seismic analysis and design of a concrete arch bridgebb/msritgerdir/trausti.hannesson.2010.pdf ·...

Trausti Hannesson

Seismic Analysis and Design of aConcrete Arch Bridge

Direct Displacement-based Design Approach toSeismic Isolation

Masters Thesis, April 2010

Seismic Analysis and Design of a Concrete Arch Bridge, Direct Displacement-basedDesign Approach to Seismic Isolation

This report was prepared byTrausti Hannesson

SupervisorsChristos Georgakis, Technical University of DenmarkBjarni Bessason, University of Iceland

Release date: April 2010Category: 1 (public)

Edition: First

Comments: This report is part of the requirements to achieve the Master ofScience in Engineering (M.Sc.Eng.) at the Technical Universityof Denmark. This report represents 30 ECTS points.

Rights: Trausti Hannesson, 2010

Department of Civil EngineeringTechnical University of DenmarkBrovej building 118DK-2800 Kgs. LyngbyDenmark

www.byg.dtu.dkTel: (+45) 45 25 17 00E-mail: [email protected]

Preface

The work presented here is part of the requirements to achieve the Masterof Science in Engineering (M.Sc.Eng.) at the Department of Civil Engi-neering at the Technical University of Denmark. This report represents 30ECTS points. The work was performed at the Faculty of Engineering at theUniversity of Iceland.

I am grateful to my supervisor professor Bjarni Bessason at the Universityof Iceland for his guidance, ideas and encouragement during my thesis work.I would also like to thank professor Christos Georgakis at the Technical Uni-versity of Denmark for his comments and for giving me the opportunity todo my studies in Iceland. Finally I would like to thank Helgi Valdimarsson,Managing Director at Almenna Consulting Engineers for providing me witha working space during my thesis work.

Akureyri, April 2010

Trausti Hannesson [s080012]

Abstract

In this thesis the response of a concrete arch bridge to seismic loads corre-sponding to the South Iceland Seismic Zone (SISZ) is calculated. The an-alyzed bridge was originally designed for non-seismic load conditions. TheSouth Iceland Seismic Zone is an active seismic zone and several times sincethe settlement of Iceland structures have collapsed and casualties been re-ported in earthquakes in that area.

The main objective of this thesis is to evaluate the effect of changing thebridge location and come up with a suitable design alternative. Linear re-sponse spectrum analysis is performed with the general purpose FE-programSAP2000 from which it is clear that the bridge is in need of redesign towithstand the seismic loads occurring in the South Iceland Seismic Zone.

A direct displacement-based design approach is employed to design an isola-tion system using lead rubber bearings. Linear response spectrum analysisis performed using equivalent linear stiffness and damping to model the non-linear behavior of the base isolation. The nonlinear behavior of the isolationsystem was then further investigated with nonlinear time history analysisusing artificial ground motions.

The study shows that by introducing lead rubber bearings as the only changeto the original design the response to seismic loads can be significantly im-proved. The direct displacement-based design approach to the design ofseismically isolated structures proved to be simple and to offer control ofthe total structural response. Simple hand calculations were verified bylinear response spectrum analysis but considerable difference was observedbetween the linear and nonlinear methods in terms of expected displacementof the bridge deck and hence section forces. Results from response spectrumanalysis were in all cases conservative.

Keywords: seismic engineering, bridge engineering, direct displacement-based de-sign, seismic isolation, lead rubber bearings

Contents

List of Figures viii

List of Tables xii

Nomenclature xiv

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Theory 5

2.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Single Degree of Freedom . . . . . . . . . . . . . . . . 5

2.1.2 Multiple Degrees of Freedom . . . . . . . . . . . . . . 6

2.2 Definition of Seismic Load . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Time Histories . . . . . . . . . . . . . . . . . . . . . . 7

2.2.2 Response Spectrum . . . . . . . . . . . . . . . . . . . 8

2.3 Seismic Isolation . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Lead Rubber Bearings . . . . . . . . . . . . . . . . . . 10

2.3.1.1 Equivalent Linear Model . . . . . . . . . . . 13

2.4 Performance Based Design . . . . . . . . . . . . . . . . . . . . 15

2.4.1 Direct Displacement-Based Design . . . . . . . . . . . 17

2.4.1.1 Design Displacement . . . . . . . . . . . . . 18

2.4.1.2 Fundamentals . . . . . . . . . . . . . . . . . 19

2.4.1.3 Elastic Stiffness of Cracked Concrete Sections 20

2.4.1.4 Displacement-Based Design of Isolated Bridges 22

2.4.1.5 Comparison With Force-based Design Ap-proach . . . . . . . . . . . . . . . . . . . . . 28

2.5 Methods of Analysis . . . . . . . . . . . . . . . . . . . . . . . 29

2.5.1 Response Spectrum Analysis . . . . . . . . . . . . . . 30

2.5.2 Nonlinear Time History Analysis . . . . . . . . . . . . 32

2.6 Eurocode 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.7 Ultimate Strength of Elements . . . . . . . . . . . . . . . . . 36

2.7.1 The Arch . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.7.2 Shear Capacity . . . . . . . . . . . . . . . . . . . . . . 38

2.7.3 Piers and Spandrel Columns . . . . . . . . . . . . . . 38

2.7.4 Bridge Deck . . . . . . . . . . . . . . . . . . . . . . . . 39

3 The Bridge and Applied Load 41

3.1 The Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.1 Structural Elements . . . . . . . . . . . . . . . . . . . 42

3.1.2 Material Properties . . . . . . . . . . . . . . . . . . . . 43

3.1.3 Isolation Devices . . . . . . . . . . . . . . . . . . . . . 44

3.1.4 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Applied Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.1 Dead Load . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.2 Traffic Loads . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.3 Temperature Load . . . . . . . . . . . . . . . . . . . . 51

3.2.4 Seismic Load . . . . . . . . . . . . . . . . . . . . . . . 52

3.2.4.1 The Design Response Spectrum . . . . . . . 52

3.2.4.2 Time Histories . . . . . . . . . . . . . . . . . 55

vii

3.2.5 Load Combinations . . . . . . . . . . . . . . . . . . . 57

4 Static and Linear Dynamic Analysis 59

4.1 Computational Model . . . . . . . . . . . . . . . . . . . . . . 59

4.2 Original Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2.1 Response to Static Loads . . . . . . . . . . . . . . . . 60

4.2.2 Response to Seismic Loads . . . . . . . . . . . . . . . 64

4.3 Base Isolated Bridge . . . . . . . . . . . . . . . . . . . . . . . 69

4.3.1 Response of Simplified Model vs. Hand Calculations . 70

4.3.2 Response to Seismic Loads . . . . . . . . . . . . . . . 72

4.4 Thoughts on Redesign . . . . . . . . . . . . . . . . . . . . . . 77

5 Time History Analysis 79

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2 Nonlinear Time History Analysis . . . . . . . . . . . . . . . . 81

6 Summary and Conclusions 85

References 87

Appendix 89

A Direct Displacement-based Hand Calculations 89

List of Figures

1.1 Recorded earthquakes in Iceland in 2003 . . . . . . . . . . . . 2

1.2 A collapsed part of the Hanshin Expressway from the Hyogo-ken Nanbu earthquake in 1995 . . . . . . . . . . . . . . . . . 3

2.1 Artificial seismic ground acceleration time history . . . . . . . 7

2.2 Response spectrum of time history in Figure 2.1 . . . . . . . 8

2.3 The effect of seismic isolation shown on acceleration and dis-placement spectrum . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 left: Lead rubber bearing with top and bottom plates vulcan-ized to the rubber [Skinner (1993)] right: Lead rubber bearingcut in half . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Bilinear hysteresis loop of a lead rubber bearing . . . . . . . . 11

2.6 Equivalent damping from Eurocode 8 and proposed improvedequation by Jara and Casas . . . . . . . . . . . . . . . . . . . 15

2.7 Influence of strength on Moment-Curvature relationship [Priest-ley (2000)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.8 Fundamentals of Direct Displacement-Based Design [Priest-ley et al. (2007)] . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.9 Effective stiffness ratio for large rectangular columns [Priest-ley et al. (2007)] . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.10 Damping for a cantilever pier with an isolated deck [Priestleyet al. (2007)] . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

x LIST OF FIGURES

2.11 Flowchart for the displacement-based design for isolated struc-tures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.12 Simple flowcharts for force-based and direct displacement-based design of isolated structures . . . . . . . . . . . . . . . 28

2.13 Modified response spectrum to consider the equivalent viscousdamping of the isolation system [Priestley et al. (1996)] . . . 32

2.14 Rayleigh damping - variation of modal damping ratios withnatural frequency [Chopra (2007)] . . . . . . . . . . . . . . . 34

2.15 Strain and stress distribution and force equilibrium in a con-crete section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.16 Simplified N-M interaction diagram for uniaxial bending . . . 39

3.1 Three dimensional view of the bridge . . . . . . . . . . . . . . 42

3.2 General cross section in bridge . . . . . . . . . . . . . . . . . 43

3.3 Predominant modeshapes of the original bridge in the longi-tudinal, transverse and vertical direction respectively . . . . . 46

3.4 Predominant modeshapes of the isolated bridge in the longi-tudinal, transverse and vertical direction respectively . . . . . 46

3.5 Details of LM1, placement of loads [EC1-2] . . . . . . . . . . 50

3.6 Assessment of groups of traffic loads [Table 4.4a in EC1-2] . . 50

3.7 Horizontal peak acceleration for Iceland [FS ENV 1998-1-1:1994] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.8 Horizontal design response spectrum. Predominant horizon-tal modes indicated with diamonds . . . . . . . . . . . . . . . 54

3.9 Vertical design response spectrum. Predominant vertical modesindicated with diamonds . . . . . . . . . . . . . . . . . . . . . 54

3.10 Displacement response spectrum used in direct displacement-based design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.11 Response spectra of artificial time histories used for time his-tory analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.12 Artificial time histories created with Simqke used for timehistory analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.1 Screen shot of computational model from SAP2000 . . . . . 60

4.2 Axial force and bending moment in arch due to static loads . 61

LIST OF FIGURES xi

4.3 Stresses in arch top and bottom respectively due to static loads 61

4.4 Shear force in arch due to static loads . . . . . . . . . . . . . 62

4.5 Simplified N-M interaction diagrams for piers and spandrelcolumns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.6 Shear force in piers and spandrel columns due to static loads.Shear strength shown with dashed line . . . . . . . . . . . . . 63

4.7 Bending moment in bridge deck due to static loads . . . . . . 63

4.8 Shear force in bridge deck due to static loads . . . . . . . . . 64

4.9 Axial force and bending moment around transverse axis inarch of original bridge due to seismic loads . . . . . . . . . . . 65

4.10 Bending moment around vertical axis in arch of original bridgedue to seismic loads . . . . . . . . . . . . . . . . . . . . . . . 65

4.11 Stresses in arch top and bottom of original bridge due toseismic loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.12 Vertical shear force in arch of original bridge due to seismicloads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.13 Transverse shear force in arch of original bridge due to seismicloads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.14 Simplified N-M interaction diagrams for piers and spandrelcolumns of original bridge. Points indicate static and seismicloaded state of columns . . . . . . . . . . . . . . . . . . . . . 68

4.15 Shear force in piers and spandrel columns of original bridgedue to seismic loads. Shear strength shown with dashed line . 68

4.16 Moment in bridge deck of original bridge due to seismic loads 69

4.17 Shear in bridge deck of original bridge due to seismic loads . 69

4.18 Axial force and bending moment around the transverse axisin the arch of base isolated bridge due to seismic loads . . . . 72

4.19 Bending moment in arch of base isolated bridge around ver-tical axis due to most severe seismic load combination . . . . 73

4.20 Stresses in arch top and bottom of base isolated bridge dueto most severe seismic load combination . . . . . . . . . . . . 74

4.21 Vertical shear force in arch of base isolated bridge due toseismic loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

xii LIST OF FIGURES

4.22 Transverse shear force in arch of base isolated bridge due toseismic loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.23 Simplified N-M interaction diagrams for piers and spandrelcolumns of base isolated bridge. Points indicate static andseismic loaded state of columns . . . . . . . . . . . . . . . . . 75

4.24 Shear force in piers and spandrel columns of base isolatedbridge due to seismic loads. Shear strength shown with dashedline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.25 Moment in bridge deck of base isolated bridge due to seismicloads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.26 Shear in bridge deck of base isolated bridge due to seismic loads 76

5.1 Displacement spectra for artificial time histories and Eurocode8. Displacements in m . . . . . . . . . . . . . . . . . . . . . . 80

5.2 Hysteresis loops for LRBs . . . . . . . . . . . . . . . . . . . . 83

5.3 Displacements of deck, pier and isolator for pier no. 2 as afunction of time . . . . . . . . . . . . . . . . . . . . . . . . . . 83

List of Tables

3.1 Physical properties of lead rubber bearings . . . . . . . . . . 44

3.2 Mechanical properties of lead rubber bearings . . . . . . . . . 45

3.3 Modal analysis results of original bridge . . . . . . . . . . . . 46

3.4 Modal analysis results of isolated bridge . . . . . . . . . . . . 47

3.5 Required number of modes to reach an effective mass ratio of90% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.6 Self weight of main structural elements . . . . . . . . . . . . . 48

3.7 LM1: Characteristic values . . . . . . . . . . . . . . . . . . . 49

3.8 Input to design response spectrum . . . . . . . . . . . . . . . 52

3.9 Parameters for Type 1 design response spectrum . . . . . . . 53

3.10 Recommended values of factors for road bridges (EC0 A.2) 57

3.11 Combination of actions . . . . . . . . . . . . . . . . . . . . . . 57

3.12 Combinations of components of seismic action . . . . . . . . . 58

4.1 Comparison of uncracked, cracked and required stiffness forpiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2 Comparison of displacements and deformations with RSA andDDBD (all values in mm) . . . . . . . . . . . . . . . . . . . . 71

4.3 Comparison of shear force and moment in columns with RSAand DDBD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

xiv LIST OF TABLES

5.1 Maximum displacements [mm] from linear time history anal-ysis (LTHA) and displacement spectra (DS) . . . . . . . . . . 81

5.2 Maximum displacements and base shear from nonlinear timehistory analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.3 Comparison of section forces in piers with RSA and NLTHAwith time history no. 6 . . . . . . . . . . . . . . . . . . . . . 84

Nomenclature

Abbreviations:DDBD Direct Displacement-Based DesignFBD Force Based DesignLTHA Linear Time History AnalysisMDOF Multi Degree Of FreedomNLTHA Noninear Time History AnalysisRSA Response Spectrum AnalysisSDOF Single Degree Of FreedomSIZS South Iceland Seismic Zone

Symbols not defined at each usage:

d Design displacement m

u Maximum displacement of lead rubber bearing m

y Yield displacement m

d Design ductility ratio

y Yield curvature 1m

yL Effective yield shear stress of lead Pa

AL Cross sectional area of lead plug m2

AR Plane area of rubber m2

GR Shear modulus of rubber Pa

Keff Effective stiffness of lead rubber bearing Nm

Ke Effective stiffness of equivalent SDOF system Nm

KLR Initial stiffness of lead rubber bearing Nm

xvi LIST OF TABLES

KR Post yield stiffness of lead rubber bearing Nm

KZ Vertical stiffness of rubber bearing Nm

me Effective mass of equivalent SDOF system kg

Qy Characteristic yield strength of lead plug N

Te Effective period of equivalent SDOF system s

tR Total thickness of rubber m

Vbase Base shear N

Chapter 1

Introduction

1.1 Background

A preliminary design of a concrete arch bridge was performed by the authorof this thesis in the course 11373 - Bridge Design, taught at the TechnicalUniversity of Denmark in the fall of 2008. The bridge was supposed to beconstructed in Finland and hence no consideration was made on seismicdesign. In this thesis the consequences of erecting the bridge in the SouthIcelandic Seismic Zone (SISZ) will be examined.

Iceland is located on the Mid Atlantic Ridge and is being split by the diver-gent plate boundary between the North American Plate and the EurasianPlate which causes regular earthquakes and eruptions. Two main seismiczones exist in Iceland, the Tjornes Fracture Zone (TFZ) and the South Ice-land Seismic Zone. The most destructive earthquakes have been recordedin these two zones.

Bridges are lifeline structures, important links in the infrastructure and theirfailure during an earthquake will seriously hamper relief and rehabilitationwork. It is therefore important that bridges exposed to severe earthquakescan survive without serious damage and be opened for traffic quickly. Fail-ures to bridges due to complete failure of their piers or other structuralelements have been observed in every major seismic event, e.g. in Hyogo-ken Nanbu in 1995 [Jangid (2004)], San Fernandi in 1971 [Dusseau and Wen(1989)], Northridge in 1994 [Priestley et al. (1996)] to name a few. Evenbridges designed specifically for seismic resistance have collapsed or havebeen severely damaged when subjected to ground shaking of an intensity of-

2 Introduction

Figure 1.1: Recorded earthquakes in Iceland in 2003

ten less than that corresponding to current code intensities [Priestley et al.(1996)].

To increase the safety of bridges during a seismic event a relatively newmethod in earthquake-resistant design has been developed called seismic iso-lation. Scientists and engineers in New Zealand, the United States, Japanand Italy have been active in developing the concept and using it in bothbuildings and bridges [Skinner (1993), Naeim and Kelly (1999)]. Many iso-lations systems have been proposed through the years (see e.g. [Naeim andKelly (1999)]) and they all aim at reducing the earthquake damage potentialby uncoupling the structure from the damaging action of the earthquake.The most commonly used isolators today are lead rubber bearings as theyprovide an economic, reliable and simple solution for protecting medium andshort span bridges [Jara and Casas (2006), Bessason and Haflidason (2004)].Lead rubber bearings have been used in all 15 seismically isolated bridgesin Iceland [Jonsson et al. (2010)].

In June 2000 two major earthquakes struck the South Iceland Seismic Zonewith magnitudes MW = 6.6 and MW = 6.5. During these earthquakes theresponse of a recently seismically upgraded bridge with seismic isolation wasrecorded. The bridge was instrumented with accelerometers monitoring theground motion as well as the structural response. The bridge survived bothearthquakes without any serious damage but numerical analysis strongly

1.1 Background 3

Figure 1.2: A collapsed part of the Hanshin Expressway from the Hyogo-kenNanbu earthquake in 1995

indicated that without the base isolation the bridge would have been severelydamaged or even collapsed [Bessason and Haflidason (2004)]. In May 2008another earthquake hit South Iceland (MW=6.3). The base isolated Oseyrarbridge was subjected to strong near-fault ground motion and experiencedsome damage but was opened for traffic only few hours after the earthquake[Jonsson et al. (2010)].

Damage of bridges during the Hyogo-ken Nanbu (Kobe) earthquake in 1995attracted considerable attention among researches realizing that strengthalone would not be sufficient for the safety of bridges [Jangid (2004)]. Thishas lead to a demand for a different approach to seismic design and to-day it is widely recognized that seismic design codes need to incorporate aperformance-based design criterion. A simple but reliable conceptual frame-work called direct-displacement based design has been proposed for achievingthe performance based design objective. It is generally agreed that deforma-tions are more critical parameters for defining performance, and as a resultit is argued that seismic design methods should largely be based on them. Incontrast to force-based design, the end result of the displacement-based de-sign procedure is the required stiffness, which is determined from the elasticspectrum by means of maximum total displacement and equivalent dampingof the system.

4 Introduction

1.2 Objectives

This thesis has two main objectives. The first is to extend the preliminarydesign of a concrete arch bridge, made in the course11373 - Bridge Design,to include a seismic design and to study how the application of seismicloading affects the design concept. Instead of the original position of thebridge in Finland, the bridge is analyzed for a code-specified seismic load insouth of Iceland.

The second main objective of this thesis is to employ the direct displacement-based design approach to design a base isolation system for the bridge andcompare it to the traditional force-based design. The accuracy of simplehand calculations, proposed by the direct displacement-based design ap-proach, is assessed with response spectrum and nonlinear time history anal-ysis.

The main chapters are as follows:

Second chapter: The basic theory used is presented. Basics of structuraldynamics, definition of seismic load with response spectra and time histo-ries, seismic isolation, performance based design, basics of Eurocode 8 andultimate strength of elements.

Third chapter: The analyzed bridge is described in detail as well as theisolation devices. The applied load is defined.

Fourth chapter: Static and linear dynamic analysis of the original and theisolated bridge. The calculation process is described and results presented.

Fifth chapter: Time history analysis of the isolated bridge is described andmain findings presented.

Sixth chapter: Summary and conclusions.

Chapter 2

Theory

2.1 Equations of Motion

2.1.1 Single Degree of Freedom

The response of a linear single degree of freedom (SDOF) system to groundmotions ug is the solution to the differential equation [Chopra (2007)]

mu+ cu+ ku = mug (2.1)

where m is the mass of the system, c is the damping constant and k isthe stiffness. Relative displacement of the system as a function of time isdenoted u(t). First derivative of the displacement is the velocity u(t) andthe second derivative is the acceleration of the system u(t).

By dividing the equation above with the mass m the normalized equationof motion is obtained.

u+ 20u+ 20u = ug (2.2)

where 0 is the natural angular frequency and is the damping ratio of thesystem.

0 =

km =

c

2km

(2.3)

6 Theory

The relation between period T and natural angular frequency 0 is given by

T =20

= 2

mk (2.4)

2.1.2 Multiple Degrees of Freedom

The dynamic behavior of most structures involves simultaneous motion ofseveral masses in shapes that are not known before the analysis. Thus, thetheory of dynamics of a single degree of freedom must be extended to dealwith several masses and systems with distributed mass like beam and framestructures, as well as complete buildings and structures such as bridges.

The first extension of the theory is from one degree of freedom involving asingle mass to multiple degrees of freedom, describing the coupled motionof several concentrated masses. This theory is called modal analysis andincludes terms such as mode shapes, modal mass and modal stiffness.

The objective of modal analysis is to determine combined motion of massesthat will retain the same combination during free vibrations and to deter-mine the corresponding frequency, loading conditions etc.

The dynamic response of a linear system with n degrees of freedom u(t)T =[u1(t), u2(t), ..., un(t)] to ground motions is described by the set of secondorder differential equations

mu + cu + ku = mIug (2.5)

The physical parameters are: the mass matrix m, the viscous damping ma-trix c and the stiffness matrix k. Effective earthquake forces are given by thevector mIug where I is the influence vector representing the displacementsof the masses resulting from static application of a unit ground displacement.

Mode shapes and periods are found by solving the generalized eigenvalueproblem

(k 2m

) = 0 (2.6)

The complete solution to the generalized eigenvalue problem consists of nsets of eigenvalues and eigenvectors, arranged as corresponding pairs of nat-ural frequency j and mode shape vector j .

j , j j = 1, 2, ..., n (2.7)

2.2 Definition of Seismic Load 7

Traditionally the mode shapes are ordered with respect to increasing magni-tude of the associated natural frequency. The generalized eigenvalue prob-lem does not determine the magnitude of the modeshape vectors. They cantherefore be normalized by multiplication with a suitable scaling factor.

Modal mass and modal stiffness corresponding to mode j are defined as

mj = Tj mj kj = Tj kj (2.8)

The eigenfrequency j can be expressed by the ratio of modal stiffness tomodal mass by premultiplication of the generalized eigenvalue equation withTj . This relation is called Rayleighs quotient and generalizes the definitionof the angular frequency for a SDOF system [Chopra (2007)].

2j =Tj kjTj mj

=kjmj

(2.9)

2.2 Definition of Seismic Load

2.2.1 Time Histories

From the equations of motion above it is clear that seismic load is intro-duced through ground motions during a seismic event. A time history is adescription of ground acceleration (or related quantities such as velocity anddisplacement) as a function of time. There are three basic options availablein terms of obtaining time histories [Bommer and Acevedo (2004)]:

1. Artificial spectrum-compatible accelerograms generated using com-puter programs

2. Synthetic accelerograms generated from seismological source modelsaccounting for path and site effects

3. Real accelerograms recorded during earthquakes

Figure 2.1: Artificial seismic ground acceleration time history

8 Theory

The first option listed above is the one used in this thesis. Artificial accelero-grams are created using the Matlab program SimSeisme [Lestuzzi (2002)]based on the well know program Simqke, created by Gasparini and Vanmar-cke. This option was chosen to obtain time histories with response spectrasimilar to the smooth EC8 design spectrum for comparison of the results ob-tained using response spectrum analysis with isolating devices representedwith linear equivalent properties and the time history analysis with nonlin-ear modeling of the isolation devices. The approach employed in Simqke isto generate a power spectral density function from the smoothed responsespectrum of EC8 (or some other target spectrum) and then derive sinusoidalsignals having random phase angles and amplitudes. The sinusoidal motionsare then summed and an iterative procedure can be invoked to improve thematch with the target response spectrum by calculating the ratio betweenthe target and actual response ordinates at selected frequencies. The powerspectral density function is then adjusted by the square of this ratio, and anew motion is generated. By doing so it is possible to obtain accelerationtime histories that are almost completely compatible with the elastic designspectrum.

2.2.2 Response Spectrum

A response spectrum describes the maximum response of a single degreeof freedom system to a ground motion. The response of such a system isgiven with the normalized equation of motion (Equation 2.2) from which itis clear that when the period and damping of a SDOF system is known theresponse can be calculated for a given time history. A response spectrumis thus the response of a single degree of freedom system as a function ofperiod for a given level of damping. In Figure 2.2 the response spectrum forthe artificially created time history in Figure 2.1 is plotted.

Figure 2.2: Response spectrum of time history in Figure 2.1

2.3 Seismic Isolation 9

2.3 Seismic Isolation

The aim of seismically isolating a structure is to increase the resistance toseismic loading. Increased resistance is achieved by increasing flexibilityand energy dissipation capacity. The damage potential of an earthquake iscaused by the unfortunate correspondence between the fundamental periodsof vibration of the majority of structures and the frequency content of theseismic input. By increasing flexibility and hence the fundamental periodsof vibration seismic forces can be significantly reduced, see Figure 2.3. FromEquation 2.2 it is also apparent that by increasing the viscous damping, theforces to be resisted by the structure decrease. This is also apparent fromFigure 2.3 which shows both an acceleration and a displacement spectrumand indicates the effect that increased flexibility and damping has on theresponse.

Figure 2.3: The effect of seismic isolation shown on acceleration and displacementspectrum

The concept of seismic isolation can be applied to a variety of complexstructures. Isolating/dissipating devices are inserted between different partsof the structural system with the purpose of dissipating energy, controllingrelative displacements and capacity-protecting structural elements againstpotentially brittle failure modes.

Many different isolation systems have been proposed through the years, but

10 Theory

systems based on elastomeric bearings are most common. In a typical iso-lated bridge, special isolation devices are used in place of the conventionalbridge bearings or monolithic connections between piers and bridge deck.The suitability of a particular arrangement and type of isolation system willdepend on many factors, including the span length, number of continuousspans, seismicity of the region, maintenance and replacement facilities. Com-mon isolation devices include elastomeric bearings, lead rubber bearings,high-damping rubber bearings and friction pendulum systems. Dampingcan either be included in the devices or provided with secondary mechanicaldevices.

2.3.1 Lead Rubber Bearings

Lead rubber bearings are low-damping laminated rubber bearings with alead plug inserted in the core of the device. The purpose of the lead plugis to increase the stiffness at relatively low horizontal force levels as well asto increase energy dissipation capacity of the bearing. The horizontal force-displacement curve is a combination of the linear response of the rubberbearing and the essentially elastic-perfectly plastic response of a confinedlead plug. A lead rubber bearing thus combines the displacement capacityof the rubber with hysteretic energy dissipation of the lead plug providingthe damping required for a seismic isolation system to be efficient. Undernormal conditions they behave like regular bearings but in the event of astrong earthquake, they add flexibility to the structure by elongating itsperiod and dissipate energy.

Lead rubber bearings were invented in New Zealand during the mid 1970sby W.H Robinson [Skinner (1993)]. Since then they have grown to becomethe most common isolation devices for bridges and up until now all the seis-mically isolated bridges in Iceland are using lead rubber bearings [Bessasonand Haflidason (2004)].

A reasonable description of the hysteresis loop of a lead rubber bearing is abilinear solid with an initial elastic stiffness of KLR followed by a post yieldstiffness of KR, see Figure 2.5. Also shown in Figure 2.5 is the characteristicyield strength of the lead plug at zero strain Qy, effective stiffness Keff ,yield and maximum displacement, y and u, as well as the correspondingyield and maximum strength, Vy and Vu.

The common way to insert the lead plug into the elastomeric bearing isto manufacture a bearing with a hole and then press an oversized lead pluginto it. However, in Iceland a non-traditional method has been used where ahole is drilled into a conventional elastomeric bearing, and subsequently thelead plug is cast directly into the hole. This results in somewhat different


Figure 2.4: left: Lead rubber bearing with top and bottom plates vulcanized tothe rubber [Skinner (1993)] right: Lead rubber bearing cut in half

Figure 2.5: Bilinear hysteresis loop of a lead rubber bearing

characteristics of the Icelandic lead rubber bearings than is documentedin literature. Dynamic tests were performed on the Icelandic bearings andbased on those tests the following parameter set is introduced [Bessason andHaflidason (2004)]

KR =ARGRtR

(2.10)

KLR = 11.6KR (2.11)

12 Theory

Qy = yLAL (2.12)

where AR is the plane area, GR is the shear modulus and tR is the totalthickness of the rubber in the bearing. AL is the cross sectional area of leadplug and yL is the estimated effective yield shear stress of the lead plug. Theparameter 11.6 in Equation 2.11 is higher than normally found in literaturewhere it has been reported to be approximately 10 (see e.g. [Skinner (1993),Jara and Casas (2006), Naeim and Kelly (1999)]). The effective yield shearstress of the lead plug was determined as yL = 8.0 MPa which is about 20%lower than documented elsewhere. The variation in these parameters canbe explained by the different method of inserting the lead plug as discussedabove. Yield displacement can be derived from the bilinear representation

y =Qy

KLR KR(2.13)

The amount of lead is often expressed as the ratio of rubber to lead, asshown in Equation 2.14. Typical values of n range from 10-20 [Jara andCasas (2006), Priestley et al. (2007)].

n =ARAL

(2.14)

Vertical load capacity of an elastomeric bearing is usually expressed by thefollowing equation [Priestley et al. (2007), Skinner (1993)]

W AGRS (2.15)

where W is the allowable weight, is the allowable shear strain, A is theminimum permitted overlap of top and bottom area of the bearing at max-imum displacement (taken as AR2 ) and S is a shape factor equal to loadedarea divided by force free area.

Allowable maximum rubber shear strain without lead plug is given as

= 0.4t (2.16)

where t is the short-duration failure strain in simple tension. Experimentssuggest this factor as a minimum for design-earthquakes and up to 0.7 forextreme earthquakes. The short-duration failure strain in simple tension can


be taken as t = 350% [Skinner (1993)], although these characteristics shouldof course be determined based on tests on a sample prior to installation. Thismeans that the allowable maximum rubber shear strain is in the interval140% - 245%. The lowest value is used here as stated in Equation 2.16.

Vertical stiffness of rubber bearings is given with [Priestley et al. (1996),Priestley et al. (2007)]

KZ =6GRS2ARkb

(6GRS2 + kb)tR(2.17)

where kb is the rubber bulk modulus, taken as 2000 MPa and the othersymbols as already defined. The vertical stiffness calculated with Equation2.17 considers the sum of the deflection due to the rubber shear strain andthe rubber volume change.

2.3.1.1 Equivalent Linear Model

The response spectrum provides some of the most important characteristicsof earthquake motion and gives the maximum elastic deformation for struc-tures over the entire range of periods. However, it is not able to predictdamage level, as damage involves inelastic deformations. Inelastic responsecan be captured through nonlinear time history analysis, but in many casesthe linear response spectrum is the preferred weapon of choice for practicingengineers. Hence approximate methods have been developed to describe theisolators with equivalent linear models using effective lateral stiffness andequivalent damping ratio. Equivalent linear models have been incorporatedin Eurocode 8 for designing bridges with passive energy dissipation systemsas well as other codes.

The behavior of an inelastic hysteretic structure subjected to ground accel-eration ug is assumed to be described by a single degree of freedom systemand the maximum inelastic response is given as [Jara and Casas (2006)]

u+ 2iiu+fs(u, u)m

= ug (2.18)

where i is the damping ratio, i is the initial circular frequency and fs(u, u)is the restoring force. In the equivalent linearization method, the maximuminelastic displacement demand ueq is approximated by

ueq + 2eqef ueq + 2efueq = ug (2.19)

14 Theory

where eq is the equivalent viscous damping ratio and ef is the effective cir-cular frequency. The value of eq and ef depend on the material hystereticbehavior, maximum displacement demand and the number of incursions intothe inelastic range, among others [Jara and Casas (2006)]. The main dif-ference of proposed methods lies in the way eq and ef are determined.Existing equations for computing eq and ef have been shown to produceinaccurate displacement predictions and an improved equation for equiva-lent viscous damping for bilinear isolators has been proposed by Jara andCasas [Jara and Casas (2006)]

eq = 0.05 + 0.05 ln(d) (2.20)

where d is the ductility ratio of the bilinear isolator. The improved equiv-alent damping value given with Equation 2.20 is obtained by equating thenonlinear displacement spectrum for a given earthquake to the linear dis-placement earthquake. Other methods of obtaining eq include deriving itfrom the harmonic response at maximum displacement and numerical min-imization of the difference between between elastic and inelastic response.In Eurocode 8 the equivalent viscous damping is given with

eq =1

2

[ED

Keffu

](2.21)

where ED is the dissipated energy in a full deformation cycle at the designmaximum displacement u and Keff is the equivalent stiffness defined inEquation 2.23. Dissipated energy in a full deformation cycle is the area ofthe hysteresis loop and given by

ED = 4Qy(u y) (2.22)

The influence of ductility on equivalent viscous damping given with Equa-tions 2.21 and 2.20 is illustrated in Figure 2.6. Note that the damping ratiodecreases as the ductility increases for d = uy > 5 using the equationlisted in Eurocode 8. This is contrary to expectations as increased ductilityis considered to lead to increased inelastic deformations and hence energydissipation. With the proposed equation, the equivalent viscous dampingincreases as ductility ratio increases.

The effective stiffness Keff of a lead rubber bearing is defined assuming thebearings deform linearly up to maximum displacement u, see Figure 2.5

2.4 Performance Based Design 15

Figure 2.6: Equivalent damping from Eurocode 8 and proposed improved equa-tion by Jara and Casas

[Naeim and Kelly (1999), Jara and Casas (2006)]

Keff =Qyu

+KR (2.23)

2.4 Performance Based Design

One of the major developments in seismic design over the past twenty yearshas been increased emphasis on limit states design, now generally termedPerformance Based Engineering [Priestley (2000)]. Several techniques havebeen developed and now three of them have developed to a stage where seis-mic assessment of existing structures, or design of new structures can be car-ried out to ensure that particular deformation-based criteria are met. Theseare the capacity spectrum approach, N2 method and direct displacement-based design. They all follow from the realization that increasing strengthmay not enhance safety nor necessarily reduce damage and focus on movingthe emphasis in design from strength to performance.

The start of performance based seismic engineering can be tracked back tothe 1970s and the development of capacity design principles in New Zealand.It was then realized that the distribution of strength through a buildingwas more important than the absolute value of the design base shear. Theweak-beam/soft-column philosophy was realized to perform better duringa seismic event as well as the importance of the shear strength exceed-ing the corresponding flexural strength. This philosophy of controlling theoverall performance of the building as a function of the design process can

16 Theory

be identified as the true start to performance based seismic design. It istoday widely recognized that seismic design codes need to incorporate aperformance-based design criterion [Priestley (2000)].

The traditional force-based design approach includes some conceptual andphilosophical problems, some of which are listed below [Priestley (2000)]

It is generally accepted that damage is strain related for structural com-ponents, or drift related for non-structural components. There is no clearrelationship between strength and damage.

Use of force-reduction or ductility factors for design results in non-uniformrisk, since ductility is a poor indicator of damage potential. Thus two differentbuildings designed to the same code and with the same force-reduction orductility factors may experience different levels of damage under a givenearthquake.

For many structures, code drift limits will be found to govern and as a con-sequence force-reduction factors will be less than code indicative limits. Thisimplies the need for iterative design and increased design complexity.

Force-based design requires the specification of initial stiffness of structuralmembers. Forces are then distributed between structural members on thebasis of initial stiffness. This is sometimes taken to be gross stiffness, andsometimes as a reduced stiffness to represent the influence of cracking inconcrete and masonry structures. This implies that the structural stiffnessis independent of strength for a given gross member dimension and thatyield displacement, or yield curvature, is directly proportional to strength,as shown in Figure 2.7(a). Detailed analysis and experimental evidence showthat this assumption is invalid, in that stiffness is essentially directly pro-portional to strength, and the yield displacement or curvature is essentiallyindependent of strength as shown in Figure 2.7(b). It has been found possibleto express the yield curvature of different reinforced concrete structural mem-bers by dimensionless relationships, see Equation 2.25 in Section 2.4.1.1. Asa consequence of this it is not possible to perform an accurate analysis of ei-ther the structural period, nor of the elastic distribution of required strengththrough the structure, until the member strengths have been defined. Sincethese aspects are the fundamental basis for force-based design, the implica-tion is that successive iteration must be carried out before an adequate elasticcharacterization of the structure is obtained.

The assumption that the elastic characteristics of the building are the bestindicator of inelastic performance, as implied by force-based design, is clearlyof doubtful validity. The reason for sticking with this approach is thought tobe historical - seismic design started on the basis of simple elastic analysisin the 1930s, and the appreciation that it is displacement capacity, ratherthan strength, that is more fundamental to seismic performance, has notbeen general for sufficient time for the fundamental basis of seismic design,namely initial stiffness characterization, to have been clearly examined anddiscarded.


Figure 2.7: Influence of strength on Moment-Curvature relationship [Priestley(2000)]

It is clear from the above that for force-based design to be adequately incor-porated into performance based seismic design, significantly increased designeffort, in the form of successive iteration of the initial elastic characteristicsmust be required. As a consequence of this, alternative design procedureshave been developed such as the Direct Displacement-Based Design proce-dure described below.

2.4.1 Direct Displacement-Based Design

Based on the discussion above in Section 2.4 it seems desirable to developa design approach that attempts to design a structure that achieves a givenperformance limit state under a given seismic intensity. The design proce-dure known as direct displacement-based design (DDBD) has been developedover the past ten years with the aim of mitigating the deficiencies in currentforce-based design [Priestley et al. (2007)].

While force-based design characterizes a structure in terms of elastic prop-erties, stiffness and damping, appropriate at first yield, direct displacement-based design characterizes the structure by secant stiffness at maximum dis-placement and a level of equivalent viscous damping appropriate to the hys-teretic energy absorbed during inelastic response. A major difference fromforce-based design is that the design utilizes a set of displacement-periodspectra for different levels of viscous damping rather than the acceleration-period spectra for 5% damping.

One of the key aspects of DDBD is the simplicity and rationality, as isapparent from the fundamental equations listed in Section 2.4.1.2.

18 Theory

2.4.1.1 Design Displacement

Determining the design displacement is for obvious reasons an importantstep in the direct displacement-based design procedure. It is selected con-sidering the actual displacement spectra which is a function of local seis-micity as well as functional requirements, see Section 3.2.4.1. Design dis-placement depends on the limit state considered and whether structural ornon-structural considerations are more critical. For any given limit statestructural performance will be governed by limiting material strains, sincedamage is strain-related for structural elements. Damage to non-structuralelements are generally drift related. Calculating design displacement fromstrain limits is generally straightforward.

The determination of design displacement depends on the structure beinganalyzed. For an isolated bridge, as is the case in this thesis, the designdisplacement consists of two terms; pier displacement and bearing displace-ment.

d = B + P (2.24)

In Equation 2.24, B is the design deformation of the bearing and P isthe displacement of the pier top. The design displacement of lead rubberbearings is limited by shear strains in the rubber as discussed in Section 2.3.1whereas the pier displacement is bounded by a choice made by the designer.An important reason for isolating a bridge is to protect the substructurefrom brittle failure modes and therefore it is desirable to limit the pierdisplacements to remain within the elastic range. Hence, pier displacementsshould be limited to a fraction of their yield displacement. This is discussedfurther in Section 2.4.1.4.

Experimental results discussed further in Section 2.4.1.3 indicate that yieldcurvature is essentially independent of reinforcement content and axial loadlevel, and is a function of yield strain and section depth alone. Yield curva-ture for a rectangular concrete column can be approximated with

y =2.10yhc

(2.25)

where y is the yield strain of the flexural reinforcement and hc is the sec-tion depth. The equation above is given for solid rectangular sections butcan be used as a reasonable estimate for hollow columns [Priestley et al.(2007)]. Hence, for a SDOF vertical cantilever the yield displacement can


be satisfactorily approximated for design purposes by the following equation[Priestley et al. (2007)]

y =y(H + Lsp)2

3(2.26)

where H is the cantilever height and Lsp is the strain penetration lengthgiven with Equation 2.27.

Lsp = 0.022fyddlb (2.27)

where fyd and dlb are the yield strength and diameter of longitudinal rein-forcement respectively.

2.4.1.2 Fundamentals

As mentioned above, direct displacement-based design characterizes thestructure to be designed by a SDOF representation of performance at peakdisplacement response, rather than by its initial elastic characteristics. Thisis based on the substitute structure approach developed by Shibata and Sozen[Priestley (2000)]. The design method is described with reference to Figure2.8, which considers a SDOF representation of a multistory building (thebasic fundamentals apply to all structural types). The bilinear envelope ofthe lateral force-displacement response of the SDOF representation is shownin Figure 2.8(b). An initial stiffness Ki is followed by a post yield stiffnessrKi. DDBD characterizes the structure by secant stiffness Ke at maximumdisplacement d and a level of equivalent viscous damping eq ( in figure),representative of the combined elastic damping and the hysteretic energyabsorbed during inelastic response.

Having determined the design displacement at maximum response and es-timated corresponding damping from the expected ductility demand, theeffective period Te can be easily read from a displacement-period spectra asshown in Figure 2.8(d). The effective stiffness Ke of the equivalent SDOFsystem at maximum displacement can be found by inverting the equationfor the period of a SDOF oscillator given by Equation 2.4 to provide

Ke =42meT 2e

(2.28)

where me is the effective mass of the structure participating in the funda-mental mode of vibration. Design lateral force, which is also the design base

20 Theory

Figure 2.8: Fundamentals of Direct Displacement-Based Design [Priestley et al.(2007)]

shear force is thus simply

Vbase = Ked (2.29)

The design concept is thus very simple. The complexity does however stillexists and relates to determination of the substitute structure characteristics,design displacement and development of displacement spectra. Also thedistribution of design base force to the different discretized mass locationsneeds careful consideration.

2.4.1.3 Elastic Stiffness of Cracked Concrete Sections

In direct displacement-based design the required stiffness is the output ofthe design, whereas in force-based design the total base shear is distributedto the structural elements in proportion to their stiffness. A fundamentalassumption in the force-based design approach is the estimated stiffness ofthe structure which is used for the determination of the natural period of the


structure and the distribution of forces to different elements. This stiffnessis not known initially so in the force-based design an estimate is made.Eurocode 8 states that the cracked stiffness can be taken as one-half ofinitial elastic stiffness if no other analysis is conducted. Uncracked stiffnessfor top transverse displacement of a cantilever pier, of height H, havingmodulus of elasticity E and moment of inertia I, subjected to transverseend force is given with [Jensen (2003)]

k =3EIH3

(2.30)

The effective or cracked section moment of inertia Ie should be used toreflect the cracked state of a concrete column. The effective stiffness EIedoes not reflect only the effect of cracking but also the state of a bridgecolumn determined at first theoretical yield of the reinforcement and canbe determined from sectional moment-curvature analysis as [Priestley et al.(1996)]

EIe =Myiyi

(2.31)

In Equation 2.31, Myi and yi represent the ideal yield moment and cur-vature for a bilinear moment-curvature approximation. Idealized moment-curvature relationship is shown in Figure 2.7.

Contrary to the common assumption made in force-based seismic design, theelastic stiffness of cracked members is essentially proportional to strengthand the concept of constant yield is valid. This has been confirmed bynumerous experiments, see e.g. [Priestley et al. (2007)] and [Neaz Sheikhet al. (2008)], in which it is clear that stiffness is proportional to strength andyield curvature is insensitive to the moment capacity. Actually, it is almostonly dependent of cross section depth and yield strength of reinforcement.In Figure 2.9 the effect of axial load ratio and reinforcement ratio on theeffective stiffness of large rectangular columns is shown. The curves shownare developed by moment-curvature analysis on a square column with sidelengths 1.6m.

22 Theory

Figure 2.9: Effective stiffness ratio for large rectangular columns [Priestley et al.(2007)]

The constant yield curvature assumption makes determination of the ef-fective stiffness possible once the yield curvature and flexural strength areknown. This can be seen from Figure 2.7(b), which shows the momentcurvature relationship, and Equation 2.31.

2.4.1.4 Displacement-Based Design of Isolated Bridges

Applying direct displacement-based design to isolated bridges seems to be alogical approach since the emphasis is on the displacement rather than theforces transmitted through the device. The superstructure is presumed tobe relatively rigid in comparison with the stiffness of piers and abutmentsand it is assumed that dynamic response of the bridge can be predicted quiteaccurately with a SDOF system. A sketch of the idealized model is shownin Figure 2.10.

Generally a preliminary design of the bridge has been performed consideringnon-seismic loading conditions and therefore a full geometry is available andpossibly a preliminary dimensioning of reinforcement. Yield displacementsof each pier can then be calculated applying Equation 2.26. The isolationsystem should be designed having a yield displacement such that the equiv-alent yield point for the pier-isolator system should correspond to a forcelevel equal to a desired fraction of pier yield to ensure elastic response of thepiers.

d = B +XyP (2.32)


In Equation 2.32, B is the design displacement of the bearing, yP is theyield displacement of the pier and X is the fraction of pier yield displacementchosen. Typically this value is chosen to be 80% to ensure that yield doesnot occur in the piers. The design process will imply the definition of auniform design displacement of the deck. This uniform displacement willcorrespond to different combinations of column and bearing displacementfor each pier. As a consequence different equivalent viscous damping valueswill also exist for each pier-isolator system.

Figure 2.10: Damping for a cantilever pier with an isolated deck [Priestley et al.(2007)]

At the limit-state response the lateral force may be essentially the sameas yield or it may be significantly larger depending on isolator properties.Pier deformation will increase in proportion to the force increase while thebearing will increase 10-15 times depending on accepted plastic deformationsassociated with the limit strains [Priestley et al. (2007)]. This is indicatedin Figure 2.10 where yS is the equivalent yield displacement of the system,yB is the yield displacement of the device, dP and dB are the designdisplacements of the pier and bearing, respectively.

Equivalent viscous damping for the pier-isolator system is a combination ofelastic damping of the pier and hysteretic damping of the isolator. Elastic

24 Theory

damping ratio is taken as el = 5% which is commonly used for concrete.

P.i =BB.i + eldP.i

d(2.33)

In Equation 2.33, B and B.i are the equivalent viscous damping ratio anddesign displacement of the bearing respectively, dP.i is the design displace-ment of the pier and d is the total design displacement.

The total equivalent stiffness is distributed to each pier-isolator system ina manner determined by the designer. This can e.g. be done in proportionto the tributary weight supported by each pier or abutment. This results inequal shear force in all piers given that the tributary weight is equal for allpiers. The stiffness, and hence total base shear, could also be distributed ininverse proportion to height which would result in equal bending momentfor the piers. The equations below assume a distribution proportional to thetributary weight.

Ki =KeMini=1

Mi

(2.34)

Vi =VbaseMini=1

Mi

(2.35)

where Mi is the tributary weight for each individual pier. Global equiva-lent viscous system damping is based on the same assumption on stiffnessdistribution as is used for Equation 2.34.

eq.sys =

ni=1

P.iMi

ni=1

Mi

(2.36)

Displacement spectra for other damping values than 5% damping are deter-mined using the Eurocode 8 reduction factor

R =(

0.100.05 + eq.sys

)0.5(2.37)


To calculate the required properties of the isolators and section forces inpiers, the stiffness of each pier-isolator system needs to be split betweenthe piers and isolators. The effective stiffness of isolators and piers can beobtained from Equations 2.38 and 2.39 [Priestley et al. (1996)]

KB.i =(

1 +XPy.id.B.i

)Ki (2.38)

KP.i = KB.idB.iXPy.i

(2.39)

where X is the fraction of yield displacement desired for pier no. i. It coulde.g. be decided to design piers for a force level equal to 1.25 times the designstrength of the corresponding isolator, which would result in X = 80%.

Shear force in each pier can then be computed by multiplying effective stiff-ness by displacement and bending moment by multiplying shear force byheight. To obtain the required flexural strength the stiffness is multipliedwith the full yield displacement.

VP.i = XPy.iKP.i MP.i = VP.iHi (2.40)

A design procedure flowchart is presented in Figure 2.11 and the step bystep procedure is described below.

1. Design displacement should be selected considering the actual displace-ment spectra, as a function of the local seismicity and the functionalrequirements. A crucial issue will be to decide whether it is acceptableto design for a displacement equal to that corresponding to the cornerperiod in the appropriately damped displacement spectrum.

2. Calculate pier yield displacements with Equation 2.26.3. Equivalent viscous damping obtainable from each type of isolation

system varies. First a value typical for the relevant isolation systemmust be assumed.

4. System equivalent stiffness is given with Equation 2.28 and is a func-tion of effective period and effective mass only.

5. Shear distribution between piers determined. It can e.g. be distributedto provide equal shear or equal moment in the piers. The distributioncan calibrated to obtain an optimal solution regarding stiffness of pier-isolator systems.

6. Determine effective stiffness of isolators and piers from Equations 2.38and 2.39.

7. Determine isolator properties (see Section 2.3.1) and compare calcu-lated equivalent viscous damping values with those assumed in step 3.

26 Theory

If the calculated value based on isolator parameters is not equal to theassumed value iteration is needed until a satisfactory result is reached.

8. Equivalent viscous damping for each pier-isolation system is deter-mined from Equation 2.36 based on assumed damping and displace-ment of piers and isolators.

9. Global equivalent viscous damping determined from Equation 2.36 anddistribution determined in step 5.

10. Reduction factor for displacement spectra determined with Equation2.37.

11. Read displacement from reduced displacement spectra. If it is the sameas assumed design displacement value in step 1, the process continues.If not, a few iterations need to be made.

12. Total base shear force and base shear coefficient determined fromEquation 2.29.

13. Distribute base shear and stiffness to piers and spandrel columns ac-cording to assumption made in step 5.

14. Check if force levels and required stiffnesses are acceptable. If so thedesign process is completed but if not some modifications have to bemade. Depending on the force levels it may be best to modify theinitial input parameters or perhaps change the design displacement orshear distribution. Obviously many options are available and this iswhere the experience of the designer comes in handy.


1. Determine design displacement d

Initial input parameters: Column dimensions, inertia mass, material properties and design spectra.

2. Calculate pier yield displacements

3. Assume equivalent viscous damping of isolators and force level for design of piers

8. Calculate equivalent viscous damping of each pier-isolator system

9. Calculate global equivalent viscous system damping

10. Calculate damping reduction factor

11. Read displacement from reduced displacement spectra. Is it the same

as design displacement?

12. Calculate total base shear force

13. Distribute base shear and stiffness to piers and spandrel columns.

Calculate section forces in piers.

5. Calculate stiffness of each pier-isolation system

6. Calculate effective stiffness of isolators and piers

4. Calculate system equivalent stiffness

7. Determine isolator properties. Does equivalent viscous damping of isolator

agree with assumed value?

14. Are force levels acceptable?

Finished

YES

NO

YES

NO

YES

Figure 2.11: Flowchart for the displacement-based design for isolated structures

28 Theory

2.4.1.5 Comparison With Force-based Design Approach

It was shown in Section 2.4.1.4 that considering, as basic design parameters,the deck, isolators and pier displacement capacities and demands lead toa simple and effective design approach which only requires the applicationof capacity design principles to protect piers. To illustrate the conceptualdifference between force-based design and direct displacement-based design,simple flowcharts are provided in Figure 2.12. Step by step guides are givenbelow.

1. Period of vibration and damper ductility chosen

1. Design displacement chosen which gives period of vibration

2. Yield displacement of piers and isolator displacement determined

3. System equivalent damping calculated

4. Base shear distributed and required stiffness of piers and isolators obtained

3. Base shear read from acceleration spectrum.

4. Base shear distributed based on assumed stiffness. Pier displacements calculated.

2. Equivalent damping calculated

5. Effective displacement of isolators determined and devices designed

Force-based approach Direct Displacement-based approach

5. Piers and isolators designed to achieve desired response

Figure 2.12: Simple flowcharts for force-based and direct displacement-based de-sign of isolated structures

Force-based Design:

1. Desired period of vibration and acceptable ductility of dampers chosen,T and d

2. Equivalent damping of system eq calculated neglecting pier responsecontribution

3. Design base shear Vbase read from acceleration spectrum with T andeq

4. Base shear distributed to pier-isolator systems and pier displacementscalculated based on assumed stiffness independent of strength. Pierdisplacements calculated.

5. Effective displacement of each pier-isolator system is e = VuKe . Isola-tion devices can then be designed.

2.5 Methods of Analysis 29

Direct Displacement-based Design:

1. Choose design displacement T2. Calculate yield displacement of piers obtain isolator displacements3. System equivalent damping calculated including pier response contri-

bution4. Distribute forces obtain required stiffness of piers and isolators5. Design piers and isolators to achieve the desired response

In force-based design the period of vibration and acceptable damping isarbitrarily chosen and will most likely need iteration. Pier displacementscalculated in step 4 are based on assumed stiffness independent of strength.Subsequently the isolator effective displacement is determined from the pierdisplacements and total effective displacement. The procedure is repeatedto convergence if the resulting system should be technically unacceptableor if the assumed ductility differs from the resulting ductility. From theabove it is clear that using this approach does not provide the designer withthe same level of control of the total structural response as in the directdisplacement-based design. The end result might nevertheless be the same,although unlikely.

2.5 Methods of Analysis

Two numerical methods were used for the seismic analysis of the bridge: Re-sponse spectrum analysis and nonlinear time history analysis. These meth-ods are described below. In addition to the seismic analysis methods, staticanalysis was performed to calculate the stresses and deformations of thebridge from non-seismic loading.

The response quantities of interest for the bridge system under considera-tion are: (1) base shear and section forces, (2) relative displacement of thebearings at the abutments and piers for the isolated bridge. Section forcesof structural elements provide an insight in the loaded state of the bridgeand the relative displacements of the bearings are crucial from the designpoint of view of the isolation system and expansion joints.

The following assumptions are made for the earthquake analysis of thebridges under consideration:

1. The bridge deck, piers and arch are assumed to remain in the elasticstate during the earthquake excitation in the base isolated bridge. Thisis a reasonable assumption, as the isolation system is designed suchthat it protects the structural elements by making sure they remain inthe elastic range.

2. The deck of the bridge is straight and is supported at discrete locations

30 Theory

along its longitudinal axis by cross diaphragms. Also the abutmentsof the bridge are assumed to be rigid.

3. The bridge piers and spandrel columns are assumed to be rigidly fixedat the foundation level and on the arch.

4. Two simultaneous horizontal components of earthquake ground mo-tion are considered in the longitudinal and transverse directions of thebridge as well as a vertical component.

5. The stiffness contribution of nonstructural elements such as curbs,parapet walls, and the wearing coat is neglected. However, their massproducing the inertial forces is considered.

6. The bridge is founded on firm rock, and the earthquake excitation isperfectly correlated at all of the supports.

7. The lead rubber bearings are isotropic, implying the same dynamicproperties in two orthogonal directions.

2.5.1 Response Spectrum Analysis

The response spectrum analysis is a procedure for dynamic analysis of astructure subjected to earthquake excitation, but it reduces the problemto a series of static analysis. The response spectrum analysis uses the vi-bration properties of the structure (natural frequencies, natural modes andmodal damping ratios) and the dynamic characteristics of the ground motionthrough its response spectrum. Time history calculations are not necessarysince the calculations have been made in developing the earthquake responsespectrum or in the case of a smooth design spectrum, the properties havebeen characterized by the spectrum.

The exact peak response of each mode can be obtained from the earthquakeresponse spectrum and subsequently the modal responses are combined usinga suitable modal combination rule. The result is not exact in the sense thatis not identical to the response history analysis. It is however considered ac-curate enough for structural design applications [Chopra (2007)]. The resultof a response spectrum analysis using the response spectrum from a groundmotion is typically different from that which would be calculated directlyfrom a linear dynamic analysis using the same ground motion directly, sincephase information is lost in the process of generating the response spectrum.

Several methods have been proposed to combine the modal responses to de-termine the peak value of the total response. A discussion of the proposedmethods is not provided here, but the interested reader is referred to e.g.[Chopra (2007)]. It is not possible to simply add up the modal responsessince they attain their peaks at different time instants and the combinedresponse attains its peak at yet at different instant. Hence, approximationsmust be introduced in combining the peak modal response because no infor-


mation is available as to when these peak modal values occur. The CompleteQuadratic Combination (CQC) rule is widely used and considered to yieldsatisfactory results and is therefore used in the work presented herein. Thetotal peak response, r0, is calculated with

r0

(Ni=1

Nn=1

inriorno

)12

(2.41)

where rio and rno are the responses of modes i and n respectively and inis the correlation coefficient for modes i and n. The correlation coefficientvaries between 0 and 1 and in = 1 for i = n. The correlation coefficient isgiven with (according to Der Kiureghian [Chopra (2007)])

in =8in(i + inn)

32in

(1 2in)2 + 4inin(1 + 2in) + 4(2i + 2n)2in(2.42)

where in is the ratio between the angular frequency of modes i and n,in = in , and i and n are the damping values of modes i and n respectively.

Response spectra are created for SDOF systems and it is assumed that theresponse is entirely elastic when the modal responses are combined. To ap-ply the response spectrum analysis on nonlinear systems such as the baseisolated bridge, the devices need to be represented by an equivalent lin-ear system, see Section 2.3.1.1. The response spectra are then modifiedto account for the increased damping in the isolation devices. It has beenobserved that the equivalent damping of the isolation system is effectiveonly for modes that involve significant yielding of the damper. These arethe fundamental periods of vibration, i.e. the modes with longest vibrationtime. A composite spectrum has therefore been proposed with a step be-tween two spectra with different damping. The step should correspond to aperiod value close to, but less than, the period of vibration of the isolatedstructure, so that only modes effectively isolated fall into the reduced ac-celeration spectrum. Higher modes of vibration that do not involve isolatordeformation are damped only by the structural viscous damping. This isillustrated in Figure 2.13

32 Theory

Figure 2.13: Modified response spectrum to consider the equivalent viscous damp-ing of the isolation system [Priestley et al. (1996)]

The general purpose finite element program SAP2000 offers the option ofdefining such a spectra but also provides the option of manually changingmodal damping ratios for selected modes. The later option was used herein.

2.5.2 Nonlinear Time History Analysis

There are situations in which the simulation of structural response using ascaled elastic response spectrum is not considered appropriate, and a fulldynamic analysis is required. These situations may include buildings de-signed for a high degree of ductility, highly irregular structures, structuresfor which higher modes are likely to be excited, critical structures and struc-tures with special features such as base isolation. By using nonlinear timehistory analysis the nonlinear behavior of materials can be modeled and theresponse can be calculated as a function of time during the seismic event.To perform such an analysis an appropriate nonlinear model of the structureis required as well as accelerograms representing the seismic excitation.

Numerical integration in structural dynamics is dominated by single step -single solve algorithms, where the response is updated one step at a timeby solving only one system of equations of the size of the similar staticequations. The classical form of these algorithms is the Newmark algorithmwhich many have tried to improve since it was originally presented.

The response of an inelastic MDOF system at time i is given with

mui + cui + (fS)i = (mIug)i (2.43)

and time-stepping methods enable the determination of the response of the


system at time i+ 1:

mui+1 + cui+1 + (fs)i+1 = (mIug)i+1 (2.44)

Newmark developed a family of time-stepping methods in 1959 based on thefollowing equations [Chopra (2007)]

ui+t = ui + [(1 )t]ui + (t)ui+t (2.45)

ui+t = ui + (t)ui + [(0.5 )(t)2]ui + [(t)2]ui+t (2.46)

where the parameters and define the variation of acceleration over atime step and determine the stability and accuracy characteristics of themethod. Typical values are = 12 and

16

14 . In this thesis the average

acceleration algorithm is used which is the Newmark method with = 12and = 14 .

Damping used in direct integration time history analysis is described withthe damping matrix, c. The damping matrix is defined as being proportionalto mass and stiffness by

c = a0m + a1k (2.47)

This is called Rayleigh damping. The damping ratio for the nth mode ofsuch a system is

n =a02

1n

+a12n (2.48)

The coefficients a0 and a1 are determined from specified damping ratios iand j for mode i and j respectively. It is reasonable to have the samedamping ratio for modes i and j and then the coefficients are given with

a0 = 2iji + j

a1 = 2

i + j(2.49)

The modes i and j with specified damping ratios should be chosen to ensurereasonable values for the damping ratios in all the modes contributing sig-nificantly to the response. This is illustrated in Figure 2.14. Modes havingangular frequency lower than mode i will be over damped and damping formodes higher than j will increase monotonically with frequency. Dampingfor modes in between i and j will have damping somewhat smaller than .

34 Theory

The Rayleigh damping represent the damping of the bridge without thelead rubber bearings. The energy dissipation of the isolation devices areautomatically included in the calculations through their nonlinear definition.

Figure 2.14: Rayleigh damping - variation of modal damping ratios with naturalfrequency [Chopra (2007)]

2.6 Eurocode 8

Seismic action on structures is dealt with in the european standard Eurocode8: Design of structures for earthquake resistance. Two parts of Eurocode 8are relevant for this thesis. Those are Eurocode 8 - part 1: General rules,seismic actions and rules for buildings (EC8-1) and Eurocode 8 - part 2:Seismic design of bridges (EC8-2). In this section a short description of themain concepts is provided.

Eurocode 8 - part 1, describes in general seismic actions on structures anddefines the seismic actions based on variables such as ground type, mag-nitude of design seismic event and peak ground acceleration. A responsespectrum is defined which has four parts, a constant acceleration part, aconstant velocity part, a constant displacement part and finally a part con-necting constant acceleration to peak ground acceleration. The spectrum isscaled with respect to peak ground acceleration, damping of the system andsoil type at the location of construction. Further the peak ground acceler-ation is multiplied with an importance factor depending on the importanceof the structure. For bridges there exists three importance classes: Greaterthan average, Average and Less than average.

Seismic loads can also be represented by time histories. In that case, EC8states that at least three pairs of horizontal ground motion time historiesshould be used. The Icelandic National Annex however has a more stringent

2.6 Eurocode 8 35

requirement in that it requires at least ten pairs of horizontal ground motiontime histories. Specific rules are given for scaling of the pairs of horizontalmotions independent from the vertical component, so as to render themcompatible to the elastic response spectrum.

Seismic loading for the bridge is described in detail in Section 3.2.4.1 wherethe design spectrum is derived and in Section 3.2.4.2 where artificial timehistories are presented.

Cracked stiffness of concrete members may be taken as equal to one-half ofthe corresponding stiffness of the uncracked elements unless a more accurateanalysis of the cracked elements is performed.

When performing a modal response spectrum analysis the response of allmodes of vibration contributing significantly to the global response shall betaken into account. That is done by either demonstrating that the sum ofthe effective modal masses for the modes taken into account amounts to atleast 90% of the total mass of the structure or by demonstrating that allmodes with effective modal masses greater than 5% of the total mass areconsidered.

Eurocode 8 - part 2, is specific for bridges and has special section dedicatedto seismically isolated bridges. EC8-2 has two basic requirements for designof bridges, the first being that the bridge should retain its structural integrityand adequate residual resistance, although at some parts of the bridge con-siderable damage may occur. The bridge should be damage-tolerant, i.e.those parts of the bridges susceptible to damage, by their contribution toenergy dissipation during the design seismic event, should be designed insuch a manner as to ensure that, following the seismic event, the structurecan sustain the actions from emergency traffic, and inspections and repaircan be performed easily. The second basic requirement is the minimizationof damage during earthquakes with a high probability of occurrence. In sucha case only secondary components and those parts of the bridge intended tocontribute to energy dissipation during the design life of the bridge shouldincur minor damage during earthquakes with a high probability of occur-rence. The non-collapse requirement for bridges under the design seismicevent is more stringent than the relevant requirement for buildings, as itcontains the continuation of emergency traffic.

If the bridge being considered is long, accounting for spatial variability maybe required. The distance beyond which ground motions may be considereduncorrelated is Lg = 600m for ground type A (Bedrock). Accounting forspatial variability is required for bridges with continuous deck when eitherthe bridge length exceeds Lg/1.5 or when more than one ground type cor-respond to the bridge supports. The bridge considered in this thesis is wellbelow the distance founded on bedrock. Hence no consideration was made

36 Theory

on spatial variability.

Three analysis methods for seismically isolated bridges are allowed:

1. Fundamental mode spectrum analysis2. Multi-mode spectrum analysis3. Time history analysis

Specific conditions need to be fulfilled in order to apply the first two methodswhereas the third method can be applied to all seismically isolated bridges.Fundamental mode spectrum analysis can be applied when all the followingconditions are met (in addition to general conditions for fundamental modespectrum analysis):

1. Distance to nearest know seismically active fault exceeds 10 km2. Ground conditions are of type A, B, C or E3. Effective damping < 30%4. Force-displacement relation of the isolation units in each direction is

essentially independent from the rate of loading, magnitude of concur-rent vertical load and horizontal load in transverse direction

5. Effective stiffness of the isolating system at the design displacement isat least equal to 13 of the stiffness at

15 of the design displacement

Multi-mode spectrum analysis can be applied when the conditions 2-5 arefulfilled.

In a response spectrum analysis the damping of the isolation system shouldbe used for appropriate modes only. Other modes should be assigned damp-ing of the nonisolated system. This is not explained any further but dividingthe spectrum into two parts as described in Section 2.5.1 is meant to coverthis.

The normal design properties of all isolator units should be assessed bymeans of special prototype tests. Excepted from this rule are normal elas-tomeric bearings, for which normal design properties and design rules aredefined by the code. Also excepted are flat siding bearings, as long as theircontribution to the damping of the isolating system is ignored.

2.7 Ultimate Strength of Elements

When evaluating the seismic response of the bridge the following calcula-tions methods were used to estimate the ultimate strength of elements. Theultimate strength was compared to the results of the FE-analyses.

2.7 Ultimate Strength of Elements 37

2.7.1 The Arch

Originally the arch was designed such that it was in compression at all times.Stresses in the arch from axial force and moments are added up with thefollowing equation [Megson (2002)]

T =P

A+Mxy

Ix+Myx

Iy(2.50)

When subjected to seismic loads significant moments arise resulting in ten-sion in the arch. Evaluation of nonlinear behavior of hollow arch rib sectionsubjected to large flexural moment under significant variation of axial forceis difficult. To estimate the load bearing capacity of the arch the initialyielding moment was calculated based on the axial force induced by thedead load only. Initial yielding moment is defined as the moment when ten-sile strain of the outmost main reinforcement reaches yield strain y [Jensen(2006)]. It is important that the steel reaches its yield strain before theconcrete crushes. The ultimate strain of concrete was taken as cu = 3.5.Yield moment was calculated assuming a reinforcement ratio of l = 1%.

Figure 2.15: Strain and stress distribution and force equilibrium in a concretesection

Neglecting compression reinforcement the neutral axis is readily found fromequilibrium of forces in Figure 2.15.

N = Fc Fs (2.51)

Where Fc and Fs are the forces in concrete and reinforcement respectively.

Fc = 0.8xbfcd and Fs = Asfyd (2.52)

Where fcd is the design compressive strength of concrete, fyd is the designyield strength of steel reinforcement, As is the amount of tensile reinforce-

38 Theory

ment and b and x are as defined in Figure 2.15. The yield moment is foundby moment equilibrium, here taken about the tension reinforcement.

My = Fc

(d 0.8x

2

)N

(d h

2

)(2.53)

2.7.2 Shear Capacity

Shear capacity was calculated according to Eurocode 2 (EN1992-1-1:2004)with the following equation

VRd,s =Aswszfywdcot() (2.54)

where Asw is the cross-sectional area of the shear reinforcement, s is thespacing of stirrups and fywd is the design yield strength of the shear re-inforcement, z is the inner lever arm and is the angle between concretecompression struts and the main tension chord. The maximum allowableshear resistance acquired from Equation 2.54 is bounded by

VRd,max =cwbwz1fcdcot() + tan()

(2.55)

Where 1 is a strength reduction factor for concrete cracked in shear and cwis a coefficient taking account of the state of the stress in the compressionchord. Equation 2.55 ensures that no crushing occurs in the compressionchord.

2.7.3 Piers and Spandrel Columns

Load bearing capacity of piers and spandrel columns is calculated by con-struction N-M interaction diagrams for uniaxial moment. The diagram issimplified by calculating only three points.

A Pure compression. Ultimate compressive strength of the piers andspandrel columns

B Balanced failure. Steel and concrete reach ultimate strains and failsimultaneously

C Pure moment. Steel yields before concrete fails in compression. Mo-ment capacity with no normal force

Forces are extracted from the FE-program and plotted on a N-M diagramfor the respective pier or spandrel column.

2.7 Ultimate Strength of Elements 39

Figure 2.16: Simplified N-M interaction diagram for uniaxial bending

2.7.4 Bridge Deck

The loaded state of the bridge deck was evaluated based on calculated mo-ment and shear capacity. Shear capacity is calculated as in Section 2.7.2 andthe moment capacity is calculated by viewing the box girder as a series ofT-beams with effective flange width estimated with methods from Eurocode2.

Moment capacity is then calculated with the same method as for the archin Section 2.7.1 and omitting the normal force N from Equations 2.51 and2.53.

Chapter 3

The Bridge and Applied Load

In this chapter the analyzed bridges are described as well as the appliedload. The bridges do not exist in reality but they are assumed to be locatedin the south of Iceland.

The chapter starts with a description of the original bridge and its struc-tural elements followed by a description of the lead rubber bearings usedin the base isolated bridge. Subsequently the results of modal analysis arepresented for the original and base isolated bridge.

In Section 3.2 the applied load is described.

3.1 The Bridge

The bridge analyzed is a reinforced concrete arch bridge with the arch po-sitioned below the bridge deck. The arch has a span of 100m and a riseof 22.3m, resulting in a span-to-rise ratio of 4.48. The superstructure iscontinuous with a total span of L = 9 20m = 180m.

In the original bridge, expansion joints are introduced immediately outsidethe main spandrel columns resting on the arch foundations. These columnsare therefore designed to be stronger than the others and able to transmitthe horizontal forces from the deck to the foundation. The horizontal forcesconsidered in the original design were mainly due to thermal actions (ex-pansion and contraction of the concrete) but also due to deformation of thestructure.

42 The Bridge and Applied Load

Figure 3.1: Three dimensional view of the bridge

3.1.1 Structural Elements

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