the failure displacements of the

regularized models with this value, it can be observed that the regularized FBE models

underestimate the failure displacement of the column by 23%, but the regularized DBE models

almost predicts the same value of failure displacement with only 4% error on average. However,

the FBE models predict the backbone of the cyclic response with accuracy, while the DBE

models overestimate the strength of the column in initial cycles by about 9% and underestimate

it in the rest of the cycles up to the failure displacement by about 9-10%.

5.3.3.4 Cyclic and Monotonic Response of the Models with Mesh Size Set Based on LP

The other method of dealing with localization and mesh sensitivity issues, which was introduced

in Section 5.2.4 is to set the mesh size and therefore the length associated to the critical

integration point (LIP) based on the empirical plastic hinge of the column (LP). Considering the

discussion in Section 5.2.4, first LIP is set equal to LP. The cyclic and monotonic responses of the

DBE and FBE models satisfying this condition are shown in Figure 5.15, along with the test

result of Column 328. In terms of the cyclic response, it is observed that both DBE and FBE

models more or less provide acceptable predictions of the actual response of the column. The

DBE model better captures the shape of the hysteresis loops, but overestimates the strength in the

80

(a) (b)

Figure 5.15 Comparison of the cyclic and monotonic responses of the (a) DBE model and (b) FBE model with

LIP=LP against the test results of Column 328

initial cycles. The situation is not the same for the monotonic response predictions. The DBE

model significantly overestimates both the failure displacement and the strength of the column,

by about 60% and 25%, respectively. As explained in the previous section, this is due to the fact

that DBE models need more elements to accurately predict the response as opposed to FBE

models. The FBE model also overestimates the failure displacement by about 46%, but

underestimates the strength in the post-peak cycles by about 13% on average.

Since the models with LIP= LP overestimates the monotonic response, it is intuitive to

choose a smaller mesh size that is not too small to significantly underestimate the response. We

will try LIP equal to 0.5 LP. This corresponds using an element size equal to the plastic hinge

length having two integration points, and follows the recommendations of PEER/ATC-72-1

(2010). The cyclic and monotonic response of the DBE and FBE models with this condition is

shown in Figure 5.16. Considering both the monotonic and cyclic responses, the DBE model

provides a reasonable prediction of the failure displacement, while the FBE model

underestimates that. However, the underestimation of the response by the FBE model is not too

significant to rule it out for design purposes. The DBE model captures the shape of the hysteresis

more closely, and the overestimation of strength in the monotonic response is less significant in

this case.

-700

-500

-300

-100

100

300

500

700

-0.15 -0.05 0.05 0.15

La

tera

l F

orc

e (k

N)

Displacement (m)

Test328PushoverCyclic

-700

-500

-300

-100

100

300

500

700

-0.15 -0.05 0.05 0.15

La

tera

l F

orc

e (k

N)

Displacement (m)


81

(a) (b)

Figure 5.16 Comparison of the cyclic and monotonic responses of the (a) DBE model and (b) FBE model with

LIP=0.5 LP against the test results of Column 328

5.3.4 Sensitivity of Monotonic Response to GFcc Value

The response of a regularized model depends on the value assumed for the crushing energy of

confined concrete, Gfcc, as evident from Equations (5.15). It is desirable however, to understand

how this value would affect the predictions of the monotonic force-displacement response of a

concrete column. To achieve so, a simple sensitivity analysis on the DBE and FBE models of

Column 328 is performed, where the confined concrete crushing energy is varied in a range,

while all the other parameters, including the unconfined concrete crushing energy are kept

constant. The mesh size for the DBE models is 0.305 m and for the FBE models is 0.610 m with

two integration points (although as demonstrated earlier, the regularization process cause models

with different mesh sizes to yield similar force-displacement responses).

Figure 5.17 shows the monotonic force-displacement response of the DBE and FBE

models for four different values of Gfcc. The minimum values for Gfcc are taken equal to the

unconfined concrete crushing energy calculated from Equation (5.12) and (5.13), which are 19.0

N/mm for the DBE models and 68.0 N/mm for the FBE models. The maximum considered

crushing energy for both of the DBE and FBE models is 180.0 N/mm, which is the value used by

Coleman and Spacone (2001) for regularizing FBE models in their work. The middle values are

calculated from Equations (5.15) using the unconfined crushing energy of Equation (5.12) and

(5.13), and 1.5 times of those values. The results in Part (a) reveal that changing Gfcc value

changes the shape of the post-peak monotonic force-displacement response of the DBE models.

-700

-500

-300

-100

100

300

500

700

-0.15 -0.05 0.05 0.15

La

tera

l F

orc

e (k

N)

Displacement (m)


-700

-500

-300

-100

100

300

500

700

-0.15 -0.05 0.05 0.15

La

tera

l F

orc

e (k

N)

Displacement (m)


82

(a)

(b)

Figure 5.17 Sensitivity of the monotonic force-displacement response of the Column 328 (a) DBE and (b) FBE

models to GFcc value

-700

-500

-300

-100

100

300

500

700

-0.15 -0.05 0.05 0.15

La

tera

l F

orc

e (k

N)

Displacement (m)

Test328Gfcc=19.0 N/mmGfcc=32.4 N/mmGfcc=48.6 N/mmGfcc=180 N/mm

-700

-500

-300

-100

100

300

500

700

-0.15 -0.05 0.05 0.15

La

tera

l F

orc

e (k

N)

Displacement (m)

Test328Gfcc=68.0 N/mmGfcc=115.6 N/mmGfcc=173.4 N/mmGfcc=180 N/mm

83

The hump in the initial part of the curves becomes smoother and eventually vanishes as the value

of Gfcc increases. The difference in response of the FBE models in Part (b) with varying Gfcc

value is subtler, although it can be seen that for Gfcc value of 68.0 N/mm the hump shape of the

post-peak response becomes visible. It should be noted that the change in the Gfcc values of the

FBE models is much smaller than the DBE models, so the resulting effect is expected to be less

significant.

Forming of the hump shapes relates to the confined concrete material model losing

strength in the post-peak region. Equation (5.15) suggests that by increasing the Gfcc value, the

ε20c value also increases, resulting in the confined concrete material model to lose strength at

higher strain values. That explains why the hump shape fades with increasing the Gfcc value,

while all other parameters are constant. The strength loss in reinforcement steel material model

does not occur until rupture, while in confined concrete it occurs right after the peak up to

crushing. The sudden significant drop of base shear in the monotonic response of the models,

which was used to define the failure displacement, occurs when both reinforcement steel and

confined concrete have reached ε20c and εsu, respectively. If confined concrete reaches to its

limiting strain earlier than reinforcement steel, the failure displacement is controlled by the

rupture of steel and vice versa.

Understanding how the value of Gfcc can considerably affect the shape of the monotonic

response of the models, the questions posed is what value of Gfcc should be used in analysis. It

seems that the Gfcc value suggested by Pugh (2012) for FBE provides reasonable predictions of

the monotonic response of concrete columns, but this is not the case for DBE. As mentioned

earlier, he recommends a much smaller value to be used with DBE to compensate for the effect

of high axial load in the critical elements. The latter needs further investigation, as the results in

this chapter do not support using smaller Gfcc values for DBE. In fact, based on the sensitivity

analysis presented in this chapter, it seems that a higher value of Gfcc, close to the value used for

FBE elements could be used with DBE, as well.

One intuitive way to answer this question is to set the Gfcc value such that a model (FBE or

DBE) with LIP = Lp, would have ε20c-PH value equal to the empirical crushing strain of confined

84

concrete εcu, obtained from Equation (5.18). To do so, Equation (5.15) may be rewritten for such

condition as:

ε20c-PH=Gfcc

0.6f'cc

LP

- 0.8f

'

cc

𝐸𝑐𝑐+εoc (5.20)

Using the above equation, the values of ε20c-PH can be calculated for varying values of Gfcc as

listed in Table 5.7 (for column 328, Lp is 376 mm and εcu is -0.018). The tabulated values shows

that the Gfcc of 135 N/mm will results in ε20c-PH= εcu= -0.018. This value is fairly close to the Gfcc

value recommended by Pugh for FBE models, which is 115.6 N/mm for Column 328, but it is

much larger than his suggested value for DBE models, which is 32.4 N/mm. Setting Gfcc value at

32.4 N/mm implies that the confined concrete at the plastic hinge of the column crushes at the

strain value of -0.0069. This strain value is unduly conservative, considering that it is much

smaller than the εcu value, and εcu itself can be conservative at least by 50% (see Section 5.3.2).

The inconsistency in the regularization process stems from the fact that there are three

separate expressions available to calculate the Gfcc, εcu, and Lp values, while using Equation

(5.20) only two of these parameters can be assigned independently. In the author’s opinion,

while the above presented method somewhat alleviates this inconsistency, this problem needs

further in depth studies, which is beyond the scope of this thesis.

Gfcc (N/mm) 19.0 32.4 48.6 68.0 115.6 135.0 173.4 180.0

ε20c-PH -0.0054 -0.0069 -0.0086 -0.0108 -0.0159 -0.0180 -0.0222 -0.0229

Table 5.7 Calculated values of ε20c-PH for the Column 328 model with LIP=LP=376 mm

5.3.5 Adjusting Local Strain Response

While regularized models produce mesh-independent global force-displacement response, their

local response, such as curvature, strain values are still mesh-dependent. To obtain the correct

values of these parameters, post-processing of outputs is required. In Section 5.2.4, the post-

processing of curvature recommended by Coleman & Spacone (2001) was outlined. However,

there have been no recommendations so far on how to perform post-processing on concrete and

reinforcing steel strain values of regularized models. This is of paramount importance for

verifying the performance of concrete structures according to CSA S6-14, since the performance

85

criteria are defined in terms of strain limits. Without strain post-processing, checking of the

strain limits of the code could be misleading. It is possible to post-process the strain response of

regularized models using the post-processed curvatures suggested by Coleman and Spacone.

However, this would require tremendous amount of post-processing effort and can become

inconvenient. In this section, a much simpler and more direct method is formulated for the post-

processing of strain values in regularized models. The proposed method is then applied to the

regularized DBE and FBE models of Column 328 to confirm validity of the derived expressions.

The Gfcc value for regularizing the DBE and FBE models is 135 N/mm, following the discussion

in the previous section.

The regularization of material models can be viewed as a mapping between the post-peak

region of a reference material model and the post-peak region of the regularized model. This is

illustrated in Figure 5.18 for the confined concrete and reinforcing steel material models. The

shaded areas belong to the reference material models before regularization and the transformed

post-peak regions belong to the regularized models. The question posed here is how to check a

strain limit defined for the reference concrete model, εc1, or reinforcing material model, εs1, in the

corresponding regularized model. One answer is to map the strain limits to the regularized

material models similar to how the ultimate strain values are mapped in the regularization

process. This is possible considering that the ratio of the post-peak energy up to the specified

strain limit to the total post-peak energy is similar for the reference and the regularized material

models. In Figure 5.18, the mapped strain limits of concrete and reinforcing steel are indicated

by εc2 and εs2, respectively. The simplifying assumption, which also has been used in deriving the

regularization expressions in the previous sections, is linear post-peak response for both concrete

and steel material models. Using this assumption, the mapped the strain limits can be obtained

using the following expressions:

- Confined Concrete:

εc2= (ε20c-PH-εoc

εcu-εoc

) (εc1-εoc)+εoc (5.21)

86

- Unconfined Concrete:

εc2= (ε20u-εo

εu-εo

) (εc1-εo)+εo (5.22)

- Reinforcing Steel:

εs2=εy+(εs1-εy)Lgage

LIP

(5.23)

In which, εu is the crushing strain of unconfined concrete, typically between -0.004 to -0.006.

The detailed derivation of the above expressions is given in Appendix B.

εoc εcu

f’cc

0.2 f’cc

Ecc

σ

ε εc1 εc2 ε20c-PH

f cc

(a)

εy εsu,exp

Es

fu

σ

fy

fs

εs1 εs2 εsu ε (b)

Figure 5.18 Mapping strain limits in the post-peak region of (a) confined concrete and (b) reinforcing steel

materials

87

To test the derived expressions, they are applied to predict the displacements of the

regularized DBE and FBE models of Column 328, which correspond to reinforcing steel strain of

0.0022 (yielding) and 0.025, unconfined concrete strain of -0.004 (cover spalling), and confined

concrete strain of -0.018 (crushing). These strain limits are defined with respect to the original

material models, with εu of -0.004, εcu of -0.018, and εsu,exp of 0.09. The first step is to find the

displacement at the top of the column from monotonic response, corresponding to the first

occurrence of these strain limits in the regularized models. In this step the strain limits are not

mapped to the regularized material models yet. These displacements are listed in Part (a) of

Table 5.8 and Table 5.9, for the DBE and FBE models of Column 328 with various mesh sizes.

By inspecting the tabulated values, it is clear that except for the yielding limit, which is not in the

post-peak region, both of the regularized DBE and FBE models predict mesh-dependent

displacements for the specified strain limits. For instance, the regularized DBE model with

0.610 m mesh size, predicts the crushing of confined concrete (εcc =-0.018) at the displacement of

0.071m, while the regularized DBE model with 0.076 m mesh size, predicts the same

displacement at 0.030 m, less than half of the first value. It should be noted that all these models

produce mesh-independent global force-displacement response, since they are regularized. So,

the mesh-dependency in the obtained displacements relates to strain limits not being post-

processed.

The second step is to map the strain limits using Equation (5.21) to (5.23). These are listed

in Part (b) of Table 5.8 and Table 5.9 for the DBE and FBE models, respectively. For instance,

the confined concrete strain limit of -0.018 for the original material model corresponds to strain

limit of -0.0215 for the regularized DBE model with 0.610 m mesh size, and -0.1481 for the

regularized DBE model with 0.076 m mesh size. This significant difference in the modified

strain limits explains why the displacements of the models in Part (a) are considerably different.

Once the strain limits are mapped, now the displacements corresponding to the first occurrence

of the mapped strain limits are found for the regularized models. These are listed in Part (c) of

Table 5.8 and Table 5.9 for the DBE and FBE models, respectively. Comparing the results in

Part (c) with Part (a) of each table shows that the discrepancy in predicting the displacements

corresponding to the post-peak strain limits is considerably reduced for the FBE models. The

DBE models results still show some level of discrepancy in the displacements corresponding to

88

Displacement (m)

εs=εy=0.0022 εs=0.025 εc=-0.004 εcc =-0.018

DBE 0.076 m 0.007 0.020 0.013 0.030

DBE 0.152 m 0.007 0.025 0.014 0.041

DBE 0.305 m 0.007 0.035 0.016 0.059

DBE 0.610 m 0.007 0.052 0.018 0.071

(a)

Mapped Strain Limits

εs=εy=0.0022 εs=0.025 εc=-0.004 εcc =-0.018

DBE 0.076 m 0.0022 0.124 -0.031 -0.148

DBE 0.152 m 0.0022 0.063 -0.016 -0.076

DBE 0.305 m 0.0022 0.033 -0.008 -0.040

DBE 0.610 m 0.0022 0.017 -0.005 -0.022

(b)

Corresponding Displacement (m)

εs=εy=0.0022 εs=0.025 εc=-0.004 εcc =-0.018

DBE 0.076 m 0.007 0.049 0.037 0.126

DBE 0.152 m 0.007 0.045 0.034 0.113

DBE 0.305 m 0.007 0.041 0.028 0.093

DBE 0.610 m 0.007 0.039 0.021 0.078

(c)

Table 5.8 Column 328 DBE models: (a) displacements corresponding to the specified strain limits, (b)

mapped strain limits based on the regularized material models, (c) displacements corresponding to the

modified strain limits

89

Displacement (m)

εs=εy=0.0022 εs=0.025 εc=-0.004 εcc =-0.018

FBE 0.610 2IP 0.007 0.041 0.018 0.071

FBE 0.914 3IP 0.007 0.025 0.014 0.045

FBE 1.828 4IP 0.007 0.025 0.014 0.044

FBE 1.828 6IP 0.007 0.017 0.013 0.027

(a)

Mapped Strain Limits

εs=εy=0.0022 εs=0.025 εc=-0.004 εcc =-0.018

FBE 0.610 2IP 0.0022 0.017 -0.005 -0.022

FBE 0.914 3IP 0.0022 0.033 -0.007 -0.040

FBE 1.828 4IP 0.0022 0.033 -0.007 -0.040

FBE 1.828 6IP 0.0022 0.078 -0.016 -0.094

(b)


εs=εy=0.0022 εs=0.025 εc=-0.004 εcc =-0.018

FBE 0.610 2IP 0.007 0.030 0.021 0.081

FBE 0.914 3IP 0.007 0.030 0.020 0.080

FBE 1.828 4IP 0.007 0.030 0.019 0.078

FBE 1.828 6IP 0.007 0.033 0.023 0.079

(c)

Table 5.9 Column 328 FBE models: (a) displacements corresponding to the specified strain limits, (b) mapped

strain limits based on the regularized material models, (c) displacements corresponding to the modified strain

limits

90

the confined and unconfined concrete strain limits. Compared to the FBE models, the DBE

models predict higher displacement values for each of the strain limits except yielding.

Considering the failure displacement of each model, the predicted displacements for the mapped

strain limits seem more reasonable. For instance, for the crushing of confined concrete at εcc =-

0.018, the mapped strain limit predicts displacement values between 0.078 to 0.126 m, and the

reference strain limit predicts values between 0.027 to 0.071 m, while the predicted failure

displacements are between 0.10 to 0.13 m. The unresolved discrepancy in the predictions of the

DBE models is expected to be related to how DBE is formulated as opposed to FBE, considering

that DBE ensures constant axial strain and curvature across the element, while FBE ensures

constant axial load and linear curvature (Section 5.1.3).

Overall, it seems that the suggested method for post-processing the local strain response of

regularized models is working very well with the FBE models and is improving the predictions

of the DBE models. Without post-processing the strain limits, the performance evaluation based

on strain values would be improper.

5.3.6 Comparison of the Options for Dealing with Localization Issues

It is worthwhile, to compare the two methods introduced in Section 5.2.4 for dealing with the

localization issues in distributed plasticity models. To do so, the FBE and DBE models of

Column 328 with LIP=LP and LIP=0.5 LP in the plastic hinge region are employed again. The

material properties of the models are not regularized (εcu =-0.018 and εsu,exp =0.09). The

monotonic force-displacement responses of these models are compared against those of the

regularized DBE and FBE models in Figure 5.19 (the models are regularized using the Gfcc value

recommended by Pugh). The results suggest that using either DBE or FBE models with LIP=LP

does not provide reasonable and accurate estimation of the response. The results also show that

surprisingly, the force-displacement response of the models with LIP=0.5 LP is fairly close to the

response of the regularized models, using both DBE and FBE. Nevertheless, the local responses

of the former differ from the latter, as they employ different material properties.

The material model regularization is a more scientific approach to deal with the

localization issue in distributed plasticity models and it allows the freedom to choose a suitable

mesh size, which ensures accuracy of the results. However, in many cases the analysis tool does

91

not provide access to the material model properties to perform material regularization. In these

cases, setting the mesh size so that LIP=0.5 LP seems to be an acceptable alternative.

(a) (b)

Figure 5.19 Comparison of the monotonic response of the regularized (a) DBE and (b) FBE models of

Column 328 with the model satisfying LP=LIP

5.3.7 Validation of the Concentrated Plasticity Model for Test 328

The monotonic response of the fibre hinge model of Column 328 in SAP2000 is shown in Figure

5.20, along with the cyclic test results. The monotonic response closely follows the backbone of

the cyclic response in the linear range and in the post-peak range up to a displacement of roughly

0.06 m, and then gradually degrades until the failure displacement. The predicted failure

displacement is very close to the value obtained from the cyclic test result.

5.3.8 Comparison of the Distributed and Concentrated Plasticity Models of Column 328

A comparison is made between the monotonic responses of the concentrated plasticity and

regularized distributed plasticity models of Column 328, which were developed in the previous

sections (Figure 5.21). The distributed plasticity models are regularized using the crushing

energy values suggested by Pugh (2012) and the adjusted confined concrete crushing energy

values, as explained in Section 5.3.4. For column 328, the failure displacements predicted by the

regularized DBE models is comparable to the fibre hinge model and they are both close to the

failure displacement of the column obtained from the cyclic test result. However, the DBE model

overestimates the strength in the first few cycles. The DBE models also overestimate the linear

-900

-700

-500

-300

-100

100

300

500

700

-0.15 -0.05 0.05 0.15

La

tera

l F

orc

e (k

N)

Displacement (m)

Test328DBE Lip=LpDBE Lip=0.5 LpDBE Reg

-900

-700

-500

-300

-100

100

300

500

700

-0.15 -0.05 0.05 0.15

La

tera

l F

orc

e (k

N)

Displacement (m)

Test328FBE Lip=LpFBE Lip=0.5 LpFBE Reg

92

Figure 5.20 Monotonic force-displacement response of the SAP2000 Fibre Hinge model of Column 328

stiffness. Nevertheless, this is related to the assumed stiffness value for the shear spring at the

base of the DBE models, and can be adjusted readily. The fibre hinge model on the other hand

uses effective section properties values for the linear elastic beam column element to correct the

stiffness value, and in this case it has provided a better estimation of the initial stiffness. The

regularized FBE models underestimate the failure displacement of the column as discussed in the

previous sections, while they closely predict the back bone of the cyclic curve. The monotonic

response of the FBE models

It is desirable to obtain the displacement values corresponding to the first occurrence of the

concrete and reinforcement steel strain limits. Similar strain limits as those used in Section 5.3.5

are considered, and the corresponding displacements of the fibre hinge model are listed in Table

5.10. These values can be compared against average values of displacements predicted by the

FBE and DBE models in Table 5.8 and Table 5.9, which are listed in Table 5.11. It is observed

that the displacements predicted by the fibre hinge model are fairly close to the predictions of the

FBE models. However, the fibre hinge model predicts smaller displacements in the post-peak

region compared to the average displacements of the DBE models. The difference in the

predicted displacements of the three types of models becomes larger at higher strain limits for

concrete, such as the crushing of core concrete.

-700

-500

-300

-100

100

300

500

700

-0.15 -0.05 0.05 0.15

La

tera

l F

orc

e (k

N)

Displacement (m)

Test328

Fiber Hinge

93

(a)

(b)

Figure 5.21 Comparison of the regularized (a) DBE and (b) FBE models with the concentrated plasticity

models of Column 328


εs=εy=0.00224 εs=0.025 εc=-0.004 εcc =-0.018

SAP2000 Fibre Hinge 0.010 0.038 0.020 0.069

Table 5.10 Column 328 fibre hinge model, displacements corresponding to the specified strain limits

Average Displacement (m)

εs=εy=0.00224 εs=0.025 εc=-0.004 εcc =-0.018

FBE models 0.007 0.031 0.021 0.079

DBE models 0.007 0.044 0.030 0.102

Table 5.11 Average displacements of Table 5.8 and Table 5.9 for the FBE and DBE models of Column 328

-1100

-900

-700

-500

-300

-100

100

300

500

700

-0.15 -0.05 0.05 0.15

La

tera

l F

orc

e (k

N)

Displacement (m)

Test328DBE 0.310 m Reg Pugh 32.4 N/mmDBE 0.310 m Reg 135 N/mmFiber Hinge

-1100

-900

-700

-500

-300

-100

100

300

500

700

-0.15 -0.05 0.05 0.15

La

tera

l F

orc

e (k

N)

Displacement (m)

Test328FBE 0.610 m 2IP Reg Pugh 115.6 N/mmFBE 0.610 m 2IP Reg 135 N/mmFiber Hinge

94

5.3.9 Comparison of the Models for the Other Tests

To extend the investigation of the previous section, the monotonic responses of the regularized

DBE and FBE models of the other selected bridge columns are compared with those of the fibre

hinge models. The DBE and FBE models are regularized using the adjusted confined concrete

crushing energy in Section 5.3.4. The results are shown in Figure 5.22 along with the cyclic test

results for Column 415, 430, 828, and 1028. These columns, as suggested by their numbers, have

higher aspect ratios (4, 8 and 10), which in turn indicates that the columns are more slender and

the flexural response is more dominant.

Considering the results of Column 415 and 430, all three models perform well in terms of

capturing the initial stiffness and the backbone of the cyclic curve. However, the regularized

FBE models underestimate the failure displacements of the columns, while the fibre hinge

models overestimate that. The regularized DBE models provide acceptable estimation of the

failure displacements.

The results for Column 828 and 1028 shows that the regularized FBE models

underestimate the failure displacements by a greater degree compared to the shorter columns

328, 415, and 430. The Fibre hinge models slightly overestimate the failure displacements of the

two. The regularized DBE model of Column 828 underestimates the failure displacement as

well, while the regularized DBE model of Column 1028 provides an acceptable prediction of the

failure displacement. All three models capture the initial stiffness and the backbone of the cyclic

curve satisfactorily.

Overall, from the investigations of the regularized distributed plasticity and concentrated

plasticity models of the selected test column, the following was observed regarding the

monotonic response of the models:

1) The regularized FBE models tend to moderately underestimate the failure displacements

of the concrete bridge columns, while they provide accurate estimations of the initial and

post-peak backbone of the cyclic response.

2) The regularized DBE models tend to provide accurate estimation of the failure

displacements of concrete bridge columns, while they may overestimate the strength.

95

3) The fibre hinge models accurately capture the initial stiffness and backbone of the cyclic

response. They may provide an acceptable estimation of the failure displacement or

overestimate that.

The above observations are made based on a few tested bridge columns. Several assumptions in

the models affect the quality of their predictions. The assumed material models for concrete and

reinforcing steel as well as the equation for estimating the plastic hinge length of the columns

directly affect the response of the columns. Moreover, different analysis tools may have slight

differences in terms of implementing the material models and the formulation of inelastic beam-

column elements. Therefore, care should be taken in extending the above observations when

other analysis tools are employed.

96

(a) (b)

(c) (d)

Figure 5.22 Comparison of the predictions of the regularized DBE and FBE models with the fibre hinge

models for Column (a) 415, (b) 430, (c) 828, and (d) 1028

-600

-500

-400

-300

-200

-100

0

100

200

300

400

-0.3 -0.1 0.1 0.3

La

tera

l F

orc

e (k

N)

Displacement (m)

Test415DBE 0.310 m 132 N/mmFBE 0.610 m 2IP 132 N/mmFiber Hinge

-800

-600

-400

-200

0

200

400

600

-0.3 -0.1 0.1 0.3

La

tera

l F

orc

e (k

N)

Displacement (m)


-400

-300

-200

-100

0

100

200

300

-1 -0.5 0 0.5 1

La

tera

l F

orc

e (k

N)

Displacement (m)


-400

-300

-200

-100

0

100

200

300

-1 -0.5 0 0.5 1

La

tera

l F

orc

e (k

N)

Displacement (m)


97

Chapter 6: Evaluating the Strain Limits of CSA S6-14 for RC Bridges

In the previous chapter, appropriate modelling alternatives for PBD of RC bridges were

examined and a number of methods for dealing with the localization of plastic strains in

distributed plasticity models were examined. This chapter is a subsequent to the previous

chapter, but pursues a different objective. The main objective of this chapter is to employ the

strain limits of CSA S6-14 and BC Ministry of Transportation and Infrastructure (MoTI)

Supplement (2016) to predict the damage to the tested bridge columns introduced in Chapter 5.

To achieve so, the regularized FBE and DBE models generated and examined in Chapter 5, are

utilized. The chapter starts by introducing the strain limits of the code and BC MoTI Supplement

for RC structures. Next, a brief review of a number of performance criteria for RC bridges that

are recommended in the literature and other design guidelines is presented. The review serves as

a reference and a point of comparison for the recommended strain limits of the code and the

supplement. Next the regularized FBE and DBE models of the tests are utilized to predict the

damage to the columns based on the strain limits of the code and BC MoTI Supplement. The

extent of the predicted damage is compared against the actual observed damage to the columns

during the tests, and the results are discussed. This comparison reveals how accurately the

combination of the established strain limits and the generated models predicts the actual damage

to the columns.

6.1 Strain Limits in CSA S6-14 and BC MoTI Supplement

An overview of the CSA S6-14 PBD approach was given in Section 2.6 with the corresponding

tables copied in Appendix A . CSA S6-14 performance criteria for RC bridges include

quantitative strain limits of concrete and reinforcing steel for specified levels of damage. The

values of these limits are summarized in Table 6.1, where εc and εcc are unconfined and confined

concrete strains, εcu is the ultimate strain capacity of confined concrete, and εs is reinforcing steel

strain. After the release of the code, the strain limits at some damage levels were found to be

unduly conservative. Meeting these criteria was particularly challenging for bridges on soft soil

sites. Modifications to the CSA S6-14 performance criteria were adopted in the BC MoTI

Supplement to CSA S6-14, which was published in late 2016. The document relaxed some of the

98

strain limits of the code and also provided additional strain limits and damage descriptions for

various performance levels, as summarized in Table 6.2.

Each strain limit in CSA S6-14 and BC MoTI Supplement represents the initiation of a

damage state in ductile concrete members, as follows (Table 6.3):

(1) Yielding of the longitudinal rebars:

This damage state is reached when the reinforcing steel tensile strain exceed yielding.

(2, 3) Cover Spalling:

This damage state is controlled by the compressive strain of unconfined concrete, and the

limiting strain values represent the onset of cover spalling.

(4) Serviceability Limit 1:

This damage state is reached when the longitudinal reinforcing steel tensile strain exceeds

0.01. This limit replaced the yielding criteria for minimal damage. The name of the

damage state is assigned based on the similarity of this damage state to the Serviceability

limit stated in Kowalsky (2000).

(5) Serviceability Limit 2:


0.015, which corresponds to residual crack width exceeding 1 mm (Kowalsky 2000).

(6) Reduced Buckling:


0.025. This limit replaced Serviceability Limit 2 for repairable damage. Since, this value

is half of the strain limit established to check the buckling of longitudinal rebars, the

damage state is referred to as reduced buckling.

(7, 8) Core Crushing:

This damage state is controlled by the compressive strain of confined concrete.

(9) Reduced Fracture 1:

99

Established by Kowalsky (Kowalsky 2000; Goodnight et al. 2013), the onset of buckling

in the longitudinal reinforcing steel is checked by their tensile strain not exceeding 0.05.

Since buckling of the longitudinal rebars is a prelude to their fracture, this damage state is

called reduced fracture.

(10) Reduced Fracture 2:

This limit represents the initiation of fracture in the longitudinal reinforcing steel for

probable replacement damage, and is controlled by their tensile strain not exceeding

0.060 or 0.075, depending on size.

Damage Level Concrete Strain Reinforcing Steel Strain

None NS NS

Minimal εc > -0.004 εs < εy

Repairable NS εs < 0.015

Extensive εcc > εcu εs < 0.050

Probable Replacement NS NS

Table 6.1 CSA S6-14 concrete and reinforcing steel strain limits (NS: not specified)

Damage Level Concrete Strain Reinforcing Steel Strain

Minimal εc > -0.006 εs < 0.010

Repairable NS εs < 0.025

Extensive εcc > 0.8 εcu εs < 0.050

Probable Replacement εcc > εcu εs < 0.075 (30 M or smaller)

εs < 0.060 (35 M or larger)

Table 6.2 BC MoTI Supplement to CSA S6-14 concrete and reinforcing steel strain limits (NS: not specified)

Damage State Strain Limit (m/m)

Yielding(1) εs < εy

Cover Spalling 1(2) εc < -0.004


Serviceability Limit 1(4) εs > 0.010


Reduced Buckling(6) εs > 0.025

80% Core Crushing(7) εcc > 0.8 εcu

Core Crushing(8) εcc > εcu

Reduced Fracture 1(9) εs > 0.050

Reduced Fracture 2(10) εs > 0.060 or 0.075

Table 6.3 Strain limits associated to the flexural damage states of reinforced concrete columns

6.2 Performance Criteria for RC Bridges in Literature and Other Design Guidelines

The flexural performance of reinforced concrete members are commonly measured by using

limits on reinforcing steel and concrete strains, section curvatures, or deformations of the

100

members. Energy-based criteria may be used as well. In the following sections, an overview of

some of the strain and deformation limits recommended in literature and design guidelines for

RC bridges is presented.

6.2.1 Strain Limits

One of the early examples of adopting strain limits for the performance-based design of bridges

can be found in ATC-32 (1996a). The limits are listed in Table 6.4 for three performance levels

of minimal, repairable, and significant damage. In comparison to both CSA S6-14 and BC MoTI

Supplement, the steel strain limits are larger, while the concrete limits are comparable to the

limits of the both documents.

Table 6.5 reproduces the strain limits recommended by Kowalsky (2000) for the two

performance levels of serviceability and damage control. These values are one of the reference

values used to establish the strain limits of CSA S6-14. The serviceability concrete strain

corresponds to initiation of spalling and the steel strain corresponds to residual crack width not

exceeding 1 mm. The damage control concrete limit corresponds to repairable damage in

concrete and is estimated using the energy balance approach developed by Mander et al. (1988)

to obtain the ultimate strain of concrete in compression. Test results have shown that this value is

consistently conservative by 50% and so it was recommended for damage control performance

level.

Hose and Seible (1999) also report reinforcing steel and concrete strain limits for

evaluating the performance of RC bridges under the five-level performance assessment

procedure of Table 2.3 (Table 6.6). The recommended values are comparable to those of CSA

S6-14 and BC MoTI Supplement. Vossoghi and Saiidi (2010) recommended limits for mean

tensile strain of longitudinal reinforcing steel based on the shake-table test of scaled models.

These values are summarized in Table 6.7. What is unique about the strain limits of this table is

the consideration of the effects of shear demand and the type of ground motion on the flexural

strain limits of reinforcing steel.

101

Minimal Damage Repairable Damage Significant Damage

Concrete Strain Limit (1/3) εcu or 0.004 (1/3) εcu or 0.008 εcu or 0.012

Steel Strain Limit Grade 420

(10-25 mm) 0.03 0.08 0.12

Steel Strain Limit Grade 420

(29-57 mm) 0.03 0.06 0.09

Table 6.4 ATC-32 (1996) strain limits (AC1 2016)

Limit State Concrete Strain Limit Steel Strain Limit

Serviceability 0.004 0.015

Damage Control 0.018, εcu 0.06

Table 6.5 Strain limits by Kowalsky (2000)

Level Description Steel Strain Concrete Strain % Drift Displacement Ductility

I Fully Operational <0.005 <0.0032 <1.0 <1.0

II Operational 0.005 0.0032 1.0 1.0

III Life Safety 0.019 0.01 3.0 2.0

IV Near Collapse 0.048 0.027 5.0 6.0

V Collapse 0.063 0.036 8.7 8.0

Table 6.6 Bridge performance/design parameters SRPH-1 by Hose and Seible (Hose and Seible 1999;

Transportation Research Board 2013)

Visual Damage State

DS-1 DS-4 DS-5

Approximate Performance State

Ductile Flexural Column Condition Fully Functional Operational Delayed Operation

Far-field ground motions

Shear stress < 4√𝑓𝑐′

0.012±0.006 0.035±0.007 0.045±0.014

Far-field ground motions

Shear stress > 6√𝑓𝑐′

0.005±0.003 0.035±0.008 0.051±0.014

Near-field ground motions

Shear stress < 4√𝑓𝑐′

0.013±0.003 0.033±0.016 0.038±0.015

Table 6.7 Mean tensile strains in longitudinal reinforcement corresponding to visual damage states as

reported by Vosooghi and Saiidi (2010; ACI 2016) (f’c is in ksi)

102

6.2.2 Curvature Limits

The FHWA Seismic Retrofitting Manual (2006) defines performance limits in terms of curvature

limits. These curvature limits are based on some reinforcing steel and concrete strain limits. The

limits are reproduced here, as follow:

- Compression failure of unconfined concrete:

ϕp=

εu

c-ϕ

y (6.1)

- Compression failure of confined concrete:

ϕp=

εcu

(c-d")-ϕ

y (6.2)

- Compression failure due to buckling of the longitudinal reinforcement:

ϕp=

εb

(c-d')-ϕ

y (6.3)

εb=2f

y

Es

(6.4)

- Longitudinal tensile fracture of reinforcing bar:

ϕp=

εsu,exp

(d-c)-ϕ

y (6.5)

- Low-cycle fatigue of longitudinal reinforcement:

ϕp=

2εap

(d-d')=

2εap

D' (6.6)

εap=0.08 (2Nf)-0.5

(6.7)

Nf=3.5 (Tn)-1/3

(6.8)

103

- Failure in the lap-splice zone:

ϕp=(μ

lapϕ+7)ϕ

y (6.9)

6.2.3 Deformation Limits

In their fundamental study on performance models for flexural damage in RC columns, Berry

and Eberhard (2003) generated expressions for predicting the drift ratios corresponding to the

onset of cover spalling and bar buckling in RC columns. They used a subset of UW-PEER

reinforced concrete column performance database (now PEER performance database) to identify

the key parameters that affect the drift ratio, displacement ductility, plastic rotation, and

longitudinal strains corresponding to these two damage states. A total of 114 rectangular-

reinforced and 52 spiral-reinforced columns were selected, which all met the following three

criteria:

1) They were flexure-critical as defined by Camarillo (2003).

2) Their aspect ratio was 1.95 or greater.

3) The longitudinal reinforcing steel was continuous with no splice.

The database of the columns used by Berry and Eberhard was employed to derive the AASHTO

Guide Specifications for Seismic Bridge Design (2011) implicit formulae to obtain the

displacement capacity of RC columns for Seismic Design Categories of B and C (Transportation

Research Board 2013). More recently, the ACI 341.4R-16 report (2016) on the seismic design of

bridge columns based on drift, utilized Berry and Eberhard`s expressions for cover spalling and

bar buckling to obtain the mean drift capacities corresponding to operational and delayed

operational performance levels, respectively. For the delayed operational performance level, the

document assumed Caltrans SDC 1.6 (2010) minimum lateral reinforcement ratio for columns

with diameters greater than 36 in (900 mm) to calculate the effective reinforcement ratio, a 0.8

factor to account for biaxial bending, and a normalized bar diameter of 0.05, to calculate the

mean drift limit, as follows:

104

DrO=1.6 (1-P

Agf'

c

) (1+Lcol

10hcol

) (6.10)

DrDO=0.8*3.25 (1.45+1.125P

Agf'

c

) (1-P

Agf'

c

) (1+Lcol

10hcol

) (6.11)

In the above equations, DrO and DrDO are drift ratios corresponding to operational and delayed

operational performance levels in %, P/Agf’c is axial load ratio, and Lcol/hcol is the shear span-

depth ratio of column6. The equations indicate the effect of axial load ratio and shear span-depth

ratio on the flexural drift capacities. As axial load increases in a column, the drift capacity of the

column reduces. The shear span-depth ratio affects the drift ratio at the onset of the damage state.

With an increase in the shear span-depth ratio, the yield displacement and also the plastic hinge

length increase.

6.3 Examining the Strain Limits of CSA S6-14 and BC MoTI Supplement

6.3.1 Damage Observed in the Tests

According to Lehman et al (2004), the sequence of the observed damage states were similar for

all of the tested bridge columns, and included concrete cracking, yielding of longitudinal

reinforcements, initial concrete cover spalling, extensive concrete cover spalling, spiral fracture,

longitudinal reinforcement buckling, and fracture. Table 6.8 lists the displacement cycles, at

which the damage states were observed for the tested columns. Table 6.9 shows the average

compressive strain of concrete and average compressive and tensile strains of reinforcing steel

corresponding to the initiation of damage states. The strain values were measured by averaging

the values along the gauge length between the instrumentation rods, while local peak strain

values were equal or greater than these values.

Lehman and colleagues presented experimental cumulative distribution functions (CDF)

for the compressive strain of concrete corresponding to initial spalling and initial core crushing,

and for the tensile strain of reinforcing steel corresponding to different residual crack width. The

test data showed large dispersions in the measured strain values. For cover spalling, the

6 The second equation is developed for heavily reinforced circular concrete columns with spirals or hoops.

The document provides also an equation for lightly reinforced columns.

105

compressive strain of concrete ranged between -0.0039 to -0.011 with a mean value of -0.00664.

Columns with larger aspect ratios had smaller spalling strains. It is argued that the strain value of

-0.004 recommended by ATC-32 for initial spalling (Table 6.4) is the mean minus one standard

deviation of the test data. For core crushing, the measured strain values ranged between -0.010 to

-0.0297. The mean tensile strain of reinforcing steel was about 0.023 and 0.024 for the residual

crack width being greater than 0.13 and 0.25 mm, respectively. For bar buckling, the authors did

not present a CDF of the strain values as they are heavily affected by the cyclic history, and so

the results would not be generally applicable. It is argued by the authors that bar buckling

damage state cannot be adequately captured by a single limiting strain value, and a more refined

model is necessary.

Test# Yielding Initial

Spalling

Initial Core

Crushing

Bar

Buckling

Failure (20%

Strength loss)

328 0.013 0.030 0.071 0.132 0.132

828 0.059 0.178 0.445 NR NR

1028 0.098 0.254 0.889 0.889 0.089

415 0.016 0.038 0.127 0.178 0.178

430 0.017 0.051 0.178 0.178 0.178

Table 6.8 Displacements cycle in meter reported for the first occurrence of the damage states in the tested

bridge columns (Lehman et al. 2004)

Initial

Spalling

Initial Core

Crushing Bar Buckling

Failure (20%

Strength loss)

Test# εc εcc εs-comp εs-ten εs-comp εs-ten

328 -0.0057 -0.0098 -0.0570 0.0860 -0.0620 0.0440

1028 -0.0043 -0.0175 -0.0310 0.0980 -0.0440 0.0470

415 -0.0068 -0.0220 -0.0470 0.0730 -0.0470 0.0500

430 -0.0110 -0.0170 -0.0510 0.0890 -0.0520 0.0440

Table 6.9 Recorded average strain values corresponding to the first occurrence of the damage states in the

tested bridge columns (Lehman et al. 2004)

6.3.2 Damage Predicted by the Models

The regularized FBE and DBE models of Chapter 3 were used to predict the damage to the test

columns. The models were subjected to the displacement time history of Figure 5.7 and Table

5.3. The strain limits of the code and BC MoTI Supplement in Table 6.3 were checked for each

7 The difference between the minimum and maximum strain values is partly due to the difference in the

confinement of the columns.

106

model by adjusting the local strain response, as explained in Section 5.3.5. The displacements

corresponding to the first occurrence of the strain limits are listed in Table 6.10. The regularized

DBE models for Column 828 and 1028, did not predict the occurrence of some of the damage

states, as designated by NA in the table.

FBE model

Test# 328 828 1028 415 430

εs > 0.00224 0.007 0.048 0.076 0.013 0.014

εs > 0.01 0.015 0.086 0.123 0.023 0.025

εs > 0.015 0.019 0.093 0.143 0.028 0.031

εs > 0.025 0.028 0.125 0.188 0.039 0.042

εs > 0.05 0.051 0.181 0.256 0.068 0.073

εc < -0.004 0.020 0.117 0.190 0.033 0.031

εc < -0.006 0.030 0.164 0.253 0.048 0.044

εcc < -0.0144 0.063 0.327 0.566 0.110 0.098

εcc < -0.018 0.075 0.269 0.403 0.115 0.114

DBE model

Test# 328 828 1028 415 430

εs > 0.00224 0.007 0.049 0.076 0.013 0.014

εs > 0.01 0.020 0.102 0.153 0.031 0.033

εs > 0.015 0.028 0.131 0.183 0.398 0.042

εs > 0.025 0.043 0.170 0.239 0.061 0.064

εs > 0.05 0.082 0.294 0.420 0.114 0.117

εc < -0.004 0.029 0.242 0.518 0.051 0.046

εc < -0.006 0.047 0.391 NA 0.082 0.073

εcc < -0.0144 0.093 0.583 NA 0.152 0.126

εcc < -0.018 0.103 NA NA 0.171 0.148

Table 6.10 Predictions of the regularized FBE and DBE models for the cyclic displacements in meter

corresponding to the first occurrence of the specified strain limits

6.3.3 Comparison and Discussion

It is now possible to compare the predictions of the regularized FBE and DBE models for

damage to the columns with the actual observed damage in the tests. This was done by

comparing the predicted drift versus observed drift ratios for the reported damage states in the

tests, including yielding of the longitudinal reinforcing steel, initial spalling, initial core crushing,

and longitudinal bar buckling, as tabulated in Table 6.11. For column 828, tests results were not

available for bar buckling drift ratio, and therefore this damage state was not evaluated for the

column. The comparison of the values in the table, suggest the following:

107

1) For yielding damage state, both DBE and FBE models provide reasonably conservative

predictions of the drift capacities using the yielding criteria of the code. The predictions

of both models are fairly close in all of the tests.

2) For initial spalling, the FBE model provides reasonable estimates of the drift capacities

using the BC MoTI Supplement criteria with the exception of Column 415, and

conservative estimates, using the code criteria. The DBE model tends to overestimate the

drift capacities using both the code and BC MoTI Supplement criteria. The difference

between the predicted and observed values increases with the aspect ratio of the columns

and is significant for Column 828 and 1028.

3) For initial core crushing, the FBE model provides conservative estimates of the drift

capacities using the code criteria, with the exception of Column 328. The underestimation

of capacities is more significant in columns with larger aspect ratios. The DBE model

tends to overestimate the drift capacities.

4) For bar buckling, both models provide considerably conservative estimates of the drift

capacities using the code criteria.

As presented in Section 6.2.1, discrete deterministic strain limits have been utilized for

evaluating RC structures in the first generation performance-based procedures. Due to

uncertainty in strain limits at the onset of damage states, typically conservative lower-bound

values are recommended. As a result, a level of conservatism is implied in the performance

evaluation utilizing these strain limits (Transportation Research Board 2013). However, the level

of added conservatism is not accurately quantified nor is it consistent across damage states. To

address this issue, it is recommended to use a probabilistic description for the strain limits at the

onset of damage states. While a probabilistic solution refines the solution, it may not completely

address the issue. A full description of damage in many instances cannot be achieved by a single

strain limit, and better models with more than one parameter are necessary. Bucking of the

longitudinal reinforcing steel is an example, as explained in Section 6.3.1.

Another consideration with using strain limits to predict damage is the numerical model

used to predict the strain values. The accuracy of damage predictions cannot be evaluated solely

based on how the strain limits are linked to the onset of damage states; rather the accuracy in

predicting damage is a combination of the accuracy in the strain limits and the numerical model

108

used to evaluate those limits. In probabilistic terms, the uncertainty in evaluating damage is a

product of the uncertainty in the strain limits and modelling uncertainty. The effect of the latter is

clearly visible by comparing the predictions of the regularized FBE and DBE models in this

section. While both models are supposed to be adequate for predicting damage, their predicted

displacements vary a lot and show significant discrepancy with the observed values in the tests.

The FBE models tend to underestimate the displacement capacities, while the DBE models tend

to overestimate the capacities of those damage states related to concrete compressive strains, in

columns with larger aspect ratios. The question is whether the predictions of the models for these

damage states can be safely used for performance evaluation. To fully understand this problem, it

should be noted that the observed displacements in the tests correspond to the first occurrence of

each damage state and not the full development of that damage state. Besides, the observed

displacements have inherent uncertainty. One way to check the predicted displacements is by

using the Berry and Eberhard`s expressions introduced in Section 6.2.2. The expressions can be

employed as the point of comparison for the element level displacement response.

109

Yielding (εs > 0.00224)

Test# Measured FBE model DBE model

328 0.013 0.007 0.007

828 0.059 0.048 0.049

1028 0.098 0.076 0.076

415 0.016 0.013 0.013

430 0.017 0.014 0.014

Initial Spalling (εc < -0.004) Initial Spalling (εc < -0.006)

Test# Measured FBE model DBE model FBE model DBE model

328 0.030 0.020 0.029 0.030 0.047

828 0.178 0.117 0.242 0.164 0.391

1028 0.254 0.190 0.518 0.253 NA

415 0.038 0.033 0.051 0.048 0.082

430 0.051 0.031 0.046 0.044 0.073

Initial Core Crushing (εcc < -0.018)


328 0.071 0.075 0.103

828 0.445 0.269 NA

1028 0.889 0.403 NA

415 0.127 0.115 0.171

430 0.178 0.114 0.148

Bar Buckling (εs > 0.05)


328 0.132 0.051 0.082

1028 0.889 0.256 0.420

415 0.178 0.068 0.114

430 0.178 0.073 0.117

Table 6.11 Comparison between the measured and predicted displacements by the regularized FBE and DBE

models, corresponding to the first occurrence of damage states (all values are in meter)

110

Chapter 7: Implementation of the Frameworks for CSA S6-14 PBD

The two frameworks introduced in Chapter 4: are applied to assess the seismic design of a two-

span steel girder concrete bridge. Through this exercise, the step-by-step implementation of the

frameworks for practical design purposes is demonstrated. The chapter is concluded by

comparing the outcomes of the two assessments and a discussion on advantages and

disadvantages of the deterministic versus the probabilistic framework options.

7.1 Description of the Bridge

The bridge considered for this study is a hypothetical major-route bridge located in

Victoria, British Columbia. The assumed coordinates of the bridge site are 48.4284, -123.3656. It

is a two-span reinforced concrete bridge with steel girders. The initial member sizing of the

bridge was achieved from force-based design principals and based on experience. A schematic

elevation view of the bridge as well as the cross-section of the bridge at the pier are shown in

Figure 7.1 and Figure 7.2. The total length of the bridge is 125 m, with the west and east

span each being 60 m and 65 m, respectively. The superstructure is comprised of three steel

girders topped with a 0.225 m concrete slab and a 0.09 m asphalt overlay. The section of the

steel girders changes along each span as shown in the

Figure 7.1, and the maximum depth of the girders is 2.9 m. The bridge bent includes two 8

m high circular reinforced concrete columns, connected at the top with a 2.1x1.8 m reinforced

concrete cap-beam. The columns are both 1.5 m in diameter and has 35-35M longitudinal rebars,

making up a 2% longitudinal reinforcement ratio. They are laterally reinforced with 20M spirals

at 0.07 m pitch in the plastic hinge region (1.5m from the top and bottom of the columns) and

0.15 m pitch, elsewhere. The thickness of the cover concrete for both columns is 0.075 m, and

their axial force ratio (Pa/f’c Ag) is 0.10. The minimum specified compressive strength of

concrete for all members is 35 MPa, with the unit weight of 24 kN/m3. The reinforcing steel

grade is 400R with minimum specified yield strength of 400 MPa and ultimate yield strength of

540 MPa. The unit weight of the reinforcing steel is 77 kN/m3. Each column has a 1.5 m deep

6x6.5 m concrete spread footing. At the abutments, the bridge has expansion bearings and it is

free to move in the longitudinal direction, until closing of the gap. At the pier, the bridge has

pinned bearings and transfers only shear.

111

7.2 Bridge Site Properties

The soil profile at the bridge site includes a soft sand layer corresponding to site class D in CSA

S6-14 (see Appendix A ). These conditions roughly correspond to a uniform sand layer with

assumed shearwave velocity of 180-200 m/s, friction angle of 32 degrees, zero cohesion,

Poisson’s ratio of 0.3, and unit weight of 18 kN/m3. For this site condition, the effects of soil-

structure interaction must be considered.

15000 mm

Section 1

20000 mm

Section 2

25000 mm

Section 3

25000 mm

Section 3

25000 mm

Section 2

15000 mm

Section 1

8000 mm

2-ϕ 1525 mm ColumnsWest Abutment East Abutment

Pier

Figure 7.1 Schematic elevation view of the bridge

10920 mm

225 mm

2900 mm

90 mm Thick Asphalt Overlay

Haunch

Railing

Parapet

3700 mm 3700 mm

10000 mm 8000 mm

1800 mm

500 mm

Capbeam, 2100 mm wide

1525 mm

1500 mm

6500 mmFootings, 6000 mm wide

Figure 7.2 Schematic cross-section of the bridge at the pier

112

7.3 Probabilistic Seismic Hazard Analysis

A probabilistic seismic hazard analysis was conducted for the city of Victoria using EZ_FRISK

(The Fugro Consultants 2015). Three distinctive sources of earthquakes are active in this region,

namely shallow crustal, and deep subcrustal sources, and Cascadia subduction zone. All three

sources contribute to the seismic hazard, depending on the fundamental period of structure and

site-to-source distance. The proposed probabilistic hazard model for Cascadia subduction zone

(Halchuk et al. 2014) enables a full PSHA for the region, and combines the contributions of all

the three sources probabilistically at once. Subsequently, uniform hazard spectrum of Victoria

was obtained for 10%/50, 5%/50, and 2%/50 hazard levels, corresponding to 475, 975, and 2475-

year return periods, respectively (Figure 7.3-a). The calculated UHS values are close to the

values obtained from the hazard calculator available online at the Natural Resources Canada

Website (2016).

The design spectrum was then obtained using the UHS at each hazard level and appropriate

site coefficients recommended by CSA S6-14 (Clause 4.4.3.3). Since the abutments were not

specifically designed for sustained soil mobilization, 5% damped spectral response acceleration

values should be used (Clause 4.4.3.5). The 5% damped design spectra of the bridge at the

specified hazard levels are presented in Figure 7.3-b. These were utilized in the response

spectrum analysis of the bridge, while the selecting and scaling of ground motion records for

time history analysis were performed using uniform hazard spectra.

(a) (b)

Figure 7.3 (a) UHS and (b) design spectra for site class D at 2%/50, 5%/50, and 10%/50 hazard levels

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 2 4 6 8 10

Sp

ectr

al

Acc

eler

ati

on

(g

)

Period (s)

2%/50

5%/50

10%/50

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 2 4 6 8 10

Sp

ectr

al

Acc

eler

ati

on

(g

)

Period (s)

2%/50

5%/50

10%/50

113

7.4 Performance Criteria and Analysis Requirements

- Seismic Performance Category

The fundamental period of the bridge in both of the longitudinal and lateral directions is

greater than 0.5 s (see Section 7.6). The seismic performance category of a major-route

bridge with T ≥ 0.5 s and S(1.0) ≥ 0.3 is SPC 3 (Clause 4.4.4, Table A.2).

- Regularity and Minimum Analysis Requirements

According to the definition of Clause 4.4.5.3.2 the case study bridge is a regular bridge. The

minimum analysis requirements of a regular major-route bridge in seismic performance

category 3 is elastic dynamic analysis at 2%/50 and 5%/50 hazard levels and is elastic static

analysis at 10%/50 hazard level (Clause 4.4.5.3.1, Table A.4 and Table A.5).

For this case study, response spectrum analysis (RSA) and nonlinear time-history analysis

were utilized to obtain the seismic demands on the bridge at the specified hazard levels. In

addition, inelastic static pushover analysis was utilized to get the sequence of plastic hinge

formation in the ductile members (i.e. columns), and the drift capacities corresponding to the

first occurrence of the considered performance criteria (drift capacities).

- Minimum Performance Levels

The minimum performance levels for major-route bridges in terms of tolerable structural

damage is “minimal” at 10%/50, “repairable” at 5%/50, and “extensive” at 2%/50 hazard

level (Clause 4.4.6.2, Table A.6). The minimum serviceability objectives for the above

performance levels, is “immediate”, “service limited”, and “service disruption”, respectively.

- Possible Failure Mechanisms

Prior to setting the performance criteria, the possible local and global failure mechanisms

should be determined. Here we consider four possible failure mechanisms as follows:

- Ductile failure of the columns in flexure (local failure)

- Brittle failure of the columns in shear (local failure)

- Unseating of the deck at the abutments in the longitudinal direction (global failure)

- Pounding between the deck and the abutments (global failure)

114

Other failure mechanisms such as foundation soil failure, abutment backfill soil failure, etc.

should also be considered, which are out of the objectives of this case study.

- Performance Criteria

I. Flexural Failure of the Columns

The strain limits of the code and BC MoTI Supplement and the corresponding damage states

were presented in Section 6.1. The ultimate strain capacity of confined concrete can be

calculated using Equation (5.18). For εfs, a value of 0.09 could be used in the equation,

following the recommendation of Caltrans SDC 1.7 (2013) for the reduced ultimate tensile

strain of Grade 400 #10 (Metric #32) rebars or smaller. The value of f’cc, can be obtained

using Mander et al. (1988) constitutive model. The confinement factor for the column cross

section in the plastic hinge region is calculated as 1.288 using Mander model, which

multiplied by the expected compressive strength of f`ce=43.75 MPa, gives f’cc=56.35 MPa.

Substituting all values in the above expression gives an ultimate compressive strain capacity

of -0.0163 for the plastic hinge region. The established strain limits and their corresponding

damage states are listed in Table 7.2.

Hazard Performance Level CSA S6-14 BC MoTI

10%/50 Minimal Damage εc >-0.004, εs < εy εc >-0.006, εs <0.010

5%/50 Repairable Damage εs <0.015 εs <0.025

2%/50 Extensive Damage εc >-0.0163, εs <0.050 εc >-0.0130, εs <0.050

Table 7.1 CSA S6-14 and BC MoTI strain limits associated to the performance levels of a major-route bridge


Yielding(1) εs > 0.0024





Reduced Buckling(6) εs > 0.025

80% Core Crushing(7) εc < -0.0130

Core Crushing(8) εc < -0.0163

Reduced Fracture(9) εs > 0.05

Table 7.2 CSA S6-14 and BC MoTI strain limits associated to the performance levels of a major-route bridge

115

II. Shear Failure of the Columns

The brittle shear failure of the columns is checked by comparing the shear demand versus

capacity of the columns. Clause 4.4.10.4.3 of CSA S6-14 defines the shear demand as either

the unreduced elastic design shear or the shear corresponding to inelastic hinging of the

columns calculated by using probable flexural resistance of the member and its effective

height. However, this has been modified in the BC MoTI Supplement to exclude the former

method. The shear capacity of concrete can be calculated using either the simplified method

with β=0.1 and θ=45 (Clause 4.7.5.2.4), or by using the general method, which modifies the

shear capacity based on the member axial strain (Clause 8.9.3.7). BC MoTI Supplement

allows using more refined methods to calculate seismic shear capacity, which modify the

shear capacity based on ductility demands.

III. Unseating and Pounding of the Deck with the Abutments

To check the last two failure mechanisms, the longitudinal displacement at the deck level

should meet the following two criteria:

Δdeck≤ Lexpansion (7.1)

Δdeck≤ N (7.2)

In the above expressions, Lexpansion is the length of the longitudinal gap and N is the provided

support length at the abutments.

7.5 Structural Analysis Models

A 3D spine model of the bridge was generated in CSI SAP2000 version 18.2.08. Expected

material properties were used in the definition of steel and concrete materials. The behaviour of

the unconfined and confined concrete was modelled with the Mander et al. (1988) constitutive

model. The program automatically calculates and applies the confinement factor to the confined

concrete material from the input information of a section. Two models were utilized for the

bridge; a nonlinear fibre hinge model for performing nonlinear time-history analysis and

8 CSI SAP2000 was selected as the analysis tool as it is one of the most widely used commercial tools in

bridge engineering along with CSI Bridge. Therefore, the implementation example would demonstrate the

applicability of the deterministic and probabilistic assessment frameworks in practice.

116

pushover analysis, and an elastic model with effective material properties for response spectrum

analysis and modal analysis. The two models differ in how they represent the nonlinear

behaviour of the substructure ductile elements (i.e. columns), but both use similar superstructure

models and boundary conditions.

- Elastic Cracked Model

The cracked section properties of the columns were calculated following Section 5.1.1. The

effective flexural stiffness was calculated from the moment-curvature response of the column

section (Figure 7.4) as the slope of the line connecting the origin to the point of first yield in the

longitudinal rebars. This gave EcIeff=0.456 EcIg. Similar stiffness modifier was applied for the

effective shear stiffness of the columns. A property modifier of 0.2 was also applied to the

torsional constant of the column, following Caltrans SDC 1.7 (2013) recommendations. The

flexural stiffness of the cap-beam was modified by a factor of 0.5.

Since the super structure steel girders were capacity protected, it was assumed that they

remain essentially elastic under seismic loading. Therefore, the steel girders and the concrete

deck slab were modelled using elastic frame elements with composite section properties as

calculated in Table 7.3. A nominal linear spring was assigned to the ends of the deck in the

lateral direction to mimic the restraining effect of shearkeys and remove the unrealistic modes of

vibration in that direction. In the longitudinal direction, the deck is free to move and simplified

roller boundary conditions were employed to model the seat-type abutments.

Section1 Section2 Section3

Equivalent Steel Area (m2) 1.61 1.65 1.82

Dead Load (kN/m) 124 127 140

Ivertical (m4) 0.82 0.91 0.96

Itransverse (m4) 8.20 8.50 9.00

Table 7.3 Composite section properties of the deck at different sections (see Figure 7.1)

- Fibre Hinge Model

For NTHA and pushover analysis, a fibre hinge model was created in SAP2000, according to

Section 5.1.2 and 5.3.2. The fibre discretization of the column section is shown in Figure 7.4.

The fibre hinges were assigned to the mid-height of the plastic hinge zone of the columns. Using

Equation (5.3), the plastic hinge length for the longitudinal direction with single curvature was

117

obtained as 1089 mm, and for the lateral direction with double curvature as 754 mm. Similar

elastic section stiffness modifiers as used with the elastic cracked model, were assumed for the

column and cap-beam sections. The deck was assumed to remain elastic.

- Modelling Foundation Flexibility to Account for SSI

For the site class D model, the soil-structure interaction (SSI) effects should be accounted for

properly. The foundation flexibility effects were captured using a set of six uncoupled equivalent

springs, recommended by FEMA-356 (2000a) for shallow rigid foundations. FEMA-356

provides expressions for calculating spring constants as well as the embedment correction factors

for shallow rigid foundations. The spring constants of the foundations were calculated using

degraded shear moduli of the soil. According to FEMA-356, the resulting static stiffness values

are sufficient to represent repeated loading conditions in seismic events. The degraded shear

modulus of the soil was taken as 0.5 to 0.2 of the maximum shear modulus, following the

commentary of CSA S6-14, and was calculated as 17 MPa for the considered site class D soil

profile. Kinematic interaction and foundation damping were not considered in the case studies,

as including their effects were not indispensable to the focus of the thesis.

7.6 Fundamental Period of the Bridge

The results of the modal analysis on the bridge model revealed that in the longitudinal direction,

the first mode with the period of T1-long=1.61 s was dominating the response with 98%

contribution. In the lateral direction two modes contributed to 90% of the response; the first

mode with the period of T1-lat=0.56 s and a contribution of 84%, and the second mode with the

period of T2-lat=0.14 s and a contribution of 11%.

7.7 Ground Motion Selection and Scaling

The commentary of CSA S6-14 recommends a minimum of 11 ground motions each containing

two horizontal components to be selected for NTHA. The selected motions should be

representative of the tectonic regime, magnitude and distances that control the seismic hazard,

and the site condition. The target spectrum for selection and scaling of ground motion records

was site class D UHS at 2%/50 hazard level. The records were linearly scaled to match to the

target spectrum by minimizing the MSE within a specified period range. MSE is the mean

squared error of the difference between the spectral acceleration of the record and the target

118

Figure 7.4 Fibre cross section of the columns in SAP2000 (left) and the moment-curvature response of

the section in the plastic hinge region (right) (the response is calculated under the dead load).

Figure 7.5 View of the bridge model in SAP2000

0

4,000

8,000

12,000

16,000

0.00 0.05 0.10 0.15

Mo

men

t (k

N-m

)

Curvature (1/m)

119

spectrum (PEER 2010). The spectral acceleration of each record was calculated by taking the

geometric mean of the two horizontal components of that record.

For the period range of interest, the code commentary recommends a period range of 0.2T1 to

larger of 2 T1 and 1.5 s, where T1 is the fundamental period of the bridge. This would yield a

period range of 0.14-3.22 s for the bridge model. The selected records should include ground

motions from crustal, subcrustal, and subduction earthquakes, with proper magnitudes and

source to site distances. This information was obtained from deaggregation of the 2%/50 UHS of

Victoria at the fundamental periods of the bridge in the longitudinal and lateral directions, as

shown in Table 7.4. Because the period range of interest spans from relatively short periods up to

much longer periods, it was decided to match the records of each type of earthquake to a portion

of the period range of interest, instead of the entire range. This is justified by considering that the

contribution of each earthquake source to the hazard varies considerably across the period range

of interest. As a result, the shape of the spectrum of the records for that type of earthquake

becomes less conforming to the shape of the UHS. This will in turn cause poor matches to the

target spectrum, if the records are to be linearly scaled in the entire period range of interest.

Earthquake Source Magnitude Distance (km)

Crustal 6 - 7.5 10 - 40

Subcrustal 6.5 - 7.5 50-100

Subduction 8.5 - 9.0 50-100

Table 7.4 Deaggregation of 2%/50 UHS of Victoria at period of 1.61 s. Results of the deaggregation at the 0.56

were very similar to the 1.61 s and are not reported separately.

Table 7.5 lists the contribution of the crustal, subcrustal, and subduction earthquakes to the

UHS of Victoria at 2%/50, 5%/50, and 10%/50 hazard levels at different periods. The tabulated

values suggest the following trends:

1) Crustal earthquakes have higher contribution to the Victoria UHS at periods less than 1 s. At

these periods the contribution is higher at higher hazard levels (less frequent events).

2) Subcrustal earthquakes have higher contribution to the Victoria UHS at periods less than 1.5

s. At these periods the contribution is higher at lower hazard levels (more frequent events).

3) Subduction earthquakes have higher contribution to the Victoria UHS at periods larger than

0.5 s. At these periods the contribution is higher at higher hazard levels (less frequent

events).

120

Based on above-mentioned trends, the following period ranges were chosen for the three types of

earthquakes:

- Crustal 0.14-0.80 s

- Subcrustal 0.20-1.50 s

- Subduction 1.00-3.22 s

For each type of earthquake, a suite of 11 ground motions (total of 33 records) were

selected to match the target spectrum at 2%/50 hazard level within the aforementioned period

ranges (the commentary of the National Building Code of Canada (2015) suggest a minimum of

5 records per suite). The selected records excluded velocity pulses and near-fault effects. The

crustal records were selected using PEER NGA-West2 database (ref). For the selection of

subcrustal and subduction records S2GM online tool (Bebamzadeh 2015; Bebamzadeh and

Ventura 2015) developed at the University of British Columbia was utilized. For the NTHA at

5%/50 and 10%/50 hazard levels, the same suites of ground motion records, which had been

already selected and scaled for 2%/50 hazard level, were rescaled. The scale factor for each

hazard level was calculated as Sa(T1-long) hazard level i / Sa(T1-long) 2% /50, in which Sa(T1-long) is the

spectral acceleration for hazard level i at the fundamental period of the bridge in the longitudinal

direction. The selected records are listed in Table 7.6, along with the year and location of their

corresponding historical event. The acceleration and displacement spectra of individual motions

are presented in Appendix C . The commentary of CSA S6-14 recommends not selecting more

than two records from the same historical event for a suite of motions. However, subcrustal and

subduction earthquakes have far less recorded historical events as compared to crustal

earthquakes that would give reasonable match to the target spectrum within the presumed period

ranges. As a result, this condition was forgone when selecting motions for these types of

earthquakes.

Another condition stipulated by the commentary of CSA S6-14, is that the mean spectrum

of each suite of motions should not fall below the target spectrum by more than 10% within the

presumed period range. Figure 7.6 shows the mean spectra of the crustal, subcrustal, and

subduction suites versus the target spectrum. The mean spectrum of each suite is calculated by

taking the arithmetic mean of the geometric mean spectra of the horizontal components of the

121

Source Contribution

Crustal Subcrustal Subduction

T=

0.2

s 2%/50 0.38 0.41 0.21

5%/50 0.31 0.47 0.22

10%/50 0.27 0.52 0.21 T

=0

.5 s

2%/50 0.21 0.42 0.37

5%/50 0.18 0.49 0.34

10%/50 0.16 0.55 0.29

T=

1.0

s 2%/50 0.18 0.22 0.60

5%/50 0.17 0.31 0.53

10%/50 0.17 0.40 0.44

T=

1.5

s 2%/50 0.13 0.19 0.69

5%/50 0.13 0.29 0.58

10%/50 0.13 0.41 0.46

Table 7.5 Source contribution to the UHS of Victoria at different hazard levels and at different periods (the

sum of the contributions in each row is equal to 1).

Type Event Year Location Record Number

Crustal Cape Mendocino 1992 California, US 1

Crustal Christchurch 2011 New Zealand 2

Crustal El Mayor 2010 Mexico 3,4

Crustal Imperial Valley 1979 California, US 5

Crustal Landers 1992 California, US 6

Crustal Loma Prieta 1989 California, US 7

Crustal Northridge 1994 California, US 8

Crustal Superstition Hills 1987 California, US 9,10

Crustal Victoria 1980 Mexico 11

Subcrustal Miyagi Oki 2005 Japan 12

Subcrustal Nisqually 2001 Washington, US 13

Subcrustal Olympia 1949 Washington, US 14

Subcrustal Geiyo 2001 Japan 15-17

Subcrustal El Salvador 2001 Guatemala 18-22

Subduction Hokkaido 1952 Japan 23-28

Subduction Michoacán

Mexico 29,30

Subduction Tohoku 2011 Japan 31-33

Table 7.6 Selected ground motion records for time history analysis

122

records in that suite. It can be observed that a considerably well match to the target spectrum is

achieved for the crustal and subduction suites and the match is satisfactory for the subcrustal

suite. Figure 7.6 also shows the mean spectrum of all 33 records across the entire period range. It

can be seen that the mean spectra for all records falls more than 10% below the target spectrum

for periods larger than 2 s. This drop in the mean spectra can be attributed to the spectral shape

of the subcrustal and crustal records, and therefore confirms the justification for using potion-

wise matching to the target spectrum.

Figure 7.6 Mean spectra for the crustal, subcrustal, subduction suites, and all 33 records vs the target

spectrum (the range over which the mean spectra are matched to the target spectrum, are shown with the

vertical lines).

123

7.8 Response Spectrum Analysis

To obtain the seismic demands on the bridge structure, response spectrum analysis (RSA) was

initially performed. 5% damped design spectra of Section 7.3 were utilized in the analysis. At

each hazard level, two load cases were considered, following Clause 4.4.9.2 of the code: “The

horizontal elastic seismic effects on each of the principal axes of a component resulting from

analyses in the two perpendicular horizontal directions shall be combined within each direction

from the absolute values to form two load cases as follows:

(a) 100% of the absolute value of the effects resulting from an analysis in one of the

perpendicular directions combined with 30% of the absolute value of the force effects

from the analysis in the second perpendicular direction.

(b) 100% of the absolute value of the effects from the analysis in the second perpendicular

direction combined with 30% of the absolute value of the force effects resulting from the

analysis in the first perpendicular direction.”

Therefore, the seismic load combination included 125%-80% dead load, 100% seismic load in

one direction, and 30% seismic load in the orthogonal direction (see Clause 3.5.1 for load

combinations). For modal combination of the seismic effects, SRSS rule was applied, since the

contributing modes were well separated.

7.9 Nonlinear Time-History Analysis

To further verify the seismic performance of the bridge model, non-linear time history analysis

(NTHA) was conducted at the three specified hazard levels. A total of 99 analyses were

performed on the bridge model (33 records x 3 hazard levels). The two horizontal components of

each ground motion were applied simultaneously to the longitudinal and lateral directions of the

bridge model. No preference was given to the individual components of a motion when assigning

to the two principal directions. A Rayleigh damping of 5% was considered for the model. P-delta

effects were included in the analyses.

7.10 Pushover Analysis

The extent of the flexural damage in the columns was predicted by checking the maximum

relative drift ratios of the columns from RSA and NTHA against the relative drift ratios

124

corresponding to the first occurrence of each of the damage states9. Separate pushover analyses

were conducted on the bridge structure in the longitudinal and lateral directions. P-delta effects

were included in the analyses. The structure was pushed to the point of failure, indicated by

significant reduction in the strength capacity of the columns. The drift ratios corresponding to the

first occurrence of each damage state in the columns were considered as the drift ratios capacities

for those damage states. This can be obtained by checking the fibre hinge strains against the

strain limits of Table 7.2. Table 7.7 lists the obtained drift ratio capacity of the columns for each

of the considered damage states.

Performance Criteria Longitudinal Drift (%) Lateral Drift (%)

Yielding 0.82 0.51

Cover Spalling 1 1.82 1.15

Serviceability Limit 1 1.80 1.31

Cover Spalling 2 2.40 1.68

Serviceability Limit 2 2.23 1.83

Reduced Buckling 3.26 2.88

80% Core Crushing 5.17 4.01

Core Crushing 6.24 4.93

Reduced Fracture 5.82 5.59

Table 7.7 Column drift ratio capacities associated to the first occurrence of the damage states

7.11 PBD Assessment Using the Deterministic Framework

The deterministic framework described in Section 4.2 is applied to evaluate the performance of

the bridge in terms of the specified performance criteria of Section 7.4. A target reserve capacity

(RC*) of 10%, 0%, and 10% was assumed for checking flexural, unseating/pounding, and shear

failure mechanisms at all hazard levels, respectively. The demand parameters were calculated

using both RSA and NTHA. For NTHA, the demand parameters values were calculated by

taking the maximum of the mean demand values of the three suites of motions.10 The mean drift

9 In NTHA, it is possible to directly check the maximum strain demands against the strain limits to predict

damage. In that case, there would be no need to check the maximum displacement demands against the displacement

capacities from pushover analysis. 10 The commentary of the NBCC 2015 recommends that “Each structural response parameter should be

taken as the mean value as computed from the three ground motions inducing the largest value of that response

parameter, for each suite of motions.” Considering that 11 motions were selected for each ground motion suites in

this example, the mean value of each suite seemed logical to be used for design assessment instead of the mean of

the three largest values in each suite. Using 11 motions guarantees acceptable estimation of the mean demand

parameter value.

125

ratios of the columns in the longitudinal and lateral directions of the bridge are reported in Table

7.8 for individual crustal, subcrustal, and subduction suites, and for all 33 records altogether. The

tabulated values suggest that in the longitudinal direction, the subduction suite induces the

maximum drift demands, while in the lateral direction the subcrustal suite induces the largest

response. This outcome could be anticipated, considering that at the fundamental period of the

bridge in the lateral direction (0.56 s), subcrustal earthquakes have the highest contribution to the

seismic hazard in Victoria. However, at the fundamental period of the bridge in the longitudinal

direction (1.61s), subduction earthquakes contribute the most.

The maximum drift ratio demands of the columns calculated from RSA and NTHA along

with the predicted level of damage under the flexural failure mechanism are summarized in

Table 7.9. Overall, the drift values predicted by the NTHA are about 20% and 40% larger than

those predicted from RSA in the longitudinal and lateral directions, respectively. Inspecting the

predicted level of flexural damage, reveals that the bridge endures less damage in the lateral

direction compared to the longitudinal direction. In the lateral direction the bridge undergoes

yielding and minor spalling, while in the longitudinal direction it endures major spalling and

wide cracks, with high plastic strains in the longitudinal rebars. This difference in performance is

due to the fact that in the lateral direction, the bridge benefits from the framing action and the

restraining effect of the shear keys. The lower period of the bridge in this direction impose lower

displacements demands on the structure as well. However, in the longitudinal direction, the

bridge essentially acts as a cantilever, and therefore the imposed displacement demands are

considerably larger.

To verify the performance of the columns under the flexural failure mechanism, the reserve

drift ratio capacity of the columns (RC) were calculated for the performance criteria listed in

Table 7.1. The drift demand to capacity ratios were obtained in the longitudinal and lateral

directions, considering both of the CSA S6-14 and BC MoTI Supplement performance criteria.

The reserve capacities were calculated using the largest drift demand to capacity ratios for the

two directions. The performance of the bridge was deemed acceptable if the calculated reserve

capacities were equal to or larger than the target reserve capacity of 10% at all hazard levels.

Considering the RSA results in Table 7.10, the following was observed:

126

Crustal Subcrustal Subduction All Records

Hazard Level Δx (%) Δy (%) Δx (%) Δy (%) Δx (%) Δy (%) Δx (%) Δy (%)

2%/50 3.84 1.48 2.66 1.60 4.65 1.41 3.72 1.50

5%/50 2.53 0.99 1.97 1.08 3.07 0.98 2.53 1.02

10%/50 1.90 0.65 1.46 0.72 2.13 0.67 1.83 0.68

Table 7.8 Mean column drift demands from NTHA in the longitudinal (x) and transverse (y) directions for

crustal, subcrustal, subduction suites, and all 33 records

CSA S6-14 BC MoTI

Hazard Level Δx (%) Δy (%) Damagex Damagey Damagex Damagey

RS

A 2%/50 3.94 1.05 SL2 Y B M

5%/50 2.58 0.74 SL2 Y SP2 M

10%/50 1.76 0.54 Y Y M M

NT

HA

2%/50 4.65 1.60 SL2 SP1 B SL1

5%/50 3.07 1.08 SL2 Y SP2 M

10%/50 2.13 0.72 SP1 Y SL1 M

Table 7.9 Column drift demands from RSA and NTHA in the longitudinal (x) and transverse (y) directions,

along with the predicted damage (M: minimal, Y: yielding of longitudinal reinforcements, SL1 & SL2:

serviceability limit 1 & 2, SP1 & SP2: cover spalling 1& 2, B: reduced buckling)

- Employing the CSA S6-14 criteria, the bridge meets the specified performance criteria at

2%/50 hazard level with acceptable reserve capacity, while it fails to meet the 5%/50 and

10%/50 performance criteria.

- Employing the BC MoTI Supplement criteria, the bridge meets the specified performance

criteria at 2%/50 and 5%/50 with reasonable reserve capacity, but fails to meet the

10%/50 performance criteria with acceptable reserve capacity.

- The controlling performance criteria using both CSA S6-14 and BC MoTI Supplement is

at 10%/50 hazard level.

- The calculated reserve capacities at different hazard levels are more uniform using the

BC MoTI Supplement criteria compared to the CSA S6-14 criteria.

The above observations are valid for the NTHA results as well, except that at 5%/50 the reserve

capacity is less than the target reserve capacity using the BC MoTI criteria.

The maximum longitudinal displacement of the deck at the three hazard levels is listed in

Table 7.11 to check the unseating and pounding failure mechanisms. The provided support

length and the longitudinal gap should be checked against these values to meet the 0% target

reserve capacity goal. The large longitudinal displacements at all hazard levels indicate the

127

possibility of pounding between the deck and the abutments. This can be rectified by either of the

following options:

1) Incorporating elastomeric bearings at the abutments to control the longitudinal

displacements of the girders.

2) Redesigning the abutment to semi-integral.

3) Reducing the longitudinal drifts of the columns, by increasing the column longitudinal

stiffness.

Finally, the shear capacity of the columns should also be checked against the shear demand to

control the shear failure mechanism. The comparison needs to be made only at 2%/50 hazard

level, which induces the largest shear demand in the columns. The details of such calculations

are not presented here, as they were carried out using standard force-based design expressions

provided by CSA S6-14 or BC MoTI Supplement.

CSA S6-14 BC MoTI Supplement

Hazard

Level

Δd/Δc

(%)-x

Δd/Δc

(%)-y RC (%)

Δd/Δc

(%)-x

Δd/Δc

(%)-y RC (%)

RS

A 2%/50 68 21 32 76 26 24

5%/50 115 40 -15 79 26 21

10%/50 217 104 -117 98 41 2

NT

HA

2%/50 80 32 20 90 40 10

5%/50 137 59 -37 94 38 6

10%/50 263 140 -163 119 55 -19

Table 7.10 Ratio of the drift demand to drift capacity of the columns in the longitudinal (x) and lateral (y)

directions, and the reserve drift capacity for each hazard level

Hazard Level Δdeck-x (m)

RS

A 2%/50 0.379

5%/50 0.248

10%/50 0.169

NT

HA

2%/50 0.423

5%/50 0.282

10%/50 0.197

Table 7.11 Maximum longitudinal and lateral displacement of the deck

128

7.12 PBD Assessment Using the Probabilistic Framework

It is desired to evaluate the performance criteria of Table 7.1 using the DCFD framework

Equation (4.19) and (4.20). Here, all 33 records are used together to obtain the median demand

values, although it is possible to repeat the same process for the individual suites of crustal,

subcrustal, and subduction records. The results of the NTHA for the maximum longitudinal and

lateral drift of the bridge columns at the three hazard levels of 2%/50, 5%/50, and 10%/50 are

shown in Figure 7.7, along with the median drift values. Using Microsoft Excel a power-law

curve was fitted to the median values of the demand in each direction, equations of which are

shown on the plots. The coefficient and the power numbers give the demand curve parameters a

and b, respectively. The median demand values as well as the record-to-record variability of the

NTHA results in the lateral and longitudinal directions are listed in Table 7.12. The latter was

obtained simply by taking the standard deviation of the natural logarithm of the drift ratio

demands. It was observed that the values of βRDx and βRDy varied with the hazard level. Therefore, the

mean values of βRDx and βRDy were employed in calculating demand factors, as listed in Table 7.13. This

table also lists the assumed values for the epistemic uncertainty in demand and the aleatory and epistemic

uncertainties in capacity based on the recommended values by ASCE/SEI 7-16 (see Table 4.1 and Table

4.2).

The next step was to obtain the hazard curve parameters. The mean annual frequency is plotted

against the spectral acceleration at the fundamental period of the bridge in the lateral and longitudinal

directions individually in Figure 7.8. The segment of the hazard curve, which spans between the three

hazard levels of 2%/50, 5%/50, and 10%/50 is indicated with the red color. A power-law curve

was fitted to the red segments of the hazard curve using Microsoft Excel. The coefficient and the

power numbers give the hazard curve parameters ko and k. The values of the obtained demand

and hazard curve parameters are tabulated in Table 7.14. Once the values of the demand and

hazard curve parameters and the uncertainties in demand and capacity were established, the

demand and capacity factors were calculated using Equation (4.15) to (4.18), as shown in Table

7.15. The confidence interval was also calculated using Equation (4.21), the value of which is

listed for confidence intervals of 95%, 90%, and 85% in Table 7.16. Implied factor of safety in

evaluating the lateral and longitudinal response of the bridge columns were obtained as in Table

7.17

129

Figure 7.7 Longitudinal and lateral drift ratio demands versus the spectral acceleration at periods of 1.61 s

and 0.56 s, respectively, using all 33 records.

Figure 7.8 Mean annual frequency versus the spectral acceleration at periods of 1.61 s and 0.56 s for Victoria

y = 4.819x0.819

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0.00 0.20 0.40 0.60 0.80

Lo

ng

itu

din

al

Dri

ft (

%)

Sa (1.61 s) (g)

2%/505%/5010%/50MedianPower (Median )

y = 1.105x1.200

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.50 1.00 1.50L

ate

ra

l D

rift

(%

)Sa (0.56 s) (g)

2%/505%/5010%/50MedianPower (Median )

y = 0.0002x-2.047

0.0001

0.001

0.01

0.01 0.10 1.00 10.00

Mea

n A

nn

ual

Fre

qu

ency

Sa (1.61s) (g)

y = 0.0007x-2.455

0.0001

0.001

0.01

0.10 1.00 10.00

Mea

n A

nn

ual

Fre

qu

ency

Sa (0.56 s) (g)

130

Median Values

Δx (%) Δy (%) βRDx βRDy Sa(1.61 s) Sa(0.56 s)

2%/50 3.25 1.38 0.38 0.28 0.628 1.221

5%/50 2.40 0.95 0.33 0.32 0.411 0.857

10%/50 1.67 0.62 0.29 0.32 0.280 0.623

Table 7.12 Median drift ratio demands of the bridge columns from NTHA in the longitudinal (x) and lateral

(y) directions, and the calculated record-to-record variabilities

βRDx 0.33 βUC 0.28

βRDy 0.31 βUD 0.24

βRC 0.60 βUT 0.37

Table 7.13 Demand and capacity aleatory and epistemic uncertainty values

Longitudinal Lateral

Hazard Parameters ko 0.0002 0.0007

k 2.047 2.455

Demand Parameters a 4.819 1.105

b 0.819 1.200

Table 7.14 Obtained values for the demand and hazard curve parameters

Longitudinal Lateral

Capacity Factors

ϕR 0.64 0.69

ϕU 0.90 0.92

ϕ 0.58 0.64

Demand Factors

γR 1.15 1.10

γU 1.07 1.06

γ 1.23 1.17

Table 7.15 Demand and capacity factors calculated for the longitudinal and lateral directions

λx

Confidence Kx Longitudinal Lateral

95% 1.65 0.64 0.62

90% 1.28 0.74 0.72

85% 1.04 0.81 0.78

Table 7.16 Confidence factor values calculated for various confidence intervals

Factor of Safety Longitudinal Lateral

γ/ϕ 2.14 1.83

γ/ϕλx 2.89 2.56

Table 7.17 Implied factor of safety in evaluating the lateral and longitudinal response of the bridge columns

in terms of drift ratio with/without considering the confidence interval

131

The final step was to perform the design checks using the factored demand and capacity

drift ratios. This is demonstrated in Table 7.18 and Table 7.19 for CSA S6-14 and BC MoTI

Supplement performance criteria, respectively. The design checks were carried out using the

following three objectives:

1) factored demand < factored capacity

2) factored demand / factored capacity < λ90%

3) median demand < median capacity

The first and second objectives correspond to the DCFD with and without considering a 90%

confidence interval. The last objective represents the situation where the design is checked using

a deterministic approach. The following is observed from the tabulated results in the tables:

- For CSA S6-14 performance criteria, the use of both frameworks suggest that the design

meets all the specified performance criteria at all hazard levels in the lateral direction,

except for the yielding criteria, where the 90% confidence is not met.

- For BC MoTI performance criteria, the use of the DCFD framework indicates that the

columns do not meet any of the specified criteria in the longitudinal direction, while the

deterministic framework suggest that they meet all the criteria, except for the

serviceability limit 1 at 10%/50.

- For BC MoTI performance criteria, the use of both frameworks suggests that the design

meets all the specified performance criteria at all hazard levels in the lateral direction.

To check the unseating and pounding failure mechanisms, the longitudinal deck displacement is

the demand variable, and the demand parameters a and b are found in the same way as was done

for the drift ratio demands of the columns. The capacity variable is the support length for

unseating and the longitudinal gap for pounding failure. The capacity in this case may vary only

minimally due to construction tolerances and unlike drift ratio capacity, it has a uniform

distribution (instead of lognormal distribution) with a small coefficient of variation. Therefore,

one may assume βRC= βUC=0 and ϕ=1, when applying the DCFD framework for checking the

unseating and pounding failure mechanisms limit states. It would be also useful to plot the MAF

of demand exceeding capacity using Equation (4.11) and setting βRC= βUC=0. This is presented

132

Longitudinal Response

Hazard Level Performance Criteria FD (%) FC (%) FD<FC FD/FC< λ90% D/C<1

10%/50 Yielding 2.32 0.47 F F F

Cover Spalling 1 2.32 1.05 F F F

5%/50 Serviceability Limit 2 3.09 1.29 F F F

2%/50 Core Crushing 4.49 3.60 F F P

Reduced Fracture 4.49 3.36 F F P

Lateral Response


10%/50 Yielding 0.32 0.33 P F P

Cover Spalling 1 0.32 0.73 P P P

5%/50 Serviceability Limit 2 0.50 1.17 P P P

2%/50 Core Crushing 0.85 3.14 P P P

Reduced Fracture 0.85 3.56 P P P

Table 7.18 CSA S6-14 performance criteria checks using factored demand and capacity drift ratios of the

bridge columns in the longitudinal and lateral direction (P: pass; F: fail; FD: factored drift ratio demand in

%, FC: factored drift ratio capacity in %; λ90%: confidence factor for 90% confidence interval; D: median

demand; C: median capacity)

Longitudinal Response


10%/50 Serviceability Limit 1 2.32 1.04 F F F

Cover Spalling 2 2.32 1.39 F F P

5%/50 Reduced Buckling 3.09 1.88 F F P

2%/50 80% Core Crushing 4.49 2.98 F F P

Reduced Fracture 4.49 3.36 F F P

Lateral Response


10%/50 Serviceability Limit 1 0.32 0.84 P P P

Cover Spalling 2 0.32 1.07 P P P

5%/50 Reduced Buckling 0.50 1.84 P P P

2%/50 80% Core Crushing 0.85 2.56 P P P

Reduced Fracture 0.85 3.56 P P P

Table 7.19 BC MoTI performance criteria checks using factored demand and capacity drift ratios of the

bridge columns in the longitudinal and lateral direction For CSA S6-14 performance criteria, the use of the

DCFD framework indicates that the columns do not meet any of the specified criteria in the longitudinal

direction, while the deterministic framework suggest that they meet the ones at 2%/50.

133

in Figure 7.9. To use this figure, one should find the displacements corresponding to the MAF of

0.000404, 0.00106, and 0.002105, which gives 0.41, 0.28, and 0.21 m, respectively. Based on

these numbers, the longitudinal deck displacement is excessive, and should be controlled using

any of the three options mentioned under the deterministic framework assessment section.

Finally, the shear failure mechanism is checked using CSA S6-14 FBD expressions similar to

Section 7.11.

Figure 7.9 MAF of exceeding limit states for longitudinal deck displacement

To this end, the assessment of the bridge design using the DCFD framework is complete.

However, it is possible to gain further insights into the design problem using the probabilistic

framework. Figure 7.10 and Figure 7.11 show the bridge column fragility curves for CSA S6-14

and BC MoTI supplement, individually. The fragility curves were calculated using the following

expression (Nielson and DesRoches 2007):

P[D>C|IM]=Φ(ln(η

D|Sa/η

C)/√β

RD

2+β

RC

2) (7.3)

where Φ is the Gaussian (or Normal) cumulative distribution function. Plotting the fragility

curves allows inspecting the two sets of performance criteria in CSA S6-14 and BC MoTI

Supplement by comparison. A spectral acceleration of 1 g may be considered as a point of

reference. At this spectral acceleration, the probability of demand exceeding capacity at each

0.0001

0.001

0.01

0.1

1

0 0.1 0.2 0.3 0.4 0.5 0.6

MA

F o

f E

xce

edin

g L

imit

Sta

te

Longidutinal Deck Displacement (m)

2%/50

10%/50

5%/50

134

performance level is considerably higher if the CSA S6-14 performance criteria is used. For

instance, in the longitudinal direction and at Sa=1 g, the probability of demand exceeding

Serviceability Limit 2 is about 87%, while the probability of demand exceeding Reduced

Buckling is 72%. It is also evident that the fragility curves of the BC MoTI Supplement

performance criteria are more evenly spaced compared to the CSA S6-14. This is in line with the

previous observation on the uniformity of reserve capacities for the two sets of criteria in Section

7.11. Similar conclusions can be made considering the response in the lateral direction.

Another valuable insight can be gained by comparing the fragility curves in the two

directions for each of the two sets of performance criteria. This comparison reveals that the

conditional probability of demand exceeding each of the specified performance criteria is much

higher in the longitudinal direction compared to the lateral direction. For instance, at Sa(1.61

s)=0.28 g (see Table 7.12) the probability of demand exceeding Cover Spalling 1 is about 45% in

the longitudinal direction, while at Sa(0.56 s)=0.62 g it is about 15% in the lateral direction. Such

information indicates that the bridge columns are performing much better in the lateral direction

compared to the longitudinal direction, and therefore suggest a revise in the design of the bridge

in the longitudinal direction to control the excessive drift demands on the columns.

One interesting representation of the DCFD outputs is to plot the ratio of factored demand

to factored capacity versus MAF exceedance for each of the performance limit states. This is

presented in Figure 7.12 to Figure 7.15. A factored demand to factored capacity ratio of one

specifies the limit of satisfactory performance, which is indicated by a vertical black dashed line

in the plots. The points on the LHS of this line define the space of satisfactory performance.

Using such plots enables to determine the MAF of exceeding a performance limit state, at which

the design of the bridge is deemed acceptable. Conversely, one may find the ratio of factored

demand to factored capacity for a given MAF of exceeding a limit state. For instance, for a MAF

of 0.002105 (corresponding to 10%/50 hazard level), the ratio of factored demand to factored

capacity for yielding criteria is 4.9 and 1.0 in the longitudinal and lateral directions, respectively.

This ratio for the relaxed Serviceability Limit 1 criteria in the BC MoTI Supplement is 2.23, and

0.39, respectively. In using these plots, it should be noted that the extrapolation of points outside

the range of 10%/50 to 2%/50 hazard levels introduces some error, since both the hazard curve

and demand curve parameters were estimated locally for this range. Nevertheless, the

135

extrapolated portions of the plots reveal the estimate trend of the plots at higher and lower hazard

levels than those specified by the code.

(a) (b)

Figure 7.10 Column fragility curves for the CSA S6-14 performance criteria in the (a) longitudinal and (b)

lateral directions.

(a) (b)

Figure 7.11 Column fragility curves for the BC MoTI Supplement performance criteria in the (a)

longitudinal and (b) lateral directions.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.5 1 1.5 2

P(D

>C

|Sa

)

Sa (1.61s) (g)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.5 1 1.5 2

P(D

>C

|Sa

)Sa (0.56 s) (g)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.5 1 1.5 2

P(D

>C

|Sa

)

Sa (1.61 s) (g)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.5 1 1.5 2

P(D

>C

|Sa

)

Sa (0.56 s) (g)

136

Figure 7.12 Factored demand to factored capacity ratio in the lateral and longitudinal directions versus the

mean annual frequency of exceeding the performance limits states of CSA S6-14.

Figure 7.13 Factored demand to factored capacity ratio in the lateral and longitudinal directions versus the

mean annual frequency of exceeding the performance limits states of BC MoTI Supplement.

0.0001

0.001

0.01

0.0 5.0 10.0 15.0

MA

F o

f E

xce

edin

g L

imit

Sta

te

Longitudinal FD/FC

0.0001

0.001

0.01

0.0 2.0 4.0 6.0

MA

F o

f E

xce

edin

g L

imit

Sta

teLateral FD/FC

0.0001

0.001

0.01

0.0 5.0 10.0 15.0

MA

F o

f E

xce

edin

g L

imit

Sta

te

Longitudinal FD/FC

0.0001

0.001

0.01

0.0 2.0 4.0 6.0

MA

F o

f E

xce

edin

g L

imit

Sta

te

Lateral FD/FC

137

Figure 7.14 Factored demand to factored capacity ratio with confidence interval of 90% in the lateral and

longitudinal directions versus the mean annual frequency of exceeding the performance limits states of CSA

S6-14.

Figure 7.15 Factored demand to factored capacity ratio with confidence interval of 90% in the lateral and

longitudinal directions versus the mean annual frequency of exceeding the performance limits states of BC

MoTI Supplement.

0.0001

0.001

0.01

0.0 10.0 20.0

MA

F o

f E

xce

edin

g L

imit

Sta

te

Longitudinal FD/(λxFC)

0.0001

0.001

0.01

0.0 5.0

MA

F o

f E

xce

edin

g L

imit

Sta

te

Lateral FD/(λxFC)

0.0001

0.001

0.01

0.0 10.0 20.0

MA

F o

f E

xce

edin

g L

imit

Sta

te

Longitudinal FD/(λxFC)

0.0001

0.001

0.01

0.0 5.0

MA

F o

f E

xce

edin

g L

imit

Sta

te

Lateral FD/(λxFC)

138

7.13 Discussion

7.13.1 Comparison of the Deterministic vs Probabilistic Frameworks

The case study demonstrated the step-by-step implementation of the deterministic framework of

Section 5.3 and the probabilistic DCFD framework. The question is which one of these

frameworks should be preferred. To answer this, it would be helpful to recognize that the DCFD

framework can be viewed as an upgrade to the deterministic framework, where the demand to

capacity ratio is less subjectively evaluated. The deterministic framework does not provide a

systematic approach to decide on the target reserve capacities for different limit states. In fact,

the code implies using a zero target reserve capacity for all limit states. This is equivalent to say

that following the code, the performance criteria are met if mean deformation demand is less

than mean deformation capacity, and that factored force demand is less than factored force

capacity. When using the DCFD framework, the discrepancy between how the deformation-

controlled and force-controlled limit states are evaluated is addressed by introducing demand and

capacity factors for the former. This enhancement though, comes with only minimal effort in

addition to what is already required to carry out the deterministic assessment. That is what makes

the DCFD a suitable framework for code-based applications and the use of the DCFD framework

should be included in the CSA S6-14 PBD approach. Nevertheless, a number of considerations

exist when using the DCFD framework for the CSA S6-14 PBD:

1) Assumed values of aleatory and epistemic uncertainties for demand and capacity:

The assumed values for aleatory and epistemic uncertainties (Table 4.1 and Table 4.2)

determine the values of calculated demand and capacity factors and therefore, directly affect

the outcome of performance evaluation as demonstrated in the implementation example. One

important number in this table is the value of aleatory uncertainty in deformation demand,

which is considerably larger than the other tabulated values. The commentary of the

ASCE/SEI 7-16 (2016) explains that: “these βC values are larger than the comparable values

for force-controlled components because the uncertainty is quite large when trying to

quantify the deformation at which loss of vertical load-carrying capability occurs.” What

ASCE/SEI 7-16 is concerned with is an accurate estimate of collapse capacity for the MCER

ground motions, and collapse is defined as loss of vertical load-carrying capability. This is a

139

substantially different philosophy than that of the CSA S6-14 PBD approach. The uncertainty

in estimation of the deformations of concrete components is larger in the strength-degraded

states. However, in CSA S6-14, the strains of concrete and steel are checked at various levels

of damage, from minimal damage to extensive damage. It is reasonable to say that the

assumed aleatory uncertainty for yielding strain capacity should be less than concrete core

crushing, or extensive spalling. Therefore, the use of large βRC value for the performance

criteria of minimal damage state may be conservative.

2) Mean vs median response:

For lognormal variables the mean and median are related through the following expression:

H=Heβ

2

2 (7.4)

where, H is mean, H is median, and β is the lognormal standard deviation of the variable.

Typically mean response values are used in performance evaluation, as was utilized with the

deterministic framework in the implantation example. The DCFD framework uses median

response values, and these two should be distinguished. For normally distributed variables,

the mean and median values coincide, while for lognormal variables, the use of Equation

(7.4) suggest that the mean is about 1.08 times the median if β=0.4, and about 1.2 if β=0.6.

3) Mean vs median strain limits:

It is not stated in the code if the values suggested for the strain limits are mean or median

values, and whether the original data followed a lognormal or normal distribution. One

reference for such information is the PEER report by Berry and Eberhard (2003). They

demonstrate that a normal distribution may provide a better fit to the experimental fragility

curves for longitudinal bar buckling and cover spalling drift ratios. They also provide mean

and coefficient of variation for the compressive strains corresponding to longitudinal bar

buckling, and cover spalling. In the implementation example, the strain limits of the code

were assumed to be mean values when using the deterministic framework and to be median

values when using the DCFD framework. While this assumption may introduce inaccuracies

for distributions other than normal, it does not compromise the validity of the outcomes. This

issue would have been solved, if there was more information on the stipulated strain limits.

140

4) The choice of confidence interval:

In FEMA-351 (2000), the 90% confidence interval is recommended for evaluating (some) of

the criteria of collapse prevention performance level, while a 50% minimum confidence

interval was recommended for evaluating the immediate occupancy performance criteria11.

Table 7.20 shows the values of confidence factor for the implementation example,

recalculated using lower confidence intervals of 75%, and 50%. It is notable that if a 50%

confidence interval is employed, the confidence factor becomes greater than 1.0. This implies

that the factored demand can be greater than the factored capacity by 19% and 15% in the

lateral and longitudinal directions. In other words, a lower safety factor is achieved. The

confidence factor would be equal to 1.0, for about 80% confidence interval. This indicates

that even when the factored demand is directly compared to factored capacity, an 80%

confidence interval is implied. Based on the above considerations, it may seem reasonable to

employ a lower confidence interval for evaluating the criteria of the minimal and repairable

damage performance levels compared to extensive damage and probable replacement levels.

λx

Confidence Kx Longitudinal Lateral

90% 1.28 0.74 0.72

75% 0.67 0.93 0.90

50% 0.00 1.19 1.15

Table 7.20 Recalculated confidence factors of Table 7.16calculated for lower confidence intervals

7.13.2 Question of Uniform Reliability (The Value of Loss Analysis)

The implementation example provided a comparison between the performance criteria in CSA

S6-14 and BC MoTI for major-route bridges. It was shown that the use of the BC MoTI criteria

will result in more uniform reserve capacities in the deterministic framework and probabilities of

exceeding limit states in the DCFD framework across the three hazard levels. The updated

criteria in the BC MoTI Supplement relaxed the strains limits at 10%/50 and 5%/50 hazard

levels, which were found to be unduly conservative, and tightened the strain limits at 2%/50

hazard level. Consequently, it is easier to satisfy the criteria of BC MoTI than CSA S6-14, and

moreover, the design is less dominated by the criteria at lower hazard levels. To this end, it

11 The choice of higher confidence interval for the collapse prevention performance level reflect more

stringent evaluation at that performance level.

141

would seem reasonable to conclude that achieving uniform reliability across hazard levels as the

performance goal is desired. Since, reliability is chosen as the performance goal, the design

approach is often referred to as reliability-based design.

However, the choice of uniform reliability has been challenged in the next generation

performance-based procedure by considering loss analysis in decision-making. Wen (2001)

explains that a cost-based design approach is more logical than a purely reliability-based design

and he suggests choosing target reliabilities by minimizing the life-cycle cost of structures.

Considering this view, one may argue that although it is easier to satisfy the relaxed criteria of

BC MoTI Supplement and they are exceeded less frequently, the consequences of exceeding

them in terms of loss is higher than the more conservative CSA S6-14 criteria. For instance, in

the implementation example the design may exceed the yielding criteria more frequently than the

serviceability limit 1, but it would be less expensive to repair the structure at the yielding damage

state than the serviceability limit 1. Consequently, to answer which set of criteria is a preferred

choice for PBD, one must first decide on the performance goals. If a uniform reliability across

hazard levels is the goal then the BC MoTI Supplement criteria is the preferred choice. However,

if minimizing cost is the goal, then further studies considering loss analysis are needed to

compare the two sets of criteria.

142

Chapter 8: Case Study-Seismic Evaluation of the Trent River Bridge

This chapter presents a case study on the PBD evaluation of the Trent River Bridge following the

provisions of CSA S6-14. The chapter aims to provide a comparison between the performance

criteria of CSA S6-14 and BC MoTI supplement when the importance category of the bridge

varies between life-line, major-route, and other. It also serves the purpose of showcasing the

importance of including SSI effects in PBD evaluation. To achieve these objectives, a number of

assumptions were made. First, the Trent River Bridge was treated as a new bridge instead of an

existing bridge. This assumption helps to keep the focus of thesis on the design of new

structures. Second, while the existing Trent River Bridge is a major-route bridge, the

performance evaluation of the bridge was carried out assuming two performance categories of

life-line and other in addition to major-route category. Third, two soil site classes were

considered for the bridge including site class C and D. For the site class C condition, SSI effects

were ignored and a fixed-based model was employed, while for the site class D condition, SSI

effects were included. From the above assumptions, it is clear that the conditions assumed for the

case study do not aim to necessarily mimic the conditions of the actual bridge. The case study

borrows the structure of the Trent River Bridge, only to fulfill the above-mentioned objectives.

8.1 Description of the Bridge

The Trent River Bridge is a major-route bridge, located at the crossing of the Trent River with

Highway 19 in Vancouver Island, British Columbia. An area map in Figure 8.1 shows the

location of the bridge relative to Victoria and Vancouver, two major cities of British Columbia

province in Canada. The bridge was originally designed using FBD according to CSA S6-88

(1988) and AASHTO (1992), with the reference seismic acceleration of A=0.4 g. The three-span

pre-stressed concrete bridge has semi-integral abutments (Figure 8.2 to Figure 8.5). The main

span of the bridge is 40-meter-long, and the two side spans are each 33 meter long. The bridge

deck is composed of three 2 m pre-stressed concrete girders, topped with a 0.25 m thick concrete

slab, and an approximately 0.05 m thick asphalt concrete wearing surface. The width of the

superstructure is 12 m and it is fixed at the pier diaphragm-pier cap connections. At the

abutments, the girders are resting on 600x425x116 mm rubber bearings pads, which are

143

reinforced with five 3 mm thick, grade 300W steel plates. The lateral movement of the girders at

the abutments is controlled with the shear keys.

The substructure of the bridge includes two reinforced concrete piers, each consisting of

two circular columns of 1.5 meter in diameter, and a pier cap-beam (Figure 8.6). The clear height

of the Pier 1 and Pier 2 columns are 14.9 m and 9.83 m, respectively. The thickness of the cover

concrete for the columns is 75 mm. The longitudinal reinforcement of the columns includes 28-

30M rebars, making up 1.1% reinforcement ratio. For the transverse reinforcement, 15M spirals

are utilized with 0.065 m pitch in the plastic hinge region and 0.15 m pitch, elsewhere. Both

piers have 1.8-meter-deep 6x12.5 m concrete spread footings. The minimum specified

compressive strength of concrete for all members is 35 MPa and the reinforcement steel grade is

400R with minimum specified yield strength of 400 MPa.

The actual bridge consists of two separate east and west bridges, carrying the northbound

and south bound lane traffics, respectively (Figure 8.3). However, only the structure of the east

bridge was considered as an individual bridge for this study.

Figure 8.1 Relative location of the Trent River Bridge with respect to Victoria and Vancouver

144

Figure 8.2 Elevation view of the Trent River Bridge

Figure 8.3 Plan view of the Trent River Bridge

Figure 8.4 Deck, pier diaphragm, and cap-beam cross-sections (dimensions in mm)

145

Figure 8.5 Connection of the deck to the end diaphragm

Figure 8.6 Pier 1 and Pier 2 elevation views (looking North)

8.2 Soil Properties

The Trent River Bridge rests on soft rock to very dense soil, which matches site class C in CSA

S6-14. These conditions roughly correspond to a uniform sand layer with shearwave velocity of

650 m/s. The sand layer has a friction angle of 32 degrees, zero cohesion, Poisson’s ratio of 0.3,

and unit weight of 18 kN/m3.

To investigate how SSI affects the performance assessment of the bridge, a softer sand

layer corresponding to site class D was also considered. The shearwave velocity in this case was

146

assumed to be around 180 m/s to 200 m/s, which represents the softest soil in this class. Similar

values were assumed for other soil properties as for the site class C sand layer.

8.3 Bridge Model

8.3.1 Choice of Analysis Tool

A 3D spine model of the bridge structure was generated in SeismoStruct. SeismoStruct is an

analysis tool developed by SEiSMOSOFT, used frequently both for commercial and academic

purposes. It offers the option of distributed plasticity beam-column elements as well as

concentrated link elements to capture the nonlinear behaviour of structures. SeismoStruct is

especially advantageous for PBD of structures, since it allows direct checking of strain limits at

the fibre level and notifies the user when those limits are reached. While being more user-

friendly compared to its counterparts, OpenSees and CSI PERFORM 3D, it does not provide the

option of manipulating material models for material model regularization discussed in Chapter

5:. Therefore, when using distributed plasticity models, the localization issues need to be

addressed by setting the mesh size based on an empirical plastic hinge length (Technique 1 in

Section 5.2.4).

8.3.2 Material Models

Expected material properties were utilized in the bridge model. The behaviour of confined and

unconfined concrete were modelled with con_ma material model that uses Mander et al. (1988)

constitutive relationship. The implemented material model takes five calibrating parameters

including, the compressive and tensile strength of unconfined concrete, the modulus of elasticity,

the strain at peak stress, and the specific weight of concrete. The strain at the peak stress of the

unconfined concrete was assumed to be 0.002 and the tensile strength of concrete was assumed

to be zero. The cyclic behaviour of the reinforcement steel was modelled using stl_mp material

model that uses Menegotto-Pinto constitutive relationship (1973) and isotropic hardening rules

proposed by Filippou et al. (1983). When defining a fibre section, the program takes the

unconfined concrete and reinforcement steel materials and the lateral and longitudinal

reinforcement arrangement, and then it automatically calculates and applies the confinement

factor to the confined portion of the section.

147

8.3.3 Structure

The bridge columns were modelled with inelastic DBE with fibre sections. Each fibre section

included 152 fibres of confined concrete at the core of the columns, unconfined concrete at the

cover of the columns, and reinforcement steel. The plastic hinge length, LP, for the Pier 1 and

Pier 2 columns was calculated as 2.5 m and 1.7 m, respectively, using Equation (5.3). The

columns were meshed into elements such that the length associated to the critical integration

point, LIP was 0.5 LP, following Section 5.3.6. The cap-beams were modelled using elastic frame

elements with cracked section properties. The connection of the columns to the cap-beams was

assumed rigid with no releases. For response spectrum analysis, elastic frame elements with

cracked section properties were utilized for the columns as well.

The pre-stressed concrete girders and the concrete slab were assumed to remain elastic,

following the recommendations of Caltrans SDC 1.7 (2013) for pre-stressed decks.

Subsequently, elastic frame elements with equivalent elastic section properties were employed to

model the deck.

8.3.4 Foundations

For the site class C model, the SSI effects at the foundations were negligible and a fixed-base

model was utilized. For the site class D model, similar to Section 7.5 the foundation flexibility

effects were captured utilizing a set of six uncoupled equivalent springs per recommendations of

FEMA-356 (2000). In calculating the spring constants, degraded shear moduli of the soil were

utilized. The maximum shear modulus of the soil for site class D was calculated as 73.4 MPa.

The degraded shear modulus of the soil was taken as 0.5 to 0.2 of the maximum shear modulus,

following the commentary of CSA S6-14, and was calculated as 14.7 MPa (Table 8.1).

To model the spring foundations, the two columns of each pier were connected at the

bottom with relatively rigid elastic frame elements. Then a link element comprised of the six

uncoupled foundation springs were assigned to the middle of the rigid elements. According to

FEMA-356, when the height of effective sidewall contact is taken larger than zero, the resulting

stiffness of the springs will include sidewall friction and passive pressure contributions.

148

Embedment Correction

Stiffness (MN/m) Footing 1 Footing 2

Translation along x-axis 5153 1.72 2.13

Translation along y-axis 5573 1.72 2.13

Translation along z-axis 6593 1.28 1.37

Rocking about x-axis 4671 2.33 2.40

Rocking about y-axis 181403 1.70 1.73

Torsion about z-axis 163872 2.45 2.45

Table 8.1 Calculated SSI spring constants for the Trent River Bridge site Class D model

8.3.5 Abutments

The abutment behaviour, embankment flexibility, and the interaction of the bridge with the

embankment soil considerably affect the seismic response of a bridge under moderate to strong

levels of shaking. The soil-structure interaction in the abutments is even more pronounced for

short span bridges with relatively stiff superstructures, where the inelastic response of the

embankment soil primarily controls the response of the bridge and intermediate column bents

(Aviram et al. 2008). Therefore, in case of the Trent River Bridge, it is necessary to employ a

appropriate model for the abutments that can adequately capture the soil-structure interaction at

the abutments.

A simplified abutment model proposed by Aviram et al. (2008) was employed to model the

semi-integral abutment and the back-fill soil behaviour. In this model, the abutment is replaced

with a rigid elastic frame element and a set of translational springs in the longitudinal, transverse,

and vertical directions. The rigid elastic frame element has a length equal to the width of the

superstructure and is rigidly connected in the middle to the superstructure. The link elements

containing the three translational springs are connected to the ends of the rigid element. In the

transverse and vertical directions, elastic springs proposed by Wilson and Tan (1990) were used

for the abutments. The static stiffness values of these springs were calculated based on the

reduced shear moduli of the soil. In the longitudinal direction, a bi-linear spring proposed by

Caltrans SDC 1.7 (2013) was utilized to mimic the response of the backfill soil. Caltrans gives

single stiffness and strength values for competent soils. These values were utilized for the

abutment springs in the site class C model. For the site class D backfill soil in the longitudinal

direction, the stiffness and strength of the longitudinal springs for site class C were reduced by

the ratio of the maximum shear moduli of the site class D soil to site class C (73.4 MPa to 775.3

149

MPa). The participating mass of the abutments and the shear resistance of the bearing pads were

not considered in the model. While these assumptions may be challenged, the simplicity of the

model makes it very convenient for practical design purposes.

Figure 8.7 Spine 3D model of the Trent River Bridge in SeismoStruct

8.4 Performance Criteria

The performance criteria of CSA S6-14 and BC MoTI supplement for a regular major-route

bridge was previously determined and tabulated in Table 7.1 and Table 6.3. Assuming

importance categories of lifeline and other for the Trent River Bridge requires that the bridge

satisfies different set of performance levels at the three hazard levels. The expected performance

levels for lifeline and other bridges and the strain limits associated to the performance criteria of

each performance level are listed in Table 8.2. The description of “no damage” performance

level in the code is unclear and does not specify a limit or a qualitative measure for allowable

damage. It is possible to assume that no damage is met if no cracking occurs in cover concrete.

But this criterion would be too conservative for a 475-year return period ground motion. A more

relaxed criterion equal to half the yielding strain of longitudinal rebars was employed instead for

verifying no damage performance level. This limit represents the “minimal” damage state.

8.5 Structural Analysis

The seismic demands on the structure were obtained by conducting NTHA and RSA at the

three specified hazard levels. NTHA was performed for only site Class D model. The design

spectra for both site class C and D models were calculated according to CSA S6-14, using the

5% damped spectral accelerations at the bridge site (Figure 8.8). The periods of the first

150

Life-line Bridges


10%/50 None εs < 0.5 εy εc >-0.006, εs <0.010

5%/50 Minimal Damage εc >-0.004, εs < εy εc >-0.006, εs <0.010


Other Bridges



5%/50 Extensive Damage εc >-0.0163, εs <0.050 εc >-0.0130, εs <0.050

2%/50 Probable Replacement εs <0.050 εc >-0.0163, εs <0.075

Table 8.2 CSA S6-14 and BC MoTI strain limits associated to the performance levels of lifeline and other

bridges.

Figure 8.8 Bridge design spectra for site class C and D at 2%/50 hazard level

longitudinal mode, and the first and second lateral modes of the bridge were obtained as 0.37,

0.27, 0.16 s on site class C, and 0.56, 0.6, 0.33 s on site class D, respectively.

For NTHA 11 pairs of horizontal ground motion records were selected following the CSA

S6-14 commentary and as previously discussed in Section 7.7. The records were linearly scaled

and matched to the site class D UHS of the bridge site in the period range of 0.1-1.5s. Table 8.3

lists the type and event of the selected records. Acceleration and displacement spectra of the

selected records along with the target spectrum are presented in C.4. Figure 8.9 shows the mean

acceleration and displacement spectra of the 11 records and how well they match the target

spectrum within the period range of interest.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 10

Sp

ectr

al

Acc

eler

ati

on

(g

)

Period (s)

class C

class D

151

Type Event Year Location Record Number

Crustal Chi-Chi 1999 Taiwan C1

Crustal El Mayor 2010 Mexico C2

Crustal Landers 1992 California, US C3

Crustal Northridge 1994 California, US C4

Subcrustal Miyagi Oki 2005 Japan SC1, SC2

Subcrustal Nisqually 2001 Washington, US SC3

Subduction Maule 2010 Chile SD1, SD2

Subduction Tohoku 2011 Japan SD3, SD4

Table 8.3 Selected motions for NTHA of the Trent River Bridge on site class D

Figure 8.9 Mean acceleration and displacement spectra for the 11 motions vs the target spectrum (the period

range over which the mean spectra are matched to the target spectrum, are shown with the vertical lines).

The drift capacities of the columns were obtained from pushover analysis of the entire

bridge system in the lateral and longitudinal directions, similar to Section 7.10. The calculated

drift capacities for the site class C and D models are listed in Table 8.4. Figure 8.10 shows the

results of the pushover analysis of the Trent River Bridge in the lateral direction on site class C.

8.6 PBD Assessment

Table 8.5 summarizes the maximum drift ratio demands of the columns from RSA in the

longitudinal and lateral directions, along with the predicted level of damage. The drift ratios are

calculated based on the relative deformation of the columns and do not include drift due to the

foundations displacements and rotations. Referring to the table, in the longitudinal direction the

152

Figure 8.10 Pushover curves for the Trent River Bridge on site class C in the lateral direction

Site class C Long. Drift (%) Lat. Drift (%)

Performance Criteria Pier1 Pier2 Pier1 Pier2

Minimal 0.37 0.28 0.32 0.23

Yielding 0.74 0.56 0.63 0.45

Serviceability Limit 1 1.52 1.19 1.51 1.12

Cover Spalling 1 2.18 1.83 1.71 1.22

Cover Spalling 2 2.75 2.19 2.35 1.73


Reduced Buckling 2.71 2.27 2.68 2.08

80% Core Crushing 3.48 2.83 2.99 2.18

Core Crushing 4.73 3.66 3.56 2.56

Reduced Fracture 4.87 4.12 4.63 3.71

Site class D Long. Drift (%) Lat. Drift (%)

Performance Criteria Pier1 Pier2 Pier1 Pier2

Minimal 0.35 0.25 0.50 0.43

Yielding 0.70 0.51 1.01 0.86


Cover Spalling 1 2.03 1.61 2.25 1.83

Cover Spalling 2 2.80 2.24 2.86 2.29


Reduced Buckling 2.73 2.29 3.16 2.65

80% Core Crushing 3.57 2.83 3.50 2.70

Core Crushing 4.35 3.43 4.13 3.20

Reduced Fracture 4.62 3.89 5.13 4.32

Table 8.4 Drift ratio capacities of the columns associated to the first occurrence of the damage states

-4000

-3500

-3000

-2500

-2000

-1500

-1000

-500

0

0 1 2 3 4 5 6

Pie

r B

ase

Sh

ear

(kN

)

Pier Lateral Drift Ratio (%)

Site class C

Pier 1 Pier 2Yielding Spalling 1Serviceability Limit 2 BucklingCrushing Fracture

153

bridge sustains minimal or no damage at all hazard levels on site class C. On site class D, Pier 2

undergoes yielding in the longitudinal direction at all hazard levels, and Pier 1 at 2%/50 hazard

level. In the lateral, the drift demands are higher. Nevertheless, Pier 1 sustains minimal or no

damage at all hazard levels, while Pier 2 undergoes yielding at 2%/50 and 5%/50 hazard levels,

on both site classes. The results of the NTHA are shown in Table 8.6 and Figure 8.11. In general

the mean drift demands from NTHA are a bit lower than the RSA demands but otherwise they

are in reasonable agreement with the RSA results.

8.6.1 Assuming Major-route Importance Category

The results of Table 8.5 indicate that the bridge satisfactorily meets all the minimum

performance requirements of major-route bridges in CSA S6-14 and BC MoTI Supplement (see

Table 7.1). One exception is the yielding criteria of the CSA S6-14 at 10%/50 hazard level,

which is not met by Pier 2 in the longitudinal direction on site class D.

The ratios of the drift demand to drift capacity and reserve capacities are listed in Table

8.7. Using the CSA S6-14 performance criteria, the reserve capacity of the design varies

considerably at different hazard levels, and is the least at 10%/50 hazard level for both of the site

class C and D models. Therefore, the performance criteria of 10%/50 hazard level control the

design. However, using the BC MoTI Supplement performance criteria, the reserve capacity of

the design is relatively even at all hazard levels. The large reserve capacities of the columns at

2%/50 hazard level are primarily due to the contribution of abutments in carrying the seismic

load, and partly due to lower displacement demands at the period of the bridge. These findings

are in line with those made in Chapter 7: for the implementation example and further confirm the

comparison of the CSA S6-14 and BC MoTI Supplement criteria for major-route bridges.

8.6.2 Assuming Lifeline Importance Category

If the importance category of the bridge is assumed to be lifeline, the bridge still meets all the

performance requirements of BC MoTI Supplement on both site class C and D. However, Pier 2

does not satisfy the CSA S6-14 criteria at 10%/50 and 5%/50 hazard levels.

The ratios of the drift demand to drift capacity and reserve capacities for the lifeline

bridge are listed in Table 8.8. The reserve capacities for CSA S6-14 criteria suggest that while

the bridge meets the requirements of 2%/50 hazard level with sufficient amount of reserve

154

Pier 1 Pier 2

Hazard Level Dx (%) Damage Dy (%) Damage Dx (%) Damage Dy (%) Damage

Sit

e

clas

s C

2%/50 0.19 N 0.48 M 0.29 M 0.68 Y

5%/50 0.13 N 0.34 M 0.20 N 0.48 Y

10%/50 0.10 N 0.25 N 0.15 N 0.35 M

Sit

e cl

ass

D 2%/50 0.86 Y 0.91 M 1.25 Y 1.25 Y

5%/50 0.60 M 0.64 M 0.87 Y 0.87 Y

10%/50 0.43 M 0.46 N 0.62 Y 0.63 M

Table 8.5 Column maximum drift demands from RSA in the longitudinal (x) and lateral (y) directions, along

with the predicted damage (N: none, M: minimal damage, Y: yielding of longitudinal reinforcements)

Pier 1 Pier 2

Hazard Level Dx (%) Damage Dy (%) Damage Dx (%) Damage Dy (%) Damage

Sit

e cl

ass

D 2%/50 0.64 M 0.87 M 0.95 Y 1.23 Y

5%/50 0.42 M 0.56 M 0.62 Y 0.77 M

10%/50 0.31 M 0.43 M 0.46 M 0.60 M

Table 8.6 Column mean drift demands from NTHA in the longitudinal (x) and lateral (y) directions, along

with the predicted damage

Figure 8.11 Drift ratios of the bridge columns from NTHA at 2%/50

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1 2 3 4 5 6 7 8 9 10 11

Lo

ng

itu

din

al

Dri

ft (

%)

Record #

P1 P2

Mean P1 Mean P20.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

1 2 3 4 5 6 7 8 9 10 11

La

tera

l D

rift

(%

)

Record #

P1 P2

Mean P1 Mean P2

155


Hazard

Level

Δd/Δc

(%)-x

Δd/Δc

(%)-y Reserve (%)

Δd/Δc

(%)-x

Δd/Δc

(%)-y Reserve (%)

Sit

e cl

ass

C 2%/50

Pier 1 4.2 13.6 86.4 5.5 16.1 83.9

Pier 2 7.8 26.4 73.6 10.3 30.9 69.1

5%/50 Pier 1 6.2 18.0 82.0 4.9 12.7 87.3

Pier 2 10.6 33.7 66.3 8.8 22.9 77.1

10%/50 Pier 1 13.3 39.3 60.7 6.4 16.4 83.6

Pier 2 26.4 77.8 22.2 12.4 31.1 68.9

Sit

e cl

ass

D 2%/50

Pier 1 19.8 22.1 77.9 24.2 25.9 74.1

Pier 2 36.4 39.1 60.9 44.3 45.6 54.4

5%/50 Pier 1 29.9 26.3 70.1 21.9 20.0 78.1

Pier 2 52.6 42.8 47.4 38.7 32.3 61.3

10%/50 Pier 1 61.6 45.6 38.4 26.9 22.8 73.1

Pier 2 122.5 72.7 -22.5 49.4 37.5 50.6

Table 8.7 Ratios of the drift demand to drift capacity of the columns the reserve drift capacities assuming

Major-route importance category for the bridge


Hazard

Level

Δd/Δc

(%)-x

Δd/Δc

(%)-y Reserve (%)

Δd/Δc

(%)-x

Δd/Δc

(%)-y Reserve (%)

Sit

e cl

ass

C 2%/50

Pier 1 9.0 25.5 74.5 7.1 18.0 82.0

Pier 2 15.4 47.9 52.1 12.8 32.5 67.5

5%/50 Pier 1 18.0 53.9 46.1 8.7 22.5 77.5

Pier 2 35.8 106.6 -6.6 16.8 42.6 57.4

10%/50 Pier 1 23.0 54.0 46.0 6.4 16.4 83.6

Pier 2 45.9 102.4 -2.4 12.4 31.1 68.9

Sit

e cl

ass

D 2%/50

Pier 1 43.1 37.8 56.9 31.5 28.7 68.5

Pier 2 75.8 61.5 24.2 55.8 46.4 44.2

5%/50 Pier 1 85.7 63.2 14.3 37.5 31.6 62.5

Pier 2 170.5 100.8 -70.5 68.7 51.9 31.3

10%/50 Pier 1 89.4 79.6 10.6 26.9 22.8 73.1

Pier 2 171.1 122.0 -71.1 49.4 37.5 50.6


Lifeline importance category for the bridge

156


Hazard

Level

Δd/Δc

(%)-x

Δd/Δc

(%)-y Reserve (%)

Δd/Δc

(%)-x

Δd/Δc

(%)-y Reserve (%)

Sit

e cl

ass

C 2%/50

Pier 1 4.1 10.4 89.6 3.4 8.9 91.1

Pier 2 6.9 18.2 81.8 5.8 14.0 86.0

5%/50 Pier 1 2.9 9.6 90.4 3.8 11.4 88.6

Pier 2 5.4 18.6 81.4 7.1 21.8 78.2

10%/50 Pier 1 4.6 13.1 86.9 3.6 9.2 90.8

Pier 2 7.8 24.6 75.4 6.5 16.7 83.3

Sit

e cl

ass

D 2%/50

Pier 1 18.7 17.8 81.3 15.8 15.7 84.2

Pier 2 32.2 28.9 67.8 25.1 23.9 74.9

5%/50 Pier 1 13.8 15.4 84.6 16.8 18.1 81.9

Pier 2 25.3 27.2 72.8 30.8 31.7 68.3

10%/50 Pier 1 21.5 19.0 78.5 15.7 14.5 84.3

Pier 2 37.8 30.9 62.2 27.8 23.8 72.2


Other importance category for the bridge

capacity, it does not meet the requirements of the other two hazard levels. The reserve capacities

for the 5%/50 and 10%/50 requirement are in the same order. Therefore, the design of the bridge

is controlled by the performance criteria of either 10%/50 or 5%/50 hazard level.

Using BC MoTI Supplement criteria, the reserve capacities are more evenly distributed

across hazard levels, with the reserve capacity at 5%/50 hazard level being slightly smaller.

Therefore, using the criteria at 5%/50 hazard level controls the design of the lifeline bridge.

8.6.3 Assuming Other Importance Category

The last scenario is to assume the importance category of the bridge as other. For this scenario,

the bridge meets all the performance requirements of both CSA S6-14 and BC MoTI Supplement

on both site class C and D.

The ratios of the drift demand to drift capacity and reserve capacities for this case are listed

in Table 8.9. Unlike the major-route and lifeline bridge scenarios, the reserve capacities for the

performance criteria of CSA S6-14 are distributed evenly across the hazard levels, with the

performance criteria at 5%/50 being slightly larger. Therefore, the design is controlled by the

criteria at either 2%/50 or 10%/50 hazard levels. Using the BC MoTI Supplement criteria,

similar to the major-route and lifeline bridge scenarios, the reserve capacities are distributed

evenly across the hazard levels. In this case, the criteria at 5%/50 hazard levels with a slight

difference control the design.

157

8.7 FBD Assessment

CSA S6-14 allows FBD as an alternative approach to PBD for regular major-route and other

bridges, but requires PBD for seismic design of life-line bridges (see Table A.3). The importance

factor, I, for major-route bridges is 1.5 and for other bridges is 1.0. The response modification

factor, R, for multiple-column ductile reinforced concrete bents is 5 and is applicable to both

principal directions of the bridge. The elastic load effects on ductile members are reduced by the

ratio of R/I, which gives 3.33 for major-route bridges and 5 for other bridges.

For FBD assessment only the major-route bridge scenario was considered. Figure 8.12

shows the axial force and bi-axial bending moment demands for the Pier 1 and Pier 2 columns at

2%/50 hazard level along with the column interaction diagram. The demands were obtained for

the site class C model. It is observed that the reduced demands are inside the capacity curve and

therefore the column is safe in flexure at 2%/50 hazard level. It is also observed that the Pier 2

columns have higher bending moment demands and consequently lower reserve strength

capacity. However, unlike the PBD, the FBD approach is not capable of predicting the level of

damage to the columns. Moreover, the interaction diagram, which serves as the capacity curve, is

calculated with the fundamental assumption that concrete crushes at strain value of 0.0035,

which is much smaller than the strain limits recommended for the PBD approach. Using the FBD

approach, if the reduced demand crosses the capacity curve, it only signifies that the force

demands have reached the strength capacity of the columns and it does not necessarily indicate

failure of the columns. Comparing the outcomes of the PBD and FBD for the major-route bridge

scenario, one may conclude that the force-based designed bridge satisfies the CSA S6-14 PBD

criteria for 2%/50 and 5%/50 hazard levels, but does not meet the yielding criteria at 10%/50

hazard level.

8.8 Conclusions

The seismic performance of a pre-stressed reinforced concrete bridge was evaluated using the

performance criteria of both CSA S6-14 and the BC MoTI Supplement, assuming three

importance category scenarios for the bridge. It was observed that the CSA S6-14 performance

criteria for the major-route and lifeline bridges are very conservative at 10%/50 and 5%/50

hazard levels, compared to 2%/50 hazard level. Consequently, the reserve capacity of the bridge

158

is uneven across the hazard levels. This is not the same for other bridge scenario as in this

scenario the CSA S6-14 criteria produce relatively even reserve capacities across the hazard

levels. In contrast to CSA S6-14 performance criteria, the criteria in BC MoTI Supplement result

in relatively similar level of reserve capacity at all hazard levels for all three importance category

scenarios. As discussed in Section 7.13.2, to be able to determine, which set of performance

criteria has preference over the other for PBD, one may need to include loss analysis and

compare the two sets of criteria in terms of incurring total loss.

The case study also highlighted the importance of including SSI effects for PBD. Soil-

structure interaction noticeably changed the displacement demands on the bridge and reduced the

reserve capacity of the columns on site class D.

Figure 8.12 Comparison of the axial force-bi-axial bending moment demand at 2%/50 hazard level with the

column interaction diagram

-100000

-80000

-60000

-40000

-20000

0

20000

0 5000 10000 15000

Axia

l F

orc

e (k

N)

Bending Moment (kN-m)

Plastic Hinge Region

P-M Intercation Curve Pier 1 Reduced Demand

Pier 2 Reduced Demand Pier 1 Elastic Demand

Pier 2 Elastic Demand

159

Chapter 9: Summary and Future Research

9.1 Summary of Thesis

The recently adopted performance-based design provisions of CSA S6-14 provides engineers

with means to accomplish more sophisticated seismic designs of conventional bridges as well as

innovative new systems, for which force-based design provisions would fall short. In addition,

the CSA S6-14 performance-based design promotes the discussion of post-earthquake

performance of bridges between owners and engineers, and facilitates asset management and

post-earthquake resiliency. All the merits of the PBD approach as envisioned by the code

committee are conditioned on whether the PBD provisions can be properly implemented in

practice. There are several challenges identified in this thesis with regards to implementing the

CSA S6-14 provisions. Aside from the general challenges that concern PBD procedures in

general, the particular challenges of implementing the CSA S6-14 approach are primarily related

to: (1) performance-based design verification framework; (2) calibration of performance criteria;

and (3) appropriate analysis models to evaluate the established performance criteria. Moreover,

there was a need expressed from the industry for case studies demonstrating the step-by-step

implementation of the CSA S6-14 PBD approach. This thesis have managed to address the

aforementioned challenges and needs within the scope of the thesis.

Two performance-based design verification frameworks, including a deterministic and a

probabilistic framework, were introduced to complement the CSA S6-14 PBD approach. The

deterministic framework reformulates the demand versus capacity checks in terms of reserve

capacity versus target reserve capacity checks. This reformulation gives a perspective on how the

reserve capacity of bridge components for each performance limit state would compare at

different hazard levels. The target reserve capacities reflect the desirable safety margin in

deterministic terms. When verifying the force-controlled limit states such as shear demand

versus capacity, the safety margin is implicitly enforced through the demand and capacity

factors. For verifying the deformation-controlled limit states such as reinforcing steel or concrete

strain limit states, however, the CSA S6-14 does not provide demand and capacity factors. This

is incompatible with the force-controlled limit states. The addition of the target reserve capacities

160

helps with resolving this incompatibility and ensures a level of safety for the deformation-

controlled limit states of the code.

For the probabilistic framework, Demand and Capacity Factored Design developed by

Jalayer and Cornell (2003) was adopted. The DCFD framework was developed with an intention

for code application and unlike many other probabilistic PBD frameworks does not rely on

fragility curves, which makes it a suitable framework option for the CSA S6-14 PBD approach.

Using the DCFD framework, engineers are able to calibrate the demand and capacity factors of

each performance limit state to achieve desirable minimum reliabilities. This could be an

important addition to the CSA S6-14 PBD approach with regards to verifying deformation-

controlled limit states. Compared to the deterministic approach, the DCFD provides a systematic

approach to include desirable levels of safety in verification of deformation-controlled limit

states. This is achieved through determining the demand and capacity factors such that each limit

state is met for an allowable mean annual frequency of exceeding. The framework also includes

a confidence interval factor to meet the limit state for the allowable mean annual frequency of

exceeding. The main formulation of the framework was borrowed in the thesis and was tailored

for adoption with the CSA S6-14 PBD approach. All the necessary inputs to the framework were

provided and discussed for further clarification.

The step-by-step implementation of each of the two frameworks was presented in a

detailed case study of a two-span steel girder reinforced concrete bridge. The performance of the

case study bridge was evaluated using both response spectrum analysis and nonlinear time-

history analysis. The selection and scaling of ground motion records for each suite of crustal,

subcrustal, and subduction motions was presented in detail. The case study demonstrated how

readily both frameworks can be implemented in practice with the current analysis tools, and no

additional cost. Moreover, a comparison of the two frameworks was presented, highlighting the

additional advantages of the DCFD framework in verifying the performance-based design

compared to the deterministic framework. It was also demonstrated that the factors calculated

from the DCFD were comparable to those recommended for deformation-controlled limit states

in ASCE/SEI 7-16 (2016).

161

With regards to calibration of performance criteria, a thorough comparison of the strain

limits of CSA S6-14 versus the BC MoTI Supplement was presented in a case study on a three-

span reinforced concrete bridge with pre-stressed concrete girders. The case study showed that

that the CSA S6-14 strain limits are unduly conservative for the minimal damage performance

level, such that this performance level controls the design. For major-route and lifeline bridges,

this means that the performance criteria at 10%/50 hazard level will control the design, while the

performance criteria at 2%/50 hazard level are the least critical. The BC MoTI Supplement on

the other hand, employs more relaxed strain limits for the minimal damage performance level

and smaller strain limits for the repairable damage performance level. It was demonstrated that

this adjustment in strain limits of the BC MoTI Supplement result in a more uniform

performance across hazard levels compared to the CSA S6-14.

Another important aspect of the performance verification addressed in the thesis was the

use of appropriate analysis models to evaluate the established performance criteria. Concentrated

plasticity models and distributed plasticity models are commonly used in practice. Distributed

plasticity models are commonly considered to be more advantageous in obtaining strain

demands. The use of distributed plasticity models with softening materials such as concrete is

challenged by the localization of plastic strains and mesh-dependency issue. Verification of both

global and local strain and curvature responses are only meaningful if the mesh-dependency

issue is addressed first. Common techniques of addressing the mesh-dependency issue in

inelastic beam-column elements include choosing a mesh size based on an empirical plastic

hinge length or alternatively applying material model regularization. The thesis provided an

insightful comparison of concentrated and distributed plasticity models and the two techniques of

dealing with mesh-dependency issue by employing them to model a number of tested reinforced

concrete bridge columns. It was shown that concentrated plasticity models with fiber-hinges

provide acceptable estimation of local strain and global displacement response of reinforced

concrete bridge columns, without the concern of mesh-dependency issue. Inelastic force-based or

displacement-based beam-column elements provide access to local strain response across the

entire element, but require dealing with the localization issue if used for reinforced concrete

components. When material model regularization is employed to address the localization issue,

verifying the strain limits of CSA S6-14 require additional post-processing effort. A technique

162

was developed in the thesis to verify the strain limits without the additional post-processing

effort and was examined on the model of a bridge column test.

A summary of the recommended strain and deformation limits in the literature and other

design guidelines was included in the thesis to serve as a point of comparison for the strain limits

of the CSA S6-14 and the BC MoTI supplement. The strain limits of the CSA S6-14 and the BC

MoTI supplement were utilized with appropriate analysis models developed in the previous

chapters of the thesis to predict the damage to the selected bridge columns tests. The predications

were compared with the test results. This comparison demonstrated how the combination of

appropriate modelling and the recommended performance criteria would predict the actual

damage to modern well-detailed bridge columns. It was brought to attention that Berry and

Eberhard’s drift limits for the spalling of concrete and buckling of longitudinal rebars maybe

used as a cross-check for the predictions of distributed plasticity models of reinforced concrete

bridge columns. This will exclude the possibility of numerical errors due to mistreatment of the

localization issue.

9.2 Main Contributions

Each chapter of the thesis elaborated on a relatively different topic and a summary of the main

takeaways was presented in the previous section. As such, the main contribution of the thesis to

the state of knowledge and practice can be highlighted as follows:

1) The challenges of implementing the CSA S6-14 PBD approach in practice were

identified and addressed within the established scope of the thesis. These challenges were

related to performance-based design verification framework, calibration of performance

criteria and appropriate analysis models to evaluate the established performance criteria.

2) A deterministic and a probabilistic verification frameworks were examined to be adopted

for the CSA S6-14 PBD approach.

3) A thorough comparison of the CSA S6-14 and the BC MoTI Supplement strain limits

was presented with regards to calibration of the established performance criteria.

4) Appropriate numerical models for verifying the local strain response of reinforced

concrete bridge components were investigated, including concentrated and distributed

plasticity models.

163

5) Detailed case-studies of reinforced concrete bridges were provided to present the step-by-

step implementation of the CSA S6-14 PBD approach and the application of the

deterministic and probabilistic verification frameworks.

In addition, this thesis serves as a reference document for practicing engineers to better

understand the challenges of implementing the CSA S6-14 PBD approach and provides

calibrated and demonstrated methods and tools through the contributions listed above.

9.3 Future Research

The addition of the PBD approach to CSA S6-14 is only the first step towards the future of

seismic design guidelines and as such, the research in this area should be a continuing effort. One

major area that is not covered in this research is the performance-based retrofit design and

seismic assessment of existing bridges. This area is of great interest as many of the bridges built

prior to 1990’s do not have adequate seismic detailing and need retrofit interventions. Both CSA

S6-14 and the BC MoTI Supplement aim to achieve similar performance-levels as new bridges

with retrofit design. Achieving those performance levels with only retrofit intervention may

increase the cost of retrofit design so high that a replacement solution becomes more viable,

considering the remaining life the bridge. In that case, the owner should decide on the expected

performance levels of the retrofit design. The challenge of this task is to associate appropriate

performance criteria to achieve the expected performance levels in the existing bridges. Research

is necessary in this area to complement the existing experience with retrofit design in practice.

Currently the public review draft of CSA S6-19 has employed more relaxed performance levels

for existing bridges with single hazard level. It should be investigated whether the associated

performance criteria would actually result in the expected performance levels in existing and

retrofitted bridges. For instance, the spalling of concrete may result in extensive damage in an

existing bridge, while it is associated to minimal damage for modern code-compliant designs.

In addition, the focus of the present research was on PBD of bridges with ductile

reinforced concrete substructures. Research is still needed to cover the PBD evaluation of other

types of bridges. A more comprehensive study may include studying archetypes of various types

of bridges. Another important topic for future studies could be the performance assessment of

bridges in aftershock events and their remaining capacity. Finally, the possibility of moving

164

towards loss-based design rather than damage-based or cost-based designs for future generations

of PBD provisions should be investigated. As mentioned in the general challenges of PBD

procedures, research is still needed in developing better damage and loss predictions models.

165

References

AASHTO. (2011). Guide Specifications for LRFD Seismic Bridge Design.

AASHTO. (2004). LRFD Bridge Design Specification.

AASHTO. (1992). Standard Specification for Highway Bridges.

ACI. (2016). "Report on the Seismic Design of Bridge Columns Based on Drift." Rep. No. ACI

341.4R-16, American Concrete Institute, Farmington Hills, Michigan.

Adams, J., and Halchuk, S. (2003). "Fourth Generation of Hazard Maps of Canada: Values for

Over 650 Canadian Localities Intended for the 2005 National Building Code of Canada." Rep.

No. Open File 4459, Geological Survey of Canada.

ASCE 41. (2006). Seismic Rehabilitation of Existing Buildings. American Society of Civil

Engineers, Reston, Virginia.

ASCE/SEI 7. (2016). Minimum Design Loads for Buildings and Other Structures. American

Society of Civil Engineers, Reston, Virginia.

Ashtari, S., and Ventura, C. (2017a). "Correlation of Damage with Seismic Intensity Measures

for Ductile Concrete Bridge Columns in British Columbia." 16th World Conference on

Earthquake Engineering, Santiago, Chile.

Ashtari, S., and Ventura, C. E. (2017b). "A Case Study on Implementing the Performance-Based

Seismic Design Provisions of the 2014 Canadian Highway Bridge Design Code." International

Workshop on Performance-Based Seismic Design of Structures, Tongji University, Shanghai,

China.

Ashtari, S., Ventura, C., and Finn, W. D. L. (2018). "Verifying the Seismic Performance of

Concrete Bridge Columns According to CSA S6-14." 11th National Conference on Earthquake

Engineering.

Ashtari, S., Ventura, C., Khan, S., and Atukorala, U. (2018). "Reinforced Concrete Bridge."

Professional Practice Guidelines for Performance-Based Seismic Design of Bridges in BC,

EGBC, Vancouver, BC, Canada, 59-70.

Ashtari, S., Ventura, Carlos, Finn, W. D. L., and Kennedy, D. (2017). "A Case Study on

Evaluating the Performance Criteria of the 2014 Canadian Highway Bridge Design Code." 39th

IABSE Symposium, Vancouver, Canada.

ATC. (1997). "Seismic Design Criteria for Bridges and Other Highway Structures." Rep. No.

ATC-18, Applied Technology Council, Redwood City, California.

166

ATC. (1996a). "Improved Seismic Design Criteria for California Bridges: Provisional

Recommendations." Rep. No. ATC-32, Applied Technology Council, Redwood City, California.

ATC. (1996b). "Seismic Evaluation and Retrofit of Concrete Buildings." Rep. No. ATC-40,

California Seismic Safety Commission, Sacramento, California.

Aviram, A., Mackie, K. R., and Stojadinovic, B. (2008). "Guidelines for Nonlinear Analysis of

Bridge Structures in California." Rep. No. PEER 2008/03, Pacific Earthquake Engineering

Research Center, University of California, Berkeley.

Baker, J., and Cornell, C. A. (2006). "Spectral Shape, Epsilon and Record Selection."

Earthquake Engineering and Structural Dynamics, 35(9), 1077-1095.

Baker, J., and Cornell, C. A. (2005). "A Vector-Valued Ground Motion Intensity Measure

Consisting of Spectral Acceleration and Epsilon." Earthquake Engineering & Structural

Dynamics, 34(10), 1193-1217.

Bazant, Z. P. (1976). "Instability, ductility, and size effect in strain-softening concrete." ASCE

Journal of the Engineering Mechanics Division, 102(2), 331-344.

Bazant, Z. P., and Cedolin, L. (1983). "Finite element modeling of crack band propagation."

Journal of Structural Engineering, 109(1), 69-92.

Bazant, Z. P., and Cedolin, L. (1980). "Fracture mechanics of reinforced concrete." ASCE

Journal of the Engineering Mechanics Division, 106(6), 1287-1306.

Bazant, Z. P., and Cedolin, L. (1979). "Blunt crack band propagation in finite element analysis."

ASCE Journal of the Engineering Mechanics Division, 105(2), 297-315.

Bazant, Z. P., and Planas, J. (1997). Fracture and size effect in concrete and other quasibrittle

materials. CRC press.

BC MoTI. (2016). "Section4-Seismic Design." Volume1-Supplement to CHBDC S6-14, BC

Ministry of Transportation and Infrastructure, Vancouver, BC.

Bebamzadeh, A. (2015). "Selection and Scaling of Ground Motions."

http://s2gm.hpcperformancedesign.com/login.php?e=1 .

Bebamzadeh, A., and Ventura, C. (2015). "New performance-based design tools developed at

University of British Columbia." LATBSDC Conference, Los Angeles, California.

Berry, M., and Eberhard, M. (2003). "Performance Models for Flexural Damage in Reinforced

Concrete Columns." Rep. No. PEER 2003/08, Pacific Earthquake Engineering Research Center,

University of California, Berkeley.

http://s2gm.hpcperformancedesign.com/login.php?e=1

167

Bertero, R. D., and Bertero, V. V. (2002). "Performance-Based Seismic Engineering: The Need

for a Reliable Conceptual Comprehensive Approach." Earthquake Engineering and Structural

Dynamics, 31 627-652.

Calabrese, A., Almeida, J. P., and Pinho, R. (2010). "Numerical issues in distributed inelasticity

modeling of RC frame elements for seismic analysis." Journal of Structural Engineering, 14(S1),

38-68.

Calderone, A. J., Lehman, D. E., and Moehle, J. P. (2001). "Behaviour of Reinforced Concrete

Bridge Columns Having Varying Aspect Ratios and Varying Lengths of Confinement." Rep. No.

PEER 2000/08, Pacific Earthquake Engineering Research Center, University of California,

Berkeley.

Caltrans. (2015). "Chapter 4-Structural Modeling and Analysis." Bridge Design Practice,

California Department of Transportation, California.

Caltrans. (2013). Seismic Design Criteria. California Department of Transportation, Sacramento,

California.

Caltrans. (2010). Seismic Design Criteria. California Department of Transportation, Sacramento,

California.

Calvi, G. M., Priestley, M. J. N., and Kowalsky, M. (2013). "Displacement-Based Seismic

Design of Bridges." Structural Engineering International, 23(2), 112-121.

Camarillo, H. R. (2003). "Evaluation of Shear Strength Methodologies for Reinforced Concrete

Columns". PhD. University of Washington.

Coleman, J., and Spacone, E. (2001). "Localization issues in force-based frame elements."

Journal of Structural Engineering, 127(11), 1257-1265.

Cordova, P. P., Deierlein, G. G., Mehanny, S. S. F., and Cornell, C. A. (2001). "Development of

a Two-Parameter Seismic Intensity Measure and Probabilistic Assessment Procedure." The

Second US-Japan Workshop on Performance-Based Earthquake Engineering Methodology for

Reinforced Concrete Building Structures.

Cornell, C. A. (1969). "A probability-based structural code." Journal Proceedings, 974-985.

Cornell, C. A., and Krawinkler, H. (2000). "Progress and Challenges in Seismic Performance

Assessment." PEER Center News, 3(2), 1-3.

CSA. (2014). S6-14 Canadian Highway Bridge Design Code. CSA Group.


168


Duan, L., and Li, F. (2003). "Seismic Design Philosophies and Performance-Based Design

Criteria." Bridge Engineering Seismic Design, W. Chen, and L. Duan, eds., CRC Press, 5-1-5-35.

Ellingwood, B. R., Galambos, T. V., MacGregor, J. G., and Cornell, C. A. (1980). "Development

of a Probability-Based Load Criterion for American National Standard A58." Rep. No. Special

Publication No. 577, National Bureau of Standards, Washington, D.C.

Ellingwood, B. R., and Wen, Y. (2005). "Risk-benefit-based design decisions for low-

probability/high consequence earthquake events in Mid-America." Progress in Structural

Engineering and Materials, 7(2), 56-70.

Elwood, K. J., and Moehle, J. P. (2005). "Drift capacity of reinforced concrete columns with

light transverse reinforcement." Earthquake Spectra, 21(1), 71-89.

Elwood, K. J., and Moehle, J. P. (2004). "Evaluation of Existing Reinforced Concrete Columns."

13th World Conference on Earthquake Engineering, Vancouver, Canada.

Fajfar, P., Tomaz, V., and Matej, F. (1990). "A Measure of Earthquake Motion Capacity to

Damage Medium-Period Structures." Soil Dynamics and Earthquake Engineering, 9(5), 236-

242.

FEMA. (2012). "Seismic Performance Assessment of Buildings." Rep. No. FEMA-P-58-1,

FEMA-P-58-2, FEMA-P-58-3, Federal Emergency Management Agency.

FEMA. (2009). "Quantification of Building Seismic Performance Factors." Rep. No. FEMA

P695, Federal Emergency Management Agency.

FEMA. (2006). "Next-Generation Performance-Based Seismic Design Guidelines." Rep. No.

FEMA-445, Federal Emergency Management Agency.

FEMA. (2000a). "Prestandard and Commentary for the Seismic Rehabilitation of Buildings."

Rep. No. FEMA-356, Federal Emergency Management Agency, Washington, D.C.

FEMA. (2000b). "Recommended Seismic Design Criteria for New Moment-Frame Buildings."

Rep. No. FEMA-350, Federal Emergency Management Agency.

FEMA. (1997). "NEHRP Guidelines for the Seismic Rehabilitation of Buildings." Rep. No.

FEMA-273, Federal Emergency Management Agency, Washington, D.C.

FEMA SAC Joint Venture. (2000). "Recommended Seismic Evaluation and Upgrade Criteria for

Existing Welded Steel Moment-Frame Buildings." Rep. No. FEMA-351, Federal Emergency

Management Agency.

169

FHWA. (2006). "Seismic Retrofitting Manual for Highway Structures: Part1-Bridges." Rep. No.

FHWA-HRT-06-032, Federal Highway Administration, US. Department of Transportation,

McLean, Virginia.

Filippou, F. C., Popov, E. P., and Bertero, V. V. (1983). "Effects of Bond Deterioration on

Hysteretic Behavior of Reinforced Concrete Joints." Rep. No. EERC 83-19, Earthquake

Engineering Research Center, University of California, Berkeley.

Foschi, R. O. (2007). "Vessel-Pier Collison Risk and Associated Forces Pitt River Bridge.".

Galambos, T. V. (1981). "Load and resistance factor design." Engineering Journal, AISC, 18(3),

78-84.

Gardoni, P., Der Kiureghian, A., and Mosalam, K. (2002). "Probabilistic Capacity Models and

Fragility Estimates for Reinforced Concrete Columns based on Experimental Observations."

Journal of Engineering Mechanics, 128(10), 1024-1038.

Ghobarah, A. (2001). "Performance-Based Design in Earthquake Engineering: State of

Development." Engineering Structures, 23 878-884.

Goodnight, J. C., Kowalsky, M. J., and Nau, J. M. (2013). "Effect of load history on performance

limit states of circular bridge columns." Journal of Bridge Engineering, 18(12), 1383-1396.

Hachem, M., Mahin, S., and Moehle, J. P. (2003). "Performance of Circular Reinforced Concrete

Bridge Columns under Bidirectional Earthquake Loading." Rep. No. PEER 2003/06, Pacific

Earthquake Engineering Research Center, University of California, Berkeley.

Halchuk, S., Allen, T. I., Adams, J., and Rogers, G. C. (2014). "Fifth Generation Seismic Hazard

Model Input Files as Proposed to Produce Values for the 2015 National Building Code of

Canada." Rep. No. Open File 7576, Geological Survey of Canada.

Haselton, C. B., Baker, J. W., and Whittaker, A. S. (2015). "ATC Webinar on Recent Advances

in Ground Motion Selection and Scaling." Projects by ATC-82 Team and BSSC Issue Team #4,

(14 October).

Haselton, C. B., Fry, A., Baker, J. W., Hamburger, R. O., Whittaker, A. S., Stewart, J. P.,

Elwood, K. J., Luco, N., Hooper, J. D., Charney, F. A., Zimmerman, R. B., and Pekelnicky, R.

G. (2014). "Response-History Analysis for the Design of New Buildings: A Fully Revised

Chapter 16 Methodology Proposed for the 2015 NEHRP Provisions and the ASCE/SEI 7-16

Standard." 10th U.S. National Conference on Earthquake Engineering.

Haukaas, T. (2008). "Unified Reliability and Design Optimization for Earthquake Engineering."

Probabilistic Engineering Mechanics, 23(4), 471-481.

170

Hose, Y., and Seible, F. (1999). "Performance Evaluation Database for Concrete Bridge

Components and Systems under Simulated Seismic Loads." Rep. No. PEER 1999/11, Pacific


Huang, Y. N., Whittaker, A. S., Luco, N., and Hamburger, R. (2009). "Scaling Earthquake

Ground Motions for Performance-Based Assessment of Buildings." Journal of Structural

Engineering, 137(3), 311-321.

Jalayer, F. (2003). "Direct Probabilistic Seismic Analysis: Implementing Non-Linear Dynamic

Assessments". Doctor of Philosophy.

Jalayer, F., and Cornell, C. A. (2003). "A Technical Framework for Probability-Based Demand

and Capacity Factor Design (DCFD) Seismic Formats." Rep. No. PEER 2003/08, Pacific

Earthquake Engineering Center.

Jansen, D. C., and Shah, S. P. (1997). "Effect of length on compressive strain softening of

concrete." Journal of Engineering Mechanics, 123(1), 25-35.

Kennedy, D., Dowling, J., and Du, H. (2017). "Opportunities in the Performance-Based Seismic

Design of Bridges in British Columbia." 39th IABSE Symposium, Vancouver, Canada.

Kowalsky, M. J. (2000). "Deformation limit states for circular reinforced concrete bridge

columns." Journal of Structural Engineering, 126(8), 869-878.

Kramer, S. L. (1995). Geotechnical Earthquake Engineering. Pearson Education.

Krawinkler, H. (1999). "Challenges and Progress in Performance-Based Earthquake

Engineering." International Seminar on Seismic Engineering for Tomorrow.

Lara, O. (2011). "The Flexural Seismic Resistant Design of Reinforced Concrete Bridge

Columns". Doctor of Philosophy. University of British Columbia, Vancouver, Canada.

Lee, Y. H., and William, K. (1997). "Mechanical properties of concrete in uniaxial

compression." ACI Material Journal, 94(6), 457-471.

Lehman, D. E., and Moehle, J. P. (2000a). "Performance-Based Seismic Design of reinforced

Concrete Bridge Columns." 12th World Conference on Earthquake Engineering, Auckland, New

Zealand.

Lehman, D. E., and Moehle, J. P. (2000b). "Seismic Performance of Well-Confined Concrete

Bridge Columns." Rep. No. PEER 1998/01, Pacific Earthquake Engineering Research Center,


171

Lehman, D. E., Moehle, J. P., Mahin, S., Calderone, A. J., and Henry, L. (2004). "Experimental

evaluation of the seismic performance of reinforced concrete bridge columns." Journal of

Structural Engineering, 130(6), 869-879.

Luco, N., and Bazzuro, P. (2007). "Does Amplitude Scaling of Ground Motion Records Result in

Biased Nonlinear Structural Drift Responses?" Earthquake Engineering and Structural

Dynamics, 36(13), 1813-1835.

Mackie, K. R., Wong, J. M., and Stojadinovic, B. (2008). "Integrated Probabilistic Performance-

Based Evaluation of Benchmark Reinforced Concrete Bridges." Rep. No. PEER 2007/09, Pacific


Madsen, H. O., Krenk, S., and Lind, N. C. (1986). Methods of Structural Safety. Prentice-Hall.

Mander, J. B., Priestley, M. J. N., and Park, R. (1988). "Theoretical Stress-Strain Model for

Confined Concrete." Journal of Structural Engineering, 114(8), 1804-1826.

Menegotto, M., and Pinto, E. (1973). "Methods of Analysis for Cyclically Loaded Reinforced

Concrete Plane Frames Including Changes in Geometry and Non-Elastic Behaviour of Elements

Under Combined Normal Force and Bending." IABSE Symposium.

Moehle, J. P., and Deierlein, G. G. (2004). "A Framework Methodology for Performance-Based

Earthquake Engineering." 13th World Conference on Earthquake Engineering.

Moller, O., Foschi, R. O., Ascheri, J. P., Rubinstein, M., and Grossman, S. (2015). "Optimization

for performance-based design under seismic demands, including social costs." Earthquake

Engineering and Engineering Vibration, 14(2), 315-328.

Natural Resources Canada. (2016). "Determine 2015 National Building Code of Canada seismic

hazard values." http://www.seismescanada.rncan.gc.ca/hazard-alea/interpolat/index_2015-en.php

2018).

NBCC. (2015). National Building Code of Canada. National Research Council Canada.

NEHRP. (2013). "Nonlinear Analysis Research and Development Program for Performance-

Based Seismic Engineering." Rep. No. NIST GCR 14-917-27, National Institute of Standards and

Technology, U.S. Department of Commerce.

Nielson, B. G., and DesRoches, R. (2007). "Seismic Fragility Methodology for Highway Bridges

Using a Component Level Approach." Earthquake Engineering & Structural Dynamics, 36(6),

823-839.

Oyen, P. (2006). "Evaluation of Analytical for Determining the Seismic Response of Reinforced

Concrete Shear Walls". Master of Science. University of Washington, Seattle, Washington.

http://www.seismescanada.rncan.gc.ca/hazard-alea/interpolat/index_2015-en.php

172

PEER. (2017). "Open System for Earthquake Engineering Simulation."

http://opensees.berkeley.edu/ 2017).

PEER. (2010). "Technical Report for the PEER Ground Motion Database Web Application."

http://ngawest2.berkeley.edu/Technical_Report.pdf. 2016).

PEER. (2003). "Structural Performance Database." https://nisee.berkeley.edu/spd/ 2015).

PEER Tall Building Initiative. (2008). "Guidelines for Performance-Based Seismic Design of

Tall Buildings." Rep. No. PEER 2017/06, Pacific Earthquake Engineering Research Center,


PEER, and ATC. (2010). "Modelling and Acceptance Criteria for Seismic Design and Analysis

of Tall Buildings;" Rep. No. PEER/ATC-72-1, Applied Technology Council and Pacific

Earthquake Engineering Research Center, Redwood City, California.

Priestley, M. J. N., and Calvi, G. M. (2007). "Displacement-Based Seismic Design of Bridges."

1st US-Italy Seismic Bridge Workshop, Eucentre; Rose School.

Priestley, M. J. N., Seible, F., and Calvi, G. M. (1996). Seismic Design and Retrofit of Bridges.

Wiley.

Priestley, M. (2000). "Performance Based Seismic Design." 12th World Conference on

Earthquake Engineering.

Pugh, J. S. (2012). "Numerical simulation of walls and seismic design recommendations for

walled buildings". Doctor of Philosophy. University of Washington, Seattle, Washington;.

Pugh, J. S., Lowes, L. N., and Lehman, D. E. (2015). "Nonlinear line-element modeling of

flexural reinforced concrete walls." Engineering Structures, 104 174-192.

Rashid, Y. R. (1968). "Ultimate strength analysis of pre-stressed concrete pressure vessels."

Nuclear Engineering and Design, 7(4), 334-344.

Riddell, R., and Garcia, J. E. (2001). "Hysteretic Energy Spectrum and Damage Control."

Earthquake Engineering & Structural Dynamics, 30 1791-1816.

Royset, J. O., Der Kiureghian, A., and Polak, E. (2001). "Reliability-based optimal structural

design by the decoupling approach." Reliability Engineering & System Safety, 73(3), 213-221.

SEAOC. (1995). Vision 2000 - A Framework for Performance Based Earthquake Engineering.

Tehrani, P. (2012). "Seismic Behaviour and Analysis of Continuous Reinforced Concrete

Bridges". Doctor of Philosophy.

http://opensees.berkeley.edu/

http://ngawest2.berkeley.edu/Technical_Report.pdf.

https://nisee.berkeley.edu/spd/

173

The Fugro Consultants. (2015). "EZ-FRISK Software for Earthquake Ground Motion

Estimation." http://www.ez-frisk.com/ .

Tothong, P., and Luco, N. (2007). "Probabilistic Seismic Demand Analysis Using Advanced

Ground Motion Intensity Measures." Earthquake Engineering and Structural Dynamics, 36(13),

311-321.

Transportation Research Board. (2013). "Performance-Based Seismic Bridge Design, A

Synthesis of Highway Practice." Rep. No. NCHRP SYNTHESIS 440, the National Academics of

Sciences, Washington, D.C.

Vamvatsikos, D., and Cornell, C. A. (2002). "Incremental Dynamic Analysis." Earthquake

Engineering and Structural Dynamics, 31 491-514.

Vosooghi, A., and Saiidi, M. S. (2010). "Seismic damage states and response parameters for

bridge columns." Special Publication, Structural Concrete in Performance-Based Seismic

Design of Bridges, 271 29-46.

Wen, Y. K. (2001). "Reliability and performance-based design." Structural Safety, 23(4), 407-

428.

Whittaker, A., Huang, Y. N., and Hamburger, R. O. (2007). "Next-Generation Performance-

Based Earthquake Engineering." 1st International Conference on Modern Design, Construction

and Maintenance of Structures, 49-57.

Wilson, J. C., and Tan, B. S. (1990). "Bridge Abutments: Formulation of Simple Model for

Earthquake Response Analysis." Journal of Engineering Mechanics, 116(8), 1828-1837.

Zhang, J., and Foschi, R. O. (2004). "Performance-based design and seismic reliability analysis

using designed experiments and neural networks." Probabilistic Engineering Mechanics, 19(3),

259-267.

http://www.ez-frisk.com/

174

Appendices

Appendix A CSA S6-14 Tables for PBD

Average properties in top 30 m

Site class Ground profile name

Shear wave average

velocity, Vs (m/s)

Standard

penetration

resistance, N60

Soil undrained shear

strength, su

A Hard rock Vs > 1500 Not applicable Not applicable

B Rock 760 < Vs ≤ 1500 Not applicable Not applicable

C Very dense soil and

soft rock

360 < Vs ≤ 760 N60 > 50 su > 100 kPa

D Stiff soil 180 < Vs ≤ 360 15 ≤ N60 ≤ 50 50 < su ≤ 100 kPa

E Soft soil Vs ≤ 180 N60 < 15 su < 50 kPa

Any profile with more than 3 m of soil with the following

characteristics:

Plastic index PI > 20;

Moisture content w ≥ 40%; and

Undrained shear strength su < 25 kPa

F Other soil Site specific evaluation required

Table A.1 CSA S6-14 site classification for seismic site response

For T < 0.5 s For T ≥ 0.5 s

Seismic performance category

Lifeline bridges Major-route and other

bridges

S(0.2) < 0.2 S(1.0) < 0.1 2 1

0.2 < S(0.2) < 0.35 0.1 < S(1.0) < 0.3 3 2

S(0.2) ≥ 0.35 S(1.0) ≥ 0.3 3 3

Table A.2 CSA S6-14 Seismic performance category based on 2475-year return period spectral values

Seismic

performance

category

Lifeline bridges Major-route bridges Other bridges

Irregular Regular Irregular Regular Irregular Regular

1 No seismic analysis is required

2 PBD PBD PBD FBD FBD FBD

3 PBD PBD PBD FBD* PBD FBD

*PBD might be required by the Regularity Authority

Table A.3 CSA S6-14 Requirements for PBD and FBD

175

Seismic

performance

category




2 EDA,

ISPA, and

NTHA

EDA and

ISPA

EDA and

ISPA

ESA EDA ESA

3 EDA,

ISPA, and

NTHA

EDA,

ISPA, and

NTHA

EDA and

ISPA

EDA EDA ESA

Table A.4 CSA S6-14 minimum analysis requirements for 2%/50 and 5%/50 hazard levels

Seismic

performance

category




2 EDA EDA EDA ESA EDA ESA

3 EDA EDA EDA ESA EDA ESA

Table A.5 CSA S6-14 minimum analysis requirements for 10%/50 hazard level

Seismic ground

motion probability

of exceedance in

50 years (return

period)


Service Damage Service Damage Service Damage

10%

(475 years)

Immediate None Immediate Minimal Service

limited

Repairable

5%

(975 years)

Immediate Minimal Service

limited*

Repairable* Service

disruption*

Extensive*

10%

(2475 years)

Service

limited

Repairable Service

disruption

Extensive Life safety Probable

replacement

*Optional performance levels unless required by the Regulatory Authority or the Owner

Table A.6 CSA S6-14 Minimum performance levels for PBD approach

176

Service Damage

Immediate

Bridge shall be fully serviceable for

normal traffic and repair work does

not cause any service disruption.

Minimal damage

• General: Bridge shall remain essentially elastic with minor damage that

does not affect the performance level of the structure.

• Concrete Structures: Concrete compressive strains shall not exceed 0.004

and reinforcing steel strains shall not exceed yield.

• Steel Structures: Steel strains shall not exceed yield. Local or global

buckling shall not occur.

• Connections: Connections shall not be compromised.

• Displacements: Pounding shall not occur. Residual displacement,

settlement, translation or rotation, of the structure or foundations, including

retaining and wing walls, shall be negligible, and not compromise the

performance level.

• Bearings and Joints: Shall not require replacement except for possible

damage to joint seals.

• Restrainers: No observable damage or loss of displacement capacity to

restraining systems or connected elements shall occur.

• Foundations: Foundation movements shall be limited to only slight

misalignment of the spans or settlement of some piers or approaches that

does not interfere with normal traffic, provided that no repairs are required.

Limited

Bridge shall be usable for emergency

traffic and be repairable without

requiring bridge closure. At least

50% of the lanes, but not less than

one lane, shall remain operational. If

damaged normal service shall be

restored within a month.

Repairable damage

• General: There may be some inelastic behaviour and moderate

damage may occur; however, primary members shall not need to be

replaced, shall be repairable in place, and shall be capable of supporting the

dead load plus full live load.

• Concrete structures: Reinforcing steel tensile strains shall not exceed

0.015.

• Steel structures: Buckling of primary members shall not occur.

Secondary members may buckle provided that stability is maintained.

Net area rupture of primary members at connections shall not occur.

• Connections: Primary connections shall not be compromised.

• Displacements: Permanent offset shall not compromise the service and

repair requirements of the bridge. No residual settlement or rotation of

main structure shall occur. There may be some movement of wing

walls, subject to performance and reparability.

• Bearings and joints: Elastomeric bearings may be replaced. If finger

joints are damaged, they shall be repairable.

• Restrainers: Restraining systems shall not be damaged.

• Foundations: Foundation movements shall be limited to only slight

misalignment of the spans or settlement of some piers or approaches

that does not interfere with normal traffic, provided that repairs can

bring the structure back to the original operational capacity.

• Aftershocks: The structure shall retain 90% of seismic capacity for

aftershocks and shall have full capacity restored by the repairs.

Table A.7 CSA S6-14 Performance criteria (continued)

177

Service Damage

Service disruption

The bridge shall be usable for

restricted emergency traffic after

inspection. The bridge shall be

repairable. Repairs to restore the

bridge to full service might require

bridge closure.

Extensive damage

• General: Inelastic behaviour is expected. Members might have extensive

visible damage, such as spalling of concrete and buckling of braces but

strength degradation shall not occur. Members shall be capable of

supporting the dead plus 50% live loads, excluding impact, including P-

delta effects, without collapse.

• Concrete structures: There might be extensive concrete spalling, but the

confined core concrete shall not crush. Reinforcing steel tensile strains shall

not exceed 0.050.

• Steel structures: Global buckling of gravity load supporting elements shall

not occur.

• Connections: There may be severe joint distortions.

• Displacements: There may be permanent offsets as long as they do not

prevent use by restricted emergency traffic after inspection of the bridge,

nor preclude return of full service to the bridge.

• Foundations: Ground lateral and vertical movements shall not exceed

those that would prevent use by restricted emergency traffic after inspection

or the bridge, nor preclude return of full service to the bridge.

• Bearings and joints: The superstructure may have loss of bearings but

shall have adequate remaining seat length and connectivity to carry

emergency traffic. Joints might require replacement.

• Restrainers: Restraining systems might suffer damage but shall not fail.

• Aftershocks: The structure shall retain 80% seismic capacity for

aftershocks and have full capacity restored by the repairs.

Life safety

The structure shall not collapse and it

shall be possible to evacuate the

bridge safely.

Probable replacement

• General: Bridge spans shall remain in place but the bridge might be

unusable and might have to be extensively repaired or replaced.

Extensive distortion of beams and column panels might occur.

Fractures at moment connections might occur but shear connections shall

remain intact. Members shall be capable of supporting the dead

plus 30% live loads, excluding impact, but including P-delta effects,

without collapse

• Displacements: Permanent offsets shall be limited such that the bridge can

be evacuated safely.

• Foundations: Ground lateral and vertical movements are not restricted

but shall not lead to collapse of the bridge superstructure

Table A.7 CSA S6-14 Performance criteria (concluded)

178

Appendix B Deriving Expressions for Post-Processing Local Strain Response

The following sequence, demonstrates the derivations of the proposed expressions for post-

processing the strain response of regularized distributed plasticity models, in Section 5.3.5. The

expressions are derived based on the assumption that the ratio of the post-peak energy up to a

specified strain limit (in the post-peak region) to the total post-peak energy is similar for the

reference and the regularized material models. Figure B.1 demonstrates this similarity and shows

how the post peak response of the reference material model (shaded area) is scaled to the

regularized model (red lines). Using similar triangles rule, the strain limits of concrete in the

reference material model, denoted by εc1 can be mapped to εc2 in the regularized model:

(fcc

-0.2fcc

'

0.8fcc

') = (

εc1-εoc

εcu-εoc

) (B.1)

(fcc

-0.2fcc

'

0.8fcc

') = (

εc2-εoc

ε20c-PH-εoc

) (B.2)

(εc1-εoc

εcu-εoc

) = (εc2-εoc

ε20c-PH-εoc

) (B.3)

εc2= (ε20c-PH-εoc

εcu-εoc

) (εc1-εoc)+εoc (B.4)

The same concept can be used to map the strain limits of reinforcing steel in the reference

material model denoted by εs1 to εs2 in the regularized model. A1 and A2 are the areas under the

post-peak region of the reference and regularized reinforcing steel models, respectively:

A1=Gs

Lgage

(B.5)

A2=

Gs

LIP

(B.6)

A1Lgage=A2LIP (B.7)

0.5 (fu+f

y) (εu,exp-εy)Lgage=0.5 (f

u+f

y) (εsu-εy)L

IP (B.8)

179

(εu,exp-εy)

LIP

=(εsu-εy)

Lgage

(B.9)

(εs2-εy)

LIP

=(εs1-εy)

Lgage

(B.10)

εs2=εy+(εs1-εy)Lgage

LIP

(B.11)

εoc εcu

f’cc

0.2 f’cc

Ecc

σ

ε εc1 εc2 ε20c-PH

f cc

(a)

εy εu,exp

Es

fu

σ

fy

fs

εs1 εs2 εsu ε (b)

Figure B.1 Mapping strain limits in the post-peak region of (a) confined concrete and (b) reinforcing steel

180

Appendix C Acceleration and Displacement Response Spectra of the Selected Records for

NTHA in Chapter 7 and Chapter 8

C.1 Crustal Suite (Chapter 7)

183

C.2 Subcrustal Suite (Chapter 7)

187

C.3 Subduction Suite (Chapter 7)

191

C.4 Suite of Motions for Chapter 8

195

Appendix D PBD Aid Using the Correlation of Damage with Seismic Intensity Measures

CSA S6-14 requires using a minimum of 11 ground motion records to assess the PBD of a bridge

with NTHA. While using a larger number of records is necessary to reduce the dispersion in

estimating mean response, it is computationally very demanding for a trial-and-error-based

design process. The additional computational effort of using 11 ground motions compared to 7 or

less, which was recommended by the previous editions of the code, is especially meaningful for

complex bridges, where PBD is the only permissible design approach. One way to address this

problem is to select a smaller number of records out of the original suite of 11 or more records,

which are likely to cause more damage, and use this smaller suite in the design process instead of

the original suite. It would be most convenient, if this smaller suite of records could be selected

without performing any additional analysis, and solely based on ground motion intensity

measures. This would be possible, if there was an adequate correlation between the intensity

measures and the response parameter such that the damage could be predicted reasonably by the

intensity measures. The main objective of this chapter is to find suitable intensity measures,

which correlate adequately with the damage of ductile concrete bridge columns, and to use this

correlation to select a smaller suite of records for NTHA required for the PBD process.

D.1 Study1: Developing the PBD Aid

D.1.1 Ground Motion Intensity Measure Candidates

The basic task of all ground motion intensity measures is to describe the important characteristics

of strong ground motions. Identifying and describing these characteristics lead to better selection

and scaling of ground motion records for the purpose of time-history analysis. Traditionally,

three main characteristics of amplitude, frequency content, and duration of motions are of

interest, and many intensity measures have been proposed to describe one or more of these

aspects. Nevertheless, due to the complex nature of ground motions, a single factor is incapable

of accurately describing all important aspects of ground motions (Kramer 1995).

A number of common and proposed intensity measures were examined in this study, to

correlate with damage of ductile concrete bridge columns. The examined parameters are as

follows:

196

1. Amplitude parameters including peak ground acceleration (PGA), peak ground velocity

(PGV), and peak ground displacement (PGD).

2. Frequency content parameters including the ratio of peak velocity to peak acceleration

(Vmax/Amax).

3. Mixed parameters, which represents more than one aspect of the important ground

motion characteristics including Arias intensity, characteristic intensity, Housner spectral

intensity, specific energy density, and RMS acceleration and velocity.

4. Proposed parameters by Fajfar et al. (1990) and Riddell and Garcia (2001). These

parameters consider a combination of peak ground parameters and significant duration of

earthquakes.

The definitions of the parameters in the first three numbered items can be found in Kramer

(1995).

D.1.2 Bridge Column Models

To investigate the correlation of the aforementioned intensity measures with the seismic damage

of ductile reinforced concrete bridges, simplified single-column models were utilized. In

reinforced concrete bridges with ductile substructure, the substructure elements such as columns,

multiple-column bents, wall-type piers, etc., are designed and detailed to incorporate the seismic

damage. In contrast, the superstructure and other elements are designed using the capacity design

concept to stay essentially damage free. Moreover, the response of multi-span ductile concrete

bridges are dominated primarily by the first mode response, as was illustrated in the previous

chapters. For these reasons, it seems justified to use simplified single-column models in place of

complete models for the purpose of this study.

Three bridge columns with fundamental periods of 0.5s, 1.0s, and 2.0s were designed and

detailed according to the CSA S6-14 FBD approach, with minimum response modification factor

of R=3.0 and importance factor of I=1.0. The three periods cover a relatively wide range of

periods for multi-span bridges, except for short period bridges. The columns were all 2 m in

diameter, and are 8.5 m, 14 m, and 21.5 m in height, respectively. The longitudinal

reinforcement ratio is 1% in all cases. The weight of the superstructure was assumed to be 200-

300 kN/m distributed along the deck. Normal strength concrete with specified compressive

197

strength of 35 MPa was assumed and the grade of reinforcement steel was 400R with minimum

specified yield strength of 400 MPa.

The bridge columns were modelled in SeismoStruct using DBE with fibre sections. The

details of modelling and utilized material models are similar to Section 8.3, and therefore are

skipped here. A concentrated mass of 800 tonne was assigned to the top node of each column.

This corresponds to mass of a 30 to 40-meter-long deck span supported by each column. The

associated dead load from the deck was applied as a concentrated gravity force at the top of the

columns. The viscous damping was modelled using Rayleigh damping with damping ratio of 3%

at the first and second periods of each column. For the boundary condition, it was assumed that

the columns are fixed at the base and there is no soil-structure-interaction.

D.1.3 Damage Criteria

Five separate damage states of yielding, cover spalling 1, serviceability limit 2, core crushing,

and reduced fracture previously introduced in Table 6.3 were employed. For convenience, the

strain limits corresponding to these damage states are summarized in Table D.1. It should be

noted that the core crushing strain is calculated for the cross-section of the columns used in this

study, and its values is different from that of Table 6.3.


Yielding εs > 0.0023

Cover Spalling 1 εc < -0.004

Serviceability Limit 2 εs > 0.015

Core Crushing εc < -0.01

Reduced Fracture εs > 0.05

Table D.1 Strain-based damage Criteria

D.1.4 Ground Motion Records for NTHA

It is assumed that the bridges under study are all located in the city of Victoria on the southern

part of Vancouver Island in British Columbia. The soil site class is assumed to be class C or firm

soil, which is the Canada-wide reference ground condition for uniform representation of seismic

hazard across the country (Adams and Halchuk 2003). Using EZ-FRISK, probabilistic seismic

hazard analysis was performed for the city of Victoria and UHS of Victoria was obtained for six

hazard levels of 50%/50, 10%/50, 5%/50, 2%/50, 1%/50, and 0.5%/50 (corresponding to 72,

475, 975, 2475, 4974, and 9975-year return periods, respectively).

198

For each type of crustal, subcrustal, and subduction earthquakes, 10 records were selected

for time-history analysis. The period ranges of interest were determined as 0.1-1.5 s, 0.2-2.0 s,

and 0.4-4.0 s for the bridge columns with fundamental periods of 0.5 s, 1.0 s, and 2.0 s,

respectively. The details of ground motion selection are similar to Section 7.7 and therefore are

skipped here for brevity. The selected records are listed in Table D.2, along with the year and

location of their corresponding historical event. Only one horizontal component of the records

was utilized for time-history analysis. For each bridge column, the records were linearly scaled

to match the 2%/50 UHS in the period ranges of interest for that column. The scale factors for

linear scaling were found by minimizing the square root of the difference between the spectral

acceleration response spectrum of each record with the target spectrum, over the period range of

scaling. The formulas for calculating the minimum square root of error and scale factors for

linear scaling can be found in the Technical Report for PEER Ground Motion Database (2010).

D.1.5 Structural Analysis

NTHA was performed to obtain the response of the bridge columns at six hazard levels of

50%/50, 10%/50, 5%/50, 2%/50, 1%/50, and 0.5%/50. For each hazard level, the same suite of

ground motion records, which had been already selected and scaled for 2%/50 hazard level, was

rescaled. The scale factor for each hazard level was calculated as Sa(T1)hazard level i /Sa(T1)2% /50, in

which Sa(T1) is the spectral acceleration of the UHS for hazard level i at the fundamental period

of the bridge column, T1. In total 540 nonlinear time-history analyses were conducted. For each

analysis the maximum drift ratio of the top of the bridge columns was extracted as the main

response parameter of interest12. The analysis was set to stop at the first occurrence of the

reduced fracture damage state. The drift ratio capacities of the columns were obtained by

pushover analysis as described in Section 7.10.

12 For first-mode-response dominated structures, maximum drift ratio is a common index for damage

prediction (Cordova et al. 2001)

199

Type Historical Event Record Year Location No. of

Records

Crustal Chi-Chi CHY028-E 1999 Taiwan 1

Crustal Imperial Valley H-DLT352 1979 California, US 1

Crustal Kern County TAF021 1952 California, US 1

Crustal Kobe SKI000 1995 Japan 1

Crustal Landers ABY090 1992 California, US 1

Crustal Loma Prieta A2E090 1989 California, US 1

Crustal Northridge UCL360 1994 California, US 1

Crustal San Fernando PDL120 1971 California, US 1

Crustal Superstition Hills B-IVW090 1987 California, US 1

Crustal Tabas BOS-T1 1978 Iran 1

Subcrustal Geiyo EHM0150103241528-EW 2001 Japan 2

EHM0160103241528-EW

Subcrustal Miyagi-Oki IWT0110508161146-NS 2005 Japan 4

MYG0060508161146-EW

MYG0100508161146-NS

MYG0170508161146-NS

Subcrustal Nisqually 0720c_a-90 2001 Washington, US 4

1032j_a-58

1416a_a-125

1423c_a-148

Subduction Hokkaido HKD0770309260450-NS 1952 Japan 4

HKD0840309260450-NS

HKD0950309260450-EW

HKD1090309260450-NS

Subduction Maule curico1002271-EW 2010 Chile 4

hualane1002271-T

stgolaflorida1002271-NS

stgopenalolen1002271-EW

Subduction Tohoku FKS0071103111446-NS 2011 Japan 2

FKS0121103111446-EW

Table D.2 Selected ground motion records for time history analysis

D.1.6 Correlation of Intensity Measures with the Response Parameter

To check the correlation of damage in bridge columns with the selected ground motion intensity

measures, the maximum drift ratios were plotted against each ground motion intensity measure.

To do so, the maximum drift ratios for the 30 records scaled at each of the six hazard levels,

were plotted against the ground motion intensity measures corresponding to that hazard level.

Subsequently, linear correlation coefficient between the intensity measures and the response

parameter at each hazard level were calculated. A positive correlation coefficient signifies an

increase in the response parameter, and thus the damage in the bridge columns, with an increase

in the intensity measure. On the other hand, a zero or a small correlation coefficient indicates that

200

the response parameter is not dependent on that intensity measure. The intensity measures, which

demonstrate the strongest positive correlation with the maximum drift ratios, are of interest.

Examples of such plots are given in Figure D.1 and Figure D.2 for intensity measures PGV,

PGD, and Vmax/Amax. The results are shown for only three hazard levels of 50%/50, 2%/50, and

0.5%/50, for the sake of clarity. The following can be observed from the figures:

1. Among the investigated intensity measures PGV, and PGD demonstrated on average

higher positive linear correlation with the response parameter, followed by Vmax/Amax, the

intensity measure proposed by Fajfar et al., and the Housner spectral intensity.

2. The correlation of the aforementioned intensity measures with the response parameter

depends on the hazard level. For lower hazard levels, where the response of the bridge

columns is essentially elastic, the linear correlation coefficients are close to zero. The

linear correlation tends to increase at higher hazard levels, and with higher levels of

damage.

3. The correlation of the aforementioned intensity measures with the response parameter

depends on the fundamental period of the bridge columns.

4. Referring to Fig.3, it can be observed that at higher hazard levels, the linear correlation of

PGV with the response parameter is significant and positive for T1=0.5s, and 1.0s, and

becomes negative for T1=2.0 s.

5. PGD demonstrates a positive linear correlation with the response parameter. The

correlation becomes stronger at larger periods (T1=2.0 s, and 1.0 s), and decreases for

T1=0.5 s.

6. Vmax/Amax shows a positive linear correlation with the response parameter, but less strong

compared to PGD, and PGV. In this case, the positive correlation tends to improve

slightly at larger periods.

7. Figure D.2 shows the maximum drift ratio of the 1.0 s period column versus PGD, for

each type of earthquakes, separately. It is observed that the positive linear correlation at

higher hazard levels tends to be stronger for the crustal, and subduction suites compared

201

Figure D.1 Maximum drift ratios versus PGV and Vmax/Amax for the three bridge columns

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 50 100 150 200

Ma

xim

um

Dri

ft (

%)

PGV (cm/s)

All-T1=0.5s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.00 0.05 0.10 0.15 0.20 0.25

Ma

xim

um

Dri

ft (

%)

Vmax/Amax (s)

All-T1=0.5s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 50 100 150 200

Ma

xim

um

Dri

ft (

%)

PGV (cm/s)

All-T1=1.0s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0.00 0.05 0.10 0.15 0.20 0.25

Ma

xim

um

Dri

ft (

%)

Vmax/Amax (s)

All-T1=1.0s

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 100 200 300

Ma

xim

um

Dri

ft (

%)

PGV (cm/s)

All-T1=2.0s

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.00 0.05 0.10 0.15 0.20 0.25

Ma

xim

um

Dri

ft (

%)

Vmax/Amax (s)

All-T1=2.0s

202

Figure D.2 (left column) Maximum drift ratios versus PGD for the three bridge columns and (right column)

maximum drift ratios versus PGD of the individual earthquake sources for the bridge column with

fundamental periods of 1.0 s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 50 100 150

Ma

xim

um

Dri

ft (

%)

PGD (cm)

All-T1=0.5s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 20 40 60 80 100 120 140 160

Ma

xim

um

Dri

ft (

%)

PGD (cm)

Crustal-T1=1.0s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 50 100 150 200

Ma

xim

um

Dri

ft (

%)

PGD (cm)

All-T1=1.0s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 10 20 30 40

Ma

xim

um

Dri

ft (

%)

PGD (cm)

Subcrustal-T1=1.0s

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 50 100 150

Ma

xim

um

Dri

ft (

%)

PGD (cm)

All-T1=2.0s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 20 40 60 80 100 120

Ma

xim

um

Dri

ft (

%)

PGD (cm)

Subduction-T1=1.0s

203

to the subcrustal suit. This can be explained considering that the response of the bridge

column to the subcrustal motions is smaller than the crustal and subduction motions, at

this period, and therefore less damage is incurred to the column by the subcrustal suite. It

was observed that for lower levels of damage, the positive linear correlation of the

response parameter with the intensity measures is weaker in comparison to higher levels

of damage.

It has been already shown in several studies that Sa(T1), when utilized as the intensity measure to

select ground motions, can be both inefficient and insufficient for predicting the response of

structures, and the extent of damage (Baker and Cornell 2005; Baker and Cornell 2006; Tothong

and Luco 2007; Luco and Bazzuro 2007; Huang et al. 2009). Based on the observations of this

study, and confirmed by the literature, Sa(T1) is sufficient at the lower hazard levels, where the

response of the bridge columns is essentially elastic. This is the reason why at lower hazard

levels, the calculated drift ratios show minimal variation. However, as the response of the bridge

columns moves into the nonlinear range at higher hazard levels, Sa(T1) becomes insufficient, and

the response becomes dependent on other intensity measures. This is evident from Figure D.1

and Figure D.2 where the variation of the drift ratios increases considerably at the higher hazard

levels.

The observed linear correlation of the response with the investigated intensity measures

could be explained theoretically as well. For instance, it is well-known that the response of

single-degree-of-freedom systems with very large periods tends to be close to PGD. This is why,

at higher levels of damage, where period elongation has occurred, and also at larger fundamental

periods, the correlation of PGD with the response of the bridge columns is stronger.

D.1.7 Selection of a Smaller Suite of Records

Following the main objective of this study, it is of interest to use the observed correlations to

sample a smaller suite of records out of the original suite of 30 motions for NTHA. For such a

suite of records, it should be examined if the predicted level of damage is comparable to the

predictions of the original suite of records. To sample the records, first the records should be

ranked based on the value of the chosen intensity measure (calculated at the 2%/50 hazard level).

Next, the mean of the maximum drift ratios for the one third, and half of the records (10 and 20

204

records) with the highest values of the intensity measure should be calculated. Subsequently, the

mean drift ratios should be compared against the drift ratio capacities of the columns to predict

the damage state of the columns. Finally, the mean drift ratio and the predicted damage state

should be compared to those obtained using the original suite of 30 records. Two scenarios are

possible:

1. A zero or insignificant linear correlation between the intensity measure and the response

parameter at lower hazard levels. In these cases, the values of the response parameter

form a small cluster (as for PGV and PGD) or a horizontal line (as for Vmax/Amax), and do

not depend on the investigated intensity measures (Sa(T1) is a sufficient intensity

measures for these cases). As a result, selecting a smaller suite of one third or half the

number of records, whether ranked based on the intensity measure or not, would yield

similar mean response values as the original suite.

2. There is a positive linear correlation between the intensity measure and the response

parameter, at higher hazard levels. In these cases, the records with higher values of the

examined intensity measure tend to have larger responses. Therefore, the mean of the

smaller suite of records selected in this way, tend to have larger mean response value as

the original suite.

The above scenarios were examined using PGV, PGD, and Vmax/Amax intensity measures.

Typically, it is required to include records from all three types of earthquakes for time-history

analysis. Therefore, it is more reasonable to choose 3 or 5 out of 10 records for each type of

earthquake, instead of choosing 10 or 15 out of 30 records, disregarding the type of earthquake.

Table D.3 shows the comparison between the damage predictions using the smaller suites with

the original suite, for the 1.0 s period column. For this case, PGD was employed as the intensity

measure. In the table, μn is the average of the maximum drift ratios for n records with the highest

PGD values. For instance, μ3 is the average of the 3 records out of 10 crustal, subcrustal, or

subduction records, with the highest PGD values. Referring to the table, it can be observed that

in almost all cases, the smaller suites give reasonable predictions of both the mean response and

damage state of the columns. In some cases, they may overestimate the mean response and the

corresponding damage. Even so, performing the design checks using the smaller suites tends to

be on the conservative side, and the final design would most likely meet the specified

205

performance criteria, if checked for the original suite of records. Similar observations were made

for the other two columns, but for brevity they are not presented here.

μn

50

%/5

0

Dam

age

10

%/5

0

Dam

age

5%

/50

Dam

age

2%

/50

Dam

age

1%

/50

Dam

age

0.5

%/5

0

Dam

age

T1

=1

.0 s

CR

μ3 0.21 M 0.98 Y 2.17 SL 2.96 C 3.29 F 3.49 F

μ5 0.21 M 0.95 Y 1.90 SL 2.59 C 3.03 F 3.42 F

μ10 0.23 M 1.06 Y 1.83 SL 2.39 C 2.83 C 3.19 F

SC

μ3 0.26 M 0.86 Y 1.39 Y 1.55 Y 2.03 SL 2.28 SL

μ5 0.24 M 0.91 Y 1.36 Y 1.62 S 1.97 SL 2.21 SL

μ10 0.25 M 0.90 Y 1.31 Y 1.60 S 1.93 SL 2.23 SL

SD

μ3 0.21 M 0.86 Y 1.99 SL 3.25 F 3.45 F 3.45 F

μ5 0.21 M 0.94 Y 1.74 S 2.75 C 3.06 F 3.18 F

μ10 0.25 M 0.92 Y 1.57 Y 2.34 C 2.64 C 2.91 C

All

μ10 0.22 M 0.97 Y 1.88 SL 2.72 C 3.07 F 3.34 F

μ15 0.22 M 0.96 Y 1.75 S 2.48 C 2.92 C 3.18 F

μ30 0.25 M 0.96 Y 1.57 Y 2.11 SL 2.47 C 2.77 C

Table D.3 Prediction of the damage to the 1.0s period bridge column at different hazard levels using smaller

suites of records. PGD was utilized as the intensity measure; μn is the average of the maximum drift ratios for

n records with the highest PGD values

* Abbreviations: M: minimal damage, Y: yielding, S: cover spalling 1, SL: serviceability limit 2, C: core

crushing, F: reduced fracture, CR: crustal, SC: subcrustal, SD: subduction

D.1.8 Conclusions

The linear correlation of a number of ground motion intensity measures with the seismic damage

of ductile concrete bridge columns in British Columbia has been studied. The examined intensity

measures were intended to be utilized to select a smaller suite of records out of the original suite

of 11 or more records, for checking performance objectives during the performance-based design

trial and error process.

The results of nonlinear time history analysis of three bridge columns with short, medium,

and long fundamental periods, demonstrated that PGV, and PGD have reasonable strong positive

linear correlation with the maximum drift ratio and damage to the columns. The correlation

increases at higher hazard levels, where higher levels of damage incur to the columns. Based on

the primary observations of this study, it seems reasonable to employ PGV for shorter period

bridge columns (0.5<T1<1.0 s), while using PGD for longer period bridges (T1>1.0 s), as the

intensity measure to select a smaller suite of records out of the original suite of motions.

206

It was also demonstrated that it is possible to employ the aforementioned intensity

measures to select a suite of one third or half the size of the original suite of records, which

would still yield a reasonable prediction of the mean response and the corresponding damage to

the bridge columns. Using a smaller number of records for the design trial and error process is

very advantageous in terms of time and computational effort for performance-based design of

bridges, where multiple performance-objectives at multiple hazard levels must be met

simultaneously. However, it should be recognized that the conclusions of this study are based on

primary observations for a limited number of bridge columns. Further studies are needed to

confirm the applicability of these outcomes to performance-based design of all ductile concrete

bridges.

207

D.2 Study2: Examining the PBD Aid for the Implementation Example of Chapter 5

The objective of this study is to further examine the PBD aid developed in the previous section

by applying the aid to the implementation example of Chapter 5. The response of the bridge in

the longitudinal direction is dominated by the first mode with period of 1.61 s and in the lateral

direction by the first two modes with periods of 0.56 and 0.14 s. Following the observations of

the previous section, three intensity measures will be tested for the bridge including PGD, PGV,

and PGA. The reason for adding PGA to the list was the contribution of the short period mode

(0.14 s) to the lateral response.

D.2.1 Sampling Records

To test the correlations of the selected intensity measures with the drift ratios of the bridge,

the response of the bridge in the two directions are treated individually. This is partly due to the

difference in the fundamental periods of the two directions and partly due to the fact that

different horizontal components act in the two directions (H1 acts in the longitudinal direction

and H2 in the lateral direction). Moreover, the correlation of the intensity measures with drift

ratios is tested for individual crustal, subcrustal, and subduction suites, as well as all records

together. Figure D.3 to Figure D.6 shows the plots of longitudinal and lateral drift ratios versus

the selected intensity measures of the two horizontal components, for individual suites and all

records. The calculated correlation coefficients at 2%/50 hazard level are listed in Table D.4. The

values of the correlation coefficients reveal that in the longitudinal direction, PGV of the

horizontal component acting in that direction has the highest positive correlation with the

longitudinal drift ratio, and in the lateral direction PGA correlates best with the lateral drift ratio.

PGD correlation with the longitudinal response is only strong for the subduction and subcrustal

suites. This is because compared to crustal motions, subcrustal and subduction motions

contribute most to the response of the modes with longer periods (compare the period range of

ground motion selection for individual suites in Section 7.7). The same reasoning explains why

the correlation of PGV with the longitudinal drift ratio is weakest for the crustal suite. The weak

correlation of PGV with the lateral drift ratio is due to the contribution of the 0.14 s mode. This

suggests that PGV may not be a suitable candidate intensity measure for periods below 0.5 s.

These observations can also be made by investigating the plots of Figure D.3 to Figure D.6.

208

What these plots demonstrate aside from the strength of the intensity measure-drift ratio

correlations, is that when PGA presents the highest positive correlation with the response, PGD

shows otherwise, and vice versa.

D.2.2 Mean Response Predictions

Based on the above discussion, H1-PGV and H2-PGA were selected as the intensity

measures to sample a smaller suite of records out of the original suite. In each direction, the

records were ranked based on the corresponding intensity measure. Next mean drift ratios for a

smaller suite of n records with the highest values of the intensity measure were calculated and

compared to the mean drift ratios obtained from the original suite of records. This comparison is

made in Table D.5 and Table D.6. The results of the two tables suggest that using the PBD aid, it

is possible to provide a reasonable estimate of the mean response with only using 3 out of 33

records. The mean drift ratios predicted by the sampled suites tend to be conservative with one

exception of the mean drift ratios predicted for the crustal suite in the longitudinal direction.

Nevertheless, even in that case the underestimation of the mean response is negligible.

D.2.3 Conclusions

The results of this study further confirm the applicability of the developed method as an

aid to facilitate PBD trial and error process. The outcomes of the study in this section and the

previous section suggest using PGA, PGV, and PGD as intensity measures to sample a smaller

suite of records out of a suite of 11 or more records that is required by the code for NTHA. PGA

is best suited for shorter period structures with T1<0.5 s, PGV for mid-range periods of

0.5<T1<1.5 s, and PGD for longer periods of T1>2.0 s. This method works best with structures

that their response is dominated with the first mode response.

209

2%/50 CR SC SD All

H1

PGD with Drx 0.19 0.57 0.44 0.49

PGV with Drx 0.31 0.58 0.69 0.63

PGA with Drx -0.18 -0.46 -0.51 -0.53

H2

PGD with Dry -0.39 -0.09 -0.58 -0.38

PGV with Dry -0.28 0.17 0.17 -0.11

PGA with Dry 0.50 0.68 0.59 0.61

Table D.4 Correlation coefficients of longitudinal and lateral drift ratios at 2%/50 hazard level with PGD,

PGV, and PGA of H1 and H2 components

PGV μn 10%/50 5%/50 2%/50

CR

μ11 3.84 2.53 1.90

μ5 3.60 2.50 1.88

μ3 3.44 2.20 1.70

SC

μ11 2.66 1.97 1.46

μ5 3.29 2.45 1.75

μ3 2.73 2.07 1.61

SD

μ11 4.65 3.07 2.13

μ5 5.60 3.49 2.16

μ3 5.98 3.77 2.26

All

μ33 3.72 2.53 1.83

μ11 4.89 3.07 2.11

μ7 4.44 2.88 1.98

μ5 3.88 2.33 1.76

μ3 4.35 2.74 1.95

Table D.5 Mean longitudinal drift ratios (%) of the smaller suites sampled based on H1-PGV correlation

PGA μn 10%/50 5%/50 2%/50

CR

μ11 1.48 0.99 0.65

μ5 1.64 1.15 0.77

μ3 1.55 1.11 0.74

SC

μ11 1.60 1.08 0.72

μ5 1.87 1.26 0.81

μ3 2.13 1.48 0.94

SD

μ11 1.41 0.98 0.67

μ5 1.73 1.22 0.86

μ3 1.87 1.30 0.87

All

μ33 1.50 1.02 0.68

μ11 1.74 1.19 0.78

μ7 1.92 1.32 0.85

μ5 2.00 1.37 0.86

μ3 1.84 1.27 0.81

Table D.6 Mean lateral drift ratios (%) of the smaller suites sampled based on H2-PGA correlation

210

Figure D.3 Maximum longitudinal and lateral drift ratios versus PGD, PGV, and PGA of H1 and H2

components of the crustal suite

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0 50 100 150

Lo

ng

itu

din

al

Dri

ft (

%)

H1-PGD (cm)

0.00

0.50

1.00

1.50

2.00

2.50

0 20 40 60 80 100

La

tera

l D

rift

(%

)

H2-PGD (cm)

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0 50 100 150

Lo

ng

itu

din

al

Dri

ft (

%)

H1-PGV (cm/s)

0.00

0.50

1.00

1.50

2.00

2.50

0 50 100 150

La

tera

l D

rift

(%

)

H2-PGV (cm/s)

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0 1 1

Lo

ng

itu

din

al

Dri

ft (

%)

H1-PGA (g)

0.00

0.50

1.00

1.50

2.00

2.50

0 1 1

La

tera

l D

rift

(%

)

H2-PGA (g)

211


components of the subcrustal suite

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

0 10 20 30 40

Lo

ng

itu

din

al

Dri

ft (

%)

H1-PGD (cm)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 10 20 30 40

La

tera

l D

rift

(%

)

H2-PGD (cm)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

0 20 40 60 80

Lo

ng

itu

din

al

Dri

ft (

%)

H1-PGV (cm/s)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 20 40 60 80 100

La

tera

l D

rift

(%

)

H2-PGV (cm/s)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

0 1 1

Lo

ng

itu

din

al

Dri

ft (

%)

H1-PGA (g)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 1 1 2

La

tera

l D

rift

(%

)

H2-PGA (g)

212


components of the subduction suite

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0 50 100 150

Lo

ng

itu

din

al

Dri

ft (

%)

H1-PGD (cm)

0.00

0.50

1.00

1.50

2.00

2.50

0 20 40 60 80 100

La

tera

l D

rift

(%

)

H2-PGD (cm)

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0 50 100 150 200

Lo

ng

itu

din

al

Dri

ft (

%)

H1-PGV (cm/s)

0.00

0.50

1.00

1.50

2.00

2.50

0 50 100 150

La

tera

l D

rift

(%

)

H2-PGV (cm/s)

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0 0 0 1 1

Lo

ng

itu

din

al

Dri

ft (

%)

H1-PGA (g)

0.00

0.50

1.00

1.50

2.00

2.50

0 1 1

La

tera

l D

rift

(%

)

H2-PGA (g)

213


components of all records

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0 50 100 150

Lo

ng

itu

din

al

Dri

ft (

%)

H1-PGD (cm)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 20 40 60 80 100

La

tera

l D

rift

(%

)

H2-PGD (cm)

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0 50 100 150 200

Lo

ng

itu

din

al

Dri

ft (

%)

H1-PGV (cm/s)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 50 100 150

La

tera

l D

rift

(%

)

H2-PGV (cm/s)

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0 1 1

Lo

ng

itu

din

al

Dri

ft (

%)

H1-PGA (g)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 1 1 2

La

tera

l D

rift

(%

)

H2-PGA (g)

evaluating the performance-based seismic design of …

Documents