performance-based seismic design and evaluation of steel

PERFORMANCE-BASED SEISMIC DESIGN AND EVALUATION OF STEEL

ECCENTRICALLY BRACED FRAMES WITH TUBULAR LINKS AS BRIDGE BENTS

by

Ahmad Shafiq

B.Sc., University of Engineering & Technology Lahore, 2014

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF APPLIED SCIENCE

in

THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES

(Civil Engineering)

THE UNIVERSITY OF BRITISH COLUMBIA

(Vancouver)

April 2018

© Ahmad Shafiq, 2018

ii

Abstract

The latest release of the Canadian Highway Bridge Design Code (CHBDC), S6-14, incorporates

performance-based design (PBD) provisions for bridges in Canada for the first time. CHBDC S6-

14 establishes two different design approaches, with the PBD being the standard method of design

and force-based design (FBD) being permitted for special cases. The focus of this study is on

ductile eccentrically braced frames (EBFs) as bridge substructure. For member proportioning, the

CHBDC S6-14 refers to the Canadian steel design standard for buildings, CSA S16-14, stating a

force reduction factor, R=4. This is a FBD, and there is need to evaluate the design in terms of the

performance descriptions and damage states by carrying out the analyses recommended by

CHBDC S6-14. For this case study, an existing bridge is considered as a Major Route bridge, and

an EBF with built-up tubular shear link has been chosen as an earthquake-resisting system (ERS).

Four different cases have been designed including two using FBD and two for PBD approach for

comparison purposes. Due to the lack of strain/rotation criteria in CHBDC S6-14 at multiple

service states for EBFs as bridge bents, different acceptance criteria for rotations and

corresponding damage states have been proposed by using fragility curves from the literature. The

link total rotation has been considered as a demand parameter and different methods of repairs

consistent with each damage state have also been provided. The response spectrum analysis

coupled with inelastic static pushover analysis is used for global displacement demands and for

demonstrating local component performance compliance of shear links. Nonlinear time-history

analysis is also used to check and provide a comparison of the first approach. The code requires

no-yielding for the 475-year return period event. This criterion governs the design and makes the

sizes large and inefficient, while the link plastic rotations corresponding to higher return period

events are very low compared to the allowable limits provided in the literature for links mainly

iii

used in buildings. Through different cases, it is demonstrated that if the links are made replaceable

and allowed to have limited yielding at 475-year earthquake, it makes the design more practical.

iv

Lay Summary

The damage caused by real-world earthquake events provides researchers with empirical data that

can be used to further develop and improve codes and standards used to design structures. In 2014,

the Canadian Standards Association introduced a new method for the first time in Canada for

earthquake design and performance evaluation of bridges, known as performance-based design

(PBD). The focus of this study is to apply PBD methodology to Eccentrically Braced Frames

(EBF) as bridge substructure using Canadian Highway Bridge Code. In an EBF, some parts of the

assembly are intentionally designed to absorb the energy of the earthquake through

deformation/damage, allowing the rest of the structure to remain undamaged. This gives the

opportunity to repair or replace the damaged part after an earthquake. This thesis evaluates the

performance of a bridge at multiple levels of earthquakes by proposing different damage

conditions and corresponding methods of repairs.

v

Preface

This dissertation is original, unpublished, independent work by the author of this thesis under the

supervision of Dr. Carlos Ventura and Mr. Saqib Khan. The author is responsible for the literature

review, design, model development and presentation of the results.

vi

Table of Contents

Abstract .......................................................................................................................................... ii

Lay Summary ................................................................................................................................ ii

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

Table of Contents ......................................................................................................................... vi

List of Tables ..................................................................................................................................x

List of Figures .............................................................................................................................. xii

List of Symbols ........................................................................................................................... xvi

List of Abbreviations ................................................................................................................. xix

Acknowledgements .................................................................................................................... xxi

Dedication ................................................................................................................................. xxiii

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

1.1 Performance-Based Design of Bridges ........................................................................... 1

1.2 Research Needs ............................................................................................................... 2

1.3 Objectives ....................................................................................................................... 4

1.4 Scope ............................................................................................................................... 4

1.5 Organization of Thesis .................................................................................................... 5

Chapter 2: Literature Review and CHBDC S6-14 Performance Criteria ...............................8

2.1 Performance-Based Seismic Design ............................................................................... 8

2.2 Preliminary Member Sizing for Performance-Based Design ....................................... 10

2.3 Need for Performance-Based Design............................................................................ 11

2.4 Performance-Based Design Criteria ............................................................................. 12

vii

2.4.1 PBD Criteria and Analysis Requirements by CHBDC S6-14 .................................. 14

2.5 Eccentrically Braced Frames as Earthquake-Resisting System (ERS) ......................... 17

2.5.1 Applicability of CHBDC S6-14 Performance Criteria for EBFs.............................. 20

Chapter 3: Seismic Performance Comparison and Design Approach ....................................22

3.1 Comparison of EBFs to CBFs and MRFs ..................................................................... 22

3.2 EBF Geometric Configurations .................................................................................... 23

3.2.1 Selected EBF Geometry ............................................................................................ 24

3.3 Case Study .................................................................................................................... 25

3.4 Force-Based Design Approach ..................................................................................... 32

3.4.1 Force-Based Design Methodology ........................................................................... 33

3.5 Performance-Based Design Approach .......................................................................... 35

3.5.1 Performance-Based Design Methodology ................................................................ 36

3.6 EBF Bent Design .......................................................................................................... 37

3.6.1 Link Sizing ................................................................................................................ 39

3.6.1.1 Link Length Calculation ................................................................................... 43

3.6.2 Capacity Design Procedure ....................................................................................... 45

3.6.3 Link Rotation Check ................................................................................................. 46

3.7 Final Designed Bents .................................................................................................... 47

Chapter 4: Numerical Modelling ................................................................................................49

4.1 Numerical Model Description....................................................................................... 49

4.2 Model Calibration ......................................................................................................... 50

4.2.1 Nonlinear Modeling of Link Member Using Plastic Hinges .................................... 53

4.2.2 Nonlinear Modeling of Link Member Using Multi-Linear Plastic Link Element .... 58

viii

4.2.3 Nonlinear Modeling of Link Member Using Fiber Hinges ...................................... 59

4.3 Summary of Modeling Techniques for Different Analyses .......................................... 60

Chapter 5: Proposed Performance Criteria and Seismic Evaluation .....................................63

5.1 Proposed Acceptance Criteria for EBFs as Bridge Piers .............................................. 63

5.1.1 Drift Ratio as a Performance Measurement Tool ..................................................... 68

5.2 Proposed Method of Repairs for Multiple Performance Levels ................................... 68

5.3 Damping Selection ........................................................................................................ 71

5.4 Modal Analysis Results ................................................................................................ 72

5.5 Response Spectrum Analysis Results ........................................................................... 72

5.5.1 Inelastic Displacement Correction ............................................................................ 73

5.6 Pushover Analysis Results ............................................................................................ 77

5.6.1 Performance Evaluation from Pushover and RSA Results ....................................... 82

5.7 Time-History Analysis .................................................................................................. 84

5.7.1 Ground Motions for Time-history Analyses ............................................................. 84

5.7.2 Seismic Hazard Deaggregation for Site Location ..................................................... 84

5.7.3 Selection of Ground Motions .................................................................................... 88

5.7.4 Linear Scaling of Scaled Ground Motions................................................................ 91

5.7.5 Spectral Matching of Scaled Ground Motions.......................................................... 92

5.8 Elastic Time-history Analysis (ETHA) ........................................................................ 96

5.9 Nonlinear Time-history Analysis (NLTHA) ................................................................ 96

5.9.1 Performance Evaluation from NLTHA Results ...................................................... 100

Chapter 6: Conclusions and Recommendations .....................................................................109

6.1 Summary and Conclusions ......................................................................................... 109

ix

6.2 Recommendations for Design ..................................................................................... 110

6.3 Limitations of this Research ....................................................................................... 112

6.4 Recommendations for Future Studies ......................................................................... 113

Bibliography ...............................................................................................................................114

Appendices ..................................................................................................................................121

Appendix A EBF Bent Design ................................................................................................ 121

A.1 Seismic Load Calculations for EBF Bent ............................................................... 121

A.2 Link Member Design Calculations ......................................................................... 122

A.3 Capacity-Protected Bent Member Capacities ......................................................... 124

A.4 Beam-Column Design for Brace and Beam Outside the Link ................................ 130

Appendix B Spectral Matching of Ground Motions ............................................................... 131

B.1 1 in 475-year Return-period Target Spectrum ........................................................ 131

B.2 1 in 975-year Return-period Target Spectrum ........................................................ 135

B.3 1 in 2475-years Return-period Target Spectrum .................................................... 138

x

List of Tables

Table 1 Performance Levels for Major Route Bridges according to CHBDC S6-14 ................... 14

Table 2 Bridge serviceability levels according to CHBDC S6-14 ............................................... 15

Table 3 Analysis required by CHBDC S6-14 for Major Route bridges for Seismic Performance

Category 3 ..................................................................................................................................... 15

Table 4 Performance Criteria for Steel Bridges as per CHBDC S6-14 (CSA Group, 2014a) ..... 16

Table 5 Considered design cases as per CHBDC S6-14 ............................................................... 31

Table 6 EBF link types, link length condition and allowable maximum inelastic link rotation... 39

Table 7 Link member forces for four design cases ....................................................................... 44

Table 8 Different cases for Sombrio Bridge substructure ............................................................ 47

Table 9 Designed EBF Bents member size (all dimensions in mm) ............................................ 48

Table 10 Nonlinear modeling parameters for EBF shear link beam (ASCE, 2014) .................... 54

Table 11 Force-Displacement backbone curve parameters for the shear hinge in SAP2000 ....... 55

Table 12 SAP2000 Models descriptions for different analyses .................................................... 61

Table 13 Proposed acceptance criteria limits for shear links in EBF bridge bents ....................... 67

Table 14 Proposed Method of Repairs (MOR) for each damage state (Gulec et al., 2011) ......... 71

Table 15 Modal analysis results for all design cases .................................................................... 72

Table 16 Bent global displacement demands from RSA .............................................................. 73

Table 17 Inelastic displacement correction for all design cases ................................................... 76

Table 18 Modified displacement and rotation demands from RSA ............................................. 78



xi


Table 22 Ground motions selection criteria from seismic hazard deaggregation ......................... 87

Table 23 Selected ground motions for 10% in 50 years hazard level (475-year return period) ... 93

Table 24 Selected ground motions for 5% in 50 years hazard level (975-year return period) ..... 93

Table 25 Selected ground motions for 2% in 50 years hazard level (2475-year return period) ... 94

Table 26 Link total rotations from RSA/Pushover and NLTHA ................................................ 103

Table 27 Performance evaluation for performance level 1: Immediate Service (1 in 475-years)

..................................................................................................................................................... 104

Table 28 Performance evaluation for performance level 2: Limited Service (1 in 975-years) .. 105

Table 29 Performance evaluation for performance level 3: Service Disruption (1 in 2475-years)

..................................................................................................................................................... 105

Table 30 Bent global displacement demands from Linear Direct Integration Time-History

(LDITH) and Nonlinear Time-History Analysis (NLTHA) and link total rotations from NLTHA

using ground motions scaled to 475-year return period.............................................................. 106



using ground motions scaled to 975-year return period.............................................................. 107



using ground motions scaled to 2475-year return period............................................................ 108

xii

List of Figures

Figure 1 Typical EBF Geometric Configuration (Chevron) ......................................................... 18

Figure 2 Possible EBF geometries (with selected EBF configuration marked)(Bruneau et al., 2011)

....................................................................................................................................................... 24

Figure 3 Sombrio Bridge elevation (WSP/MMM Group, 2013) .................................................. 26

Figure 4 Sombrio Bridge superstructure typical section (WSP/MMM Group, 2013) .................. 26

Figure 5 EBF bent (used as replacement for the 2-column reinforced concrete bent) ................. 27

Figure 6 Selected EBF geometry with girders orientation for design 1, 2 and 3 .......................... 29

Figure 7 Selected EBF geometry with girders orientation for design 4 ....................................... 30

Figure 8 5% damped Sombrio UHS for 475,975 and 2475-year return period ............................ 31

Figure 9 Effect of e/L over relative frame stiffness (Popov & Engelhardt, 1988) ....................... 38

Figure 10 Deformed shape of an EBF bent under lateral load Vb using the rigid-plastic mechanism.

Adapted from: (Berman & Bruneau, 2007) .................................................................................. 40

Figure 11 Free-body diagram of half EBF bay with lateral load Vb ............................................. 40

Figure 12 Typical built-up tubular cross-section with external stiffeners (Berman & Bruneau,

2007) ............................................................................................................................................. 41

Figure 13 Half-bay model for capacity design using SAP2000 ................................................... 46

Figure 14 Calibration model and loading history in SAP2000 ..................................................... 52

Figure 15 Proof-of-concept test setup of EBF with Built-up Tubular Shear Links (Berman &

Bruneau, 2007) .............................................................................................................................. 52

Figure 16 Backbone curve (Force - Deformation) according to ASCE41-13 .............................. 54

Figure 17 Numerical model calibration results using shear plastic hinge .................................... 57

xiii

Figure 18 Experimental hysteresis calibration using shear plastic hinge ..................................... 57

Figure 19 Experimental hysteresis calibration using a multi-linear plastic link element ............. 58

Figure 20 Experimental hysteresis calibration using fiber hinges in SAP2000............................ 60

Figure 21 Deformed shape of an EBF frame under lateral using the rigid-plastic mechanism.

Adapted from: (Berman & Bruneau, 2007) .................................................................................. 66

Figure 22 Fragility functions for shear links corresponding to each method of repair (lognormal

distribution) (Gulec et al., 2011) ................................................................................................... 70

Figure 23 Steps for ATC- 40 capacity spectrum, FEMA 356 & 440 coefficient method, and FEMA

440 equivalent linearization methods. Adapted from: CSI Knowledge Base (Powell, 2013) ...... 75

Figure 24 Design 1 pushover curve with modified RSA demands and acceptance criteria limits 78



Figure 27 Design 4 pushover curve with modified RSA demands and acceptance criteria limit 81

Figure 28 Satellite image and sitemap of Sombrio Bridge one, Port Renfrew ............................. 85

Figure 29 Magnitude-distance (M-R) deaggregation of 2475-year return period hazard level (T=

0.4 sec) .......................................................................................................................................... 85

Figure 30 Magnitude-distance (M-R) deaggregation of 975-year return period hazard level (T=0.4

sec) ................................................................................................................................................ 86

Figure 31 Magnitude-distance (M-R) deaggregation of 475-year return period hazard level (T=0.4

sec) ................................................................................................................................................ 86

Figure 32 Original acceleration-time-histories for 2475-year target spectrum (subduction records)

....................................................................................................................................................... 90

xiv

Figure 33 Original acceleration-time-histories for 2475-year target spectrum (subduction records)

....................................................................................................................................................... 91

Figure 34 Matched acceleration-time-histories for 2475-year target spectrum (subduction records)

....................................................................................................................................................... 94

Figure 35 Original spectra of scaled selected GMs and 2475 years target spectrum (subduction

records) ......................................................................................................................................... 95

Figure 36 Matched spectra of scaled selected GMs and 2475 years target spectrum (subduction

records) ......................................................................................................................................... 95

Figure 37 Total link rotation for each time-history record from NLTHA for 475-year return period

event .............................................................................................................................................. 98


event .............................................................................................................................................. 99


event ............................................................................................................................................ 100

Figure 40 475-year GM Records (Subduction) .......................................................................... 131

Figure 41 475-year GMs Spectrally Matched Accelerograms (Subduction).............................. 132

Figure 42 475-year GM Records (Subcrustal) ............................................................................ 132

Figure 43 475-year GMs Spectrally Matched Accelerograms (Subcrustal) ............................... 133

Figure 44 475-year GM Records (Crustal) ................................................................................. 134

Figure 45 475-year GMs Spectrally Matched Accelerograms (Crustal) .................................... 134

Figure 46 975-year GM Records (Subduction) .......................................................................... 135

Figure 47 975-year GMs Spectrally Matched Accelerograms (Subduction).............................. 135

Figure 48 975-year GM Records (Subcrustal) ............................................................................ 136

xv

Figure 49 975-year GMs Spectrally Matched Accelerograms (Subcrustal) ............................... 136

Figure 50 975-year GM Records (Crustal) ................................................................................. 137

Figure 51 975-year GMs Spectrally Matched Accelerograms (Crustal) .................................... 137

Figure 52 2475-year GM Records (Subduction) ........................................................................ 138

Figure 53 2475-year GMs Spectrally Matched Accelerograms (Subduction)............................ 139

Figure 54 2475-year GM Records (Subcrustal) .......................................................................... 139

Figure 55 2475-year GMs Spectrally Matched Accelerograms (Subcrustal) ............................. 140

Figure 56 2475-year GM Records (Crustal) ............................................................................... 140

Figure 57 2475-year GMs Spectrally Matched Accelerograms (Crustal) .................................. 141

xvi

List of Symbols

∆𝑒𝑙 Elastic Storey Drift

𝛾𝐸 Elastic Rotation

𝛾𝑇 Total Rotation

𝛾𝑝 Plastic Rotation

𝜀𝑠𝑡 Steel Strain

𝜀𝑦 Yield Strain

𝜃1 Total Rotation for Minimal Damage

𝜃2 Total Rotation for Repairable Damage

𝜃3 Total Rotation for Extensive Damage

𝜃𝑝 Storey Plastic Rotation

𝜃𝑝 Link Plastic Rotation

∆ Link Vertical Displacement

A Gross Area

Aw Area of Web

b Overall Width

d Overall Depth

e Link length

Fy Yield Strength

Fyf Yield Strength of Flange

Fyw Yield Strength of Web

xvii

H Bent Height

IE Importance Factor

L Bay Width

ML Link End Moment

𝑀𝑝 Plastic Moment Capacity

R Response Modification Factor

Rd Ductility-related Force Modification Factor

RD Damping Modification Factor

Ro Overstrength-related Force Modification Factor

Ry Overstrength Factor

S(T) Design Spectral Response Acceleration

Sa Spectral Acceleration

T Fundamental Period

tf Thickness of Flange

tw Thickness of Web

Vb Lateral Base Shear

VL Link Shear Force

Vn Nominal Shear Resistance

𝑉𝑝 Plastic Shear Capacity

𝑉𝑙𝑖𝑛𝑘 Link Probable Shear Resistance

W Dead Load

Z Plastic Section Modulus

xviii

𝜉 Damping Ratio

xix

List of Abbreviations

AASHTO American Association of State Highway and Transportation Officials

ASCE American Society of Civil Engineers

CBF Concentrically braced frame

CHBDC Canadian Highway Bridge Design Code

CP Collapse prevention

CQC Complete quadratic combination

CSA Canadian Standard Association

DBD Displacement-based design

EBD Energy-based design

EBFs Eccentrically braced frames

EDA Elastic dynamic analysis

ERS Earthquake-resisting system

ETHA Elastic time-history analysis

FBD Force-based design

FEMA Federal Emergency Management Agency

IO Immediate occupancy

ISPA Inelastic static pushover analysis

LDITH Linear direct integration time-history

MOR Method of repair

MRF Moment resisting frame

xx

MSE Mean squared error

NBCC National Building Code of Canada

NLTHA Nonlinear time-history analysis

PBD performance-based design

PEER Pacific Earthquake Engineering Research

RSA Response spectrum analysis

SDOF Single degree of freedom

SLS Serviceability limit sate

UHA Uniform hazard spectrum

ULS Ultimate limit sate

WF Wide-flange

xxi

Acknowledgements

There are many people to whom I am grateful for their assistance throughout this project. First and

foremost, I would like to express my sincere appreciation and thanks to my supervisor, Dr. Carlos

Ventura, Professor, UBC Civil Engineering Department, for his continuous support of my masters’

study and research, and for his depth of knowledge, patience and guidance throughout the course

of this work. I would also like to express my deepest gratitude to my co-supervisor and second

reader of my thesis, Mr. Saqib Khan, of McElhanney Consulting Services Ltd., whose support has

allowed me to explore my field openly. His vast knowledge and encouragement gave me the

direction to question the applications and implications of my work and gave me the motivation to

work hard and achieve more. His patience and kindness to me will always be remembered. I also

thank Dr. Donald Anderson for taking time to review my thesis. His valuable input and

suggestions have helped make this a successful research endeavour.

Furthermore, I thank Dr. Charles King, of COWI, for patiently answering my endless questions

and providing feedback for the duration of my research. I must also acknowledge Dr. Armin

Bebamzadeh, Mike Fairhust, and Kuldeep Kaushik from the EERF UBC, as a source of practical

guidance in working out with me the numerous issues in this research. I extend my sincere

appreciation to Brook Robazza and Kristin Greinacher for their valuable feedback and help at

different stages of this project. Furthermore, I would like to express my great appreciation to

Charlene Hails and Alexander Dacko who kindly gave their time and effort to proofread my thesis.

Many thanks to the administrative staff of UBC Civil Engineering Department, particularly Ms.

Terry Moser, for her help throughout my time in this research group.

xxii

The funding for this research was provided by Canadian Institute of Steel Construction (CISC)

and is greatly appreciated.

Above all, I would like to thank my parents, who have supported me throughout my life by

providing me with unconditional love, showing outstanding amounts of patience, and giving me

the space to study and pursue my interests. Special thanks to my sister, Iram, for always believing

in me and providing me with a kind ear and comforting word when it was needed most. Thank

you to my younger brothers, Umair and Zain, for keeping the laughs coming despite the challenges

of time zones. I express my utmost gratitude to my wife, Kiran, whose judgment was invaluable,

and patience near unlimited through challenging periods of this undertaking—thank you for this

continuing encouragement and steadfast belief in me. Happiness has little value without good

people around to share it.

I would like to thank my Grandpa, Muhammad Ameen, for always being a source of inspiration

for me to study mathematics and continue my education, my Grandma, Sakina Ameen, for teaching

me how to be kind and strong, with her immense love. I am exteremly grateful to my dadi, Zanib

Bibi, for her love throughout my life. Equally, I am thankful to my whole family and friends back

home for all their best wishes.

Last but not least, I must also thank all my friends, who have continued to support me throughout

all my years of study and made my time in Vancouver much more enjoyable. I could not have

made it this far without you all.

xxiii

Dedication

For my parents, with love:

my mother, Kausar Shafiq,

and

my father, Muhammad Shafiq Akram,

who serve as a source of constant inspiration

and motivation in my life.

1

Chapter 1: Introduction

1.1 Performance-Based Design of Bridges

The latest release of the Canadian Highway Bridge Design Code (CHBDC), S6-14, incorporates

performance-based design (PBD) provisions for bridges in Canada for the first time. Until recently,

the main design goal has been life safety with designs mostly based on strength criteria which is

the main concept of force-based design (FBD) approach. There has been a gradual shift from

‘strength-based design’ to ‘performance-based design’ and a recognition that strength is not always

equal to better performance. Moreover, the increase in strength does not essentially mean higher

safety, nor does it imply less damage (EGBC, 2018). In fact, a large increase in strength can be

unfavorable and strength without ‘ductility’ is ineffective in a seismic event (Priestley, Seible, &

Calvi, 1996). However, better performance means more ductile, robust and predictable behaviour

of bridges in an earthquake.

The owners now increasingly want their structures to be serviceable after small and

moderate earthquakes and FBD approach does not clearly incorporate this. This shortcoming has

acted as one of the main reasons behind this significant evolution to performance-based design

(PBD) to emphasize and incorporate the interaction between bridge owners and structural

engineers to consider the post-earthquake performance of new and existing bridges. Recent events

like the February 2011 Christchurch earthquake and March 2011 Japan earthquake played a vital

role for such motivation, as a lack of collaboration was observed among various stakeholders

including owners, structural engineers, and the public (Kennedy, Dowling, & Du, 2017).

In the CHBDC S6-14, PBD is the primary design approach, with FBD allowed for certain

cases based on structure’s seismic performance category (SPC) and structural regularity. The FBD

2

approach which includes designing for required ductility using reduction factors is fundamentally

different from the PBD approach, which includes consideration of bridge serviceability, damage

quantification, acceptable bridge closure for repairs, and returning to full traffic usage (Kennedy

et al., 2017).

The underpinning of the PBD approach is the displacement-based design (DBD)

philosophy. While quantifying earthquake demands primarily as deformations rather than forces,

the anticipated performance of the bridge can be measured at the component level by relating the

damage conditions with deformations such as displacements, plastic rotations and material strains.

These damage conditions can be linked to the operational condition of the bridge as well as repair

costs and interruption in the serviceability of the bridge after an earthquake. The connection

between engineering demand parameters (deformations) with the functionality of the bridge has

provided an opportunity to assure the use of structure after an earthquake (Marsh & Stringer,

2013). Hence, PBD can assist bridge engineers to decide a target performance and use different

earthquake-resisting systems (ERS) to achieve the required performance at different levels of

earthquakes.

1.2 Research Needs

In CHBDC S6-14, multiple performance objectives are required to be met for numerous

performance levels corresponding to earthquake events at a variety of return periods. However,

there is a general lack of specific engineering demand parameters for these performance objectives

(Sheikh & Légeron, 2014). The performance criteria specified in CHBDC are very qualitative and

broad. In addition, the damage states assigned to these performance levels seem very generic and

do not provide enough guidance to demonstrate performance of different code-recommended

3

earthquake-resisting systems (ERS), such as steel eccentrically braced frames (EBFs) and steel

moment-resisting frames (MRFs). The inelastic action in bridges is anticipated to happen in the

piers, not superstructure or foundations. Hence, the seismic response of a bridge is mainly

dependent on the inelastic capacity of the bridge piers (Sheikh & Légeron, 2014).

The focus of this study is on the use of ductile eccentrically braced frames (EBFs) for the

bridge substructure. For member proportioning, CHBDC (S6-14) refers to the Canadian steel

building code S16-14, using a force reduction factor, R=4. This design approach is force-based

and post-earthquake performance of the bridge cannot be quantified using this approach. There is

a need to assess the design in terms of performance descriptions and damage states by carrying out

the analyses recommended by CHBDC S6-14. Moreover, CHBDC S6-14 does not provide clear

guidelines to check whether such performance objectives are achieved, and there is limited

literature available on this issue so far, especially for bridges with ductile steel sub-structures.

Therefore, practising engineers face many challenges associated with the implementation of PBD

for seismic design of steel substructure bridges. The emphasis here is to apply the PBD

methodology to bridges with steel EBF substructure and determine the necessary steps and

additional analyses that need to be conducted by a bridge designer to assess the performance

criteria required by the new code. Appropriate application of performance-based seismic design

rules should decrease the loss of lives, lessen economic losses in earthquakes, and reduce the need

for emergency services after an earthquake (Floren, A., and Mohammadi, 2001).

4

1.3 Objectives

The primary goal of this research is the performance assessment of eccentrically braced frame

(EBF) substructure bridges designed using FBD and PBD approach according to CHBDC S6-14.

The goal of the study is decomposed into specific objectives as follows:

• Seismic behaviour comparison between EBF substructure bridges designed using FBD and

PBD approach according to CHBDC S6-14

• Use of PBD approach to evaluate the design of the bridge substructure in accordance with

the performance descriptions and damage states by carrying out the analyses recommended

by CHBDC S6-14

• Demonstrate the use of simple existing tools and analyses to implement the PBD approach

• Evaluate multiple performance objectives and determine the performance objectives

governing the design

• Propose more specific damage states corresponding to multiple performance levels and

method of repairs consistent with each damage state

1.4 Scope

The scope of this research is focused on steel eccentrically braced frames (EBFs) with built-up

tubular shear links as bridge piers. For the case study considered in this thesis, the tributary seismic

mass of the existing superstructure is applied to a single bent. The structure is therefore modelled

as a single-degree-of-freedom (SDOF) system as it has been confirmed from practical projects that

this approach gives a reasonable estimate of the dynamic behaviour of a bridge when compared

with the detailed model including the superstructure (Gerin & Khan, 2017).

5

A literature review of FBD and PBD approaches is carried out to start the research work.

For the case study designs, previous guidelines and literature for EBFs are also reviewed. Two

different design approaches, each based on FBD and PBD method are considered, for a total of

four cases. A review of performance criteria for a range of performance levels and damage states

for different categories of bridges is carried out as per CHBDC S6-14 (CSA Group, 2014a). Link

plastic rotation is used as the demand parameter for damage evaluation of EBFs. Link rotations

corresponding to required seismic return-period events are evaluated using analyses recommended

by CHBDC S6-14. The performance of four designs is evaluated and compared with both code-

based performance criteria as well as proposed damage states obtained from previous literature. A

specific method of repair has been proposed corresponding to each damage state for EBFs with

tubular shear links.

1.5 Organization of Thesis

This thesis consists of a total of six chapters. Chapter 1 contains an introduction to the

performance-based design of steel substructure bridges, need for research, objectives, and scope

of this research.

In Chapter 2, a literature review on performance-based seismic design approach and its

need has been carried out, and different methods for preliminary member sizing to conduct PBD

have been identified. This chapter also summarizes the performance criteria for steel substructure

bridges as per Canadian Highway Bridge Design Code (CHBDC) S6-14 along with its

applicability to EBFs as bridge bents. A literature review on EBFs has also been provided.

In Chapter 3, a more detailed introduction to EBFs and their selection as an earthquake-

resisting system has been provided. A Major Route bridge has been taken as a case study to explain

6

and apply both FBD and PBD approaches. A total of four cases have been designed including two

each employing the FBD and PBD approach, respectively, for comparison and evaluation

purposes. A step-by-step procedure to adopt each method has been explained followed by sizing

of the link member and capacity design as per Steel Building Code CSA S16-14.

In Chapter 4, designs from previous chapters have been modeled in SAP 2000. A

calibration procedure with actual test results has been performed by comparing different modelling

approaches to capture the nonlinear behaviour of the link member. A summary of all the models

developed for analysis has been provided.

In Chapter 5, specific damage states and demand parameters for EBFs as bridge bents have

been proposed based on previous literature, and different methods of repair for each damage state

have been explained. Different damping assumptions have been stated. Modal analysis has been

performed on all four bents. To determine the displacement demands, response spectrum analysis

(RSA) has been performed for multiple hazard levels, and pushover analysis has been carried out

for seismic performance evaluation of all designed bents. Inelastic displacement modification has

been applied using four different methods to correct RSA displacement demands. The required

number of ground motions has been selected and matched with the target spectrum to carry out

time-history analyses. Elastic time-history analysis (ETHA) is used to confirm the displacement

demands from RSA and ensure that ground motions properly match with target uniform hazard

spectrum (UHS). Nonlinear time-history analysis (NLTHA) has been performed to evaluate the

performance of EBF bents and compare the results from pushover and RSA. Performance

evaluation has been carried out by using proposed performance criteria in this chapter, and

applicable methods of repair are provided.

7

Lastly, Chapter 6 provides findings of the research along with recommendations for future

research.

8

Chapter 2: Literature Review and CHBDC S6-14 Performance Criteria

2.1 Performance-Based Seismic Design

For continued development of a modern society, a sound transportation system plays a significant

role, and in a global transportation system, bridges are vital elements. In case of an earthquake

event, the damage or collapse of important bridges can interrupt the whole transportation system

of a city by disconnecting specific areas from hospitals, relief and aid camps, etc. After the 1971

San Fernando earthquake, considerable research was carried out on seismic performance of

reinforced concrete bridges. Even in developed countries like New Zealand and Japan with modern

structural design codes, recent earthquakes have resulted in a substantial loss in economy and

human lives. Several defects in the design and detailing of existing bridges have been observed in

previous earthquakes (Saiidi, 2011). The lessons learned from these events generated a motivation

for considerable modification and improvements in seismic design practices of concrete bridges.

However, the previous research on the seismic performance of bridges with steel substructures is

still insufficient. Some previous research work is based on specific case studies for large steel

substructure bridges such as the use of EBFs as the earthquake resisting system for San Francisco–

Oakland Bay Bridge and the Richmond–San Rafael Bridge (Dusicka P, Itani AM, 2002; Itani AM.,

1997).

Currently, seismic design of new bridges in Canada is based on performance-based seismic

design guidelines that interrelate the serviceability of the bridge to the damage and performance

of a bridge after an earthquake event. This design approach differs from force-based design (FBD)

approach in which bridge designer does not have clear understanding of the seismic behaviour and

performance of the bridge at multiple earthquake levels.

9

In the FBD approach, one does not explicitly consider the damage and functionality of the

bridge after an earthquake event. During the 2011 Christchurch Earthquake, many well-designed

structures designed using the latest code satisfied the collapse-prevention objective, but many

structures needed considerable repair, and in some cases decommissioning of the structures was

required as the seismic induced loading was much greater than the code-based design loading.

The PBD approach gives a choice to consider multi-hazard levels, or different ground

motion records, along with different serviceability levels, such as immediate-use, limited-use or

service disruption after an earthquake event. These performance levels could facilitate the instant

availability of bridge for traffic and allow the traffic to use the bridge while it is being repaired at

the same time without disrupting the regular traffic, emergency vehicle access or both. The PBD

approach helps structural engineers and bridge owners to design and construct bridges whose

behaviour during an earthquake would be more controlled and predictable.

Since 2005, several bridge projects in Canada have been completed using the PBD method,

such as the Port Mann Bridge in Vancouver, BC (Jones, Semyon Treyger, Pence, & Shama, 2013),

the Golden Ears Bridge in Maple Ridge, BC (Kennedy et al., 2017) and the Vancouver Evergreen

Line Rapid Transit Project (Khan & Jiang, 2015). As PBD has already been in practice for

significant projects in British Columbia in addition to its ongoing development internationally, the

CHBDC S6-14 required the PBD approach as a compulsory design and evaluation method for

seismic design of important bridges in Canada for the first time.

Section 4 of CHBDC S6-14 establishes two different design approaches, with the PBD

being the standard method of design, and FBD approach being permitted for special cases (CSA

Group, 2014a). Multiple performance levels are prescribed to meet the required service and

damage states. In accordance with CHBDC S6-14, bridges are required to meet specific

10

performance criteria defined regarding minimal structural damage, repairable damage, extensive

damage and probable replacement at multi-hazard levels with 10%, 5% and 2% probability of

exceedance in 50 years corresponding to 475-, 975-, and 2475-year return period event

respectively (CSA Group, 2014a). Different seismic analysis procedures have been specified to

predict the performance of structures at the three hazard levels. These seismic analyses depend on

the seismic performance category, the importance of the bridge, and whether the bridge is regular

or irregular (Mitchell, 2017). The performance of structures designed using the FBD approach is

expected to be consistent with PBD at the 2475-year return-period (CSA Group, 2014c). The

owners now increasingly want their structures to be serviceable after small and moderate

earthquakes and FBD approach does not clearly incorporate this. The PBD approach comes into

play to address this issue.

2.2 Preliminary Member Sizing for Performance-Based Design

Under the umbrella of the PBD framework, there are many design methodologies for preliminary

member sizing such as direct displacement-based design (DDBD), force-based design (FBD), and

energy-based design (EBD) (Leelataviwat, S., Goel, & Stojadinovic, 2002). By comparing all

these design methodologies, it can be concluded from previous literature that DDBD appears to be

the most promising design approach for bridges. It provides an opportunity for a designer to control

the deformations and consequently the damages, directly (Chopra & Goel, 2001; Dwairi &

Kowalsky, 2006; Priestley, M., Calvi, G., & Kowalsky, 2007).

Unfortunately, the DDBD approach is limited to structures having a simple, predictable

deformed shape, this makes it difficult to apply to long-span and irregular bridges (Ayala, Paulotto,

& Taucer, 2007; Sullivan, Calvi, Priestley, & Kowalsky, 2003). Over the years, researchers have

11

investigated and applied the DBD approach to a large number of reinforced concrete bridges,

including many actual projects, such as continuous concrete bridges (Kowalsky, 2002), long-span

bridges (Adhikari, Petrini, & Calvi, 2010), and reinforced concrete arch bridges (Khan, E.,

Sullivan, T. J., Kowalsky, 2013). However, to date, the research on the application of the DDBD

approach for steel bridges is insufficient. Therefore, one of the primary objectives of this study is

to use simple existing tools and analyses such as response spectrum analysis (RSA) and inelastic

static pushover analysis (ISPA) to design and evaluate the performance of EBF substructures.

2.3 Need for Performance-Based Design

Under earthquake shaking, bridges may experience structural damage mainly because of

deformation beyond the elastic limit of structural members. The FBD approach is based on the

strength of the structure and does not incorporate these deformations. Until recently, the main

design goal has been life safety with designs mostly based on strength criteria which is the main

concept of the FBD approach. There is a gradual shift from ‘strength’ to ‘performance’ and a

recognition that strength is not always equal to performance. Moreover, the increase in strength

does not necessarily mean enhanced safety, nor does it imply less damage. In fact, a large increase

in strength can be detrimental and strength without ‘ductility’ is ineffective in a seismic event.

However, better performance means more ductile, robust and predictable behaviour of bridges in

an earthquake. The owners now increasingly want their structures to be serviceable after small and

moderate earthquakes and the FBD methodology does not incorporate this (EGBC, 2018). The

current FBD approach has many shortcomings in obtaining a specific performance of structure

(Priestley, M., Calvi, G., & Kowalsky, 2007). With the latest release of CHBDC S6-14, PBD is

required for important and irregular bridges by fulfilling the specific performance objectives

12

defined by the code. With that, the bridge owners now have the choice to select the target

performance of a bridge that bridge engineers can achieve by fulfilling different performance

objectives and prescribed damage states. PBD has certain advantages over the FBD approach

(Mitchell, 2017), such as:

• Design of structure is based on functional objectives of the service and damage states with

a clear demonstration of meeting performance criteria.

• PBD provides reliable expectations of structural performance for different levels of seismic

events as well as the flexibility of selecting materials and a variety of design options.

• In addition to life safety consideration, PBD includes post-earthquake behaviour prediction

and may decrease economic losses.

2.4 Performance-Based Design Criteria

In PBD, the performance of structures at different return period earthquakes is linked with specific

design criteria, such as plastic rotations and strains. Structures are needed to be designed for several

criteria which primarily include safety, serviceability, and economy (Bertero, 1996). Many

specifications and codes consider multi-hazard levels for seismic design of structures (CSA Group,

2014a; PEER, 2010a).

In general, one performance level corresponding to a particular seismic hazard level

governs the design, although other levels do need to be checked for performance compliance. The

primary goal at lower seismic hazard levels is usually the functionality of structure without

significant traffic interruption, and reparability with reduced traffic functionality to probable

replacement at higher seismic hazard levels.

13

Each design code or specification has different minimum-level and highest-level

earthquakes with corresponding design criteria. As per The Pacific Earthquake Engineering

Research (PEER) guidelines for Performance-Based Seismic Design of Tall Buildings, at a lower

seismic hazard level (less than 43-year return-period event), the structure is intended to remain elastic

with minor damage (PEER, 2010a).

CHBDC S6-14 classifies bridges into three importance categories: Lifeline bridges, Major

Route bridges and Other bridges. In CHBDC S6-14, the lowest seismic hazard level is a 475-year

return period event without any yielding of members, and bridge is intended to be essentially elastic

for Major Route and Lifeline bridges. The highest seismic hazard level in CHBDC S6-14 is a 2475-

year return-period which matches FEMA-350 maximum considered earthquake (MCE) ground-

shaking level. However, the design spectrum is two thirds of MCE spectrum for design earthquake

(DE) ground-shaking level as per FEMA-350 (Venture, S. J., Committee, G. D., & Venture, 2000).

On the other hand, FEMA-356 allows limited yielding for steel moment-resisting frames at lower

design hazard level (FEMA, 2000b).

In the past, many bridges such as the Tacoma Narrows Bridge, the San Francisco Oakland Bay

Bridge, the Vancouver Evergreen Line, and the Gerald Desmond Bridge were designed for multiple

hazard levels with minimum hazard level of less than 100 years and an essentantially elastic target

performance with minor inelastic behaviour (Zhang, 2015). From all this literature review, it can be

concluded that CHBDC S6-14 is most stringent for lower hazard level of shaking by implementing

no-yielding criterion at the 475-years return period for Major Route and Lifeline bridges.

14

2.4.1 PBD Criteria and Analysis Requirements by CHBDC S6-14

For each of three bridge importance categories (Lifeline bridges, Major Route bridges and Other

bridges), CHBDC S6-14 specifies service and damage levels required to be fulfilled for multiple

hazard levels including 475-, 975-, and 2475-year return period events, which correspond to 10%,

5%, and 2% probability of exceedance in 50 years, respectively. As the considered bridge in this

study is taken as a Major Route bridge (which characterizes most highway structures), the

performance objectives for Major Route bridges according to CHBDC S6-14 are provided in Table

1.

Table 1 Performance Levels for Major Route Bridges according to CHBDC S6-14

Return-Period Performance

Level

Service Damage

475-years 1 Immediate Minimal Damage

975-years 2 Limited Repairable Damage

2475-years 3 Service Disruption Extensive Damage

Furthermore, for each of these service and damage performance levels, the code has specified

performance criteria that qualitatively consider the performance of the bridge after an earthquake

by considering the type and duration of repairs. A summary of serviceability levels for Major

Route Bridges is provided in Table 2.

15

Table 2 Bridge serviceability levels according to CHBDC S6-14

Service Fully

Serviceable

Normal

Traffic

Emergency

Traffic

Repair

Works

Bridge

Closure

Lanes

Open

Immediate Yes Yes Yes Yes No All

Limited No No Yes Yes Limited 50% lanes,

minimum one

Service

Disruption No No

Restricted

emergency

traffic only

Yes Yes Inspection

required

For each hazard level, CHBDC S6-14 specifies required analyses corresponding to each seismic

performance category (SPC) and importance category of bridge. For Major route bridges in SPC

of 3, the required analyses by CHBDC S6-14 are given in Table 3.

Table 3 Analysis required by CHBDC S6-14 for Major Route bridges for Seismic Performance

Category 3

Hazard Level Minimum Seismic Analysis Requirement

475-years return-period event EDA

975-year and 2475-year return period events EDA, ISPA

Here,

EDA = Elastic dynamic analysis including multi-mode elastic response spectral

ISPA = Inelastic static push-over analysis

For this study, in addition to analyses provided in Table 3, a complementary nonlinear

time-history analysis (NLTHA) will be used to check and provide a comparison of EDA and ISPA.

16

Even though the PBD approach provides the option to choose from a variety of earthquake-

resisting systems (ERS), one challenge in the implementation of the PBD approach is the selection

and definition of demand or engineering design parameters such as plastic rotations and strains.

These parameters correlate the various damage states with bridge performance, and hence repair

works. This is a significant reason why PBD is not part of codes around the world (Gerin & Khan,

2017).

The performance criteria for multiple performance levels and damage conditions are

provided in Table 4.16 in CHBDC S6-14 (CSA Group, 2014a). For each of these damage-level

criteria in CHBDC S6-14, in addition to defining quantitative limits for concrete and reinforcement

strains, the code specifies the limits for overall displacement and bridge components, such as

primary structural members, connections, bearings and joints, restraints, and foundations. As the

focus of this study is mainly steel substructure bridges, specified performance criteria for these

bridges are given in Table 4.

Table 4 Performance Criteria for Steel Bridges as per CHBDC S6-14 (CSA Group, 2014a)

Level Service Damage Criteria

1 Immediate Minimal

damage

- The bridge shall remain fully elastic with only

insignificant damage that does not change the

performance level of the bridge

- Steel strains (𝜀𝑠𝑡) ≤ yield strain (𝜀𝑦)

- No local or global buckling

2 Limited Repairable

damage

- Full dead plus live load-carrying capability

- No buckling of primary members

- Secondary members may buckle without causing

instability

17

Level Service Damage Criteria

- Primary member connections shall not experience net

area rupture

- 90% seismic capacity retention for aftershocks; full

capacity restoration after repairs

3 Service

Disruption

Extensive

damage

- Full dead plus 50% live load-carrying capability

- No global buckling of gravity-load-supporting elements

- 80% seismic capacity reserved for aftershocks with full

capacity restoration after repairs

4 Life Safety Probable

replacement

- May be unusable and need replacement but collapse to

be prevented

- Bridge to be able to carry full dead plus 30% live load

without impact including P-delta effects

2.5 Eccentrically Braced Frames as Earthquake-Resisting System (ERS)

It is required to have an identifiable earthquake-resisting system (ERS) with sufficient energy

dissipation capability. In addition, the ERS must be able to provide a reliable and continuous load

path for transfer of seismic forces to the foundation. AASHTO (AASHTO, 2013) and CHBDC

S6-14 specify some options for global design strategies including ductile substructure with

essentially elastic superstructure.

EBFs can be used as ERS for bridges in areas where seismic demands are significant. EBFs

are lateral-load-resisting systems whose primary purpose is to dissipate energy in the event of an

earthquake through yielding of a small segment called a link element, usually between the ends of

two braces as shown in Figure 1. The link elements are short segments in the frame designed to

undergo plastic deformation under high loading conditions to form a collapse mechanism allowing

the remainder of the structure to remain elastic. The level of ductility that an EBF can demonstrate

18

before failure is measured by the amount of deformation that the ERS can undergo before the onset

of collapse. The remainder of the frame is designed using capacity design principles, where they

contain a higher capacity than the link elements, and thus will remain elastic. Energy dissipation

is ensured with the help of links in an EBF as marked in Figure 1. These links act as fuses that can

be designed to be repaired or replaced after an earthquake event.

Figure 1 Typical EBF Geometric Configuration (Chevron)

While EBFs have been used in applications to resist wind loading, studies started at the University

of California in the late 1970s provided information regarding the cyclic behaviour of EBFs under

seismic loading (Popov & Engelhardt, 1988). As a result, EBFs in steel frames became more

prevalent in the 1980s, with many testing programs taking place. In the past two decades, further

developments and full-scale testing of links for EBFs have provided information regarding the

19

applicability of EBFs not only in building structures, but also for bridges (Dusicka P, Itani AM,

2002; McDaniel, Uang, & Seible, 2003; Sarraf & Bruneau, 2004).

EBFs with wide flange (WF) or I shaped link beams have been designed, tested, and

implemented for some large projects, such as the San Francisco–Oakland Bay Bridge and the

Richmond–San Rafael Bridge (Dusicka P, Itani AM, 2002; Itani AM., 1997). In these projects,

due to the need for provision of lateral bracing for out-of-plane buckling of the link member,

special arrangements were designed and implemented resulting in a considerable increase in the

cost of projects. To resolve this issue, a new form of tubular link section made of built-up steel

plates was tested and validated; it was found that tubular sections do not need lateral bracing

against lateral torsional buckling (Berman & Bruneau, 2007). This new advancement made it

possible to use EBFs as bridge bents for general bridge types. EBF towers made of built-up tubular

links were implemented as temporary towers in San Francisco Oakland Bay Bridge with world’s

largest self-anchored suspension bridge (SAS) span (EGBC, 2018).

With the latest research, different codes specify built-up sections in addition to I-shaped

rolled sections. These include I-shaped built-up sections, tubular (boxed) sections (Berman &

Bruneau, 2007), and back-to-back channel sections (Mansour, Shen, Christopoulos, & Tremblay,

2008).

With the current trend towards transition from FBD to PBD approach, tests on replaceable

links have also been conducted during recent years (Dusicka & Lewis, 2010; Ghobarah &

Ramadan, 1994; Mansour et al., 2008; Stratan, Dubina, & Dinu, 2003). The developments in the

usage of EBFs have continued through to the present, with recent testing including combinations

of the structural systems of EBFs and buckling-restrained frames (Bruneau, Uang, & Sabelli,

2011).

20

The Canadian steel design standard CSA S16-14 does not give any recommendations regarding

replaceable links made of built-up tubular sections. Past research on EBFs with built-up tubular

beams only addressed the case of beams with constant cross-section over the full frame width and

no research has been conducted on modular (replaceable) links made of built-up tubular sections.

Due to the requirement of CSA S16-14 for continuous link beam for the built-up tubular section,

i.e., same link section as beam outside, it might require a more significant link section for EBFs as

bridge bents, causing a considerable increase in all other capacity-protected members designed for

forces generated by fully yielded and strain-hardened link.

2.5.1 Applicability of CHBDC S6-14 Performance Criteria for EBFs

As previously mentioned, only the link beam is expected to deform inelastically in an EBF, while

all other frame members are intended to remain within elastic limits. Due to this reason, the

performance criteria given in Table 4 by CHBDC S6-14 seem ambiguous or inapplicable for EBFs

as bridge bents. There is a need to specifically define the damage states for nonlinear behaviour of

link elements. For instance, the criteria underlined in Table 4 refer to load-carrying capacity of a

substructure. In fact, all the capacity-protected members resisting gravity loading will probably

remain in the elastic range and would not buckle until the P-delta effect caused huge drifts in the

structure. Thus, there is a need to define more specific engineering demand parameters for the

assessment of bridge performance. This is one of the primary objectives of this study further

described in Chapter 5.

The link beams in an EBF can be designed to either yield in shear, flexure, or combination

of both known as shear links, flexural link and intermediate links, respectively. From previous

literature, it is recognized that the link beam plastic rotation is an important parameter to quantify

21

the damage in an EBF and evaluate the nonlinear behaviour of link elements. The maximum

allowable link beam plastic rotation by CSA S16-14 is 0.08 radians for shear links and 0.02 radians

for flexural links, corresponding to life safety performance objective (CSA Group, 2014b). Linear

interpolation is recommended for intermediate links. Linear interpolation is recommended for

intermediate links. These link rotations are the same regardless of section type (WF, tubular etc.).

22

Chapter 3: Seismic Performance Comparison and Design Approach

Eccentrically braced frames (EBFs) are a hybrid of moment-resisting frames (MRFs) and

concentrically braced frames (CBFs), where one bay of the frame consists of a beam, columns and

a bracing system. What distinguishes an EBF is that at least one end of the brace is eccentrically

connected to the frame (R. Becker & M. Ishle, 1996) as shown in Figure 2. This lateral-force-

resisting system provides a structure with stiffness that is similar to a CBF, while keeping the

desirable ductile behaviour of an MRF. A comparison between EBFs, CBFs, and MRFs is carried

out to demonstrate the motivation behind the selection of EBFs as ERS.

3.1 Comparison of EBFs to CBFs and MRFs

This section provides a summary of seismic performance comparisons of EBFs with CBFs and

MRFS.

Comparison to concentrically braced frames (CBFs):

• The yielding mechanism of CBFs and EBFs are different. CBFs rely on axial yielding of

braces, whereas the links of EBFs are designed to yield in flexure or shear.

• CBFs are limited by the rapid deterioration of braces under cyclic loading causing poor

energy dissipation capacity (Popov & Engelhardt, 1988). In contrast, EBFs are designed to

keep the braces elastic, while only a link is designed to yield. Some CBFs are designed

with a weak link in part of the brace, and the rest of the brace prevents buckling. Thus, this

type of CBFs somewhat perform like an EBF but providing a probably stiffer structure than

an EBF.

• CBFs are stiffer structures that have a reduced story drift when compared to EBFs (Popov

& Engelhardt, 1988).

23

• Yielding elements in CBFs are not part of the gravity system but are required to provide

lateral stability to the structure. In such cases, the post-earthquake damage is typically

easier to repair than links that are integral in the gravity system in EBFs (such as cap beams

in bridge bents). Some EBF geometries isolate the link from the gravity system.

Comparison to moment-resisting frames (MRFs):

• MRFs are much more flexible than EBFs, with sizes often increased dramatically to

meet drift limits (as opposed to strength limits).

• MRFs as bridge bents are designed on the principle of the weak-column, strong-beam

approach in accordance with the bridge code. This is due to the reason that the beams are

highly loaded in bending to support the superstructure, and the column moment capacities

are generally lower as a result. In contrast, the building code takes the opposite approach;

it relies on the strong-column, weak-beam concept to spread plasticity through various

stories in a building.

3.2 EBF Geometric Configurations

The use of EBFs is popular due to their performance under seismic loading conditions. Although

there is literature available on the topic of designing EBFs, the choice of frame configuration to

best suit the application at hand may be unclear to the designer. Of course, no single configuration

suits all applications. Instead, the choice of system depends on many factors, including link and

brace configuration requirements.

Some general characteristics of EBFs are noted below:

• Lateral stiffness quickly degrades as link length (eccentricity) is increased (Popov &

Engelhardt, 1988).

24

• Small links limit the ability of the link to deform (Popov & Engelhardt, 1988).

• Ductility, when a link yields in shear, is higher than in flexure (Özhendekci & Özhendekci,

2008), but there is very little interaction between moment and shear within a link (Bruneau

et al., 2011)

Different possible EBF geometries are shown in Figure 2. Links in the frames are shown with

hatching.

Figure 2 Possible EBF geometries (with selected EBF configuration marked)(Bruneau et al.,

2011)

3.2.1 Selected EBF Geometry

The selected EBF configuration is a chevron design that has the yielding link located in the centre

of the beam, between two braces, as marked in Figure 2. For the considered case study bent, a

single-storey EBF is studied with all pinned-base restraints. All other connections are fixed. The

25

benefits of this design include symmetry of the frame to reduce design detailing (Bruneau et al.,

2011).

The inelastic behaviour of a link in an EBF is mainly dependent on its length. The links

designed to yield in shear are known as short links, whereas, flexure-dominated links are known

as long links, and they are designed to yield in flexure. Shear-dominated short links are preferred

because of their uniform distribution of plastic shear strains along the web of the link. This makes

the link capable of sustaining significant inelastic rotations without causing extreme local strains

(Engelhardt & Popov, 1989).

To obtain significant inelastic rotations in flexure-dominated links, it is required to have

high bending moments at both ends of the link member. These bending moments will result in

high local plastic deformations and will prevent desired repeated inelastic behaviour of the link

member, leading to lesser inelastic rotation of the link (Esmaili, 2015). The maximum plastic

rotation reported from experimental tests is around 0.1 radians for short links (Whittaker, Uang, &

Bertero, 1987), as opposed to 0.02 radians for long links (Engelhardt & Popov, 1989).

3.3 Case Study

The two-span Sombrio Bridge located on Vancouver Island, British Columbia, with a total span

of 122 m is selected as a case study, and an eccentrically braced frame (EBF) has been chosen as

the substructure to replace a two-column concrete bent as shown in Figure 3 and Figure 4. This

existing bridge consists of two unequal spans of 40 m and 82 m, with one concrete bent comprising

two circular columns and a bent cap. For this case study, the bridge is considered as a regular

Major Route bridge. Based on existing drawings, site class C has been considered. It is confirmed

that the contribution of bent to longitudinal restraint is minor, and abutments are mainly

26

contributing for longitudinal restraint. The substructure EBF bent is therefore designed for seismic

loads in the transverse direction only.

Figure 3 Sombrio Bridge elevation (WSP/MMM Group, 2013)

Figure 4 Sombrio Bridge superstructure typical section (WSP/MMM Group, 2013)

27

Figure 5 EBF bent (used as replacement for the 2-column reinforced concrete bent)

The energy dissipation in EBFs is mainly based on the link beam that is designed either to fail in

shear, flexure, or a combination of both, between the eccentric braces. All other members including

beams outside the link, braces, and columns, are designed as capacity-protected elements based on

forces imposed by the fully-yielded and strain-hardened link. Therefore, the seismic performance

of a bridge with EBF as a substructure is directly controlled by the link member.

Usually, the link beam in EBFs is an I-shaped or wide-flange (WF) cross-section that needs

lateral bracing to avoid lateral torsional buckling. Based on previous work (Berman & Bruneau,

2007), CSA S16-14 introduces EBF with built-up tubular sections that do not require out-of-plane

bracing for lateral torsional buckling. This requirement makes it possible to use the EBFs as bridge

piers without any issues of lateral bracing. To obtain full advantage of larger allowable plastic

28

rotations by maintaining shear yielding behavior with larger link lengths, built-up hybrid tubular

link members made of steel plates of specific thicknesses are required rather than HSS sections.

The steel material selected for ductile EBFs is CSA G40.21, Grade 350W, with specified

minimum yield stress Fy of 350MPa as permitted by Clause 4.8.2.1 of CHBDC S6-14 for steel

structures (CSA Group, 2014a). The probable yield stress has to be equal to RyFy. Here, Ry is the

ratio of expected yield stress to minimum yield stress Fy. For built-up tubular sections it is taken

as 1.1 according to Clause 27 in CSA S16-14 and Table A3.1 in AISC 341 (AISC, 2016; CSA

Group, 2014d). Usually, tubular built-up sections are made out of 300W or 350W plates which

have predictable over strength (CSA Group, 2014d). To account for the appropriate overstrength

of the steel section, a maximum yield strength should be specified on the drawings, which needs

to be determined from coupon tests for the material being ordered.

For this study, the two exterior girders are placed directly above the columns, while the

two interior girders are placed at the ends of the shear link, i.e., the connection of braces with the

cap beam as shown in Figure 5. This geometry helps transfer dead load (SLS for each girder =

3470 kN) to foundations using columns and braces without putting high demands on cap-beams

from gravity loading as shown in Figure 6 and Figure 7.

Superstructure bridge geometry (location of girders) affects the member sizing. High

gravity loading on a beam outside the link will require a more significant section. Due to the

requirement of CSA S16-14 for continuous link beam for the built-up tubular section, i.e., same

link section as beam outside, it might require a more significant link section, causing a considerable

increase in all other capacity-protected members designed for forces generated by fully yielded

and strain-hardened link. Therefore, it is decided to consider one design case with replaceable link

beam for this study.

29

The center to center (c/c) distance between girders is 3 m. This dimension is a constraint

due to the geometry of the existing structure. Due to this restriction, shear link length is taken as 3

m for the case where the link section is the same as the beam outside (continuous link), and 2.4 m

for a case where the link is made smaller than the outside beam portion (replaceable link).

Figure 6 Selected EBF geometry with girders orientation for design 1, 2 and 3

30

Figure 7 Selected EBF geometry with girders orientation for design 4

There is a lack of literature for studies related to the design of EBFs for bridge piers according to

Canadian design provisions (CSA Group, 2014a, 2014c; NRCC, 2015a). According to Clause

4.4.5.3.1 in CHBDC S6-14, FBD is required for regular Major Route bridges with seismic

performance category of 3 with a condition that, PBD might be required by the Regulatory

Authority for this case. Another clause states that PBD may be used for all cases. Therefore, the

bent will be designed using both design approaches, i.e. PBD and FBD for comparison purpose.

Table 5 shows a summary of all considered design cases along with the corresponding design

approach and hazard level. A step-by-step procedure for the design of an EBF as bridge pier

according to CHBDC S6-14 (CSA Group, 2014a) is provided for both FBD and PBD approach.

31

Table 5 Considered design cases as per CHBDC S6-14

Design Case Design Approach Hazard Level

D1 FBD 2475-year return period event

D2 FBD 2475-year return period event

D3 PBD 475-, 975-, and 2475-year return period event

D4 PBD 475-, 975-, and 2475-year return period event

The Natural Resources Canada (NRC) provides 5% damped spectral acceleration values

for different hazard levels and firm ground conditions (site class C) using the online seismic hazard

calculator tool (NRC, 2016). For the selected case study site location with geographical

coordinates of 48.4952 ºN and 124.2584 ºW located on Highway 14 between Jordan River and

Port Renfrew, BC, uniform hazard spectrum (UHS) values crossponding to 475-, 975-, and 2475-

year return period event are provided in Figure 8.

Figure 8 5% damped Sombrio UHS for 475,975 and 2475-year return period

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0 2 4 6 8 10

Spec

tral

Acc

eler

atio

n S

a, g

Period T, sec

1 in 2475 years 1 in 975 years 1 in 475 years

32

3.4 Force-Based Design Approach

According to CHBDC S6-14, ductile seismic performance can be obtained by providing ductile

substructure elements such as ductile MRFs as bridge bents, ductile single-column structures,

ductile CBFs, ductile EBFs, and buckling restrained braced frames (BRBs). For performance

category 2 and 3, there is a need to conduct inelastic static pushover analysis (ISPA) to demonstrate

the performance of multi-tiered or multi-storey braced frame bents.

Clause 4.8.4.4.4 of CHBDC S6-14 refers to ductile eccentrically braced frames. It, however,

further refers to Clause 27.7 of the Canadian steel design standard for buildings CSA S16-14 for

member proportioning, mentioning an R=4 for the design (previously, R=5) based on AASHTO

(AASHTO, 2013). CHBDC S6-14 mentions capacity design as a robust design principle and

requires capacity design to ensure the desired yielding mechanism in bridge piers without having

unwanted failure in other structural members.

The following considerations need to be taken into account while designing using the FBD

approach as per CHBDC S6-14:

➢ The design of the substructure needs to be carried out corresponding to the 2475-year return

period event. This is not allowed for Lifeline bridges as PBD is compulsory for Lifeline

bridges according to CHBDC S6-14.

➢ The Importance factor incorporated in the response spectrum analysis (IE = 1.5 for Major

Route bridges, 1.0 for Other bridges) for determining forces; IE = 1.0 for determining

displacements.

➢ The forces (base shear) from the spectrum are reduced by R-factors provided in Table 4.17

of CHBDC S6-14 (CSA Group, 2014a).

33

➢ Code-based detailing is mandatory to ensure ductility and energy absorption including the

capacity design of the seismic force resisting system.

3.4.1 Force-Based Design Methodology

The following methodology is used to carry out FBD for two cases:

1. Select a seismic performance category according to Table 4.10 in CHBDC

S6-14.

2. Select EBFs as Earthquake-Resisting System (ERS).

3. Determine the 2475-year return-period uniform hazard spectrum (UHS).

4. Estimate the preliminary period of vibration of the structure and

corresponding spectral acceleration from UHS in previous step.

5. Calculate minimum lateral earthquake force for elastic static analysis using

Clause 4.4.7.4 in CHBDC S6-14.

6. Modify the lateral earthquake force calculated in step 5 by dividing by

response modification factor R given in Table 4.17 in CHBDC S6-14.

7. Select the link length, link type (shear, flexure, or intermediate), and size the

link member according to the forces calculated in previous step.

8. Size beams outside the link, columns, and braces for forces applied by fully

yielded and strain-hardened link using capacity design approach.

9. Model the member sizes and carry out modal analysis and determine the

period of vibration and forces.

34

As the selected case study bridge is located on Vancouver Island (Sa(0.2) = 1.413 g), according to

Table 4.10 in CHBDC S6-14, a seismic performance category of 3 has been selected. According

to Clause 4.4.7.4 of CHBDC S6-14 (CSA Group, 2014a), the minimum lateral elastic earthquake

force, V, known as base shear force is calculated as follows:

𝑉 = 𝑆(𝑇)𝐼𝐸𝑊 3-1

S(T) = design spectral response acceleration from 2475-year return period uniform hazard

spectrum

T = fundamental period of bent in the transverse direction

IE = importance factor for FBD, taken as 1.5 for Major Route Bridges

W = dead load from superstructure mainly

According to clause 4.4.10.4.2 of CHBDC S6-14, for the design of structures in the seismic

performance category 3, the seismic design forces for ductile substructure element(s) (link member

in an EBF) shall be calculated by dividing the elastic base shear with appropriate response

modification R, i.e. R=4 for EBFs. These design forces are called modified design forces for FBD

approach. All other capacity-protected elements such as superstructure and substructure (beams

outside the link, braces, and columns in an EBF) are designed to have factored resistance equal to

10. Re-check all member sizes to ensure appropriate force capacities. Calculate

the link rotation using elastic analysis as per CSA S16-14 Commentary Clause

27.7.5 and confirm that it is within allowable limits provided by code CSA S16-

14.

11. Done.

35

or greater than maximum forces that can be applied by a ductile substructure element achieving

their probable resistance (CSA Group, 2014a).

As a starting point, its required to know the period of the vibration of the bent to determine

minimum lateral earthquake force on the EBF. CHBDC S6-14 does not provide any guidance to

estimate the fundamental period for EBFs as bridge bents. Therefore, it is required to assume the

initial period of the bent and then calculate preliminary design forces using spectral acceleration

from the 2475-years return period uniform hazard spectrum (UHS) as shown in Figure 8Figure 7.

From these forces, preliminary member sizes are determined. However, an iteration procedure is

carried to determine actual seismic demands on the substructure after calculating the fundamental

period from modal analysis using initial member sizes. A back-and-forth iteration procedure with

the revised design is required until all the variables are satisfied. Example calculations to determine

the modified lateral forces have been provided in Appendix A.1 for Design 1.

3.5 Performance-Based Design Approach

In CHBDC S6-14, basic seismic design rules are mainly based on FBD principles with the

requirement of additional analyses for PBD approach such as inelastic static pushover analysis

(ISPA) and nonlinear time-history analysis (NLTHA).

For PBD, one needs to determine the performance levels and damage states by carrying

out the analyses recommended by CHBDC S6-14. The approach comprises determining

displacement demands from response spectrum analysis corresponding to the multiple hazard

levels and comparing these with the displacement capacities of the structure obtained from

pushover analysis. In other words, during seismic design, it is verified that displacements

36

corresponding to a given damage level are higher than displacements that a structure will

experience from design level earthquake (Gerin & Khan, 2017).

The displacement demands can be obtained by using site-specific UHS by conducting a

response spectrum analysis referred in the code CHBDC S6-14 as the elastic dynamic analysis

(EDA).

3.5.1 Performance-Based Design Methodology

The following methodology is used to carry out a PBD for two cases:

1. Conduct preliminary design of EBF bent.

2. Demands: Perform RSA considering multiple earthquake levels (475-, 975-

and 2475-year return period) to obtain displacement demands.

3. Capacity: Carry out pushover analysis for structural capacity and determine

plastic rotations of link member corresponding to displacement demands from

RSA.

4. Check if rotational demand < capacity and the structure meet the performance

criteria.

5. If yes, check other performance criteria. Otherwise, revise member sizes.

6. If required, carry out NLTHA by using scaled or spectrally matched ground

motions with target UHS to confirm results obtained from pushover and RSA.

Yes No

Revise

b

37

3.6 EBF Bent Design

Once the seismic design forces have been determined by both FBD and PBD approaches, the next

step is to size the link member in an EBF which will behave as a structural fuse. The rest of the

members are designed afterwards to stay elastic for the forces applied by the link in its entirely

yielded and strain-hardened state.

To meet required strength, stiffness, and ductility demands, specific factors such as link

length, link member size, and bracing arrangement play a significant role. Of all of these, length

of the link has the greatest impact on the elastic lateral stiffness of an EBF. Figure 9 explains the

effect of e/L ratio on the elastic lateral stiffness of an EBF. An EBF having small link length would

have greater lateral stiffness. For e = L, the frame will be an MRF with less elastic stiffness,

whereas with e = 0, the frame will act like a CBF with maximum stiffness. An EBF lies between

these two limits with the advantage of having a link member that will act as a structural fuse (Popov

& Engelhardt, 1988).

Another important consideration is the bracing arrangement in an EBF, as the angle

between the beam outside the link affects the axial forces in the beam adjacent to the link. A very

small angle between the brace and beam outside the link, i.e. less than 40 degrees results in large

axial forces in the beam and causes strength and stability issues in the beam (Popov & Engelhardt,

1988). The angle between the brace and beam outside the link is around 66 degrees for selected

cases in this study.

38

Figure 9 Effect of e/L over relative frame stiffness (Popov & Engelhardt, 1988)

According to CSA S16-14, three different types of behaviour are possible for a link member in an

EBF, i.e., shear yielding (shear links), flexural yielding (flexural links), and some combination of

shear and flexural yielding (intermediate links).

The link length plays a crucial role for all member sizes as well as the inelastic response of

an EBF. Shear links are preferred due to their high energy dissipation capacity as well as ductility.

Corresponding to each type of yielding , CSA S16-14 gives inelastic rotation limits (𝛾𝑝) as shown

in Table 6 with a deformed shape of an EBF in Figure 10. These are ultimate rotation limits

corresponding to Life Safety (LS) performance objective as per CSA S16-14. The CHBDC S6-14

requirement for the Major Route bridges is Repairable Damage and Extensive Damage, not Life

Safety; however, the bridge code does not provide any plastic rotation limits for EBFs.

39

Table 6 EBF link types, link length condition and allowable maximum inelastic link rotation

Link Type Link Length Condition Maximum Inelastic Link Rotation by

CSA S16-14 (radians)

Shear Links 𝑒 ≤

1.6 𝑀𝑝

𝑉𝑝

0.08

Flexural Links 𝑒 ≥

2.6 𝑀𝑝

𝑉𝑝

0.02

Intermediate Links 1.6 𝑀𝑝

𝑉𝑝< 𝑒 <

2.6 𝑀𝑝

𝑉𝑝

Linear interpolation

3.6.1 Link Sizing

The deformed shape of an inverted chevron type EBF with link length e is shown in Figure 10.

The link is designed to resist the lateral forces (Vb) inelastically whereas all other members are

designed to stay elastic. All connections in EBF bent are considered as fixed except joint base

restraints; these are usually designed as pinned joints. The reason for the rigid connection between

the brace and the beam outside the link is to attract more moment in the brace than the beam outside

the link. The forces in the link member are calculated using static equilibrium according to current

industry practice and available literature (Bruneau et al., 2011). The free body diagram of half-bay

EBF bent in Figure 11 shows the transfer of design base shear Vb to the link beam. Therefore,

shear force in the link, VL, can be calculated using equation 3-2.

40

Figure 10 Deformed shape of an EBF bent under lateral load Vb using the rigid-plastic

mechanism. Adapted from: (Berman & Bruneau, 2007)

Figure 11 Free-body diagram of half EBF bay with lateral load Vb

41

𝑉𝐿 = 𝑉𝑏𝐻

𝐿

3-2

Here, H is the storey height, and L is bay width. It is clear from Equation 3-2 that strength of the

link is mainly dependent on the geometry of the EBF.

It was decided to use a built-up tubular member section as shown in Figure 12, where d, b,

tf, and tw correspond to the overall depth of the section, the overall width of the section, the

thickness of flange and thickness of web respectively. Fyw and Fyf represent the yield strength of

webs and flanges respectively.

Figure 12 Typical built-up tubular cross-section with external stiffeners (Berman & Bruneau,

2007)

42

According to CSA S16-14 Clause 27.7, the plastic shear capacity of a link member with a tubular

cross-section is:

Here, 𝑡𝑤 is web thickness, d is the overall depth, and Fy is the yield strength of the cross-section.

As per Clause 27.7.3.1 of CSA S16-14, the factored shear resistance of the link should be taken as

the minimum of ϕV'p, and 2ϕM'p/e. V'p and M'p are plastic shear capacity and moment capacity

of link member corresponding to the axial force in the link member.

𝑉′𝑝 = 𝑉𝑝√1 − (𝑃𝑓

𝐴 𝐹𝑦)

2

3-4

𝑀𝑝′ = 1.18 𝑀𝑝√1 −

𝑃𝑓

𝐴𝐹𝑦≤ 𝑀𝑝

3-5

The plastic moment strength of the section is calculated using the equation,

𝑀𝑝 = 𝑍𝐹𝑦 3-6

In the above equations, A is the gross area of the link beam, and Z is plastic section modulus of a

tubular section. According to CSA S16-14 Commentary, the axial force in the link is usually very

low and ignored in most of the cases; therefore, 𝑉𝑝= 𝑉𝑝′and 𝑀𝑝= 𝑀𝑝

′ (CSA Group, 2014b). The

interaction between shear force and bending moment in link member has been found minimal in

pervious literature and therefore not taken into account (CSA Group, 2014b).

𝑉𝑃 = 0.55(2 𝑡𝑤)𝑑𝐹𝑦 3-3

43

Nominal shear resistance (Vn) of the link is the lesser of Vp and 2Mp/e. Probable link resistance of

a fully yielded and strain-hardened link with tubular cross-section is calculated by:

𝑉𝑙𝑖𝑛𝑘 = 1.45 𝑅𝑦 𝑉𝑛 3-7

Here, Ry is the ratio of expected yield stress to minimum specified yield stress Fy. For built-up

tubular sections it is taken as 1.1 according to Clause 27 in CSA S16-14 and Table A3.1 in AISC

341(AISC, 2016; CSA Group, 2014d). The effect of strain hardening is considered using a factor

of 1.45 in Equation 3-7 that was determined by tests on tubular links (Berman & Bruneau, 2007).

They also demonstrated that the strain-hardening effect in built-up tubular links is usually 11%

higher than wide-flange (WF) links.

3.6.1.1 Link Length Calculation

The following conditions are checked to determine the link length according to CSA S16-14 (CSA

Group, 2014d):

• For shear links 𝑒 ≤1.6 𝑀𝑝

𝑉𝑝

• For yielding in shear, ∅𝑉𝑝′ <

2∅𝑀𝑝′

𝑒 → 𝑒 <

2𝑀𝑝′

𝑉𝑝′ → 𝑒 <

2 𝑍𝑥𝐹𝑦

0.55 𝐴𝑤 𝐹𝑦

→ 𝑒 < 3.6 𝑍𝑥

𝐴𝑤

• The length of the link (e) should be larger than the depth of the section selected, i.e., e ≥ d

Here, Aw is an area of web and calculated as (d-2tf) (2tw) for tubular cross-section links. By

considering these limits to select the link length and taking into account the superstructure

geometry, two link lengths were selected, i.e. 3 m and 2.4 m. The link length of 2.4 m was selected

44

to have a smaller link section as compared to the beam outside the link to maintain the shear

yielding mechanism.

After determining the seismic loads using either the FBD or PBD approach, the procedure

for preliminary member sizing is same for both approaches. Once the link length has been decided,

the link section is designed according to the calculated forces. A spreadsheet with detailed

calculations for Design 1 bent has been provided in Appendix A.2. Table 7 shows the link member

forces for all four links. Vp is the plastic shear force capacity of the link calculated using Equation

3-3 and Mp is the plastic moment capacity of the link member obtained using Equation 3-6. Vlink

is the probable link resistance of a fully yielded and strain-hardened link. All other capacity-

protected members are designed corresponding to Vlink.

Table 7 Link member forces for four design cases

Design Design Approach e

(m)

Vp

(kN)

Mp

(kN-m) Vlink (kN)

1 FBD 3 5775 11966 9211

2 FBD 3 3927 7481 6264

3 PBD 3 7219 14212 11514

4 PBD 2.4 3465 5414 5527

It is assumed that full-depth web stiffeners have been provided as per the guidance of CSA S16-

14 Clause 27.

45

3.6.2 Capacity Design Procedure

Once we have the link member size, the next step is the capacity design of all other framing

members including the beam outside the link, braces, and columns. In the capacity design

procedure, all other framing members are designed for the forces applied by the fully yielded and

strain-hardened link member. The purpose of this is to ensure that only the link member will go

into the inelastic range, keeping all other members in the elastic state.

To carry out the capacity design, the probable shear resistance of link member (Vlink)

calculated from Equation 3-7 is applied to the half-bay model in SAP2000 at the centre of the link

member (e/2) where the moment is taken as zero. Gravity loads from the superstructure of the

bridge are applied on this capacity design model. The base supports are taken as pinned and the

edge joint at the top of the column is restrained against the lateral motion to provide stability to

run the analysis and to attain the balanced reactions in half-bay model.

The member forces in all framing members are determined corresponding to the forces

applied by the fully yielded and strain-hardened link member. If the demand/capacity ratios for all

framing members are less than 1, it means that they will remain elastic and only the link member

will yield.

46

Figure 13 Half-bay model for capacity design using SAP2000

Sample calculations to determine capacities of capacity-protected members, the beam-column

design of the link, the design of the outside beam portion, and the beam-column design of the brace

and the column has been provided in Appendix A.3 and A.4 for Design 1.

3.6.3 Link Rotation Check

After the capacity design is done, the next step is to calculate the inelastic rotation of the link

member. The inelastic part of a rotation of link member with the beam outside the link is known

as inelastic link rotation, and it can be calculated using the rigid-plastic mechanism of EBF as

shown in Figure 10.

47

A response spectrum analysis (RSA) at multiple seismic hazards is used to determine the link

rotations. The plastic rotation of link member is simply the link vertical displacement (Δ) divided

by the length of the link (e) as follows:

𝛾𝑝 =∆

𝑒

3-8

3.7 Final Designed Bents

A total of four different designs of bents were carried out for comparison purposes. Table 8 shows

the design details and differences between designs. All member sizes for designed bents are given

in Table 9.

Table 8 Different cases for Sombrio Bridge substructure

Designs Design Characteristic Comments

Design 1

FBD

• Link initial sizing using

equivalent static force procedure

• R = 4

• IE =1.5 For a Major Route Bridge

by CHBDC S6-14

• Link section same as beam

outside the link

• Does not fulfil the level-1

performance-based criteria

that there should not be any

yielding at immediate service

(475-year return period).

Design 2

FBD

• Same criteria as design 1, except

IE = 1

• Does not fulfil the level-1

performance-based criteria

that there should not be any

yielding at immediate service

(475-year return period).

48

Designs Design Characteristic Comments

Design 3

PBD

• Fulfils the level-1 performance-

based criteria of no yielding at

475-year return period given in

Chapter 2

• Link beam is continuous—the

same section is used for the beam

outside the link as required by

CSA S16-14

• It is the revision of the first

design and fulfils the level-1

performance-based criteria of

no yielding at immediate

service (475-year return

period).

Design 4

PBD

• Shear link is replaceable; i.e., link

member cross-section is smaller

than the beam outside

• Smaller sections for capacity-

protected members as

compared to other designs

• Limited yielding at 475-year

return period

Table 9 Designed EBF Bents member size (all dimensions in mm)

Design 1 Design 2 Design 3 Design 4

Link Length e (mm) 3000 3000 3000 2400

Link Type shear shear shear shear

Link Type Non-replaceable Non-replaceable Non-replaceable Replaceable

Braces Intersection e' (mm) 3000 3000 3000 3000

Link Size (Built-up Tubular) (d x w x tf x tw) 750x800x54x20 600x700x50x17 750x900x57x25 600x600x40x15

Beam Size (Built-up Tubular) (d x w x tf x tw) 750x800x54x20 600x700x50x17 750x900x57x25 750x700x55x25

Brace Size (Built-up Tubular) (d x w x tf x tw) 650x650x40x40 700x700x40x40 800x800x55x55 600x600x25x25

Column Size (Built-up Tubular) (d x w x tf x tw) 600x250x30x30 600x250x30x30 700x350x30x30 500x400x25x25

49

Chapter 4: Numerical Modelling

This chapter describes the analytical models developed for the four design cases from Chapter 3

using both FBD and PBD approaches. To precisely capture the monotonic and dynamic behaviour

of an EBF under applied loading, finite element models have been developed. To estimate the

nonlinear behaviour of the shear link member, different modeling approaches are described in the

following sections and calibration of each method is done with actual experimental test results

available from literature. Then, by considering different factors, one nonlinear modeling approach

is selected and used for all nonlinear analyses in subsequent chapters. In the end, a summary of

characteristics of different models that have been used for analyses is provided.

4.1 Numerical Model Description

SAP2000 version 20 is used as the platform for numerical modelling and analyses of EBFs.

SAP2000 is a powerful tool for capturing nonlinear structural behaviour providing various options

for modelling nonlinear elements and carrying out different types of nonlinear analyses such as

nonlinear direct integration time history analysis (CSI, 2017). Three different approaches are

considered in the link member, such as, plastic hinges, multi-linear plastic link/support elements,

and fiber hinges, to capture the nonlinear behaviour of the shear link. Each method has its benefits

and limitations. For verification of these numerical model approximations, calibration with real

existing test data is carried out.

Girders have been arranged such that gravity loads from superstructure are applied on

beams outside the link member. Only material nonlinearity is considered, and geometric

50

nonlinearity such as P-Delta effects are ignored for this study due to high bent stiffness. Pinned-

base restraints are provided as they simplify the yielding mechanism.

4.2 Model Calibration

To validate the numerical modelling approach for prediction of nonlinear link behaviour, a

calibration procedure is carried out. Berman and Bruneau (2007) reported the results of a proof-

of-concept test setup consisting of full-scale single-panel EBF bent with a built-up hybrid tubular

cross-section (Berman & Bruneau, 2007). The term hybrid means that a link cross-section has

different web and flange yield strengths. The full-scale single panel EBF bent is modelled in SAP

2000 with the same geometry, and member sizes and same loading protocol is applied on the bent

as shown in Figure 14. Similar to bents designed for Sombrio Bridge, Berman and Bruneau (2007)

considered all connections as moment resisting except brace-to-column connection that is

considered pinned for Sombrio bents. These test results are the only data available for a full-storey

test of an EBF using tubular links, as most of the testing has been done on W-shape links in the

past.

For this test setup, the steel specified for the link was A572 Grade 50, with a nominal yield

strength of 345 MPa (Berman & Bruneau, 2007). From coupon test results of both web and flange

plates, a higher yield strength of 448 MPa was obtained for web plate (with 8 mm of thickness)

whereas a yield strength of 393 MPa was reported for flange material (with 16 mm of thickness)

that is closer to the specified yield strength of 345 MPa. According to the proposed equation by

Berman and Bruneau, for calculation of plastic shear strength of a link member (Vp), only

contribution from the web is considered. AISC Seismic provisions (AISC, 2016) also specify a

similar equation for calculation of plastic shear strength of tubular links. SAP2000 does not

51

provide the option to consider different yield strengths for webs and flanges within a section while

using plastic hinges and multi-linear plastic link/support elements. Therefore, while assigning the

backbone curve for calibration model, the actual yield strength of 448 MPa is considered for plastic

hinges and multi-linear plastic link/support elements. For the fiber hinges, the user has the option

to assign different material properties for webs and flanges for a particular section. The whole test

setup is modelled in SAP2000, and the quasi-static loading protocol based on ATC-24 (ATC,

1992) is applied as per actual test procedure. The three different approaches used for nonlinear

modelling of the shear link are explained in the following sections. Capacity design is a core

requirement for an EBF; therefore, all other members are modelled as elastic elements as they are

not expected to experience plastic deformation.

The loading history values are provided in terms of cycle number and storey drift (mm) in

a time-history function in SAP2000. Berman and Bruneau (2007) provided the values of

percentage drift corresponding to each loading cycle. From percentage drift, storey drift is

calculated as:

Storey drift (mm) = [% drift x storey height (mm)]/100

52

Figure 14 Calibration model and loading history in SAP2000

Figure 15 Proof-of-concept test setup of EBF with Built-up Tubular Shear Links (Berman &

Bruneau, 2007)

53

4.2.1 Nonlinear Modeling of Link Member Using Plastic Hinges

The first approach considered is to model plastic hinges in the link member to capture nonlinear

behaviour by using the nonlinear modelling parameters as recommended by ASCE41-13 (ASCE,

2014). This approach has been well calibrated previously, and ASCE 41-13 provides

recommendations for nonlinear modelling of plastic hinges as well as acceptance criteria for

different damage states (immediate occupancy (IO), life safety (LS) and collapse prevention (CP))

in terms of link plastic rotation angle (ASCE, 2014). These acceptance criteria limits however are

mainly for EBFs in buildings. A deformation-controlled shear (V2) hinge with force-displacement

type is assigned in the middle of the link member where shear is expected to be maximum. All

other bent elements being designed as capacity-protected elements are modelled using elastic

elements as they are expected to behave linearly.

The force-deformation backbone curve is assigned based on nonlinear modelling

parameters given in ASCE41-13 for shear links in EBFs, with the backbone curve shown in Figure

16 and modelling parameters provided in Table 10. A default kinematics hysteresis model is

considered for this case that does not require any additional parameters input in SAP2000 (CSI,

2017).

54

Figure 16 Backbone curve (Force - Deformation) Adapted from: ASCE41-13

Table 10 Nonlinear modeling parameters for EBF shear link beam (ASCE, 2014)

In Table 10, plastic rotation values are 0.15 and 0.17 radians corresponding to the parameter “a”

and “b” in the force-deformation curve respectively as shown in Figure 16. As it is required to

input values for deformation in the form of displacement in SAP2000, link plastic rotation is

multiplied by link length to get displacement at a particular link rotation value using:

∆= 𝛾𝑝. 𝑒 4-1

The next step is to figure out the shear forces corresponding to points A, B, C, and D in the force-

deformation curve shown in Figure 16. SAP2000 uses a similar backbone curve for force-

Shea

r Fo

rce

55

deformation input and requires shear force as well as displacements for parameters A, B, C, and

D.

Table 11 Force-Displacement backbone curve parameters for the shear hinge in SAP2000

Point of Backbone

Curve Force Displacement

A 0 0

B Yield Force

(Vp)

Negligible

(elastic displacement)

C Ultimate Capacity (Vlink) a

D c = 0.8 x Yield Force

(Residual strength)

b

E 0 d

Table 11 provides information regarding the understanding of each parameter for the backbone

curve in SAP2000. For example, point A is an origin; point B corresponds to yield shear force

where deformation is only elastic and plastic deformation is negligible and not taken into account.

Point C represents the shear capacity of fully yielded and strain-hardened link and its deformation

can be determined using plastic rotation value given by parameter “a” in Table 10. Point D shows

the residual strength of link member that is equal to 80 % of yield capacity according to parameter

“c” as given in Table 11 with displacement value determined through plastic rotation parameter

“b” in Figure 16. Lastly, point E represents the collapse of the structure; it is recommended to

specify a considerable deformation value for point E as beyond this point, the hinge will drop the

56

load. (CSI, 2017). The displacement corresponding to point E is represented by “d” with a value

greater than parameter “b” as given in Table 11 and shown on backbone curve in Figure 16.

By using shear link cross-sectional dimensions from the experiment, parameters are

calculated for backbone using equations provided in CSA S16-14. The value of plastic shear force

calculated using Equation 3-3 of Chapter 3 is 460 kN whereas, ultimate shear capacity, i.e.,

probable link shear resistance from Equation 3-7 of Chapter 3 is 736 kN. Berman and Bruneau

(2007) reported a yield shear of 490 kN and maximum link shear of 742 kN from actual test results.

As previously mentioned, the measured yield strength for web plates of the actual test was high as

compared to expected overstrength. This increase might have caused a slightly higher yield

strength relative to the calculated value. The difference between the calculated plastic yield

strength and actual value from the test is only 6%, however, and is in an acceptable range. The

calculated probable or ultimate shear capacity value is very close to the actual value.

One drawback of using the plastic hinge method is that while defining the hysteresis model

in SAP2000, the user does not have the option to change parameters to exactly match the hysteresis

behaviour of actual test results. Regardless of that, this method gives a good match to the actual

backbone curve from the test as shown in Figure 17 and Figure 18. Of all three options to capture

nonlinear behavior of link, the shear hinge option requires straightforward input and provides

quick results.

57

Figure 17 Numerical model calibration results using shear plastic hinge

Figure 18 Experimental hysteresis calibration using shear plastic hinge

-1000

-800

-600

-400

-200

0

200

400

600

800

1000

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

Lin

k S

hea

r (K

N)

Link Rotation (rad)

Berman Hysteresis Backbone (Actual data) Shear Hinge Model SAP2000

-1000

-800

-600

-400

-200

0

200

400

600

800

1000

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

Lin

k S

hea

r (K

N)

Link Rotation (rad)Berman Hysteresis Berman Hysteresis Backbone (Actual data)Shear Hinge Model SAP2000

58

4.2.2 Nonlinear Modeling of Link Member Using Multi-Linear Plastic Link Element

A zero-length multi-linear (ML) plastic link element with specific mass and weight is assigned in

the middle of the link member. This element is assigned with nonlinear properties in shear (U2

direction) with all other degree of freedoms considered as fixed. A multi-linear force-deformation

curve is assigned as per actual experimental results by Berman and Bruneau (2007). A kinematic

hysteresis model is considered for this case. As previously mentioned for plastic hinges, SAP2000

does not provide the option to change parameters for this hysteresis model in case of ML-Plastic

link element as well. Therefore, even though the model follows the backbone curve, it does not

exactly match the hysteresis behaviour of the actual test as shown in Figure 19.

Figure 19 Experimental hysteresis calibration using a multi-linear plastic link element

59

4.2.3 Nonlinear Modeling of Link Member Using Fiber Hinges

SAP2000 gives the option to model fiber hinges using any cross-sectional shape or type with

Section Designer tool with multiple points or fibers over the entire cross-section, each having

corresponding material properties. The fiber hinges are mainly utilized to capture the coupled axial

force as well as bi-axial bending behaviour for a frame member of a specific length (CSI, 2017).

After modelling the section in Section Designer tool in SAP2000, material nonlinear stress-strain

curves can be provided from actual coupon test results. SAP2000 provides biaxial moment-rotation

relationship by adding up the behaviour of all fibers defined at a cross-section and multiplying it

by the fiber hinge length (CSI, 2017).

A user-defined fiber hinge is assigned with hinge length equal to the length of the shear

link member. Then, the material is assigned to each fiber defined in section designer with actual

material nonlinear stress-strain curves. This fiber hinge is assigned at each end of link member in

an EBF. As a result, the moment-rotation curve from defined stress-stains curves of fibers is

obtained. Link end moments are converted into link shear using shear-moment equilibrium

equation: VL = 2 ML/e.

Using fiber hinges a good match with actual test hysteresis provided by Berman and

Bruneau (2007) is obtained from the calibration model. The shear behaviour of elements is not

considered in fibers, rather, it is directly calculated by using linear section modulus in SAP2000.

It can be seen from the hysteresis shown in Figure 20 that the numerical model with fiber hinges

gives the best match with the backbone curve as well as hysteresis of the actual test results.

However, fiber hinges take much more time to converge as compared to plastic hinges. In

consideration of number of ground motions to run, it was decided to incorporate shear hinges for

pushover and nonlinear time-history analyses.

60

Figure 20 Experimental hysteresis calibration using fiber hinges in SAP2000

4.3 Summary of Modeling Techniques for Different Analyses

To carry out all analyses recommended by CHBDC S6-14, multiple models were developed in

SAP2000 by using the EBF bent designs. The author decided to use a plastic hinge approach for

nonlinear modelling as it is based on ASCE 41- 13 from cyclic loading results. Factors taken into

account while selecting the modeling approach are the results of calibration with actual experiment

tests, ability to predict the actual inelastic behaviour, and processing time for each analysis. Table

12 provides a summary of each model that is developed for a particular analysis.

-1000

-800

-600

-400

-200

0

200

400

600

800

1000

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

Lin

k S

hea

r (K

N)

Link Rotation (rad)

Berman Hysteresis Berman Hysteresis Backbone (Actual data) Fiber Hinges

61

Table 12 SAP2000 Models descriptions for different analyses

Analysis SAP2000 Model Characteristics

Modal Analysis - Elastic model

- Lumped masses from superstructure

- Base restraints: pinned with no soil-structure interaction consideration

- A single mode spectral method in the transverse direction

- Outcome: fundamental period

Multi-mode

Elastic Response

Spectral Analysis

(RSA)

- Load Combination: 1.2 DL + 1.0 EQ

- Mass source: Bent self-mass + superstructure mass (SLS)

- Earthquake: 5% damped UHS for three hazard levels (475-, 975-, and

2475-year return periods)

- 5% constant modal damping

- Elastic model


- Outcome: displacement demands at multiple hazard levels

Inelastic Static

Pushover

Analysis (ISPA)

- Material nonlinearity: link member with a deformation controlled

plastic shear hinge in the middle as per ASCE 41-13

- Hysteresis type: kinematic

- All other members modelled as elastic elements


- Geometric nonlinearity (P-delta) ignored due to high bent stiffness

- Effect of dead load considered before pushing the bent

- Displacement-controlled load application

- Outcome: bent capacity curve (link shear force-displacement), link

plastic rotations at required bent displacements, confirmation of

capacity design

62

Analysis SAP2000 Model Characteristics

Elastic Time-

History Analysis

(ETHA)

- Load Case: Linear Direct Integration Time History


- Earthquake: 5% damped spectrally matched time histories for three

hazard levels (475-, 975-, and 2475-year return period)

- Elastic model


- 5% constant modal damping

- Outcome: displacement demands for each time-history record

Nonlinear Time

History Analysis

(NLTHA)

- Load Case: Nonlinear Direct Integration Time History

- Material nonlinearity: Link member with a deformation controlled

plastic shear hinge in the middle as per ASCE 41-13

- Hysteresis type: kinematic


- Earthquake: 5% damped spectrally matched time histories for three

hazard levels (475-, 975-, and 2475-year return period)

- All other members modelled as elastic elements


- Geometric nonlinearity (P-delta) ignored due to high bent stiffness

- Stressed state from dead loads considered before running nonlinear

time history analysis

- Material nonlinearity: link member with a plastic shear hinge in the

middle as per ASCE 41-13

- Rayleigh damping (mass and stiffness proportional damping) with 2%

damping in the first two modes

- Outcomes: link plastic rotations at required bent displacements, design

checking, confirmation of results from pushover/RSA

63

Chapter 5: Proposed Performance Criteria and Seismic Evaluation

In this chapter, specific damage states and demand parameters for EBFs as bridge bents have been

proposed based on previous literature, and different methods of repair for each damage state have

been explained. The results of the analyses required by CHBDC S6-14 including response

spectrum analysis (RSA), inelastic static pushover analysis (ISPA) and nonlinear time-history

analysis (NLTHA) have been presented. The models developed in Chapter 4 have been used to

perform the noted analyses. Seismic hazard deaggregation is used to determine the criteria to select

the ground motions and spectral matching of ground motions for different target spectrum is done.

The analyses results have been used for seismic evaluation of designed bents using CHBDC S6-

14 performance criteria and the proposed performance criteria described in the next section.

Different methods of repair have also been provided corresponding to each damage state.

5.1 Proposed Acceptance Criteria for EBFs as Bridge Piers

There is a need to carry out performance evaluation of EBFs mainly based on repair cost and

business interruption, by relating the damage limits with demand parameters such as link plastic

rotation in an EBF (Gulec, Gibbons, Chen, & Whittaker, 2011). Damage states can be directly

related to the failure mode of a steel member, such as yielding of webs and flanges, yielding of

stiffeners, local buckling of stiffeners, fracture of webs and flanges, and lateral torsional buckling

(Gulec et al., 2011).

Based on the work done by Gulec et al. (2011) related to damage states and fragility

functions for shear link beams in EBFs, shear link total rotations for a variety of damage states are

reviewed. These fragility functions can be used to correlate the probability of exceeding a damage

state to the demand parameter, such as plastic rotations in EBFs. Gulec et al. (2011) developed the

64

fragility functions for shear and flexural critical links in EBFs by doing a statistical evaluation of

test results of 82 links gathered from literature review. The researchers used the plastic link rotation

as a demand parameter for damage evaluation of EBFs as shown in Figure 21. They correlated the

experimental test results with different damage states and recommended different suitable methods

of repair. Damage states are characterized in most instances by direct indicators of damage to steel

components such as web and flange local buckling, and fracture.

For minimal damage, Gulec et al. (2011) did not provide a fragility curve and reported that

even though shear links experiences significant web, flange, or intermediate stiffener yielding,

structural repair is unnecessary. Berman & Bruneau (2007) tested one full-scale EBF bent with a

tubular shear link to determine the web and flange compactness limits for laterally unbraced links.

The link specimen experienced the first yield in the form of web shear yielding at a yield rotation

of 0.014 radians. After that, they conducted tests on a total of twelve built-up tubular links with

varying stiffener requirements and compactness ratios. Out of these twelve specimens, six links

were designed as shear links based on three categories: unstiffened links, links with stiffeners along

webs and flanges according to the recommended compactness ratio limits, and links with stiffeners

along webs and flanges with compactness ratio limits well below the recommended limits.

The hysteresis results from all these six specimens were reviewed, and a yield rotation of

around 0.015 radians was obtained as an average value from all the tests. It was decided to adopt

acceptance criteria rotation for minimal damage as 0.015 radians. Even though this is a very

conservative approach, this value is close to ASCE 41 immediate occupancy (IO) value (θT = 0.010

radians) that can be obtained by adding elastic rotation and plastic rotation limit given by ASCE

41, such as elastic link rotation (θE) + IO plastic rotation (θp) of 0.005 radians (ASCE, 2014). Here,

the elastic link rotation is taken as 0.005 radians which represents most of the design cases in this

65

case study. Therefore, this acceptance criteria limit allows limited yielding to occur in the link

member at the immediate-service level.

For a consistent approach, total link rotation is used for acceptance criteria limits instead

of plastic rotation. The total link rotation can be calculated as:

ᵞT = ᵞE + ᵞP 5-1

Here 𝛾𝑇 is the total link rotation, i.e., the ratio of total relative vertical displacement at the

end of a link to the length (e) of the link member. 𝛾𝐸 is the elastic link rotation that can be calculated

using theoretical equations provided in FEMA 356 (FEMA, 2000a) based on Timoshenko beam

theory as shown in Appendix A.2. 𝛾𝑝 is the link plastic rotation, which is the inelastic component

of the rotation of the link member relative to the beam outside the link as shown in Figure 21. The

link plastic rotation can be calculated from the rigid plastic mechanism using Equation 5-2.

As shown in Figure 21, L is the total frame bay width, e is the link member length and 𝜃𝑝 is the

storey plastic rotation angle.

As all other framing members are designed to remain elastic, deformations from beam

outside the link are not considered by assuming that it will stay principally elastic while the link is

subjected to large plastic deformations (Berman & Bruneau, 2007). The results from all analyses

are provided in the form of total rotation for comparison purposes.

𝛾𝑝 =𝐿

𝑒 𝜃𝑝

5-2

66

Figure 21 Deformed shape of an EBF frame under lateral using the rigid-plastic mechanism.

Adapted from: (Berman & Bruneau, 2007)

The proposed acceptance criteria limits with total link rotation as a demand parameter obtained

from literature are provided in Table 13 and will be used for performance evaluation of different

EBF bridge bents for this study. The literature reference corresponding to each selected rotation

limit is also provided. The fragility curves developed by Gulec et al., 2011, used to select the total

link rotation limits for Limited and Service Disruption serviceability levels are shown in Figure

22.

67

Table 13 Proposed acceptance criteria limits for shear links in EBF bridge bents

Level Service Damage

Shear Link

Total

Rotation

(radians)

Reference for

Acceptance Criteria

Damage

States

1 Immediate Minimal

damage 0.015

Average yield rotation

from actual tests of six

tubular shear links

(Berman, 2006)

- Web yielding

- Flange yielding

- Stiffener

yielding


damage 0.06

The median value of

method of repair (MOR) -

2 fragility curve

(Gulec et al., 2011)

- Web local

buckling

- Flange local

buckling

3 Service

Disruption

Extensive

damage

0.08

(maximum

allowable

rotation by CSA

S16-14 for shear

links)

The median value of

method of repair (MOR)-3

fragility curve

(Gulec et al., 2011)

- Web fracture

- Flange fracture

- Lateral

torsional

buckling


replacement 0.1

70% of ASCE-41 for LS,

NBCC 2015 Commentary

(NRCC, 2015)

- Connection

fracture

- Unusable

bridge

5 Collapse

Prevention Fracture 0.12

70% of ASCE-41 for CP,

NBCC 2015 Commentary

(NRCC, 2015)

—

68

5.1.1 Drift Ratio as a Performance Measurement Tool

Drift ratio of the EBF bent can also be used as a parameter to limit the displacements under lateral

load. For a deformed shape of an EBF under lateral force as shown in Figure 21, the story plastic

rotation angle is equal to the drift ratio from geometry as follows:

Also,

It can be seen from Equation 5-4 that drift ratio is dependent on link length i.e., greater link lengths

correspond to larger drift ratios as compare to EBF bents with smaller link member.

5.2 Proposed Method of Repairs for Multiple Performance Levels

Once the damage states have been defined and related to the demand parameters, the next step in

the PBD approach is to evaluate the damage states. Different repair types are linked with visible

damage type to estimate the serviceability of a bridge after an earthquake.

In shear-critical links, web buckling followed by fracture occurs mainly due to web

proportions; unlike flexure-critical links where buckling occurs locally at the flanges (Gulec et al.,

2011). For a full-storey EBF bent with built-up tubular shear link, web shear yielding was

experienced before yielding of flanges from flexure at yield point (Berman, 2006). However,

failure of the link at a plastic rotation of 0.151 radians was mainly due to fracture of the bottom

flange. The fracture was a result of multiple causes including a high degree of constraint by the

𝜃𝑝 =∆𝑝

ℎ

5-3

𝜃𝑝 =𝑒

𝐿 − 𝑒 𝛾𝑝 5-4

69

gusset plates, stiffeners, brace-beam welds, and heat-affected-zone (HAZ) brittleness close to

gusset-stiffener weld (Berman & Bruneau, 2007).

As per guidance provided by Gulec et al., (2011) related to the method of repairs (MOR)

for each damage states, there is no need to provide structural repair to regain before-earthquake

strength and stiffness of the shear link in case of web yielding, flange yielding, or intermediate

stiffener yielding. This is MOR-0, named as ‘cosmetic repair’ and corresponds to performance

level 1 with minimal damage as described in Table 14 when total link rotation is less than or equal

to 0.015 radians. For cosmetic repair, it might be required to repaint the link beam and replace the

spray-on thermal fire protection, if applicable.

As shown in Figure 22, MOR -1 is related to concrete slab replacement. As this is an issue

in buildings and does not directly apply to bridges, MOR-1 is not considered.

The next repair method is MOR-2, and it includes heat straightening when total link

rotation is less than or equal to 0.06 radians. As there is no floor framing or concrete slabs in

bridges, it might be easier to directly provide heat straightening to the link member between the

girders.

Lastly, when there is extensive link damage with irrecoverable strength loss (0.06<θT ≤

0.08 radians), it is required to replace the link. The replacement of the link member depends upon

the type of link section used, i.e., continuous link member with the same section as beam outside,

or a replaceable (modular) link section having bolted or welded connection to the outside beam.

For a continuous link member, to avoid instability, it would be required to lift the girders to remove

the damaged link member with flame cutting. It might also be required to replace the short sections

of beam outside the link and braces, depending upon the connection between the link, brace, and

70

beam outside the link (Gulec et al., 2011). After removing the damaged link section, a new link

section needs to be welded followed by application of superstructure loads on the bents.

Figure 22 Fragility functions for shear links corresponding to each method of repair (lognormal

distribution) (Gulec et al., 2011)

71

Table 14 Proposed Method of Repairs (MOR) for each damage state. Adapted from: (Gulec et

al., 2011)

Level Service Damage Damage States Method of

Repair Repair Action

1 Immediate Minimal

damage

- Web yielding

- Flange yielding

- Stiffener yielding

Cosmetic

repair

MOR-0

No structural repair required;

repaint structural steel

𝛾𝑇 ≤ 0.015 radians


damage

- Web local buckling

- Flange local buckling

Heat

straightening

MOR-2

Provide heat straightening in

the immediate area of web and

flange local buckling


3 Service

Disruption

Extensive

damage

- Web fracture

- Flange fracture

- Lateral torsional

buckling

Link

replacement

MOR-3

Replace link by flame cutting

and weld new link section



replacement

Human life survival, not structural.

5.3 Damping Selection

Usually, for steel structures, 2% damping is considered. However, for this study, due to the

presence of a concrete deck and semi-integral abutments, 5% damped response acceleration values

are considered for RSA to have more realistic results. In addition to the above, due to soil-structure

interaction and foundation damping effects, higher damping can be expected (FEMA, 2005). For

NLTHA, Rayleigh damping (mass and stiffness proportional damping) with 2% damping in the

first two modes is used.

72

5.4 Modal Analysis Results

Previous studies (Gerin & Khan, 2017) have demonstrated that for most of highway and transit

structures, the majority of the mass comes from superstructure that is supported on single or

multiple columns bents with dynamic behaviour similar to a single-degree-of-freedom system

(SDOF) in both transverse and longitudinal directions. For a bridge primarily responding in the

first mode, a simplified bent model can be used with lumped masses to capture the behaviour of

the bridge without modelling the superstructure. This assumption provides a reasonable estimation

for the fundamental period of the bridge.

A single-mode spectral method is used for modal analysis by considering each bent as

SDOF system. A model according to properties provided in Table 12 is used to carry out modal

analysis. The fundamental periods of all designed bents are given in Table 15.

Table 15 Modal analysis results for all design cases

Bent Design Case D1 D2 D3 D4

First Mode Period T, (sec) 0.46 0.53 0.39 0.57

5.5 Response Spectrum Analysis Results

One of the primary purposes of this study is the design calibration between FBD and PBD.

Therefore, elastic dynamic (response spectrum) analysis (EDA) coupled with inelastic static

pushover analysis (ISPA) is carried out for global demands and for demonstrating local component

performance compliance of shear link.

The modal damping is considered as 5% for RSA. The considered bridge is a regular bridge

with the fundamental mode of vibration governing the response and well-separated frequencies.

The effect of directional combination rule is found to be negligible and hence ignored in this case.

73

The displacement demands at the top right edge of the bent are monitored for bridge site-specific

response spectrum. The displacement demands obtained from each hazard level are shown in Table

16.

Table 16 Bent global displacement demands from RSA

Load Case

D1 RSA

Demands

(mm)

D2 RSA

Demands

(mm)

D3 RSA

Demands

(mm)

D4 RSA

Demands

(mm)

Yield Point 26 25 25 28

1.25D+EQ_475 29 36 21 40

1.25D+EQ_975 45 56 33 63

1.25D+EQ_2475 68 87 50 98

5.5.1 Inelastic Displacement Correction

For these design cases with a low fundamental period, the structure lies in the acceleration-

sensitive zone of the spectrum. Even though CHBDC S6-14 does not require any correction, the

equal displacement principle is of doubtful validity in this zone; in fact, it has been shown that the

corresponding inelastic displacements based on the equal displacement principle for such a case

are usually grossly underestimated form linear analysis-based models (ATC, 1996). FEMA 356

(FEMA, 2000a) is used to calculate the characteristic period, Ts, of the UHS that indicates a

borderline between the constant-acceleration and constant-velocity portions. This period helps to

identify whether the structure lies in short-period or long-period range. The calculated

characteristic period, Ts, for Sombrio UHS is around 0.58, 0.52 and 0.45 sec for 2475-, 975-, and

475-year return period hazard levels calculated as per FEMA 356 (FEMA, 2000a) .

74

Four different methods are used to determine the inelastic displacement from the elastic

analysis for the 475-, 975- and 2475-year earthquakes. These include FEMA 440’s improved

capacity spectrum method (FEMA, 2005), FEMA 440’s displacement modification (coefficient)

method (FEMA, 2005), FEMA 356’s coefficient method (FEMA, 2000a) and ATC- 40’s capacity

spectrum method (ATC, 1996). The methodology adopted for these methods is shown in Figure

23.

The results of inelastic displacement correction are provided in Table 17 from all four

methods. From previous studies (Powell, 2013), it is clear that FEMA 440’s methods provide more

reasonable estimate as compared to FEMA 356 and ATC 40. These modified displacements are

compared with the displacement demands from NLTHA, and it is observed that FEMA 440

displacement modification method provides the closest results to those of NLTHA. Hence,

modified displacement demands from FEMA 440 displacement modification method are used for

further performance evaluation of EBF bents.

75

Figure 23 Steps for ATC- 40 capacity spectrum, FEMA 356 & 440 coefficient method, and FEMA

440 equivalent linearization methods. Adapted from: CSI Knowledge Base (Powell, 2013)

76

Table 17 Inelastic displacement correction for all design cases

Hazard

Level

Return-

period

Response

Spectrum

Analysis

(RSA)

mm

FEMA 440

Displacement

Modification

mm

FEMA 440

Equivalent

Linearization

mm

FEMA 356

Coefficient

Method

mm

ATC-40

Capacity

Spectrum

mm

Design 1

475-years 29 32 34 32 30

975-years 45 51 50 53 45

2475-years 68 83 109 82 101

Design 2

475-years 36 41 40 41 33

975-years 56 68 65 65 59

2475-years 87 112 117 100 143

Design 3

475-years 21 23 26 22 21

975-years 33 37 38 39 30

2475-years 50 60 73 63 63

Design 4

475-years 40 47 42 46 38

975-years 63 76 72 72 70

2475-years 98 127 145 112 188

77

5.6 Pushover Analysis Results

Inelastic static pushover (ISPA) analysis is carried out by using the modelling approach described

in Chapter 4 to obtain the capacity of designed bents. ASCE 41-13 nonlinear modeling parameters

are used to define the plastic shear hinge in the middle of the shear link for nonlinear static

pushover analysis. The RSA displacement demands from 475-, 975-, and 2475-year return period

response spectrum analyses obtained from Section 5.5 are superimposed on pushover capacity

curve after applying the displacement correction to evaluate the performance of bents at required

performance levels. Link plastic rotations have been calculated corresponding to RSA

displacement demands and presented as total link rotations by adding the elastic rotation

component.

Figure 24 through Figure 27 show the pushover curves with displacement demands from

475-, 975-, and 2475-year return period response spectrum analyses. The pushover curves P1, P2,

P3, and P4 corresponds to Design 1, Design 2, Design 3, and Design 4 respectively. On the

pushover curves, acceptance criteria limits for damage states concerning minimal damage,

repairable damage, and extensive damage have also been provided. These acceptance criteria

limits are presented in Section 5.1 and were chosen following the literature review to evaluate the

performance of designed bents at multiple performance levels.

78

Figure 24 Design 1 pushover curve with modified RSA demands and acceptance criteria limits

Table 18 Modified displacement and rotation demands from RSA

Design 1: FBD (IE = 1.5)

Load Case

Modified

Displacement

Demands

from RSA

(mm)

Total

Link

Rotation

(radians)

Acceptance

Criteria

Rotation from

Literature

(radians)

Yield Point 26 0.005 —

1.25D+EQ_475 32 0.007 0.015

1.25D+EQ_975 51 0.013 0.060

1.25D+EQ_2475 83 0.022 0.080

0

2000

4000

6000

8000

10000

0 25 50 75 100 125 150 175 200 225 250 275 300

Bas

e S

hea

r (K

N)

Displacement (mm)

P1

Yield

475

975

2475

Minimal

Repairable

Extensive

79



Design 2: FBD (IE = 1.0)

Load Case

Modified

Displacement

Demands

from RSA

(mm)

Total

Link

Rotation

(radians)

Acceptance

Criteria

Rotation from

Literature

(radians)


1.25D+EQ_475 41 0.010 0.015

1.25D+EQ_975 68 0.018 0.060

1.25D+EQ_2475 112 0.031 0.080

0

2000

4000

6000

8000

10000

0 25 50 75 100 125 150 175 200 225 250 275 300

Bas

e S

hea

r (K

N)

Displacement (mm)

P2Yield4759752475MinimalRepairableExtensive

80



Design 3: PBD (No yielding at 475-year return period)

Load Case

Modified

Displacement

Demands

from RSA

(mm)

Total

Link

Rotation

(radians)

Acceptance

Criteria

Rotation from

Literature

(radians)


1.25D+EQ_475 23 0.005 0.015

1.25D+EQ_975 37 0.009 0.060

1.25D+EQ_2475 60 0.016 0.080

0

2000

4000

6000

8000

10000

12000

14000

0 25 50 75 100 125 150 175 200 225 250 275 300

Bas

e S

hea

r (K

N)

Displacement (mm)

P3Yield4759752475MinimalRepairableExtensive

81

Figure 27 Design 4 pushover curve with modified RSA demands and acceptance criteria limit


Design 4: PBD (Replaceable link)

Load Case

Modified

Displacement

Demands

from RSA

(mm)

Total

Link

Rotation

(radians)

Acceptance

Criteria

Rotation from

Literature

(radians)


1.25D+EQ_475 47 0.014 0.015

1.25D+EQ_975 76 0.026 0.060

1.25D+EQ_2475 127 0.046 0.080

0

2000

4000

6000

8000

10000

12000

14000

0 25 50 75 100 125 150 175 200 225 250 275 300

Bas

e S

hea

r (K

N)

Displacement (mm)

P4 Yield

475 975

2475 Minimal

Repairable Extensive

82

5.6.1 Performance Evaluation from Pushover and RSA Results

For Design 1, it can be seen from Figure 24 that yield displacement is less than displacement

demand at the 475-year earthquake event. Hence, the shear link has yielded at 475-year hazard

level and does not fulfill the no-yielding criteria for immediate serviceability of bridge by CHBDC

S6-14. An interesting observation is that link total rotations corresponding to the 975-year and

2475-year return period events are 0.013 and 0.022 radians, respectively, which are significantly

less than the code allowable shear link rotation values (0.08 radians). Hence, the bent performance

at the 475-year return period event governs the design. However, by comparing the total rotation

at the 475-year return period event with proposed acceptance criteria, it can be seen that rotation

demands meet the acceptance criteria limit for minimal damage limit (0.015 radians). For 975- and

2475-year return period events, rotation demands are much less than the proposed allowable limits

for repairable damage (0.06 radians) and extensive damage (0.08 radians) based on fragility curves

provided by Gulec et al. (2011). An important thing to note here is that, CSA S16-14 has a plastic

rotation limit of 0.08 radians for shear links corresponding to a performance level of Life Safety.

This shows that code’s acceptance limits for plastic rotations may be conservative.

Design 2 performs similarly to Design 1, with the 475-year return period event governing

the design. It meets the acceptance criteria limits derived from literature for all service states, but

it does not fulfil the no-yielding criterion (immediate service) for the 475-year return period event

per CHBDC S6-14. Also, similar to Design 1, rotation demands of 975- and 2475-year return

period events are much less than the proposed allowable limits for repairable and extensive

damage.

83

Hence, both Design 1 and 2 based on FBD approach do not fulfil the no-yielding

performance criteria of CHBDC S6-14 but satisfy the proposed acceptance criteria limits.

For Design 3, which was designed to remain elastic for the 475-year return period event,

the link does not yield at this hazard level as required by CHBDC S6-14. The maximum rotation

is 0.016 radians for 2475-year hazard level, which is considerably less than the allowable rotation

of 0.08 radians set out by steel design standard CSA S16-14. It is clear from Design 3 that to meet

no-yielding criterion at 475-year return period event, a considerable increase in member size is

required. This design meets the proposed acceptance criteria limit for all three performance levels

for the Major Route Bridges.

For Design 4, which includes a replaceable link, the link yields at 475-year return period

event and has larger total rotations for 975-year and 2475-year return period events than for all

other designs. However, all rotations are within the proposed allowable values.

Based on the above discussion, the 475-year return period event governs the design in all

cases and makes the sizes large and inefficient, while the link plastic rotations corresponding to

higher return period events are very low compared to the allowable limits provided in the literature

for links mainly used in buildings. Therefore, it can be concluded that if the links are made

replaceable and allowed to have limited yielding at 475-year return period ground shaking levels,

the design becomes more practical.

According to CHBDC S6-14, the designer has the option to adopt either FBD approach

using an importance factor (IE) of 1.5 or PBD approach for Regular Major Route Bridges. The

performance of structures designed using the FBD approach is expected to be consistent with PBD

at the 2475-year return period as per CHBDC S6-14. The code does not require any design

checking corresponding to lower hazard levels (475-,975-year return period) for structures

84

designed using FBD approach. It has been observed from the analysis results that the member sizes

for Design 3 (PBD) are quite different than Design 1 (FBD). Moreover, both FBD cases (Design

1 and Design 2) do not fulfil minimal damage performance criteria set out by CHDBC S6-14.

5.7 Time-History Analysis

5.7.1 Ground Motions for Time-history Analyses

According to Clause 4.4.3.6 of CHBDC S6-14, for each hazard level, a total of eleven or more sets

of ground motions are required to carry out NLTHA, without selecting more than two sets of time

histories from any given earthquake event. A method known as seismic hazard deaggregation is

used to determine the relative contribution of various earthquake types to the rate of exceedance

of a given ground motion intensity (Bazzurro & Allen Cornell., 1999). This method was employed

for this study as described in the next section.

5.7.2 Seismic Hazard Deaggregation for Site Location

To select ground motions for time-history analyses, a seismic hazard deaggregation is carried out

for three hazard levels including 475-, 975- and 2475-year return period events for the bridge

location shown in Figure 28. The purpose of deaggregation for the target location is to determine

which sources contribute most significantly to the total seismic hazard. EZ-FRISK software (EZ-

FRISK Version 7.52, 2011) is used for magnitude-distance (M-R) hazard deaggregation for the

475-, 975- and 2475-year return period events by considering a fundamental period of 0.4 sec as a

reference period for the designed bents. For representation of probability density distribution, the

three-dimensional graphs of magnitude-distance (M-R) deaggregation are plotted in Figure 29,

85

Figure 30 and Figure 31 for UHS corresponding to 475-, 975-, and 2475-year return period events

respectively.

Figure 28 Satellite image and sitemap of Sombrio Bridge one, Port Renfrew

Figure 29 Magnitude-distance (M-R) deaggregation of 2475-year return period hazard level (T=

0.4 sec)

Subduction Subcrustal

Crustal

86

Figure 30 Magnitude-distance (M-R) deaggregation of 975-year return period hazard level

(T=0.4 sec)

Figure 31 Magnitude-distance (M-R) deaggregation of 475-year return period hazard level

(T=0.4 sec)

Subduction

Subduction

Subcrustal

Subcrustal

Crustal

Crustal

87

The geography of the Pacific Southwest region of Canada may be subject to earthquakes with

different source types, namely crustal, subcrustal and subduction earthquakes. Crustal and

subcrustal ground motions contribute to the lower period end of the seismic risk spectrum while

the Cascadia subduction quakes contribute more to the longer period end of the spectrum. The

primary cause of these sources is the plate movement between the Pacific Plate and Juan de Fuca

Plate (Li, 2016) located just offshore of Vancouver Island.

Three different clusters of probability density can be seen in Figure 29, Figure 30 and

Figure 31. The hazard is dominated by three events, first being the subduction earthquake with

highest probability density and magnitude of 8-9.5 and longest distance range of 0-150 Km. It is

interesting to note that subduction event is governing at a low period of 0.4 sec because, the bridge

location is close to Cascadia fault. The second contribution is from the subcrustal earthquakes with

medium probability density and magnitude from 5.5-7.5 with 0-100 Km distance range. Last, the

crustal earthquakes contribute with the smallest probability density and magnitude from 5.0-7.5

with a short epicentre distance range of 0-50 Km. These results from seismic hazard deaggregation

are summarized in Table 22 and are used as criteria for the selection of ground motions.

Table 22 Ground motions selection criteria from seismic hazard deaggregation

4

Source Distance Range Magnitude Range

No. of Ground

Motions for 2% in

50 yrs

No. of Ground

Motions for 5% in

50 yrs

No. of Ground

Motions for 10%

in 50 yrs

Crustal 0 - 50 Km 5.0 - 7.5 3 2

3

Subcrustal 0 - 100 km 5.5 - 7.5 2 4 4

Subduction 0 - 150 Km 8 - 9.5 6 5

88

5.7.3 Selection of Ground Motions

According to the deaggregation results, selection criteria for all three types of earthquakes are

summarized and listed in Table 22. The ground motions used for seismic evaluation will be

selected based on the magnitude-distance ranges shown in Table 22. There is a need to select a

period range to cover the periods that mainly participate in the seismic response of bridge. The

period range of 0.2T1 to 1.5T1 but not less than 1.5 sec (which is 0.08sec to 1.5sec in this case) is

used to match the spectrum, where T1 is period of the first mode period as per recommendation of

CHBDC S6-14 Commentary Clause 4.4.3.6 (CSA Group, 2014c).

A novel web-based database developed at University of British Columbia, Vancouver,

named S2GM (Bebamzadeh, Ventura, & Fairhurst, 2015) is used for selection and linear scaling

of crustal, subcrustal and subduction sources of ground motions. The S2GM tool is available online

in the form of a website and uses a robust selection and scaling algorithm with the option to

download worldwide time histories records corresponding to a target UHS. For crustal ground

motions, S2GM provides a scale factor for linear scaling and refers to The Pacific Earthquake

Engineering Research (PEER) Center NGA-West 2 database to download the time-history records.

Therefore, crustal time-history records are obtained from PEER database (PEER, 2010b), and

subcrustal and subduction time-history records are directly downloaded from S2GM. For this

study, only horizontal components of ground motions are considered for transverse analyses of

bridge bents. As an example, original time-histories of subduction ground motions for the 2475-

year return period earthquake are shown in Figure 32 and Figure 33 with specific record numbers,

time history name and year.

Once the ground motions have been selected based on seismic hazard deaggregation

results, the next step is to scale the time-history records to match the target hazard level properly

89

in the period range of interest. CHBDC S6-14 provides two methods for matching the ground

motions including linear scaling, and spectral matching.

Time (sec)

161 - Hokkaido-2003

172 - Hokkaido-2003

723 - Tohoku-2011

Record Number

Record Name

Year

Acc

eler

atio

n (

g)

Acc

eler

atio

n (

g)

Acc

eler

atio

n (

g)

Time (sec)

Time (sec)

90

Figure 32 Original acceleration-time-histories for 2475-year target spectrum (subduction

records)

842 - Tohoku-2011

1064 - Maule-2010

1072 - Maule-2010

Acc

eler

atio

n (

g)

Time (sec)

Time (sec)

Acc

eler

atio

n (

g)

Acc

eler

atio

n (

g)

Time (sec)

91

Figure 33 Original acceleration-time-histories for 2475-year target spectrum (subduction

records)

5.7.4 Linear Scaling of Scaled Ground Motions

In linear scaling, spectral acceleration values of the time-history record are simply scaled up and

down to obtain the best match with a target spectrum for a specific period range. In S2GM, the

Mean Squared Error (MSE) between the ground motion spectrum and target spectrum over the

selected period range is minimized. Characteristics of a total of 11 seed ground motions for each

hazard level are described in Table 23, Table 24 and Table 25 showing seismic source or faulting

mechanism, magnitude, source-to-site distance, and scale factor from linear scaling. The crustal

earthquakes are listed as Normal, Reverse or Reverse Oblique in these tables.

Acc

eler

atio

n (

g)

Time (sec)

161 - Hokkaido-2003

172 - Hokkaido-2003

723 - Tohoku-2011

842 - Tohoku-2011

1064 - Maule-2010

1072 - Maule-2010

92

5.7.5 Spectral Matching of Scaled Ground Motions

Spectral matching provides matching for a larger period range by changing the frequency content

of the selected time histories in either time or frequency domain to match the target spectrum. The

spectral matching of selected ground motions was done using Seismomatch (Seismosoft, 2016).

While providing the input for each time-history record in Seismomatch, the user also has the option

to define a scale factor for each record. Therefore, the scale factor obtained from linear scaling in

Section 5.6.4 is used to simply scale the spectral acceleration values up and down followed by

spectral matching to obtain a good match in selected period range. The selected 33 ground motions

are spectrally matched and shown in Appendix B marked with record number for each time-

history. As an example, matched time-histories of subduction ground motions for the 2475-year

return period earthquake are shown in Figure 34. Additionally, Figure 35 and Figure 36 show the

spectra of both original and matched time-histories respectively for comparison with target

spectrum shown bold and in red. These spectrally matched time-history records are used for

analysis.

93

Table 23 Selected ground motions for 10% in 50 years hazard level (475-year return period)


Record

No.

Database

NumberEarthquake Mechanism Year Recording Station Magnitude Distance (Km) Scale Factor Vs30 (m/s)

1 78 San Fernando Reverse 1971 Palmdale Fire Station 6.61 28 1.937 452

2 291 Irpinia, Italy-01 Normal 1980 Rionero In Vulture 6.9 30 2.4253 574

3 302 Irpinia, Italy-02 Normal 1980 Rionero In Vulture 6.2 22.69 2.212 574

4 769 Loma Prieta Reverse Oblique 1989 Gilroy Array #6 6.93 18 1.9271 663

5 646 Miyagi-Oki, Japan Subcrustal 2005 MYG0130 7.2 105 1.72 535

6 825 Nisqually, US Subcrustal 2001 1422a 6.8 67 2.1 463

7 10 Geiyo, Japan Subcrustal 2001 YMG0180 6.4 60 2.42 501

8 258 Geiyo, Japan Subcrustal 2001 EHM0080 6.4 83 3.09 654

9 161 Hokkaido, Japan Subduction 2003 HKD084 8 92 1.02 510


11 1074 Maule, Chile Subduction 2010 TALCA 8.8 35 0.79 640

Record

No.

Database

NumberEarthquake Mechanism Year Recording Station Magnitude Distance (Km) Scale Factor Vs30 (m/s)

1 125 Friuli, Italy-01 Reverse 1976 Tolmezzo 6.5 15 1.4711 505.23

2 741 Loma Prieta Reverse Oblique 1989 BRAN 6.93 10.7 0.699 476.54

3 963 Northridge-01 Reverse 1994 Castaic - Old Ridge Route 6.69 20.7 0.785 450.28

4 825 Nisqually, US Subcrustal 2001 1422a 6.8 45 3.39 463





9 723 Tohoku, Japan Subduction 2011 MYC016 9 59 0.63 693

10 1064 Maule, Chile Subduction 2010 CURICO 8.8 37 1.06 537

11 1089 Michoacan, Mexico Subduction 1985 AZIH85 8.1 93 2.79 720

94


Figure 34 Matched acceleration-time-histories for 2475-year target spectrum (subduction

records)

Record

No.

Database

NumberEarthquake Mechanism Year Recording Station Magnitude

Distance

(Km)

Scale

FactorVs30 (m/s)

1 125 Friuli, Italy-01 Reverse 1976 Tolmezzo 6.5 15 2.323 505.23

2 741 Loma Prieta Reverse Oblique 1989 BRAN 6.93 10.7 1.105 476.54

3 801 Loma Prieta Reverse Oblique 1989 San Jose - Santa Teresa Hills 6.93 14.6 2.3119 671.77





8 723 Tohoku, Japan Subduction 2011 MYC016 9 59 1.01 693

9 842 Tohoku, Japan Subduction 2011 MYG0161 9 114 1.72 580

10 1064 Maule, Chile Subduction 2010 CURICO 8.8 37 1.71 537

11 1072 Maule, Chile Subduction 2010 STGOPENALOLEN 8.8 150 2.31 452

Time (sec)

Acc

eler

atio

n (

g)

161 - Hokkaido-2003

172 - Hokkaido-2003

723 - Tohoku-2011

842 - Tohoku-2011

1064 - Maule-2010

1072 - Maule-2010

95

Figure 35 Original spectra of scaled selected GMs and 2475 years target spectrum (subduction

records)

Figure 36 Matched spectra of scaled selected GMs and 2475 years target spectrum (subduction

records)

Period (sec)

Acc

eler

atio

n (

g)

Period (sec)

Acc

eler

atio

n (

g)

161 - Hokkaido-2003

172 - Hokkaido-2003

723 - Tohoku-2011

842 - Tohoku-2011

1064 - Maule-2010

1072 - Maule-2010

161 - Hokkaido-2003

172 - Hokkaido-2003

723 - Tohoku-2011

842 - Tohoku-2011

1064 - Maule-2010

1072 - Maule-2010

96

5.8 Elastic Time-history Analysis (ETHA)

An elastic time-history Analysis (ETHA) is also carried out as a linear direct integration time-

history load case for all designed bents to ensure that ground motions have been appropriately

matched to the target spectrum in selected period range. The bent global maximum displacements

obtained from ETHA are shown in Table 30, Table 31 and Table 32. The average of displacement

demands from 11 ground motions for each hazard level is compared with actual displacement

demand from RSA at the same hazard level and found to be in close agreement.

5.9 Nonlinear Time-history Analysis (NLTHA)

The nonlinear dynamic analysis is carried out by using the spectrally matched time-histories

presented in Table 23 through Table 25 to evaluate the performance of four designed EBF bents

at different hazard levels and to compare the results with pushover and response spectrum analyses.

Like all other analyses, serviceability limit state (SLS) loads from the superstructure are applied

at each girder location to capture the seismic mass in SAP2000 models. The damping is assigned

as 2% Rayleigh damping in first two modes. A plastic shear hinge is assigned in the middle of the

link member to capture the nonlinear behaviour of the yielded link member. All other members

are capacity-protected and modelled as elastic elements. All modeling assumptions and procedures

are provided in Table 12.

The link plastic or inelastic rotation is the most appropriate component-level demand

parameter for EBFs and is obtained easily from nonlinear time-history analyses of EBFs. The drift

ratio of an EBF bent can also be used as a parameter to limit the deformations and is mainly

dependent on its link length. A bent with a greater link length will have more drift ratio as

compared to a bent with smaller link member as described in Section 5.1.1. For example, for 3 m

97

long link member, the drift ratio for extensive damage (𝛾𝑇= 0.08 radians) is around 4% as compare

to a drift ratio of 3% for a bent with link length of 2.4 m.

From NLTHA, total rotations corresponding to each time-history record are shown in

Figure 37, Figure 38, and Figure 39. Each of these graphs has three acceptance criteria limits

corresponding to minimal damage, repairable damage and extensive damage with link total

rotation of 0.015 radians, 0.06 radians, and 0.08 radians respectively. For comparison purposes,

bent global modified displacement demands from RSA, link total rotations (θE + θp) obtained from

pushover capacity curve corresponding to the RSA demands as well as link total rotations from

NLTHA are provided in Table 26. The total link rotation provided in Table 26 is an average of 11

ground motion responses at each hazard level. To evaluate these results, proposed acceptance

criteria limits for minimal, repairable, and extensive damage have also been provided. The link

maximum plastic rotations for each time-history from NLTHA are shown in Table 30, Table 31

and Table 32.

98

Figure 37 Total link rotation for each time-history record from NLTHA for 475-year return

period event

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0.080

0 1 2 3 4 5 6 7 8 9 10 11 12

Lin

k T

ota

l R

ota

tion (

rad)

Ground Motion Record Number

Design 1

Design 2

Design 3

Design 4

Minimal

Damage

Repairable

Damage

Extensive

Damage

99


period event

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0.080

0 1 2 3 4 5 6 7 8 9 10 11 12

Lin

k T

ota

l R

ota

tion (

rad)


Design 1

Design 2

Design 3

Design 4

Minimal

Damage

Repairable

Damage

Extensive

Damage

100


period event

5.9.1 Performance Evaluation from NLTHA Results

The results from NLTHA analysis have been compared with total link rotations obtained from

pushover analysis corresponding to modified RSA displacement demands and found to be in close

agreement. By evaluating the total link rotations from pushover/RSA and NLTHA and comparing

them with CHBDC S6-14 acceptance criteria and proposed acceptance criteria, some observations

are:

• Designs 1 and 2 do not meet the no-yielding criterion by CHBDC S6-14 corresponding to

the 475-year return period event. Although the seismic demands increase for the 975-year

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0.080

0.090

0 1 2 3 4 5 6 7 8 9 10 11 12

Lin

k T

ota

l R

ota

tion (

rad)


Design 1

Design 2

Design 3

Design 4

Extensive

Damage

Repairable

Damage

Minimal

Damage

101

and 2475-year return period events, the increase is such that the 475-year hazard and the

corresponding performance criteria govern the design.

• Design 1 and Design 2 fulfil the proposed acceptable limit for immediate service; in

addition, the total link rotation is well below the proposed limits for repairable damage and

extensive damage acceptance criteria as shown in Figure 37. This means that the shear

link will experience yielding either of the web, flange, or stiffener at performance level 1.

By comparing this with the damage states and methods of repair proposed in this chapter

based on the work by Gulec et al., (2011), MOR-0 cosmetic repair is required for this

damage state.

• Design 3, which is based on the elastic link performance corresponding to the lowest

return-period event, shows a considerable increase in member sizes to avoid plastic

behaviour at 475-year return period. This design fulfills the no-yielding criterion of

CHBDC S6-14.

• Design 3 fulfils the proposed acceptable limit for immediate service without having

yielding at 475-year return period event. No repair work is anticipated for this design at

475-year return period as the link is fully elastic at this hazard level.

• Design 4, which is also based on satisfying PBD performance criteria with a replaceable

link, meets the proposed acceptance criteria limit for minimal damage with the total link

rotations well below the limits for repairable and extensive damage for higher return period

events. This design does not meet the no-yielding criterion by CHBDC S6-014. By

comparing this with the damage states and methods of repair proposed in this chapter based

102

on the work by Gulec et al., (2011), MOR-2 heat straightening will be required for this

damage state.

• Design 4 demonstrates that if the links are made replaceable and allowed to have limited

yielding at 475-year return period, the design is more practical.

• According to CHBDC S6-14, the designer has the option to adopt either FBD approach

using an importance factor (IE) of 1.5 or PBD approach for Regular Major Route Bridges.

The performance of structures designed using the FBD approach is expected to be

consistent with PBD at the 2475-year return period as per CHBDC S6-14. The code does

not require any design checking at lower hazard levels (475-,975-year return period) for

structures designed using FBD approach. It has been observed that the member sizes for

Design 3 (PBD) are quite different than Design 1 (FBD). Moreover, both FBD cases

(Design 1 and Design 2) and the PBD case (Design 4) do not fulfil minimal damage

performance criteria by CHDBC S6-14.

Table 27, Table 28, and Table 29 show the evaluation of performance criteria based on

proposed acceptance criteria as well as CHBDC S6-14 acceptance criteria. A summary of

different methods of repair (MOR) has been provided corresponding to each damage state.

103

Table 26 Link total rotations from RSA/Pushover and NLTHA

Load Case

Modified

Displacement

Demands

from RSA

(mm)

Total Link

Rotation

Pushover/RSA

(radians)

Total

Link

Rotation

NLTHA

(radians)

Acceptance

Criteria

Rotation

from

Literature

(radians)

Meets

Proposed

Acceptance

(Yes or No)

Design 1

1.25D+EQ_475 32 0.007 0.009 0.015 Yes

1.25D+EQ_975 51 0.013 0.014 0.060 Yes

1.25D+EQ_2475 83 0.022 0.022 0.080 Yes

Design 2

1.25D+EQ_475 41 0.010 0.011 0.015 Yes

1.25D+EQ_975 68 0.018 0.015 0.060 Yes

1.25D+EQ_2475 112 0.031 0.025 0.080 Yes

Design 3

1.25D+EQ_475 23 0.005 0.006 0.015 Yes

1.25D+EQ_975 37 0.009 0.010 0.060 Yes

1.25D+EQ_2475 60 0.016 0.017 0.080 Yes

Design 4

1.25D+EQ_475 47 0.014 0.015 0.015 Yes

1.25D+EQ_975 76 0.026 0.025 0.060 Yes

104

Load Case

Modified

Displacement

Demands

from RSA

(mm)

Total Link

Rotation

Pushover/RSA

(radians)

Total

Link

Rotation

NLTHA

(radians)

Acceptance

Criteria

Rotation

from

Literature

(radians)

Meets

Proposed

Acceptance

(Yes or No)

1.25D+EQ_2475 127 0.046 0.046 0.080 Yes

Table 27 Performance evaluation for performance level 1: Immediate Service (1 in 475-years)

Design Proposed Performance Criteria & MOR

Table 13 & Table 14

CHBDC S6-14 Performance

Criteria

Table 4

D1 FBD • Meets the criteria

• MOR-1: cosmetic repair

• Does not meet the criteria




D3 PBD • Meets the criteria

• No repair required

• Meets the criteria




105

Table 28 Performance evaluation for performance level 2: Limited Service (1 in 975-years)


Table 13 & Table 14


Criteria

Table 4


• MOR-2; heat straightening






• MOR-1; cosmetic repair





Table 29 Performance evaluation for performance level 3: Service Disruption (1 in 2475-years)


Table 13 & Table 14


Criteria

Table 4













106

Table 30 Bent global displacement demands from Linear Direct Integration Time-History (LDITH) and Nonlinear Time-History

Analysis (NLTHA) and link total rotations from NLTHA using ground motions scaled to 475-year return period

LDITH

Disp.

(mm)

NLTHA

Disp.

(mm)

NLTHA

Rotation

(Rad)

LDITH

Disp.

(mm)

NLTHA

Disp.

(mm)

NLTHA

Rotation

(Rad)

LDITH

Disp.

(mm)

NLTHA

Disp.

(mm)

NLTHA

Rotation

(Rad)

LDITH

Disp.

(mm)

NLTHA

Disp.

(mm)

NLTHA

Rotation

(Rad)

1 78 San Fernando Reverse 29 33 0.007 36 45 0.012 25 30 0.007 50 58 0.018

2 291 Irpinia, Italy-01 Normal 29 32 0.007 27 33 0.009 23 26 0.006 40 48 0.015

3 302 Irpinia, Italy-02 Normal 31 39 0.009 34 44 0.011 23 26 0.006 38 45 0.013

4 769 Loma Prieta Reverse Oblique 27 31 0.007 36 47 0.012 22 25 0.006 38 45 0.013

5 646 Miyagi-Oki, Japan Subcrustal 29 32 0.007 33 43 0.011 24 28 0.006 41 51 0.015

6 825 Nisqually, US Subcrustal 32 45 0.011 40 51 0.013 23 29 0.007 46 56 0.018

7 10 Geiyo, Japan Subcrustal 27 37 0.008 30 41 0.010 23 29 0.007 32 42 0.013


9 161 Hokkaido, Japan Subduction 28 56 0.014 39 48 0.013 21 25 0.006 35 44 0.013


11 1074 Maule, Chile Subduction 30 36 0.008 40 47 0.013 21 29 0.007 40 51 0.016

29 38 0.009 34 44 0.011 22 27 0.006 41 50 0.015

MechanismEarthquake Database

Number

Record

Number

Average:


107



LDITH

Disp.

(mm)

NLTHA

Disp.

(mm)

NLTHA

Rotation

(Rad)

LDITH

Disp.

(mm)

NLTHA

Disp.

(mm)

NLTHA

Rotation

(Rad)

LDITH

Disp.

(mm)

NLTHA

Disp.

(mm)

NLTHA

Rotation

(Rad)

LDITH

Disp.

(mm)

NLTHA

Disp.

(mm)

NLTHA

Rotation

(Rad)

1 125 Friuli, Italy-01 Reverse 45 53 0.014 55 58 0.015 32 40 0.010 63 70 0.023


3 963 Northridge-01 Reverse 45 54 0.013 45 51 0.013 35 41 0.010 52 64 0.021

4 825 Nisqually, US Subcrustal 54 76 0.020 68 71 0.019 32 47 0.012 80 93 0.032





9 723 Tohoku, Japan Subduction 45 41 0.010 45 47 0.006 40 52 0.014 63 74 0.025


11 1089 Michoacan, Mexico Subduction 42 58 0.014 59 62 0.017 30 34 0.008 74 84 0.029

46 57 0.014 54 59 0.015 32 40 0.010 65 75 0.025

Record

Number

Database

NumberEarthquake Mechanism

Average:


108



LDITH

Disp.

(mm)

NLTHA

Disp.

(mm)

NLTHA

Rotation

(Rad)

LDITH

Disp.

(mm)

NLTHA

Disp.

(mm)

NLTHA

Rotation

(Rad)

LDITH

Disp.

(mm)

NLTHA

Disp.

(mm)

NLTHA

Rotation

(Rad)

LDITH

Disp.

(mm)

NLTHA

Disp.

(mm)

NLTHA

Rotation

(Rad)

1 125 Friuli, Italy-01 Reverse 70 46 0.011 80 84 0.023 30 40 0.010 80 99 0.035











69 85 0.023 89 103 0.025 54 66 0.018 106 129 0.046


MechanismEarthquake Database

Number

Record

Number

Average:

109

Chapter 6: Conclusions and Recommendations

This chapter summarizes the findings of this study, recommendations for design of EBFs as bridge

bents, limitations of the research and potential areas for further research.

6.1 Summary and Conclusions

This thesis presented the work done on the performance assessment of eccentrically braced frame

(EBF) substructure bridges designed using FBD and PBD approaches according to CHBDC S6-

14. An existing bridge is considered as a Major Route bridge, and an EBF has been chosen as an

earthquake-resisting system (ERS). Four different cases have been designed including two using

FBD approach and two for PBD approach to evaluate the design in accordance with the

performance descriptions and damage states by carrying out the analyses recommended by

CHBDC S6-14. Due to the absence of strain/rotation criteria in CHBDC S6-14 at multiple service

states for EBFs as bridge bents, different acceptance criteria for rotations and corresponding

damage states have been proposed by using fragility curves from the literature. Link total rotation

has been considered as a demand parameter and different methods of repairs consistent with each

damage state have been provided. The drift ratio of an EBF bent can also be used as a parameter

to limit the deformations and is mainly dependent on its link length. Due to the difficulty of

providing lateral bracing against lateral torsional buckling of EBF wide-flange (WF) link beams

in bridges, a built-up tubular link section has been selected, which does not require lateral bracing.

Multiple hazard levels with 475-,975-, and 2475- year return period events have been used

for PBD, whereas, the single hazard level of 2475-year return period is used for FBD as required

by CHBDC S6-14. Elastic dynamic (response spectrum) analysis coupled with inelastic static

pushover analysis is used for global displacement demands and for demonstrating local component

110

(shear link) performance compliance. Nonlinear time-history analysis is also used to check and

provide a comparison of the first approach. Multiple performance objectives have been evaluated

and governing performance objective has been determined. The designed bents have also been

evaluated using proposed criteria and comparison is made with CHBDC S6-14 performance

evaluation.

The code requires no-yielding for the 475-year return-period event. To meet the no-

yielding criterion, a considerable increase in member sizes is observed. The results show that this

criterion governs the design and makes the sizes large and inefficient, while the link plastic

rotations corresponding to higher return-period events are very low compared to the allowable

limits provided in the literature for links mainly used in buildings. Through different cases, it is

demonstrated that if the links are made replaceable and allowed to have limited yielding at 475

years earthquake, it makes the design more practical.

The performance of structures designed using the FBD approach is expected to be

consistent with PBD at the 2475-year return period as per CHBDC S6-14. However, the code does

not require any design checking at lower hazard levels (475-,975-year return period) for structures

designed using the FBD approach. It is observed that the member sizes for a PBD case are quite

different than the FBD case.

6.2 Recommendations for Design

The following design recommendations can be drawn from this study:

1. The geometry and girder orientation of superstructure play a very important role for use of

EBFs as bridge substructure. To avoid the gravity load being directly resisted by the link

beam, bridge girders should be placed at the ends of the link member, i.e., at the intersection

111

of braces and link beam, and, in alignment with the columns. With an EBF bent having

girders placed within the link beam region will result in high bending and shear demands

from gravity loading on the link beam. Additional seismic demands will require a larger

section for the link beam to satisfy the performance criteria. This will have a significant

impact on capacity protected member sizes.

2. Usually for girder bridges having three or more girders, the center-to-center distance

between girders could typically vary from 2.5m to 3.5m. Avoiding girder placement

directly on the link beam would results in long link lengths potentially leading to flexural

links, with significantly lower allowable rotational limits, e.g. a maximum rotation limit of

0.02 radians. It may be difficult to satisfy the performance criteria with lower rotation limits

and it may be advantageous to size up the links as shear links.

3. It is preferable to have a shear yielding link as compared to flexural link because shear

links can dissipate more energy with well predictable and stable hysteretic behaviour as

compared to flexure links. Moreover, shear links are capable of withstanding around four

times more plastic rotational demands (0.08 radians) as compare to flexural links with a

much lower allowable maximum plastic rotation capacity (0.02 radians).

4. With an EBF bent having girders being placed at the locations other than ends of the beam

outside the link, result in high bending demands from gravity loading when lateral loads

are applied. These high bending demands require a larger section for beam outside the link

and hence a greater continuous link section (same section as beam outside the link) as well.

This will have a significant impact on capacity protected member sizes. This provides a

need for replaceable links in certain cases.

112

5. The deformed EBF bent might impact the deck and adjacent girders due to the vertical

movement of the outside beam if it is part of link beam (continuous link). Therefore, the

behaviour of superstructure should also be considered for conventional bridges. One option

could be to provide a separate elastic element such as a truss between substructure and

superstructure; this might be helpful to reduce the differential movement of adjacent

girders such as used in the temporary towers for San Francisco–Oakland Bay Bridge

(EGBC, 2018). Another option could be to use a replaceable link member that is smaller

than the beam outside the link.

6.3 Limitations of this Research

This study has the following limitations:

• This study is carried out by taking the superstructure of an existing bridge with the as-

designed geometry and girder orientation. To avoid the gravity load being directly placed

on the link beam, it is placed at the ends of the link member, i.e., at the intersection of

braces, link beam, and outside cap beam. However, this strategy might not work for a

bridge having different geometry with more than four girders.

• Due to the limitation of experimental test data for EBFs with larger links in bridge bents,

the proposed acceptance criteria limits are mainly based on tests carried out for EBFs used

in buildings.

• As box sections are considered very stiff in torsion, it has been assumed that bracing against

lateral torsional buckling is not required for a replaceable built-up tubular links. No

experimental tests have been done so far to determine the requirement of bracing to avoid

113

lateral torsional buckling of link members smaller than the beam outside the link. There is

a need to verify this assumption.

6.4 Recommendations for Future Studies

Following are some recommendations for future research work:

• There is a need to recalibrate the 475-year return period related no-yielding criteria in

CHBDC S6-14 for obtaining practical designs. Also, more guidance needs to be provided

for rotational limits for higher return period events corresponding to required member sizes

for EBFs supporting bridge superstructures that are significantly larger than those usually

required for buildings.

• The majority of previous tests have been done on EBFs with WF links mainly used in

buildings. Test data for EBFs for tubular (box) links is very limited. There is a need to

conduct more testing programs using built-up tubular bents. This information will be

helpful in the evaluation of damage states at multiple performance levels.

• The connection between the brace and the beam/shear-link will need some careful thought

to identify satisfactory load paths for box sections.

• There is also a need to carry out tests on EBFs with replaceable links. A link beam smaller

than the outside beams results in small member sizes from capacity design forces.

• The effect of greater steel plate thicknesses and its impact on EBF link performance should

be examined through experimental testing. Attention will be needed for weld connections

needed to connect the plates of a built-up box section together.

114

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121

Appendices

Appendix A EBF Bent Design

A.1 Seismic Load Calculations for EBF Bent

EBF

C

IE = 1.5

H = 6.8 m

Ta = 0.462 sec

L = 9 m

R = 4

Elastic Modulus Es = 200000 MPa

Yield Strength Fy = 350 Mpa

T, sec 0 0.2 0.5 1 2 5 10

Sa(T), g - 1.413 1.32 0.824 0.497 0.158 0.055

Site Class Factor F(T) - 1 1 1 1 1 1

S(T), g 1.413 1.413 1.32 0.824 0.497 0.158 0.055

W= 14207.96 KN ← SLS load per each girder = 3470KN

V = S(T) IE W 18922 KN

Vb = (S(T) IE W)/ R 7096 KN

Vb= 7096 KN

VL = Vb. H/L 5361 KN

Total Weight

Elastic Base Shear

Modified Base Shear

Bent

Site Class

Importance Factor

Story Height

Fundamental Time Period

Total Frame Bay Width

Response Modification Factor

Sombrio UHS (2% in 50 years)

Required Link Strength

122

A.2 Link Member Design Calculations

Shear Link Section =

b= 800 mm

d= 750 mm

Flange thickness tf= 54 mm

Web thickness tw= 20 mm

Link length e= 3000 mm

Material Properties

Yield strength Fy = 350 MPa

Elastic Modulus Es= 200000 MPa

Section Properties

Gross Area Ag = b.h - (b-2tw).(h-2tf) Ag = 112080 mm2

Web Area Aw = 2.(h - 2tf). tw Aw = 25680 mm2

Plastic Shear Strength of Link Vp = 0.55 (2w)dFy Vp = 5775 KN

Link Demand/Capacity, Vu/Vp= 0.93 Ok

Z = tf . (b - 2.tw).(h - tf) + (tw. h^2)/2 Z = 34188840 mm3

Plastic Moment Strength Mp = Fy . Z Mp = 11966 KN-m

a = a= 7977 KN = 0.9

Ultimate Moment Mu = Mu = 10769 KN.m

Nominal Shear Vn= Vn = 5775 KN

Vu= Vu = 5198 KN

Vlink :- 1.45 Ry Vn Vlink = 9211 KN

ev= ev = 3.315 m

Link Type = Shear Link

rad 4.58 degreesMaximum Link rotation for shear links

750 x 800 x 54 x 20

Width

Ultimate shear

Probable Link Resistance

Depth

Plastic Section Modulus

For Shear Links

2 𝑀

𝑒0

𝑉

𝑀

If 𝑝 ≤2 𝑀𝑝

𝑒 𝑉

2 𝑀𝑝

𝑒)

𝑛

1.6𝑀

𝑝

𝛾𝑝 = 0.08

123

λ1 = 14.07

λf = 15.30

check_flange = OK

λ2 = 32.1

λw1 = 39.92

λw2 = 15.30

check_web = stiffner required

= OK

Maximum Shear Link Rotation 4.58 degress

Maximum Bent Rotation θp 1.526 degress ← L = 9m , e = 3m

Maximum story drift Δp 151 mm ← H = 6.8 m

Story drift from Response Spectrum Analysis 68 mm ← 2475 years return period UHS

OK

Shear Stiffness Ks = G Aw / e 659120 KN/m

Flexural Stiffness Kb = 12 EI/e3 1101663.256 KN/m

Ke = Ks Kb / Ks + Kb 412389.3629 KN/m

Ke = 412 KN/mm

Δ = Vy/Ke 14.0 mm

Yield Rotation θ = Δ/e 0.0047 rad

Compactness limits for flange

Elastic Stiffness of Link Beam

Link Rotation

Stiffness of Link Beam

Yield Displacement

Compactness limits for web

1 = − 2 𝑡𝑤

𝑡𝑓

𝑓 = 0.64 𝑠𝐹𝑦

2 = − 2 𝑡𝑓

𝑡𝑤

1 = 1.6 𝑠𝐹𝑦

𝑓 = 0.64 𝑠𝐹𝑦

𝛾𝑝 = 0.08

𝜃𝑝 = 𝛾𝑝𝑒

𝐿

𝑝 = 𝐻 𝜃

124

A.3 Capacity-Protected Bent Member Capacities

Beam Outside the Link:

=

b= 750 mm

h= 800 mm



Aw= 0.032 m2

Material Properties

Fy = 350000 KN/m2

Es= 200000000 KN/m2

G= 7.70E+07 KN/m2

Plastic Section Modulus Z = tf . (b - 2.tw).(h - tf) + (tw. h2)/2 Z= 35001640 mm3 0.03500164 m3

Ag= 108680 mm2 0.10868 m2

Ixx= 0.012393712 m4

Iyy= 0.008515 m4

Moment of Inertia Check Iyy ≥ 0.67 Ixx Iyy/Ix 0.7 Ok

Gross Area Ag = b.h - (b-2tw).(h-2tf)

Torsional Constant J:-= 0.012 m4

Beam-Outside-Link Length Lb= 3 m

Total Bay Width L= 10.97 m

K= 1 ← Pin-pin connection (conservative approach)

r= 0.2756 m ← Minimum of rx and ry

n= 1.34 ← HSS Class C sections

ϕ= 0.9

λ = 1.45E-01

Cr Beam= 34091 KN

1.22* Cr Beam 41591 KN [CSA S16-14, Clause 27.7.9.2 ]

Link Beam is Continuous as Beam-Outside-the-Link

Elastic Modulus

Shear Modulus

Axial Compressive Resistance

750 x 800 x 54 x 20

Yield strength

CSA S6-14 Clause 10.9.3 Axial Compressive Resistance

Width

Depth

Web area

Beam Section

= 𝑠𝐴 𝐹𝑦 1 2𝑛 1 𝑛

125

Mp= ZFy 12251 KN-m

= 0.67 Mp 8208 KN-m

2.5 ← Manually calculated

Cw = 0 ← For hollow rectangular sections

For doubly symmetric sections 3282401.57 KN-m

12666.09 KN-m 11026

When Mu > 0.67Mp

Mr Beam= 11026 KN-m

1.2Mr Beam= 13231 KN-m [CSA S16-14, Clause 27.7.9.2 ]

Kv= 5.34

h/w= 1.07

1.961 OK

Ft = 0

Fcr = 2.02E+05 KN/m2

Fs = 2.02E+05 KN/m2

5816.16 KN-m

Vr Beam = 5816 KN-m

Link Beam is Continuous as Beam-Outside-the-Link 1.2 Vr Beam = 6979 KN-m [CSA S16-14, Clause 27.7.9.2 ]

CSA S6-14 Clause 10.10.5.1 Factored Shear Resistance

CSA S6-14 Clause 10.10.2.3 Moment resistance for Laterally unbraced members

2 =

≤𝑀 = 1.15∅𝑠𝑀𝑝 1 −

0.28𝑀𝑝

𝑀 ≤ ∅𝑠𝑀𝑝

ℎ𝑒 ℎ

≤ 502

𝐹𝑦

𝐹 = 0.5 𝐹𝑦𝐹𝑡 = 0

502 𝑉𝐹𝑦

𝑉 = ∅𝑠𝐴𝑤𝐹𝑠

𝑀 = 2

𝐿 𝑠𝐼𝑦 𝑠

𝑠𝐿

2

𝐼𝑦 𝑤

126

Brace:

=

b= 650 mm

h= 650 mm



Aw= 0.052 m2

Material Properties

Fy = 350000 KN/m2

Es= 200000000 KN/m2

G= 7.70E+07 KN/m2

Plastic Section Modulus Z = tf . (b - 2.tw).(h - tf) + (tw. h2)/2 Z= 22358000 mm3

Ag= 97600 mm2 0.0976 m2

Ixx= 0.006078853 m4

Iyy= 0.006078853 m4

Gross Area Ag = b.h - (b-2tw).(h-2tf)


Brace Length L= 7.43 m




ϕ= 0.9

λ = 3.96E-01

Link Beam is Continuous as Beam-Outside-the-Link Cr Brace= 28959 KN

Mp= ZFy 7825 KN-m

= 0.67 Mp 5243 KN-m




8073.45 KN-m 7042.8

When Mu > 0.67Mp

Mr Brace= 7043 KN-m



Axial Compressive Resistance

Brace Section 650 x 650 x 40 x 40

Width

Depth

Web area

Yield strength

Elastic Modulus

Shear Modulus

2 =

≤

= ∅𝑠𝐴𝐹𝑦 1 2𝑛 1𝑛

𝑀 = 2


𝑠𝐿

2

𝐼𝑦 𝑤

𝑀 = 1.15∅𝑠𝑀𝑝 1 −0.28𝑀𝑝

𝑀

≤ ∅𝑠𝑀𝑝

127

Kv= 5.34

h/w= 1

1.961 Ok

Ft = 0

Fcr = 2.02E+05 KN/m2

Fs = 2.02E+05 KN/m2

9451 KN-m

Vr Brace = 9451 KN-m


502𝑘 𝐹𝑦

=

ℎ𝑒 ℎ

≤ 502

𝐹𝑦

𝐹 = 0.5 𝐹𝑦𝐹𝑡 = 0


128

Column:

=

b= 250 mm

h= 600 mm



Aw= 0.036 m2

Material Properties

Fy = 350000 KN/m2

Es= 200000000 KN/m2

G= 7.70E+07 KN/m2

Plastic Section Modulus Z = tf . (b - 2.tw).(h - tf) + (tw. h2)/2 Z= 8649000 mm3

Gross Area Ag= 47400 mm2 0.0474 m2

Ixx= 0.00200682 m4

Iyy= 0.000472595 m4

Ag = b.h - (b-2tw).(h-2tf)


Column Section L= 6.8 m




ϕ= 0.9

λ = 9.05E-01

Cr Column= 9767 KN


Column Section 600 x 250 x 30 x 30

Width

Depth

Web area

Yield strength

Elastic Modulus

Shear Modulus

Axial Compressive Resistance = ∅𝑠𝐴𝐹𝑦 1 2𝑛 1𝑛

129

Mp= ZFy 3027 KN-m

= 0.67 Mp 2028 KN-m




3098.25 KN-m 2724

When Mu > 0.67Mp

Mr Column= 2724 KN-m

Kv= 5.34

h/w= 2

1.961

Ft = 0

Fcr = 2.02E+05 KN/m2

Fs = 2.02E+05 KN/m2 ← Conservative approach

6543 KN-m

Vr Column = 6543 KN-m



2 =

≤

502𝑘 𝐹𝑦

=

ℎ𝑒 ℎ

≤ 502

𝐹𝑦

𝐹 = 0.5 𝐹𝑦𝐹𝑡 = 0

𝑀 = 2


𝑠𝐿

2

𝐼𝑦 𝑤

𝑀 = 1.15∅𝑠𝑀𝑝 1 −0.28𝑀𝑝

𝑀

≤ ∅𝑠𝑀𝑝


130

A.4 Beam-Column Design for Brace and Beam Outside the Link

Angle of brace = 1.155 rad

Axial force in brace = 13857 KN

= 5593 KN

= 13817 KN-m

Moment distribution factor (DF) for brace = 0.165

= 2284 KN-m

Remining Moment taken by Beam = 11533 KN-m

Interaction Equation for Beam

Cf = 5593 KN

Cr = 41591 KN

= 1 [CSA S6-14 Clause 10.9.4.2]

ce = 2715490

U1 = 0.9979

Mf = 11533

Mr = 13231

= 1.00 Ok

Interaction Equation for Brace

Cf = 13857 KN

Cr = 28959 KN

= 1 [CSA S6-14 Clause 10.9.4.2]

ce = 217137

U1 = 1

Mf = 2284

Mr = 7043

= 0.782 Ok

Axial force in beam

Moment in beam

Moment in Brace M beam x DF Brace

Brace will take a portion of moment depending upon the relative stiffness (I/L) of the brace and the beam segment outside the link.

Forces in Beam and Brace due to Fully Yielded and Strain Hardened Link (Capacity Design)

1

1

𝑇𝑏 𝑒 = 𝑉

(

𝐿

𝐿 𝑒)

𝑏𝑒 = 𝑇𝑏 𝑒 𝜃

𝑀𝑏𝑒 = 𝑉𝑙𝑖𝑛𝑘𝑒

2

+ 𝑥 𝑀 𝑥

𝑀 𝑥+ 𝑦 𝑀 𝑦

𝑀 𝑦≤ 1.0

+ 𝑥 𝑀 𝑥

𝑀 𝑥+ 𝑦 𝑀 𝑦

𝑀 𝑦≤ 1.0

131

Appendix B Spectral Matching of Ground Motions

B.1 1 in 475-year Return Period Target Spectrum

Figure 40 475-year GM Records (Subduction)

132

Figure 41 475-year GMs Spectrally Matched Accelerograms (Subduction)

Figure 42 475-year GM Records (Subcrustal)

133

Figure 43 475-year GMs Spectrally Matched Accelerograms (Subcrustal)

134

Figure 44 475-year GM Records (Crustal)

Figure 45 475-year GMs Spectrally Matched Accelerograms (Crustal)

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performance-based seismic design and evaluation of steel

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