an analytical model for simulating the critical …

The Pennsylvania State University

The Graduate School

College of Engineering

AN ANALYTICAL MODEL FOR SIMULATING

THE CRITICAL BEHAVIOR OF ELASTOMERIC

SEISMIC ISOLATION BEARINGS

A Dissertation in

Civil Engineering

by

Xing Han

© 2013 Xing Han

Submitted in Partial Fulfillment of the Requirements for the Degreee of

Doctor of Philosophy

December 2013

ii

The dissertation of Xing Han has been reviewed and approved* by the following:

Gordon P. Warn Assistant Professor of Civil Engineering Chair of committee Dissertation Advisor Andrew Scanlon Professor of Civil Engineering Swagata Banerjee Basu Assistant Professor of Civil Engineering Charles E. Bakis Distinguished Professor of Engineering Mechanics Peggy Johnson Professor of Civil Engineering Department Head *Signatures are on file in the Graduate School

iii

Abstract

Elastomeric bearings are one type of seismic isolation devices widely used in practice. Under

simultaneous lateral displacement and vertical compressive load, the shear force of an individual bearing

might reach a maximum beyond which the tangential horizontal stiffness becomes negative. This

behavior, referred to herein as “critical behavior,” has implications for the earthquake response and

stability of isolation systems composed of elastomeric and lead-rubber bearings and yet is not adequately

considered by widely used bearing models for numerical earthquake simulation. Semi-empirical bearing

models that are able to simulate the critical behavior have been developed and shown to simulate the

experimentally observed response of elastomeric bearings with reasonable accuracy. However these

models rely upon numerous empirical parameters that must be experimentally calibrated and/or have

complex solutions making the models impractical for design purpose.

This study aims to evaluate existing semi-empirical bearing models to elucidate the underlying

mechanism(s) leading to critical behavior and to use this knowledge as guidance for the development of

an improved analytical model for simulating such behavior that does not rely upon experimentally

calibrated parameters. Experimental data from past and present physical testing are being used to evaluate

the capabilities of the improved analytical model for simulating the behavior of elastomeric bearings and

isolation systems.

iv

Table of Contents

List of Notations ......................................................................................................................................... vii

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

List of Tables .............................................................................................................................................. xv

Acknowledgment ....................................................................................................................................... xvi

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

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

1.2 Motivation ............................................................................................................................................... 3

1.3 Objective ................................................................................................................................................. 3

1.4 Methodology ........................................................................................................................................... 3

1.5 Organization ............................................................................................................................................ 4

Chapter 2 Literature Review and Background Theory ................................................................................. 5

2.1 Experimental evidence of critical behavior ............................................................................................ 5

2.2 Existing models for simulating the bearing behavior ............................................................................. 6

2.2.1 Two-spring model .................................................................................................................... 6

2.2.2 Adaptation of the two-spring model for simulating critical behavior ...................................... 7

2.2.3 Adaptation of the two-spring model for simulating shear force response ............................... 8

2.3 Critical behavior of elastomeric bearings ............................................................................................... 8

2.3.1 Definition of critical behavior .................................................................................................. 8

2.3.2 Design requirements ................................................................................................................ 9

2.3.3 Critical load capacity in laterally undeformed configuration................................................. 10

2.3.4 Critical load capacity in laterally deformed configuration .................................................... 11

2.3.5 Experimental studies on the critical load of laterally deformed elastomeric bearings ........... 12

2.3.6 Finite element studies on the critical load of elastomeric bearings ........................................ 15

2.4 Summary ............................................................................................................................................... 16

Chapter 3 Evaluation of existing semi-empirical bearing models ........................................................ 17

3.1 General .................................................................................................................................................. 17

3.2 Scope ..................................................................................................................................................... 17

3.3 Analysis procedures for evaluating the existing models ....................................................................... 18

3.3.1 Existing semi-empirical models ............................................................................................. 18

3.3.2 Evaluation of existing model simulations .............................................................................. 23

3.3.3 Global sensitivity analysis ..................................................................................................... 24

v

3.4 Results for model evaluation................................................................................................................. 25

3.4.1 Evaluation of existing model ................................................................................................. 25

3.4.2 Global sensitivity analysis – identifying controlling parameters for simulation.................... 27

3.5 Discussion ............................................................................................................................................. 31

Chapter 4 Development and evaluation of an improved analytical model .......................................... 33

4.1 General .................................................................................................................................................. 33

4.2 Scope ..................................................................................................................................................... 33

4.3 Analytical model ................................................................................................................................... 34

4.3.1 Analytical model description ................................................................................................. 34

4.3.2 Convergence study on improved analytical model ................................................................ 38

4.3.3 Evaluation of analytical model using experimental data ....................................................... 38

4.4 Finite element model ............................................................................................................................. 41

4.4.1 Finite element model description ........................................................................................... 42

4.4.2 Constitutive model ................................................................................................................. 43

4.4.3 Mesh sensitivity ..................................................................................................................... 45

4.4.4 Validation of FE method ........................................................................................................ 48

4.5 Sensitivity analysis on FE models ........................................................................................................ 49

4.5.1 Method of Morris ................................................................................................................... 50

4.5.2 Results of sensitivity analysis ................................................................................................ 51

4.5 Assessing simulation capability of analytical model comparing to the FE model ................................ 53

4.6 Discussion ............................................................................................................................................. 54

Chapter 5 Numerical and experimental earthquake simulation of an isolation system composed of elastomeric bearings ................................................................................................................................. 56

5.1 General .................................................................................................................................................. 56

5.2 Scope ..................................................................................................................................................... 56

5.3 Dynamic model ..................................................................................................................................... 56

5.3.1 Formulation of the equations of motion ................................................................................. 58

5.3.2 Analytical bearing model ....................................................................................................... 59

5.3.3 Stepwise solution procedure .................................................................................................. 62

5.4 Physical earthquake simulation testing ................................................................................................. 63

5.4.1Test setup and instrumentation ............................................................................................... 63

5.4.2 Test bearing specimen ............................................................................................................ 65

5.4.3 Test program .......................................................................................................................... 66

vi

5.5 Evaluation the dynamic response of bearing model ............................................................................. 68

5.5.1 Comparison between numerical and experimental simulation .............................................. 68

5.5.2 Simulation of experimental tests by Sanchez et al. and comparison of results ...................... 77

5.6 Further demonstration of the analytical model ..................................................................................... 79

5.7 Discussion ............................................................................................................................................. 82

Chapter 6 Conclusion ............................................................................................................................... 83

6.1 Summary ............................................................................................................................................... 83

6.2 Specific conclusions .............................................................................................................................. 84

6.3 Significance of study ............................................................................................................................. 84

6.4 Recommendations for future research .................................................................................................. 85

References .................................................................................................................................................. 87

Appendix A Instrumentation used in dynamic test ..................................................................................... 90

Appendix B Additional test specimen ........................................................................................................ 93

Appendix C Earthquake simulation testing program .................................................................................. 94

Appendix D Parameter set used in method of Morris ................................................................................. 96

vii

List of Notations

a ratio of inner bearing diameter to outer bearing diameter

Ab the bonded rubber area of elastomeric bearing

Ar the overlapping area between the top-bonded and bottom-bonded elastomer area at a

given lateral displacement

As effective area of elastomeric bearing

b distance between the center of the left and right bearings

C10, C20 constants for constitutive relationship of rubber material

Cs dimensionless shear spring constant

Cθ dimensionless rotational spring constant

cx, cz equivalent viscous damping coefficient in x and z direction

Do outer diameter of elastomeric bearing

Di inner diameter of elastomeric bearing

D1 constants for constitutive relationship of rubber material

Eb bending modulus of an individual rubber layer

Ec compressive modulus of rubber

F shear force in the bearing

Fix, Fiz inertia force in x and z direction

FL, FR bearing shear force in the left and right bearings

FDxL, FDxR horizontal damping force for left and right bearings

FDzL, FDzR vertical damping force for left and right bearings

fi ith factor in sensitivity analysis

G effective shear modulus of elastomeric bearing

Go initial shear modulus of elastomeric bearing

h total height of elastomeric bearing

viii

hc distance between center of mass and top of bearing

I moment of inertia of bonded rubber area

I0 moment of inertia of the lump mass

1I first strain invariant of the Cauchy–Green deformation tensor

2I second strain invariant of the Cauchy–Green deformation tensor

Jel elastic volume ratio for constitutive relationship of rubber material

Kh tangential horizontal stiffness

Kθ reduced rotational spring stiffness

Kθo initial rotational spring stiffness

Ks reduced shear spring stiffness

Kso initial shear spring stiffness

kv tangential vertical stiffness

kvo initial vertical stiffness

k total number of factors being investigated in sensitivity analysis

Miy inertia moment in y direction

Ms internal moment

Mu ultimate moment of rotational spring

My yield moment of rotational spring

m mass of the lump mass

N number of rubber layers in elastomeric bearing

n number of values obtained from the analytical models

P total vertical load applied on the elastomeric bearing

Pcr critical load of elastomeric bearing in laterally deformed configuration

Pcro critical load of elastomeric bearing in laterally undeformed configuration

PD unfactored vertical dead load

ix

PE Euler buckling load

PL*, PR

* dynamic vertical forces on the left and right bearings

PLL unfactored vertical live load

POT vertical load due to seismic live load

PSL vertical load due to overturning moment effect

p numbers of interval used in Method of Morris

pP target prototype pressure

pM target model pressure

Qs internal shear spring force

R evaluation index

r rotational spring parameter

rk rank of the method of Morris

S shape factor

Si first order index in Sobol’ method

STi total index in Sobol’ method

s shear spring deformation

s1, s2 nonlinear shear spring parameter

Tr total rubber thickness

TP target prototype period

TM target model period

tu rubber layer of unit inch thickness

tr individual rubber layer thickness

U volumetric strain energy

u lateral displacement of the bearing model

ucr critical displacement

ucr,m critical displacement value obtained from model

x

ucr,t critical displacement value obtained from experiment

um lateral displacement in x-direction of the dynamic model

ug horizontal component of ground motion

V(y) total variance of y

V~i the average variance from all factors but ith factor

Vi the portion of the total variance only due to the ith factor

Vij the portion of output variance due to the interaction between ith factor and jth factor

v height reduction of the bearing model

vg vertical component of ground motion

vk height reduction due to kinematics

vm vertical displacement in z-direction of the dynamic model

vP height reduction due to vertical load P

vst height reduction due to static weight

W static weight

Z section modulus

α dimensionless constant with a value of Tr

β damping ratio

εs strain in vertical spring

εcr critical shear strain

θ rotation in the bearing model

θy yield rotation angle of the bearing model

σ nominal stress

σs stress in the vertical spring

σy tensile yield stress

rotation in dynamic model

g rotation component of ground motion

xi

φ rotation of the dynamic isolation system model

xii

List of Figures

Fig. 1-1 Seismic isolation ............................................................................................................................. 1

Fig. 1-2 Critical behavior of elastomeric bearings: (a) elastomeric bearing under simultanous lateral

displacement and vertical load; (b) bearing shear force response; (c) stability curve. ................................. 2

Fig. 2-1 Shear force-lateral displacement relationship for individual bearing from test conducted by

Sanchez et al. (2013) ..................................................................................................................................... 5

Fig. 2-2 Two-spring model ........................................................................................................................... 7

Fig. 2-3 Illustration of shear factor, S ......................................................................................................... 11

Fig. 2-4 Illustration of the reduced area for circular bearing in the laterally deformed configuration ....... 12

Fig. 2-5 Comparison between critical load data obtained from past experimental data and simulation from

reduced area method ................................................................................................................................... 15

Fig. 3-1 Nonlinear spring properties from Nagarajaiah’s model for Bearing 1: ......................................... 20

(a) shear spring; (b) rotational spring .......................................................................................................... 20

Fig. 3-2 Moment-rotation relationship for rotational spring proposed by Iizuka for Bearing 1 ................. 21

Fig. 3-3 Critical displacement comparion between model simulation and experimental testing for Bearing

1: (a) Nagrajaiah’s model; (b) Iizuka’s model ............................................................................................ 26





Fig. 3-6 Sobol’s sensitivity indices for Bearing 1 ....................................................................................... 28



Fig. 3-9 Regions of controlling parameters illustrated on critical load simulations from Iizuka model: (a)

Bearing 1; (b) Bearing 2; (c) Bearing 3 ...................................................................................................... 31

xiii

Fig. 4-1 Analytical model for elastomeric bearing ..................................................................................... 34

Fig. 4-2 Uniaxial constitutive model for vertical springs in analytical model ............................................ 37

Fig. 4-3 Convergence study for mechanistic model: shear force - lateral displacement results for different

numbers of vertical springs: (a) n = 6 springs; (b) n = 12 springs; (c) n = 20 springs. ............................... 39

Fig. 4-4 Shear force response comparison between test result (Sanchez et al. 2013) and analytical model

result: (a) experimental test; (b) analytical model simulation ..................................................................... 40

Fig. 4-5 Critical load comparison between test result and analytical model result ..................................... 41

Fig. 4-6 Illustration of FE model: (a) elevation view; (b) plain view ......................................................... 43

Fig. 4-7 Comparison of shear stiffness between 3D FE model and test result for Bearing 1: (a) Neo-

Hookean; (b) Mooney-Rivlin ...................................................................................................................... 45

Fig. 4-8 Comparison of vertical stiffness between 3D FE model and test result for Bearing 1: (a) Neo-

Hookean; (b) Mooney-Rivlin ...................................................................................................................... 45

Fig. 4-9 Cross sections of different meshes for Bearing 1 .......................................................................... 46

Fig. 4-10 Simulated critical displacements of Bearing 1 for different meshes ........................................... 47

Fig. 4-11 Simulated critical displacement of Bearing 1 in percentage difference for different meshes ..... 47

Fig. 4-12 Converged mesh for Bearing 2 .................................................................................................... 48

Fig. 4-13 Shear force response - lateral displacement from FE simulaiton for various vertical compressive

force levels .................................................................................................................................................. 49

Fig. 4-14 Comparison of critical load-displacements from FE model simulation with experimetnal data:

(a) Bearing 1; (b) Bearing 2 ........................................................................................................................ 49

Fig. 4-15 Sensitivity indices for input parameters at different P/Pcro ratio ................................................. 52

Fig. 4-16 Further evaluation result for analytical models in critical displacement ..................................... 54

Fig. 4-17 Further evaluation result for analytical models in peak shear force ............................................ 55

Fig. 5-1 Illustration of the dynamic analytical model: (a) undeformed; and (b) deformed configurations 57

Fig. 5-2 Undeformed and deformed configurations for analytical model for elastomeric bearing ............. 60

Fig. 5-3 Dynamic shake table test setup ..................................................................................................... 64

xiv

Fig. 5-4 Plan view of bearing setup ............................................................................................................ 65

Fig. 5-5 Details for test bearings ................................................................................................................. 66

Fig. 5-6 Comparison of lateral displacement history at the center of mass for the Erzincan ground motion

at 50% intensity........................................................................................................................................... 69

Fig. 5-7 Comparison of shear force history in the individual bearings for the Erzincan ground motion at

50% intensity .............................................................................................................................................. 70

Fig. 5-8 Comparison of vertical force history in the individual bearings for the Erzincan ground motion at

50% intensity .............................................................................................................................................. 70

Fig. 5-9 Comparison of individual and total shear force - lateral displacement response for Erzincan

ground motion at 50% intensity .................................................................................................................. 71

Fig. 5-10 Comparison of lateral displacement history at the center of mass for the Erzincan ground motion

at 95% intensity........................................................................................................................................... 73

Fig. 5-11 Comparison of shear force history in the individual bearings for the Erzincan ground motion at

95% intensity .............................................................................................................................................. 73

Fig. 5-12 Comparison of total vertical force history in the individual bearings for the Erzincan ground

motion at 95% intensity .............................................................................................................................. 74

Fig. 5-13 Comparison of individual and shear force - lateral displacement response for the 95% intensity

test. .............................................................................................................................................................. 75

Fig. 5-14 Comparison of experimental and model shear force - lateral displacement response for test

resulting in failure of bearings .................................................................................................................... 77

Fig. 5-15 Comparison of experimental and model shear force - lateral displacement response for

simulations with the Erzincan ground motion scaled to 150% intensity. ................................................... 79

Fig. 5-16 Comparison of model shear force - lateral displacement response from numerical simulation of

Sanchez et al. tests with Erzincan ground motion scaled to 150% intensity and varying hc ...................... 81

xv

List of Tables

Table 3-1. Details of experimentally tested bearings .................................................................................. 18

Table 3-2 Evaluation index for analytical models ...................................................................................... 27

Table 3-3 Varied parameter and parameter ranges used for global sensitivity analysis ............................. 27

Table 4-1 Evaluation index for analytical models ...................................................................................... 40

Table 4-2 Material constant used in FE constitutive model ........................................................................ 44

Table 4-3 Detail of three meshes used in mesh convergence study for Bearing 1...................................... 47

Table 5-1 Details of experimentally tested bearings ................................................................................... 66

Table 5-2 Result from dynamic characterization test ................................................................................. 67

Table 5-3 Program for earthquake simulation test ...................................................................................... 68

Table 5-4 Relative error of model simulation result comparing to experimental test result ....................... 76

xvi

Acknowledgment

I would like to gratefully acknowledge the people who have helped me during the process of my

PhD study. It is hard to imagine that I could finish this dissertation without all of you.

My advisor, Professor Gordon Warn has been a great mentor and friend. His generous guidance and

support make the whole process a thoughtful and memorable journey. It was a pleasure working with him.

I would also like to thank my other committee members, Professor Andrew Scanlon, Professor Swagata

Banerjee Basu and Professor Charles Bakis for their time and advices on my study.

The financial support of the National Science Foundation through award number CMMI-1031362 is

gratefully acknowledged. The support of the faculty and staff of the George E. Brown Jr., Network for

Earthquake Engineering Simulation (NEES) equipment site at the University at Buffalo during the

experimental portion of this study is gratefully acknowledged as well.

I would like to express my gratitude to the brothers and sisters in State College Chinese Alliance

Church. Their prayers and unconditionally love have kept me moving forward.

I would like to give my grateful thanks to my parents, Defang and Rongna. I could not step on the

journey of my PhD study without the consistent support from you. Thank you for raising me up and

letting me make all the important decisions in my life.

The deepest gratitude is for my gorgeous wife, the love of my life, Lijiao. I would like to say more

but none of words in the world can express that. May our Lord Jesus Christ keep blessing our family and

especially to our soon-to-be-born son/daughter.

Chapter 1

Introduction

1.1 Background

Seismic isolation is a technique used to protect important structures, such as hospitals, data centers,

emergency response centers, historical structures, and bridges from the damaging effects of horizontal

earthquake ground shaking. Seismic isolation is achieved by introducing horizontally flexible yet

vertically stiff elements that decouple the superstructure from its supporting foundation (Fig. 1-1a). The

low horizontal stiffness of the isolators shifts the fundamental horizontal natural period of the structure

into long period range, e.g. 2.5 to 4 seconds, thereby reducing absolute acceleration and drift demands

above the plane of isolation during earthquake ground shaking. The reduced acceleration and drift

demands minimize the likelihood of damage to acceleration sensitive and displacement sensitive

nonstructural systems, equipment and content in buildings thereby minimizing economic losses and loss

of operational functionality of the facility. Elastomeric bearings are one type of seismic isolation devices

(a) Illustration of isolated structure (b) Bearing cross section (taken from Warn and Whittaker 2006)

Fig. 1-1 Seismic isolation

Earthquake ground motion

2

widely used in practice. A typical elastomeric seismic isolation bearing consists of a number of rubber

layers (natural or synthetic) bonded to intermediate steel shim plates as shown in the photograph in Fig.

1-1b. The lateral flexibility is achieved by providing a specific total thickness of rubber for a given

bonded rubber area and shear modulus. The close spacing of steel shim plates provide a large vertical

stiffness, relative to the horizontal.

When an elastomeric bearing subjected to simultaneous vertical compressive load, P, and relative

lateral displacements, u, as illustrated in Fig. 1-2a, the shear force might pass through a maximum value

beyond which the bearing exhibits negative tangential horizontal stiffness as illustrated in Fig. 1-2b.

However this behavior is not adequately considered by widely used bearing models yet it has implications

on the stability and earthquake response of the individual bearings and global isolation system. Numerical

earthquake simulation is an important tool for the design of seismically isolated structures. However,

numerical earthquake simulation software widely used to analyze isolated structures, for example 3D-

Basis (Nagarajaiah et al. 1991), OpenSees (PEER 2013), SAP2000 (CSI 2012), and others, utilize

coupled-plasticity (Ryan et al. 2005) or modified Bouc-Wen (Nagarajaiah et al. 1991), models that

assume a constant, positive, second-slope stiffness irrespective of bearing lateral displacement and/or

vertical load.

Fig. 1-2 Critical behavior of elastomeric bearings: (a) elastomeric bearing under simultanous lateral displacement and vertical load; (b) bearing shear force response; (c) stability curve.

Rubber layer

u

P

FM

Steel shim plate

P

F M

a. b.

ucru

F P = 0

P > 0

c.

Pcr

ucr

3

The point where the shear force passes through a maximum, corresponding to zero horizontal

tangential stiffness, is the point of neutral equilibrium that is considered the stability limit of the bearing

and the displacement referred to as the critical displacement. The vertical load, P and corresponding

critical displacement, ucr, represent a point on a curve that describes the critical load capacity of the

bearing at a given lateral displacement as illustrated in Fig. 1-2c.

1.2 Motivation

The underlying physical mechanism(s) causing the behavior illustrated in Fig. 1-2 are not well

understood. Semi-empirical bearing models (Iizuka 2000, Yamamoto et al. 2009 and Kikuchi et al. 2010)

exist that are able to simulate this behavior. However the number of experimentally determined empirical

parameters precludes a physical understanding of the bearing behavior. The conventional method for

estimating the critical load capacity of an elastomeric bearing at a given lateral displacement lacks a

rigorous theoretical basis and provides an inconsistent estimate of the critical load by comparison to

experimental data. Furthermore, the widely used bearing models do not adequately account for the

reduction in tangential horizontal stiffness with increasing vertical load and lateral displacement that has

been experimentally observed (Sanchez et al. 2013).

1.3 Objective

The overarching objectives of this dissertation are to develop a fundamental understanding of the

mechanism(s) leading to the reduction in tangential horizontal stiffness observed in elastomeric bearings

with increased lateral displacement and to develop an analytical model for simulating the behavior of

elastomeric bearings. Furthermore, the analytical model will be employed for numerical earthquake

simulation to simulate the responses of individual bearing and global isolation system and evaluated using

data from physical earthquake simulation testing of an isolation system composed of elastomeric bearings.

1.4 Methodology

To accomplish the overarching objectives, this study will progress as follows:

4

1. Evaluate the simulation capability of existing semi-empirical bearing models of elastomeric

bearings using experimental data from past studies.

2. Evaluate semi-empirical bearing models using state-of-the-art global sensitivity techniques to

determine the relative importance of the model parameters for simulating the critical behavior

of the bearings.

3. Develop an improved, analytical bearing model that requires only material and physical

parameters based on the knowledge gained from the sensitivity studies.

4. Employ the improved analytical bearing model for numerical earthquake simulation of an

isolation system composed of elastomeric bearings.

5. Evaluate the models using existing experimental data from past studies and data generated

from physical earthquake simulation testing performed as part of this study.

6. Expand the parameter set beyond that available from existing experimental data using three-

dimensional finite element modeling.

1.5 Organization

The dissertation contains six chapters as follows. Chapter 2 presents the literature review and

background theory of elastomeric bearings. Chapter 3 presents the evaluation of two existing semi-

empirical models and the global sensitivity analysis on the most accurate model. Chapter 4 presents the

development and evaluation of an analytical bearing model. Chapter 5 presents a dynamic model that

consists of the proposed analytical bearing model, for simulating the earthquake response of isolation

systems composed of elastomeric bearings. Chapter 6 presents the key conclusion, impact of this study

and recommendations for future study.

5

Chapter 2

Literature Review and Background Theory

2.1 Experimental evidence of critical behavior

Sanchez et al. (2013) tested bearings with four different geometries to investigate the critical

behavior of elastomeric bearings. Figure 2.1 presents the shear force- lateral displacement relationship for

Bearing # 11795 tested in this study. The bearing was first applied on a specific vertical compressive load,

and then sheared with increasing lateral displacement while keeping the vertical load constant. From the

results shown in Fig. 2-1, the initial horizontal stiffness reduced with increasing vertical load applied on

the bearing. For each test that the bearing was applied on a non-zero vertical load, the tangential

horizontal stiffness reduced with increasing lateral displacement. For tests of vertical compressive load, P

= 178 kN, 267 kN and 356 kN, the shear force reached a maximum value at some lateral displacement

beyond which the tangential horizontal stiffness became negative.

Fig. 2-1 Shear force-lateral displacement relationship for individual bearing from test conducted by Sanchez et al. (2013)

Sanchez et al. (2013) also tested an isolation system composed of four elastomeric bearings on shake

table to investigate the dynamic behavior of elastomeric bearing and global system. During the test, shear

force response of individual bearing and isolation system reached maximum after which the tangential

horizontal stiffness went to negative. Since the current widely used models (Nagarajaiah et al. 1991, Ryan

0 25 50 75 100 125 1500

5

10

15

20

25

30Sanchez et al. 2013

P = 0 kNP = 89 kNP = 178 kNP = 267 kNP = 356 kN

Lateral displacement: u (mm)

She

ar fo

rce:

F (

kN)

6

et al. 2005) assumes a constant, positive, second-slope stiffness irrespective of bearing lateral

displacement and/or vertical load, this finding shows that those models are not able to adequately simulate

the influence of vertical load and lateral displacement on the shear force response of elastomeric bearings.

2.2 Existing models for simulating the bearing behavior

To simulate the behavior of elastomeric bearings, both analytical and semi-empirical models are

developed in past studies. Koh and Kelly (1988) introduced a two-spring mechanical model to simulate

the bearing behavior. However, the two-spring model is not able to simulate the bearing shear force

reaching a maximum value, i.e. the critical behavior. Subsequent models based on two-spring model were

developed to simulate: (1) critical behavior (Nagarajaiah and Ferrell 1999) or (2) the influence of vertical

load on the shear force response of elastomeric bearings specifically at high shear strains (Iizuka 2000,

Yamamoto et al. 2009, Kikuchi et al. 2010). However, all these subsequent models include semi-

empirical parameters that must be calibrated from experiment testing. This section reviews these existing

models.

2.2.1 Two-spring model

Koh and Kelly (1988) introduced a two-spring mechanical model (Fig. 2-2) to simulate the reduction

in initial horizontal stiffness, and reduction in height, v, with increasing vertical load, P, and simultaneous

lateral displacement, u. The two-spring model shown in Fig. 2-2 consists of a shear spring and a rotational

spring that are connected through rigid elements and frictionless rollers. A rigid tee is supported by two

frictionless rollers and connected to a rocking base by a shear spring. The rocking base consisting of a

rigid element, a frictionless hinge, and a rotational spring supports the tee-roller assembly. Under

simultaneously vertical, P, and shear force, F, the top of model translates with a lateral displacement, u,

and height reduction, v. The global deformation (u and v) are the result of local deformation, s, and θ, in

the shear and rotational springs, respectively, due to the internal shear spring force, Qs, and internal

rotational spring moment, Ms. Considering equilibrium and compatibility of the two-spring model in the

deformed configuration (see Fig. 2-2), the following equations are derived:

7

Fig. 2-2 Two-spring model

cos sinu s h (2-1)

sin (1 cos )v s h (2-2)

( )sM Pu F h v (2-3)

sin cossQ P F (2-4)

Specifying linear properties for the shear spring stiffness, Ks = Qs/s, and rotational spring stiffness,

Kθ =Ms/θ, the two-spring model was found to simulate the reduction in initial horizontal stiffness,

reduction in total height with increasing vertical load (Koh and Kelly 1988), and reduction in vertical

stiffness with increasing lateral displacement (Warn et al. 2007) using simple linear properties for both

the shear and rotational springs. However, the two-spring model proposed by Koh and Kelly (1988) is not

able to simulate the critical behavior due to the linear spring properties (Warn and Weisman 2010).

2.2.2 Adaptation of the two-spring model for simulating critical behavior

Nagarajaiah and Ferrell (1999) proposed a model to simulate the critical behavior of elastomeric

bearings. The model proposed by Nagarajaiah and Ferrell is an adaptation of the two-spring model that

uses empirical relationships to reduce the tangential stiffness of the shear and rotational springs in place

of linear spring properties (provided details in Section 3.3.1). The Nagarajaiah model was shown to

simulate a reduction in critical load capacity with increasing lateral displacement and a non-zero value of

critical load at a displacement equal to the bearing diameter but did not simulate the value of the critical

load at a given lateral displacement with a high degree of accuracy by comparison to the experimental

P

F

hshear spring

rotational spring

uv

s

x

y

x’

y’

8

data (Buckle et al. 2002). Furthermore, no rational theoretical basis was provided for the tangential

stiffness reduction of the spring properties used in Nagarajaiah model.

2.2.3 Adaptation of the two-spring model for simulating shear force response

Other researchers also proposed semi-empirical models to capture the shear force response of

elastomeric bearings at high shear strains and large vertical loads. Iizuka (2000) proposed a nonlinear

model, based again on the two-spring model, for the purpose of simulating the effect of large vertical

loads on shear force response at large shear strains (see details in Section 3.3.1). The model proposed by

Iizuka uses semi-empirical relationships for the shear and rotational springs. Yamamoto et al. (2009)

proposed a more complex semi-empirical model to simulate the influence of vertical load on the shear

force response of elastomeric bearing at large shear strains. Yamamoto model consists of a shear spring at

mid-height and a number of axial springs at the top and bottom boundaries. Later, Kikuchi et al. (2010)

extended the Yamamoto model to a three-dimensional model. The Yamamoto and Kikuchi models were

shown to agree well with experimental data. However these models require a number of empirical

parameters and Bessel functions resulting in a complex solution procedure. The reliance of these models

on a number of empirical parameters that must be experimentally calibrated make them impractical for

the purpose of design since the values of these parameters are not known a priori.

2.3 Critical behavior of elastomeric bearings

An important consideration for the design of isolation systems composed of elastomeric bearings is

assessing the stability limit of the isolation system and the individual bearings. The current design

requirements and past research of critical behavior of elastomeric bearing are reviewed in this section.

2.3.1 Definition of critical behavior

When elastomeric bearings are subjected to simultaneous lateral displacement, u, and large vertical

compressive force, P, as illustrated in Fig. 1-2a, the shear force might pass through a maximum value

beyond which the bearing exhibits negative tangential horizontal stiffness. The stability limit of the

9

individual bearing can be defined as the point of neutral equilibrium where the variational total potential

energy is zero (Tauchert 1974). The point of neutral equilibrium corresponds to the point where the shear

force, F, reaches a maximum value or the tangential horizontal stiffness is equal to zero, i.e.:

0H

FK

u

(2-5)

where F is the shear force and u is the lateral displacement as illustrated in Fig. 1-2b. The displacement

and vertical force at the point of neutral equilibrium are considered the critical displacement, ucr, and

critical load, Pcr. Determination of these critical points (ucr, Pcr) for a range of vertical force levels

provides a locus of points representing the "stability" curve shown in Fig. 1-2c. Applying the chain rule,

as identified by Sanchez et al. (2013), to Eq. (2-5) yields:

0H

F F PK

u P u

(2-6)

where P is the vertical compressive force (see Fig. 1-2b). There is no requirement that ∂F/∂P must equal

to zero. Therefore in order for Eq. (2-6) to hold the following must be true:

P

u 0 (2-7)

Eq. (2-7) suggests the point of neutral equilibrium also corresponds to the point where the derivative of

the vertical force with respect to the lateral displacement is equal to zero. Inspection of Eq. (2-5) and (2-7)

indicates the point of neutral equilibrium (i.e. critical points) can be determined using two different

methodologies in the experimental testing.

2.3.2 Design requirements

Currently stability of elastomeric bearings is assessed using procedures that have been adopted from

the American Association of State Highway and Transportation Officials (AASHTO) Guide

Specifications for Seismic Isolation Design (2010). The guide specifications require that all the vertical

load-carrying elements of the isolation system satisfy the following service loading criteria:

10

3cro

D LL

P

P P

(2-5)

where Pcro is the critical load in laterally undeformed configuration (i.e. u = 0), PD is the unfactored

vertical dead load, and PLL is the unfactored vertical live load. Under MCE loading,

1.2cr D SL OTP P P P (2-6)

where Pcr is the reduced critical load at 1.1 times the total design lateral displacement , PSL is the vertical

load due to seismic live load, and POT is the vertical load due to overturning moment effect.

2.3.3 Critical load capacity in laterally undeformed configuration

In the laterally undeformed configuration (zero lateral displacement) an elastomeric bearing is

essentially a short composite column with low shear rigidity. Estimation of the critical load capacity of

elastomeric bearings in the laterally undeformed configuration, Pcro, is based on theoretical work of

Haringx (1948, 1949a, b) for simulating the critical load of rubber rods accounting for shear deformations

assuming geometric and material linearity. Subsequent experimental tests (Gent 1964, Derham and

Thomas 1981) showed that experimentally obtained critical load data for rubber springs agreed well with

the theoretical simulations from Haringx. However, to utilize the results of Haringx's theory (derived for

homogenous rubber rods) the elastomeric bearings geometric properties are used to determine an

"effective" rubber rod for which the critical load is estimated using:

24

2s E s s

cro

GA P GA GAP

(2-7)

where PE is the Euler buckling load; G is the effective shear modulus and As is the shear area accounting

for composite material, which is defined as:

s br

hA A

T (2-8)

where Ab is the bonded rubber area; h is the total height of the bearing and Tr is the total rubber thickness

of the bearing. When PE is much greater than GAs, Eq. (2-7) can be simplified to:

11

cro E sP P GA (2-9)

for elastomeric bearings when shape factor, S > 5 (Kelly 1997). The dimensionless geometric parameter,

shape factor, S, is generally defined as the ratio of the loaded area to the area free to bulge. For example,

for a circular bearing shown in Fig. 2-3, the shape factor, S, can be expressed in Eq. (2-10).

2Loaded area 4

Area free to bulge 4r r

D DS

Dt t

(2-10)

A detail calculation of the shape factor for an annular bearing can be found in Constantinou et al. (1992).

Fig. 2-3 Illustration of shear factor, S

2.3.4 Critical load capacity in laterally deformed configuration

There is no codified guidance on how to estimate Pcr. In section 17.2.4.6, ASCE 7 (2010) states that

“Each element of the isolation system shall be designed to be stable under the design vertical load where

subjected to a horizontal displacement equal to the total maximum displacement”, but does not provide

any procedure to achieve the criteria. Though not codified, the reduced area method (Buckle and Liu 1994)

is the generally accepted procedure for estimating critical loads at a given lateral displacement. Using the

reduced area method, the critical load at a given lateral displacement, u, is estimated using:

rcr cro

b

AP P

A

(2-11)

where Ar is the overlapping area between the top-bonded and bottom-bonded elastomer area at a given

lateral displacement, u, as illustrated on the circular bearing shown in Fig. 2-4; and Pcro is the critical load

at zero lateral displacement [Eq. (2-7)]. The reduced area method effectively treats the elastomeric

Bearing diameter, D

Individual rubber layer thickness, tr Area free to bulge

Bonded rubber area, Ab

12

bearings as a simple column, which provides a zero critical load capacity when Ar = 0, corresponding to a

lateral displacement equal to the bearing diameter.

Fig. 2-4 Illustration of the reduced area for circular bearing in the laterally deformed configuration

2.3.5 Experimental studies on the critical load of laterally deformed elastomeric bearings

A few experimental studies have been designed to collect critical load data for elastomeric bearings

(Buckle et al. 2002, Weisman and Warn 2012, Sanchez et al. 2013) while other experimental studies have

provided information on the critical behavior though not the focus of the experimental testing (Kelly 1991,

Yamamoto et al. 2009, Kikuchi et al. 2010).

Two methods are used to experimentally obtain the point of neutral equilibrium (critical load) for

elastomeric bearings. The first method is based on Eq. (2-5), referred to herein as the constant vertical

load method (CV), involves subjecting the bearing to a constant vertical load, P, then shearing the bearing

to a specified lateral displacement u (see Fig. 1-2b). The point of equilibrium is determined directly from

the recorded shear force - lateral displacement response as the point where the slope equals to zero. The

second method is based on Eq. (2-7), referred to herein as constant lateral displacement method (CL),

involves shearing the bearing to a target lateral displacement, u, holding this displacement constant while

applying increasing vertical force P and monitoring a reduction in the shear force F. Repeating this

procedure for different target lateral displacement levels provides unique equilibrium trajectories (F

versus P) from which the point of neutral equilibrium, thus critical load, can be indirectly obtained.

u Do

Ab

Ar

13

Further details on each of these two methods can be found in Weisman and Warn (2012) and Sanchez et

al. (2013).

Kelly (1991) tested four Bridgestone high damping rubber bearings to observe the dynamic failure

modes of the bearings. A series of quasi-static and dynamic cyclic force - displacement tests were

conducted on each bearing followed by a monotonic shear test to failure. The monotonic shear failure

tests were conducted by subjecting each bearing to a specified (but different) vertical compressive load

then shearing them to failure. Though not the objective of Kelly's tests, points of neutral equilibrium were

observed for two bearings having shape factors of 25.2. The failure strains in those tests were between

470% and 550% and ratios of Pcr/Pcro were between 5% and 8%. Therefore, the range of critical load data

obtained from this test is limited however it is one of the few studies to provide critical load data for a

bearing having a high shape factor.

Buckle and Liu (1993, 1994) used the CL method to test stability behavior of elastomeric bearings.

Data presented in Buckle and Liu (1993, 1994) was used to evaluate the nonlinear model proposed by

Nagarajaiah and Ferrell (1999) and is presented in Buckle et al. (2002). As such, this data will be referred

to herein as Buckle et al. (2002). The bearings tested by Buckle et al. (2002) were square in plan

dimension and had relatively low shape factors ranging from 1.67 to 5. The geometry of these bearings is

more typical of non-seismic bridge bearings. The critical load data obtained by Buckle et al. clearly

demonstrated: (1) the critical load capacity of elastomeric bearings reduces with increasing lateral

displacement; (2) the critical load capacity at a lateral displacement that equals to the bearing diameter is

not equal to zero as presented by reduced area method, but a significant percentage of the Pcro.

Additional experimental tests used the CL method to obtain critical load data was conducted by

Weisman and Warn (2012). One type of elastomeric bearing (S = 10.2) and one type of lead rubber

bearing (S = 12.2) were tested by Weisman and Warn (2012). The lead rubber bearing in this test has the

same geometry with the elastomeric bearing other than the lead core in the center mandrel hole. From the

result, the lead core was shown not to affect the critical load capacity of the bearing over the displacement

range considered, e.g. from 150% to 280% shear strain (Weisman and Warn 2012). The shear strain of the

14

elastomeric bearing is defined as the ratio of lateral displacement, u, divided by the total rubber thickness,

Tr.

Sanchez et al. (2013) tested the same bearing specimen with Weisman and Warn (2012), and two

additional elastomeric bearings (S = 10.6 and S = 5.5) using both the CV and CL methods. They also

evaluated the dynamic stability of an isolation system composed of four elastomeric bearings subjected to

earthquake motions on a shake table. The critical load results obtained from quasi-static test, i.e. CV and

CL method agreed well with that from dynamic test. Furthermore, no significant difference was observed

in critical load data obtained using the CV and CL methods for the same bearing.

Other researchers also obtained data of the critical load behavior while experimentally testing on

elastomeric bearings though again the focus of their studies was not to obtain critical load data.

Yamamoto et al. (2009) tested two types of bearings (S = 31.3) to investigate shear force behavior at large

shear strain under different vertical load levels. They suggested that at large shear deformations,

elastomeric bearings exhibited stiffening behavior under low vertical load and critical behavior under

high vertical load. Kikuchi et al. (2010) tested one type of square bearing (S = 32.5) under large shear

deformations and high axial loads. The test results showed that the shear response of elastomeric bearings

at a large lateral displacement could be simulated using a proposed semi-empirical analytical model.

The capability of critical load simulation from reduced area method is evaluated by comparing to

critical load data from experimental test (Kelly 1991; Buckle et al. 2002; Weisman and Warn 2011;

Sanchez et al. 2013). However, the reduced area method provides inaccurate and inconsistent simulation

of critical load behavior. The comparison between the critical load data with the simulated value using the

reduced area method, Eq. (2-10) and that from existing experimental result is presented in Fig. 2-5. The

critical load in the laterally undeformed configuration, Pcro, was calculated using Eq. (2-6) and the

effective shear modulus, corresponding to 25% shear strain as recommended by Sanchez et al. (2013).

Also identified in each plot is the bearing shape factor, S. The comparison presented in Fig. 2-5 shows

that area method is, for the most part, conservative. However, the reduction in critical load with

increasing lateral displacement was not well simulated for all bearings. Furthermore, the simulation

15

capability of reduced area method is not consistent for bearings with similar shape factors as observed by

comparing plots of S = 5 to 5.5 and S = 10.2 to 10.6.

Fig. 2-5 Comparison between critical load data obtained from past experimental data and simulation from reduced area method

2.3.6 Finite element studies on the critical load of elastomeric bearings

The finite element method has been used to investigate the stability behavior of elastomeric bearings

(Simo and Kelly 1984; Kelly and Takhirov 2004; Warn and Weisman 2011).

Simo and Kelly (1984) used two-dimensional FE model to capture the instability behavior of

elastomeric bearings under small shear strain, i.e. laterally undeformed configuration, which concluded

that the instability behavior was not controlled by nonlinear material properties. The boundary conditions

enforced at the top and bottom of the bearing were key to assess the elastic instability of the bearings.

Buckle et al. (2002) used two-dimensional FE model to simulate the two tested square bearings by

imposing plane strain restriction in order to minimize the computational effort. Mooney-Rivlin model was

0 0.5 1 1.5 2 2.5 30

50

100

150

200Reduced AreaExperimental

Buckle 2002S=1.67

Shear strain

Pcr

(kN

)

0 0.5 1 1.5 2 2.5 30

50100150200250

Buckle 2002

S=2.5

Shear strain

Pcr

(kN

)

0 0.5 1 1.5 2 2.5 30

100200300400500

Buckle 2002

S=5

Pcr

(kN

)

0 0.5 1 1.5 2 2.5 30

50

100

150

200Sanchez 2013

S=5.5

Shear strain

Pcr

(kN

)

0 0.5 1 1.5 2 2.5 30

100200300400500

Weisman 2012Sanchez 2013

S=10.2

Shear strain

Pcr

(kN

)

0 0.5 1 1.5 2 2.5 30

50100150200250300

Sanchez 2013

S=10.6

Shear strain

Pcr

(kN

)

0 0.5 1 1.5 2 2.5 30

200

400

600

800Sanchez 2013

S=12.2

Shear strain

Pcr

(kN

)

0 1 2 3 4 5 60

5000

10000

15000

20000

S=25

Kelly 1991

Shear strain

Pcr

(kN

)

16

used to simulate the rubber material in Buckle et al. study. The result from FE models shows that the two-

dimensional FE models are able to simulate the reduction in critical load with increasing lateral

displacement.

Kelly and Takhirov (2004) also used two-dimensional FE model to determine both compression and

tension buckling load of elastomeric bearings in the laterally undeformed configuration with various

shape factors from 3 to 14. Two different constitutive models, Neo-Hookean model and Polynomial

model were used to simulate the rubber material in Kelly and Takhirov study. The simulation observed

from FE model showed good agreement with the calculation from Eq. (2-7). The critical load observed

from FE models were corresponding to shear strain less than 35%, which is not typical for shear strain

level of elastomeric bearings under MCE, e.g. 150% to 300%.

Warn and Weisman (2011) verified the FE model for simulating critical load at a given displacement

by comparing to experimental result, and further used the FE method to investigate the dependency of

critical load in strip elastomeric bearings on the bearing geometry. Neo-Hookean constitutive model was

used to simulate the rubber material in Warn and Weisman study. The result showed that the critical load

varies significantly with different individual rubber layer thicknesses at a given lateral displacement.

2.4 Summary

The existing models for simulating the bearing behavior, current design procedure and past research

of critical behavior of elastomeric bearing are reviewed in this chapter. Existing models are: (1) not able

to simulate the critical behavior of elastomeric bearings or (2) including semi-empirical parameters that

must be experimentally tested. The widely used design method for stability of elastomeric bearing, i.e.

reduced area method provides inconsistent simulation for the critical load of elastomeric bearings. An

analytical model that does not utilize empirical parameters for simulating the behavior of elastomeric

bearings would be more practical for design and provide greater insight into the underlying physical

mechanism controlling bearing behavior.

17

Chapter 3

Evaluation of existing semi-empirical bearing models

3.1 General

While the reduced area method provides a simple means of estimating the critical load capacity at a

given lateral displacement it lacks a rigorous theoretical basis and has been shown not able to simulate the

trends observed from experimental data (see Fig. 2-5). In this chapter, two existing analytical models

(Nagarajaiah and Ferrell 1999; Iizuka 2000) of elastomeric bearings are used to simulate the point of

neutral equilibrium when subjected to simultaneous vertical load and lateral displacement. In this study

the Iizuka model is used for the first time to specifically simulate the critical behavior of elastomeric

bearings. The simulated critical displacement from each model is compared to data from past

experimental studies. A global variance-based sensitivity analysis is performed on the analytical model

showing the best simulation capability to identify the model parameters to which the model simulation is

most sensitive. The results of the global variance-based sensitivity analysis will guide the development of

a mechanics based model and improve the simulation of critical behavior of elastomeric bearings.

3.2 Scope

In this chapter, two semi-empirical models, i.e., Nagarajaiah model and Iizuka model, are evaluated

for critical displacement simulations of elastomeric bearings. The variance-based global sensitivity

technique is applied on the model output showing the best simulation capability to identify the controlling

parameter of critical behavior simulation. Data from three bearings (Buckle et al. 2002; Weisman and

Warn 2012; Sanchez et al. 2013) was selected and used in evaluation of analytical models and global

sensitivity analysis. These three bearings are significantly different in material properties (shear modulus)

and bearing geometries (shape factor, plan geometry) so that the evaluation can be as robust as possible.

Details of these bearing including shear modulus, construction, and dimensions are provided in Table 3-1.

Bearing 1 was tested both by Weisman and Warn (2012), and Sanchez et al. (2013). Bearing 2 and

Bearing 3 were tested by Sanchez et al. (2013), and Buckle et al. (2002), respectively.

18

Table 3-1. Details of experimentally tested bearings

Description Symbol Unit Bearing Id.

1 2 3 Shape – – Annular Round Square

Effective shear modulus G MPa 1.08 0.95 1.38 Outer diameter / width Do mm 152 140 127 Inner diameter / width Di mm 30 – –

Individual rubber layer thickness tr mm 3 6.35 6.35 Individual shim layer thickness ts mm 3 3 1.4

Number of rubber layers n – 20 12 8 Shape factor S – 10.2 5.51 5

Critical load at undeformed configuration Pcro kN 442 153 422

3.3 Analysis procedures for evaluating the existing models

The following procedures are used to diagnose the existing semi-empirical models:

1. Two semi-empirical models (Nagarajaiah and Ferrell 1999; Iizuka 2000) proposed from

previous study are described to simulate the critical displacement of elastomeric bearings.

2. An evaluation index is proposed to identify the model with best simulation capability of the

critical displacement of an elastomeric bearing under a given vertical compressive load.

3. Global sensitivity technique is applied on the model with best simulation capability to

investigate the controlling parameters.

3.3.1 Existing semi-empirical models

Existing models for simulating the critical displacement in elastomeric bearings considered in this

study are based on two-spring model introduced by Koh and Kelly (1988). The model description as well

as compatibility and equilibrium equations were presented in Section 2.2.1. Koh and Kelly (1988)

demonstrated that the two-spring model was able to simulate the reduction in initial horizontal stiffness,

increased mechanical damping, and reduction in height, v, with increasing vertical load, P. However, the

two-spring model was not able to simulate the critical behavior with linear spring properties (Nagarajaiah

and Ferrell 1999).

Nagarajaiah and Ferrell (1999) proposed nonlinear relationships for the tangential stiffness of the

shear and rotational springs in order to simulate the critical behavior in the laterally deformed

19

configuration. The shear spring stiffness, Ks, is assumed to reduce with increasing shear spring

deformation, s, according to:

1 tanhs so sK K C s (3-1)

where Cs is the dimensionless constant equal to 0.325; Kso is initial shear stiffness, that is given by:

bso

r

GAK

T (3-2)

where Tr is the total rubber thickness; Ab and G are defined previously. The stiffness of the rotational

spring, Kθ, is assumed to reduce according to:

1or

sK K C

T

(3-3)

where Cθ is a dimensionless constant and Kθo is the initial rotational stiffness calculated according to:

( )u r ooC t D t D (3-4)

2

bo

r

E IK

T

(3-5)

where Do is the the outer bonded bearing diameter or bearing width; α is a dimensionless constant with

value of Tr; tr is the individual rubber layer thickness; Eb is the bending modulus of an individual rubber

layer; I is the moment of inertia of bonded rubber area. From Eq. (3-3), the value of Kθ drops to zero

when the shear spring deformation, s, is equal to the bearing width/outer diameter. Relationships of the Ks

and Kθ shown in Eqs. (3-1) and (3-3), respectively are plotted in Fig. 3-1 for Bearing 1 to illustrate the

reduction in stiffness properties proposed by Nagarajaiah and Ferrell (1999). Equations (2-1) through (2-

4) can be solved simultaneously to obtain lateral displacement, u, versus vertical load, P, equilibrium

trajectories for different shear force values from which the critical load, Pcr, and displacement, ucr, can be

obtained using Eq. (2-7).

20

(a) (b)

Fig. 3-1 Nonlinear spring properties from Nagarajaiah’s model for Bearing 1: (a) shear spring; (b) rotational spring

Iizuka (2000) proposed a nonlinear model, based again on the two-spring model (Koh and Kelly

1988), for the purpose of simulating the effect of large vertical loads on the shear force response of

elastomeric bearings at large shear strains. The model proposed by Iizuka uses semi-empirical

relationships for the shear and rotational springs. The yield moment, My, and ultimate moment, Mu, for an

individual rubber layer are considered in the relationship for the rotational spring. However, the

relationship between Ms and θ is semi-empirical. The tangential rotational stiffness (dMs/dθ) in the elastic

range is given as:

so y

dMK

d

(3-6)

where θ is the rotation in rotational spring (see Fig. 2-2); Kθo is calculated according to Eq. (3-5) and θy is

the yield rotational angle calculated using:

2 y

yo

Z

K

(3-7)

where Z is the elastic section modulus of bonded rubber area; σ is the nominal stress (σ = P/Ab); and σy is

the tensile yield stress of rubber (assumed to be equal to 3G (Gent 2001)). For θ θy, dMs/dθ is assumed

to be:

1

1 13

s oyr

r

y

dM K

dr

(3-8)

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

s (mm)

Ks /

Kso

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

s (mm)

Kθ /

Kθo

21

where r is an empirical dimensionless parameter. The moment-rotation relationship for Bearing 1

obtained from the Iizuka model normalized by My and θy, is plotted in Fig. 3-2 for values of r ranging

from 1 to 8 to illustrate the moment-rotation relationship. From Fig. 3-2 it can be seen that parameter r

controls the rotation at which the ultimate moment is achieved. The larger the value of r is, the more

rotation that is required to achieve Mu. Iizuka reported values for r ranging from 1.2 to 3.5 determined

empirically from characterization tests performed on elastomeric bearings. However, no guidance for

determinations for r was provided by Iizuka.

Fig. 3-2 Moment-rotation relationship for rotational spring proposed by Iizuka for Bearing 1

Iizuka proposed the following relationship for the tangential stiffness of the shear spring:

2

1 21 1s

sso

r

dQ ss s K

ds T

(3-9)

where s1 and s2 are empirical dimensionless parameters; Kso is defined previously in Eq. (3-2). The shear

spring stiffness is linear when specifying s1 = 0. Similar to r, s1 and s2 are parameters whose values were

empirically obtained from the results of material characterization testing. Iizuka specified values for s1

ranging from 0.0068 to 0.01 and a value of 3 for s2.

The Iizuka model requires an incremental form of the two-spring model equilibrium and

compatibility equations shown, i.e. Eqs. (2-1) through (2-4) that necessitates a stepwise solution

procedure. Rearranging incremental equilibrium and compatibility equations [Eqs. (2-1) to (2-4)], the

unknown incremental response is solved by using:

0 20 40 60 80 1000

1

2

3

4

θ / θy

M /

My

r = 1

r = 2

r = 4

r = 8

22

1sin cos sin 1 0

cos sin cos 0 0 0

10

0cos sin 0 cos

i i i i

i i i ii

sii i

ii

sii i i i

i

h

h

dMF h

d P

F dQ

s

s

sv u

v

P Fds

(3-10)

where incremental force and deformation quantities i(Δθ), i(Δs), i(Δv) and i(ΔF) are obtained for step i

from Eq. (3-10) for a specified vertical force, P, and incremental lateral displacement, Δu. The

incremental response quantities obtained from the solution of Eq. (3-10) for the ith step are added to the

previous step response values to move the solution to (i+1)th step using following equations.

1i i i (3-11)

1i i is s s (3-12)

1i i iv v v (3-13)

1i i iF F F (3-14)

The incremental analysis procedure is repeated for a constant vertical load, P, and increasing values of

lateral displacement, u.

Specifying s = 0 and θ = 0 for the initial condition results in an ill-conditioned matrix for Eq. (3-10),

such that a solution can not be obtained. Therefore, linear properties for the shear and rotational springs

are assumed for the initial step in each stepwise analysis to initiate the solution procedure. Initial values

for the response quantities are determined again from equilibrium and compatibility of the two-spring

model using the following equations:

1 1s

E r

GA Ps

P T

(3-15)

2

1 1

s E

E r

GA P PF s

P T

(3-16)

1 1 1 1cos sinu s h (3-17)

23

1 1 1 1sin 1 cosv s h (3-18)

where PE is the Euler buckling load of the bearing defined as:

2

2b s

E

E IP

h

(3-19)

where h and Eb are defined previously; Is is the effective moment of inertia to account for the composite

construction of the elastomeric bearing calculated as:

sr

hI I

T (3-20)

The analysis is terminated when height reduction v is larger than the total rubber thickness Tr. The

shear force response, F(u) is obtained from the solution for a specified vertical load P from which the

critical displacement ucr is determined by numerically identifying the maximum value, i.e. dF/du=0. The

stepwise solution procedure is repeated for several values of P to obtain a series of critical points (i.e. Pcr

and ucr) used to construct a stability curve.

3.3.2 Evaluation of existing model simulations

In order to identify the model that best simulates the critical displacement of an elastomeric bearing

under a given vertical load, the model simulations were evaluated by comparison to experimental data

using the root mean square deviation method (Ledolter and Hogg 2010). The evaluation index R is

calculated using:

2

, ,1

1 n

cr m cr ti

o

u un

RD

(3-21)

where n is the number of critical displacement values obtained from the analytical models; ucr,m is the

critical displacement value obtained from the analytical models; ucr,t is the critical displacement value

obtained from experiment; and Do is the outer bonded bearing diameter or bearing width. For the

evaluation purposes of this study, the model with the smallest values of R is assumed to have the best

simulation capability.

24

3.3.3 Global sensitivity analysis

Sensitivity analysis quantifies how variability in a given model output, y = f(x1, x2, …, xk), can be

attributed to variability in model factors x1, x2, …, xk (Saltelli et al.2004). Sobol’s method (Sobol’ 1993), a

global, variance-based, sensitivity analysis technique is used in this study to identify model factor (e.g.

parameters, variables, and initial conditions) to which model parameter is most sensitive to the model

output (i.e. ucr).

Sobol’s sensitivity analysis method (Saltelli et al.2008) is unique among the sensitivity methods in

that it decomposes a models output variance V(y) into the relative contributions from individual factors

and factor interactions through numerical integration in a Monte Carlo framework. The term factor is used

as an umbrella term including model parameters (those that do not vary with time) and variables.

Assuming the factors are statistically independent, the model’s output variance can be expressed as:

12( ) i ij ki i j i

V y V V V

(3-22)

where k is the number of factors (e.g. parameters); y is the distribution of model output; V(y) is the total

model output variance; Vi is the output variance due to the ith parameter; Vij is the output variance due to

interaction of factors xi and xj; V1,2, …k is the output variance due to higher order interactions. A given

parameter’s sensitivity is quantified by a ratio of variance contributing to the total (i.e. due to all

parameters) output variance, resulting in an index value ranging from 0 to 1. The “First order” Sobol’

sensitivity index used in this study reflects the effect of the ith parameter alone and is defined as:

i

iSy

V

V (3-23)

The “total” Sobol sensitivity index, also used in this study, reflects the effect of the ith parameter alone

plus interactions of the ith parameter with all other parameters and is defined as:

~1Ti

iSV

V y (3-24)

25

where V~i is the average variance from all factors but fi. The total order index STi is the sum of the

interactions index and the first order index Si.

3.4 Results for model evaluation

3.4.1 Evaluation of existing model

Critical point simulations (ucr, Pcr) were obtained from the two analytical models for three different

bearings and compared to the associated experimental data (Buckle et al. 2002; Weisman and Warn 2012;

Sanchez et al. 2013) to evaluate the simulation capabilities of the individual models.

Critical displacement, ucr, simulations were obtained from each model using the properties of

Bearings 1, 2 and 3 (see Table 3-1). However, values for parameters r, s1 and s2 of Iizuka model had to be

assumed in lieu of characterization data for each of these bearings. Based on values provided in Iizuka

(2000), s1 and s2 were assumed as s1 = 0.01 and s2 = 3 respectively for all three bearings. Values of the

parameter r reported in Iizuka (2000) were 1.2, 1.5 and 3.5 for three different bearings. With no guidance

provided by Iizuka for determination of the value of the r parameter, an average value of 2 was used for

Bearings 1, 2 and 3.

A graphical comparison of the critical point (ucr, Pcr) simulations from the two models and the

experimental data are presented in Figs. 3-3, 3-4, and 3-5 for Bearings 1, 2 and 3 respectively. From Fig.

3-3a, the critical point simulations from the Nagarajaiah model (solid triangle) do not agree well with the

experimental data for Bearing 1 over the range of displacements for which experimental data is available.

Relative error in critical displacement between the simulated and experimentally data range from 3.9% to

57%. The Iizuka model (solid diamond), shown in Fig. 3-3b, better simulates the general trend observed

from the experimental data as well as the critical displacement values with relative error ranging from 3.0%

to 19% for Bearing 1. Comparisons presented in Fig. 3-4 and 3-5 for Bearings 2 and 3 show similar trends

in that simulations from the Iizuka model are in better agreement with the experimental data both in terms

of trend and critical displacement values for the range of lateral displacement considered by comparison

to those obtained from the Nagarajaiah model. The largest relative error between the Iizuka model and

26

experimental data for all three bearings is less than 20%. As discussed previously, an average value of 2

was used for r in the analysis.

Fig. 3-3 Critical displacement comparion between model simulation and experimental testing for

Bearing 1: (a) Nagrajaiah’s model; (b) Iizuka’s model

Fig. 3-4 Critical displacement comparion between model simulation and experimental testing for

Bearing 2: (a) Nagrajaiah’s model; (b) Iizuka’s model

0 50 100 150 2000

100

200

300

400

Lateral displacement: ucr

(mm)

Crit

ical

load

: Pcr

(kN

)

TSM NagarajaiahExperimental

(a)

Sanchez 2012 (#11795)Sanchez 2012 (#11800)Weisman 2012

0 50 100 150 2000

100

200

300

400


(mm)

Crit

ical

load

: Pcr

(kN

)

TSM IizukaExperimental

(b)


0 50 100 150 2000

50

100

150

200

ExperimentalSanchez 2012

TSM Nagarajaiah(a)


(mm)

Crit

ical

load

: Pcr

(kN

)

0 50 100 150 2000

50

100

150

200


TSM Iizuka(b)


(mm)

Crit

ical

load

: Pcr

(kN

)

0 50 100 150 2000

100

200

300

400

ExperimentalBuckle 2002

TSM Nagarajaiah(a)


(mm)

Crit

ical

load

: Pcr

(kN

)

0 50 100 150 2000

100

200

300

400


TSM Iizuka(b)


(mm)

Crit

ical

load

: Pcr

(kN

)

27

Fig. 3-5 Critical displacement comparion between model simulation and experimental testing for Bearing 3: (a) Nagrajaiah’s model; (b) Iizuka’s model

Values of the R index calculated per Eq. (3-23) for each model and bearing are presented in Table 3-

2. The Iizuka model produced the lowest values of R for all three bearings. These results considered with

the comparisons presented in Figs. 3-3 through 3-5 suggest the Iizuka model provides better simulation of

the critical points of the two models considered. For this reason a global, variance-based, sensitivity

analysis was performed only on the Iizuka model.

Table 3-2 Evaluation index for analytical models

Bearing Id. Evaluation index (%)

Iizuka Nagarajaiah and Ferrell 1 12.5 20.5 2 18.9 32.9 3 9.7 11.2

3.4.2 Global sensitivity analysis – identifying controlling parameters for simulation

Sobol’s method was applied on the Iizuka model to compute sensitivity indices for the four

independent model parameters, specifically G, r, s1 and s2, to identify the relative importance of the

individual parameters in the simulation of the critical displacement, ucr. The sensitivity analysis was

performed on an ensemble of ucr simulations generated for various levels of vertical load, P. The process

was repeated for Bearings 1, 2 and 3 (see Table 3-1). Table 3-3 presents the range of values considered in

the sensitivity analysis for the four model parameters (i.e. G, r, s1 and s2). The range for the shear

modulus is based on a percentage of the nominal value for each bearing listed in Table 3-1. Each

parameter was assumed to be uniformly distributed over the ranges presented in Table 3-3. The range of

values for parameters r, s1 and s2 were kept as broad as possible while avoiding non-physical

combinations leading to nonrealistic simulations, noticing s1 = 0 was considered in the parameter range.

Table 3-3 Varied parameter and parameter ranges used for global sensitivity analysis No. Type Description Symbol Unit Range 1 Material Effective shear modulus G % of nominal value 100±5 2

Model Rotational spring constant r n.a. 0.5–6

3 Shear spring constant s1 n.a. 0–0.01 4 Shear spring constant s2 n.a. 0–5

28

Total indices (individual contribution plus interactions) and first order indices (individual) are

plotted for Bearings 1, 2 and 3 in Figs. 3-6, 3-7 and 3-8, respectively. Figure 3-6 includes four bar graphs,

one for each parameter, with total and first order sensitivity indices plotted for vertical load values P

normalized by the corresponding Pcro (see Table 3-1) ranging from 0.3 to 0.9. Similar trends were

observed for each bearing as shown in these figures. Specifically, the total and first order indices for

parameters s1 and s2 are near zero, indicating variability in these parameters has no impact on the critical

load simulated by the Iizuka model over the range of vertical load values considered. By considering s1 =

0 in the parameter range, these results demonstrate the relationship assumed for the shear spring (i.e.

linear vs. nonlinear) has no effect on the simulated critical displacements. Therefore, the Iizuka model

could be simplified by assuming zero values for s1 and s2, equivalent to assuming linear spring properties,

with little to no impact on the simulation capability.

Fig. 3-6 Sobol’s sensitivity indices for Bearing 1

0.9 0.8 0.7 0.6 0.5 0.4 0.30

0.2

0.4

0.6

0.8

1

First orderTotal

Sen

sitiv

ity in

dice

s G

0.9 0.8 0.7 0.6 0.5 0.4 0.30

0.2

0.4

0.6

0.8

1

r

0.9 0.8 0.7 0.6 0.5 0.4 0.30

0.2

0.4

0.6

0.8

1

P / Pcro

Sen

sitiv

ity in

dice

s s1

0.9 0.8 0.7 0.6 0.5 0.4 0.30

0.2

0.4

0.6

0.8

1

P / Pcro

s2

29



0.9 0.8 0.7 0.6 0.5 0.40

0.2

0.4

0.6

0.8

1

First orderTotal

Sen

sitiv

ity in

dice

s G

0.9 0.8 0.7 0.6 0.5 0.40

0.2

0.4

0.6

0.8

1

r

0.9 0.8 0.7 0.6 0.5 0.40

0.2

0.4

0.6

0.8

1

P / Pcro

Sen

sitiv

ity in

dice

s s1

0.9 0.8 0.7 0.6 0.5 0.40

0.2

0.4

0.6

0.8

1

P / Pcro

s2

0.7 0.6 0.5 0.4 0.30

0.2

0.4

0.6

0.8

1

First orderTotal

Sen

sitiv

ity in

dice

s G

0.7 0.6 0.5 0.4 0.30

0.2

0.4

0.6

0.8

1

r

0.7 0.6 0.5 0.4 0.30

0.2

0.4

0.6

0.8

1

P / Pcro

Sen

sitiv

ity in

dice

s s1

0.7 0.6 0.5 0.4 0.30

0.2

0.4

0.6

0.8

1

P / Pcro

s2

30

The results presented in Figs. 3-6 through 3-8 show that both total and first order indices for the

shear modulus, G, decrease as P/Pcro decreases. This result suggests variation in the shear modulus only

has an appreciable impact on the ucr simulation when P is a significant percentage (60% to 90%) of Pcro.

There is a difference between the total and first order indices for G when P/Pcro is large. This suggests

there is interaction between G and r for large P/Pcro. Sensitivity indices for parameter r show that as P/Pcro

decreases, both first and total order indices increase approaching a value of nearly 1 for P/Pcro equal to or

less than 0.6. The importance of parameters G and r is better illustrated when considered alongside

simulated critical points (Pcr, ucr). Fig. 3-9 presents plots of the Pcr and ucr simulations from the Iizuka

model for Bearings 1, 2 and 3. In Fig. 3-9, the critical load simulations have been normalized by the

corresponding Pcro value (see Table 3-1) and plotted as a function of rubber shear strain defined as the

lateral displacement, u, divided by the total thickness of rubber, Tr. Threshold P / Pcro values of 0.75, 0.7

and 0.55 were obtained from the total sensitivity indices for G presented in Figs. 3-6 through 3-8. The

threshold values were determined as the P / Pcro for which the total index fell below 0.05 (judged as value

indicating insensitive). Horizontal lines corresponding to these threshold values of P / Pcro are plotted in

Fig. 3-9 to identify corresponding threshold values of shear strain where threshold value of P / Pcro

intercepts the simulations at a shear strain between 1 and 1.25 for each bearing. The shaded region in the

upper left quadrant therefore represents the region of the critical points where the ucr simulated is

sensitive to both parameters G and r. The shaded region on the lower right quadrant indicates the region

of the critical points where the ucr simulated is sensitive only to parameter r. The results presented in Fig.

3-9 suggest that for all three bearings, the shear modulus and rotational spring behavior are controlling

parameters for high vertical loads or for rubber shear strains below 1. For shear strains greater than 1.25,

the rotational spring behavior controls the critical displacement, ucr, simulation and the relationship

between Ms and θ is controlling stability at moderate to high shear strains i.e. those expected during an

earthquake event.

31

Fig. 3-9 Regions of controlling parameters illustrated on critical load simulations from Iizuka

model: (a) Bearing 1; (b) Bearing 2; (c) Bearing 3

3.5 Discussion

The capabilities of two semi-empirical models for simulating critical displacement in elastomeric

bearings were evaluated using critical load data from past experimental studies (Buckle et al. 2002;

Weisman and Warn 2012; Sanchez et al. 2013). Critical displacement simulation obtained from the Iizuka

model compared well with experimental critical displacement data both in terms of the general trends and

the estimated values. From the result obtained from the global variance-based sensitivity analysis, the

critical displacement simulated from the Iizuka model is most sensitive to the variations in the nonlinear

rotational spring parameter, r, whose value is determined from experimental calibration. The nonlinear

rotational spring parameter, r, controls the moment-rotation behavior of the rotational spring, specifically,

the rotation at which point the ultimate moment capacity of the spring is reached suggesting the rotational

0

0.2

0.4

0.6

0.8

1

Pcr

/ P

cro

G,r r

(a)

0

0.2

0.4

0.6

0.8

1

Pcr

/ P

cro

G,r r

(b)

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

Shear strain

Pcr

/ P

cro G,r

r

(c)

32

behavior of the individual rubber layers control the critical behavior of elastomeric bearings. Based on

this finding, an approach to simulate the rotational behavior using a number of parallel vertical spring and

a bilinear elastic constitutive relationship is explored in the following chapter.

33

Chapter 4

Development and evaluation of an improved analytical model

4.1 General

The Iizuka model has the best capability for simulating the critical displacement of elastomeric

bearings. Applying global sensitivity analysis on Iizuka model, the nonlinear rotational spring parameter

is identified as the key parameter to simulate the bearing behavior. However, Iizuka model relies on semi-

empirical parameters that should be experimentally calibrated. An improved analytical model that

contains only material and physical parameters is proposed in this chapter,. The simulation capability of

the analytical model will be evaluated by comparison to the existing experimental data. Due to the limited

existing experimental test data, three-dimensional (3D) finite element (FE) models are used to further

identify whether the analytical model has similar fidelity as highly detailed 3D FE model.

4.2 Scope

An improved analytical bearing model is proposed in this chapter for simulating the critical behavior

of elastomeric bearing. The simulation capability, in terms of the estimated critical displacement, of the

analytical model is evaluated by a graphical comparison to experimental data for the three bearings(see

Table 3-1) presented in Chapter 3 and by calculating the relative error between the results from the model

simulation and experimental tests. Because of the limited quantity of experimental data, fifty-four 3D FE

annular bearing models were developed and used to further compare to the analytical model for a broader

parameter set that can be achieved using existing experimental data. The parameter set for the parametric

finite element study is established using a one-at-a-time sensitivity technique that also allows the results

of the FE simulations to be used to determine sensitivity indices to understand the relative importance of

the material and physical parameters of elastomeric bearings for simulating critical behavior.

34

4.3 Analytical model

In this section, an improved analytical bearing model is proposed. The analytical model contains

only material and physical parameters without the need of experimental calibration. The simulation

capability of the analytical model is evaluated by comparison to existing experimental data.

4.3.1 Analytical model description

Based on the findings of sensitivity analysis presented in Chapter 3, a more practical, mechanistic

bearing model for simulating the critical behavior is developed that uses a number of parallel vertical

springs and a bilinear elastic constitutive relationship, illustrated in Fig. 4-1. The approach is similar to a

fiber element model, whereby the cross-section is discretized into individual fibers as shown in Fig. 4-1.

Fig. 4-1 Analytical model for elastomeric bearing

Simultaneous application of vertical force, P, and shear force, F, at the top of the model produce a

global lateral displacement, u, and height reduction, v. The total height reduction, v, in Eq. (4-1) is the

sum of the vertical deformation resulting from initial application of P, vp, and a component of

deformation, vk, resulting from the kinematics of the bearing model.

hshear

spring

vertical

springs

u

v

s

θ

x

y

1 2 n... ...i

D

rigid

vPF

vk

P

bearing cross section

ds1 x

netural axis

35

P kv v v (4-1)

The initial vertical deformation, vp, due to the application of, P, is calculated according to:

rP

c b

PTv

E A (4-2)

where Tr and Ab are defined previously; and Ec is the compression modulus of an individual rubber layers.

For a solid circular bearing, Ec is calculated according to (Constantinou et al. 1992):

26c oE G S (4-3)

where Go is the initial shear modulus, S is the shape factor defined previously; F is a coefficient

determined from the geometry of the bearing to account for the annular cross-section of the bearing

(Constantinou et al. 1992). The expression for the compression modulus presented in Eq. (4-3) assumes

incompressible material behavior.

The global deformation (u and v) are the result of local deformation, s, in the shear spring due to

the internal shear spring force, Qs, and vst and in the vertical springs due to the combination of the

vertical force, P, and moment in the vertical springs, Ms. By considering the bearing model in the

deformed configuration (see Fig. 4-1), the equilibrium and compatibility equations presented in Eqs. (4-4)

– (4-7) are derived.

sin cosu h s (4-4)

sin (1 cos ) Pv s h v (4-5)

( )s PM P u F h v v (4-6)

sin cossQ P F (4-7)

Equations (4-4) through (4-7) are differentiated with respect s and θ and solved using the incremental

analysis procedure described in Iizuka (2000) using Eq. (3-10). The differential quantity dMs/dθ at each

step i is estimated using a backward finite difference approximation according to Eq. (4-8).

( 1)

( 1)

s si is

i i i

M MdM

d

(4-8)

36

The differential quantity dQs/ds represents to the tangential stiffness of the shear spring as:

bs

r

G u AdQ

ds T (4-9)

where G(u) is shear modulus at a given lateral displacement. The nonlinear relationship proposed by

Nagarajaiah and Ferrell (1999) shown in Eq. (4-10) is adopted for this study to simulate the reduction in

the shear modulus with bearing lateral displacement, u.

1 0.325tanhor

uG u G

T

(4-10)

The force in the individual vertical springs is determined at each step in the incremental analysis by

considering equilibrium and compatibility of the system of vertical springs. Equilibrium of the vertical

springs also facilitates calculation of the dMs/dθ term in Eq. (12). The force and moment equilibrium

equations for the vertical springs are:

sj jj

P A (4-11)

s sj j sjj

M A d x (4-12)

where σsj is the stress in the jth spring element, dsj is the distance between the center of jth spring element

and the center of bearing cross-section (see Fig. 4-1), x is the distance between the neutral axis and the

center of bearing cross-section; Aj is the area of jth spring element. Assuming the element above the

vertical springs is rigid the following compatibility condition can be imposed to relate the strain in one

spring element to another:

1 2

1 2

sj ss s s s sn s

s s sj sn

ll l l

d x d x d x d x

(4-13)

where εsi is the strain ith spring element; ls is the initial length of vertical spring element.

Figure 4-2a presents a plot of the tensile force-elongation response obtained from experimental

testing of an elastomeric bearing under cyclic tensile loading (Warn and Whittaker 2006). The tensile

force-elongation response plotted in Fig. 4-2a shows a complex response that is nonlinear including

37

Mullin’s effect (Mullin 1966). In an effort to retain a level of practicality, a simple bilinear force-

deformation relationship, shown in Fig. 4-2b, was assumed for the vertical spring elements intended to

approximate the backbone curve of the cyclic response shown in Fig. 4-2a. The bilinear relationship is

defined by two engineering parameters, specifically the modulus, Ec and the tensile “yield” stress of the

rubber, y. The modulus, Ec is assumed equal in compression and tension until the stress in the vertical

spring element reaches y, beyond which the spring behavior is purely plastic. The unloading paths, both

in tension and compression, follow the loading path.

(a) Tension test result for Bearing 1 (b) Uni-axial constitutive model for vertical springs in analytical model

Fig. 4-2 Uniaxial constitutive model for vertical springs in analytical model

The stress in an individual vertical spring element is determined according to the following

constitutive law:

/

/

c sj sj y c

sjy sj y c

E if E

if E

(4-14)

where σy is defined previously. The equilibrium and compatibility equations (4-11) through (4-14) are

solved simultaneously to determine the moment, Ms, and equilibrium in vertical springs. The initial length

of vertical spring element, ls, is predetermined by setting the initial rotational stiffness of the parallel

vertical springs equal to the initial rotational stiffness of the bearing according to:

0 25 50 75 100−20

0

20

40

60

80

100

Elongation (mm)

Ten

sile

forc

e (k

N)

ε

σ

σy

Ec

Tension

Compression

38

2

2c bj sj

js

E E IA d

l h

(4-15)

where Eb is the bending modulus of elastomeric bearing, which is one third of the value of Ec (Kelly

1997).

4.3.2 Convergence study on improved analytical model

Prior to simulating the critical behavior of the experimental tests, a convergence study of the number

of vertical springs is performed to ensure that the estimated critical displacement values from the

analytical model are converged solutions. For all models it was determined that 12 vertical springs

provided converged solutions. Figure 4-3 presents shear force - lateral displacement responses obtained

from simulations of the analytical model with 6 (Fig. 4-3a), 12 (Fig. 4-3b) and 20 (Fig. 4-3c) vertical

springs at four different compressive force levels for Bearing 1. From the results presented in Fig. 4-3, the

shear force response for P = 89 kN with number of springs, n = 6 and n = 20 show a small difference in

both peak value and corresponding displacement which is ucr. Specifically, for P = 89 kN, the peak shear

force for n = 6 is 26.4kN in comparison to 25.4 kN for n = 20 model, that is a 3.9% difference.

Furthermore, the critical displacement for n = 6 is 145 mm in comparison to 135 mm for n = 20 that is a

7.4% difference. For n =12 the peak shear force is 25.4 kN occurring at 136 mm of lateral displacement,

the difference between n =20 model being negligible. For the two other bearings, Bearings 2 and 3, n = 12

was also sufficiently large to obtain a converged solution for all vertical load levels.

4.3.3 Evaluation of analytical model using experimental data

The converged analytical model with 12 vertical springs is used to simulate the shear force lateral

displacement response from experimental tests performed by Sanchez et al. (2013). Figure 4-4 presents a

side-by-side comparison of the shear force – lateral displacement response from the Sanchez et al. tests

(Fig. 4-4a) and the analytical bearing model (Fig. 4-4b) denoted Bearing 1 in Table 3-1. The qualitative

comparison presented in Fig. 4-4 shows the analytical model is capable of simulating the reduction in

horizontal tangential stiffness with increasing lateral displacement for vertical force levels greater than 0.

39

The results from the analytical model agree with the general trend observed in the experimental data of

deceasing maximum shear force and critical displacement with increasing vertical load. However

analytical model over-estimates the shear force response for all vertical load levels most notably when P

is greater than 89 kN.

Fig. 4-3 Convergence study for mechanistic model: shear force - lateral displacement results for different numbers of vertical springs: (a) n = 6 springs; (b) n = 12 springs; (c) n = 20 springs.

The analytical bearing model was used to simulate the shear force-lateral displacement response of

all three bearings (see Table 3-1). The critical displacement, ucr, and corresponding vertical load, referred

to as the critical load, Pcr, from the model simulations were compared to the experimental data. Figure 4-5

presents graphical comparison of the model and experimental critical load results for Bearings 1, 2 and 3.

For Bearing 1, Fig. 4-5a, the relative error in critical displacement between model and experimental

results ranges from 1.7% to 18.9%. For Bearing 2, Fig. 4-5b, the relative error ranges from 0.6% to 18.5%.

0 25 50 75 100 125 1500

5

10

15

20

25

30 P = 0kNP = 89kNP = 178kNP = 267kNP = 356kN

a.

n = 6


She

ar fo

rce:

F (

kN)

0 25 50 75 100 125 1500

5

10

15

20

25

30b.

n = 12


She

ar fo

rce:

F (

kN)

0 25 50 75 100 125 1500

5

10

15

20

25

30c.

n = 20


She

ar fo

rce:

F (

kN)

40

For Bearing 3, Fig. 4-5c, the relative error ranges from 2.6% to 13.4%. From the comparison of results

presented in Fig. 4-5, the model is capable of simulating the general trend observed in the experimental

displacement and provides a reasonable, i.e. < 20% error, accurate estimate of the critical displacement

for a given vertical load.

Fig. 4-4 Shear force response comparison between test result (Sanchez et al. 2013) and analytical

model result: (a) experimental test; (b) analytical model simulation

Values of the model evaluation index R for the analytical and Iizuka models are presented in Table

4-1. For the Iizuka model, the R index values were taken from the evaluation presented Table 3-2. From R

index values presented in Table 4-1, the analytical model provides a similar or improved estimate of the

critical displacement by comparison to the Iizuka model suggesting the fidelity is retained in the

analytical model without the need for empirical parameters.

Table 4-1 Evaluation index for analytical models

Bearing Id. Evaluation index (%)

Iizuka Analytical model 1 12.5 14.1 2 18.9 18.2 3 9.7 7.1

0 25 50 75 100 125 1500

5

10

15

20

25

30a.

P = 0kNP = 89kNP = 178kNP = 267kNP = 356kN


She

ar fo

rce:

F (

kN)

0 25 50 75 100 125 1500

5

10

15

20

25

30b.


She

ar fo

rce:

F (

kN)

41

Fig. 4-5 Critical load comparison between test result and analytical model result

4.4 Finite element model

Three-dimensional FE half-space bearing models are developed to evaluate the critical load

simulations from 3D FE analysis. Three-dimensional FE analysis is conducted in this study using the FE

analysis package ABAQUS/Standard (DSSC 2011). For the evaluation of the FE method, critical load

from converged 3D FE models of two different types of bearings are compared to existing experimental

data. The relative error between the results of the FE simulation and the existing experimental data is less

than 20%. A one-at-a-time sensitivity analysis method is performed on the FE model to investigate the

relative importance of physical and material parameters for simulating the critical behavior.

0 50 100 150 2000

100

200

300

400


(mm)

Crit

ical

load

: Pcr

(kN

)

Analytical modelExperimental

a.


0 50 100 150 2000

50

100

150

200


Analytical model

b.


(mm)

Crit

ical

load

: Pcr

(kN

)

0 50 100 150 2000

100

200

300

400


Analytical model

c.


(mm)

Crit

ical

load

: Pcr

(kN

)

42

4.4.1 Finite element model description

Past research (Warn and Weisman 2011) demonstrated the finite element method is able to simulate

the critical behavior of elastomeric bearings in the laterally deformed configuration with reasonable

accuracy. However, only one bearing geometry was considered in this study. To more broadly validate

the FE method for simulating critical behavior, 3D FE models of additional bearings from experimental

tests were established and used to simulate the tests to facilitate the evaluation.

The symmetry of the bearings and loading is exploited so that only half of the bearing needs to be

modeled in an effort to minimize computation time. The modeled half bearing is constrained against out-

of-plane bulging since the other half of bearing is not modeled. Boundary conditions are used to simulate

the restraint provided to the bearing end-plates during the experimental tests. Specifically, the bottom end

plate is fixed in all degrees of freedom and the top end plate is fixed against rotation but allowed to

translate laterally and vertically.

An illustration of the 3D FE model for Bearing 1 (see Table 3-1) is presented in Fig. 4-6. For all

bearing models, the rubber components, i.e. rubber layers and cover are modeled using 8-node solid

elements (C3D8H) with first order hybrid formulation to prevent volumetric locking due to the nearly

incompressible material behavior, i.e. Poisson’s ratio, ν ≈ 0.5. The intermediate steel shim plates and end

plates are also modeled using C3D8H elements as well to prevent volumetric locking due to material

plasticity. For Bearing 2, since it is a solid circular bearing, 6-node solid elements (C3D6H) for both the

rubber and steel components at the center of the model. All the other elements in the model of Bearing 2

are C3D8H elements.

43

Fig. 4-6 Illustration of FE model: (a) elevation view; (b) plain view

4.4.2 Constitutive model

The steel material is represented using a bilinear, isotropic, material model with Poisson’s ratio, υ =

0.3, elastic Young’s modulus E = 200 GPa and yield strength σy = 448 MPa. A post-yield modulus of 2%

of the initial modulus is specified. Two types of constitutive models were considered for the rubber

material, specifically the Neo-Hookean and Mooney-Rivlin models. Both the Neo-Hookean and Mooney-

Rivlin model assumes the material is hyper-elastic, isotropic, and incompressible or nearly incompressible

if a large value of the bulk modulus, K, is specified. The strain energy potential for the Mooney-Rivlin

constitutive models is defined as:

2

10 1 20 21

1( 3) ( 3) ( 1)elU C I C I J

D (4-16)

where U is the volumetric strain energy; C10 and C20 are material constant; D1 is equal to 2/K; 1I and 2I

are the first and second strain invariant of the Cauchy–Green deformation tensor, respectively; and Jel is

the elastic volume ratio. The Neo-Hookean model is obtained by setting C20 = 0 in Eq. (4-16). The

a.

b.

44

parameters of the strain energy relationship are determined using data collected from standard material

tension tests of rubber coupons according to ASTM D412 method A. The rubber used in the material test

is made from a single sheet of vulcanized rubber provided by the bearing manufacturer with the Bearing 1

specimens. Table 4-2 presents the material constant determined from the material test. The constants are

determined using a least square fit of the material test result to the constitutive model. Insufficient

material was available to perform all the material tests required to reliably determine the parameters of a

higher order constitutive model. A typical value of 2000 MPa was assumed for the bulk modulus, K, for

all the models. Warn and Whittaker (2006) showed the results of the FE simulation were insensitive to the

value of the bulk modulus varying between plus and minus 20% of the assumed value.

Table 4-2 Material constant used in FE constitutive model Material model C10 C20

Neo-Hookean 0.43 - Mooney-Rivlin 0.39 0.04

Figure 4-7 presents shear force – lateral displacement results from simulations using the FE models

with the Mooney-Rivlin and Neo-Hookean constitutive relationships are compared to the result from

experimental test performed by Warn and Whittaker (2006). Both models are converged models using a

mesh sensitivity procedure described in Section 4.4.3. From the comparison of result presented in Fig. 4-7,

both FE models show good agreement to the experimental data the lateral displacement range from 0 to

60 mm. Similarly, Fig. 4-8 presents vertical force – vertical displacement results from the FE simulations

with the two constitutive rubber models each compared to experimental results from Warn and Whittaker

(2006). Again, simulation results using both constitutive models show good agreement with the

experimental data for vertical displacement ranging from 0 to 0.5 mm. The finite element simulations

using both constitutive relationships overestimate the vertical load after the vertical displacement reaches

0.5 mm. Based on the comparisons presented in Figs. 4-7 and 4-8 the Neo-Hookean model is selected for

the FE parametric study, in part, because the parameters of the Neo-Hookean are directly related to the

material properties, i.e. shear modulus, G and bulk modulus, K.

45

Fig. 4-7 Comparison of shear stiffness between 3D FE model and test result for Bearing 1: (a) Neo-Hookean; (b) Mooney-Rivlin

Fig. 4-8 Comparison of vertical stiffness between 3D FE model and test result for Bearing 1: (a)

Neo-Hookean; (b) Mooney-Rivlin

From the simulation results of lateral and vertical stiffness, both models provide similar simulation

capability. Since parameters of the Neo-Hookean model have direct relationship with the engineering

properties, shear modulus, G and bulk modulus, K, the Neo-Hookean model is used in the FE model.

4.4.3 Mesh sensitivity

In order to ensure to the models used in the comparison with experimental result provided converged

solutions, different meshes are generated for both Bearings 1 and 2. Cross sections of different meshes,

M21093, M28124 and M42186 for Bearing 1 are plotted in Fig 4-9. Details of each mesh are shown in

Table 4-3.

0 20 40 60 800

5

10

15

20

Lateral displacement (mm)

She

ar fo

rce

(kN

)

Experimental

FE Neo−Hookean

a.

0 20 40 60 800

5

10

15

20


She

ar fo

rce

(kN

)

Experimental

FE Mooney−Rivlin

b.

0 0.25 0.5 0.75 10

25

50

75

100

125

150

Vertical displacement (mm)

Ver

tical

load

(kN

)

Experimental

FE Neo−Hookean

a.

0 0.25 0.5 0.75 10

25

50

75

100

125

150

Vertical displacement (mm)

Ver

tical

load

(kN

)

Experimental

FE Mooney−Rivlin

b.

46

(a) M21093 (b) M28124

(c) M42186

Fig. 4-9 Cross sections of different meshes for Bearing 1

The mesh naming protocol is defined by the number of elements in an individual rubber layer in

each direction. The first two numbers indicate the number of elements in the circumferential direction.

The third and fourth numbers indicate the number of elements in the radial direction. The last number

indicates the element number in the direction of the individual rubber layer thickness. For example,

M28124 has 28 elements in the circumferential direction, 12 in the radial direction, 4 elements per rubber

layer. The model is considered to be converged when critical displacement, ucr, simulation of the model is

within 2% difference from the finest mesh model.

The results of the mesh convergence study for Bearing 1 are shown in Fig. 4-10. Critical

displacements ucr at three different vertical compressive load levels (267 kN, 178 kN, 89 kN) were

evaluated to assess convergence for each bearing geometry. The normalized differences which are based

on the result of the finest mesh model, i.e. M42186 are shown in Fig. 4-11. From the result, difference in

ucr simulation between M21093 and M42186 ranges from 4.2% to 7.2%, so that M21093 is not a

converged model according to the 2% criteria defined previously. For the results from M28124, an

a. b.

c.

47

acceptable difference (0% to 1.5%) is found by comparing with that from M42186. Therefore, M28124 is

considered to be a converged model for Bearing 1.

Table 4-3 Detail of three meshes used in mesh convergence study for Bearing 1

Notation Element number in each direction

for an individual rubber layer Total number of elements Circumferential Radial Vertical

M21093 21 9 3 17,010 M28124 28 12 4 36,764 M42186 42 18 6 106,596

Fig. 4-10 Simulated critical displacements of Bearing 1 for different meshes

Fig. 4-11 Simulated critical displacement of Bearing 1 in percentage difference for different meshes

0 3 6 9 120

50

100

150

200

Pcr

= 267 kN

Pcr

= 178 kN

Pcr

= 89 kN

No. of Elements (104)

Crit

ical

Dis

plac

emen

t: u

cr (

mm

)

0 3 6 9 12

0

2

4

6

8P

cr = 267kN

Pcr

= 178kN

Pcr

= 89kN

No. of Elements (104)

Crit

ical

Dis

plac

emen

t Diff

eren

ce: (

%)

48

The above procedure was repeated for models for Bearing 2. For Bearing 2, M30144 is found to be a

converged model with the lowest computational demands and will therefore be used for simulation and

the evaluation.

Fig. 4-12 Converged mesh for Bearing 2

4.4.4 Validation of FE method

A series of analyses with different levels of vertical compressive force are performed using the

converged 3D FE models of two circular bearings for which there is experimental data, specifically

Bearings 1 and 2. During each analysis, the bearing is first subjected to a vertical compressive load to a

specific value, and then sheared to the target lateral displacement by imposing the a displacement type

boundary condition for the node at the centroid of top end-plate surface while keeping the vertical load

constant. The point of neutral equilibrium (i.e. ucr) is obtained by numerically differentiation the shear

force, F, lateral displacement, u, response obtained from the FE solution.

The lateral force-displacement responses for Bearing 1, with P ranging from 0 to 356 kN, is

presented in Fig. 4-13. The results plotted in Fig. 4-13 show the finite element model is capable of

simulating the critical behavior of elastomeric bearings by comparison to the experimental observations.

The critical load and critical displacement data from the FE simulations for Bearings 1 and 2 are

plotted in Figs. 4-14a and 4-14b with the corresponding experimental data. For Bearing 1, Fig. 4-14a, the

relative error between the FE and experimental values range from 2% to 15% using the Sanchez et al.

data (2013). For Bearing 2, Fig. 4-14b, the relative error ranges from 5% to 19%. For Bearing 2, the trend

from the FE results do not coincide with the experimental data for each vertical force level though there is

reasonable agreement for the higher and lower vertical load levels. However, two different critical load

49

values corresponded to critical displacement at 153 mm in the experimental data. The bearing itself might

be damage during the test, which may cause the difference between test data and FE result.

Based on the evaluation, the FE modeling approach used here is able to simulate the critical

displacement with reasonable accuracy based on the relative error values being < 20% by comparison to

the experimental data.

Fig. 4-13 Shear force response - lateral displacement from FE simulaiton for various vertical compressive force levels

Fig. 4-14 Comparison of critical load-displacements from FE model simulation with experimetnal

data: (a) Bearing 1; (b) Bearing 2

4.5 Sensitivity analysis on FE models

In this section, a one-at-a-time sensitivity analysis technique, specifically the Method of Morris

(Morris 1991), is used to establish the parameter range for the parametric finite element study to: (1)

0 25 50 75 100 125 1500

5

10

15

20

25

30 P = 0kNP = 133kNP = 178kNP = 267kNP = 356kN


She

ar fo

rce:

F (

kN)

0 50 100 150 2000

100

200

300

400


(mm)

Crit

ical

load

: Pcr

(kN

)

Finite elementExperimental

a.


0 50 100 150 2000

50

100

150

200b.


Finite element


(mm)

Crit

ical

load

: Pcr

(kN

)

50

obtain critical load behavior for a broad range of parameters and (2) understand how variations in

physical and material parameters affect the critical behavior of elastomeric bearings to support the

findings from the global sensitivity analysis results presented in Chapter 3.

4.5.1 Method of Morris

The method of Morris is a randomized one-at-a-time sensitivity analysis technique to identify and

rank the importance of input parameters (Morris 1991). The method of Morris assumes the input

parameter changes with the same relative amount at each time, the parameter that causes the largest

variation in the output y is the most important parameter. Method of Morris is typically employed when

there are a large number of parameters and computationally demanding models. The computational

demand of the 3D FE model is sufficiently large such that the method of Morris is chosen over global

sensitivity technique in an attempt to limit the number of models and computational time. For a model

with k input parameters, the elementary effect of the ith parameter is defined as:

1 2 1 1 1 2, , , , , , , , , , /i i i i k kd x y x x x x x x y x x x (4-17)

where Δ is the magnitude of each step that is a multiple of 1/(p-1); and p is the value of the number of

input parameters samples. For the sampling rule, only one parameter is changed from parameter set to

parameter set. Therefore, the method requires k+1 models to calculate one elementary effect for each of

the k input parameters. The rank rk is defined as the number of parameter set that is needed to calculate

the one elementary effect for each input parameters. Morris (1991) proposed two sensitivity indices

namely the mean, μ, and the standard deviation, δ, of the set of elementary effects for each input

parameter that are calculated according to Eqs. (4-18) and (4-19)

1( )kr

ini

k

d x

r (4-18)

2

1

1 kr

i i ink

d xr

(4-19)

51

The sensitivity index, μi, calculated from Eq. (4-18), assesses the relative importance of the ith parameter

of the output response due to all first and high order effects. The sensitivity index, δi, calculated from Eq.

(4-19), indicates the degree of potential interactions between the ith parameter and other parameters,

and/or that the ith parameter has a nonlinear effect on the output (Campolongo and Braddock 1999).

In this study, six physical and two material parameters are chosen to investigate the relative

importance of the input parameters for FE models. The parameters and corresponding ranges are listed in

Table 4-3. Therefore the rank of the analysis rk is 6 since the suggested value of rk ranges from 4 to 10.

Selecting rk = 6 associated with 8 input parameters results in totally of 54 FE models required for the

sensitivity analysis. For each FE model the two-step analysis procedure describe previously was

performed for vertical load levels corresponding to 30%, 40%, 50%, 60%, 70%, 80% of Pcro. The value of

Pcro is derived from FE model using buckling analysis process described in Kelly and Takhirov (2004).

The output of the model is εcr, which is the shear strain level at the critical displacement calculated

according to Eq. (4-20).

crcr

r

u

T (4-20)

Table 4-4 Parameter range for sensitivity analysis No. Type Parameter Symbol Unit Range 1

Physical

Number of rubber layers N - 8-40 2 Single layer rubber thickness tr mm 5-25 3 Single layer shim thickness ts mm 1.5-5.5 4 Bearing cover thickness tc mm 5-35 5 Outer bearing diameter Do mm 250-1250 6 Inner bearing diameter Di mm 5-45 7

Material Shear modulus G MPa 0.414-0.965

8 Bulk modulus K MPa 689-3447

4.5.2 Results of sensitivity analysis

The sensitivity indices, μi, and δi, for the eight parameters at each vertical load level are presented in

Fig. 4-15. From the μi values presented in Fig. 4-15, the number of rubber layers, N, the single layer

thickness tr, and the outer bearing diameter, Do, are the three parameters to which the normalized output,

εcr, is most sensitive at each vertical load levels. The result support the findings from the previous global

52

sensitivity analysis presented in Chapter 3, in that parameters related to the rotational behavior of the

bearing control the simulation of the critical displacement, since all three parameters can be related to the

initial rotational stiffness of the bearing.

Fig. 4-15 Sensitivity indices for input parameters at different P/Pcro ratio

The sensitivity indices for these three parameters decrease with increasing P/Pcro, which suggests

changing the same amount of these three parameters causes less effect on the model output, εcr, at higher

vertical load level. Each of these three parameters have a significant amount of standard deviation of

elementary effect, which suggests higher order interaction between parameters exists and/or the bearing

have a nonlinear effect in the output. The sensitivity values for the other five parameters, specifically the

0.8 0.7 0.6 0.5 0.4 0.30

0.5

1

1.5

2

2.5

δμ

Sen

sitiv

ity in

dex

N

0.8 0.7 0.6 0.5 0.4 0.30

0.5

1

1.5

2

2.5tr

0.8 0.7 0.6 0.5 0.4 0.30

0.5

1

1.5

2

2.5

Sen

sitiv

ity in

dex

ts

0.8 0.7 0.6 0.5 0.4 0.30

0.5

1

1.5

2

2.5tc

0.8 0.7 0.6 0.5 0.4 0.30

0.5

1

1.5

2

2.5

Sen

sitiv

ity in

dex

Do

0.8 0.7 0.6 0.5 0.4 0.30

0.5

1

1.5

2

2.5D

i

0.8 0.7 0.6 0.5 0.4 0.30

0.5

1

1.5

2

2.5

P / Pcro

Sen

sitiv

ity in

dex

G

0.8 0.7 0.6 0.5 0.4 0.30

0.5

1

1.5

2

2.5

P / Pcro

K

53

intermediate shim plate thickness, ts, shear modulus, G, bulk modulus, K, inner bearing diameter, Di, and

bearing rubber cover, tc, suggests variations in these parameters has a negligible effect on the critical

shear strain output from the FE simulations.

4.5 Assessing simulation capability of analytical model comparing to the FE model

The 3D FE model has been evaluated for simulating the critical behavior of elastomeric bearings by

comparison of the critical displacement and critical load results with existing experimental data. From this

comparison, the FE model is capable of simulating the critical behavior, though the results from the FE

models do not agree equally well for different models. In light of this, the FE model is further used to

assess whether the analytical model has a similar simulation capability comparing to the FE models for a

broader range of parameters than is possible with existing experimental data. The relative error between

the results of the analytical and finite element models for the critical displacement, ucr, and maximum

shear force are used to evaluate the analytical model where the results from the FE model are considered

the “true” values in the relative error calculation.

The relative errors for the critical displacements at different vertical load levels are plotted in Fig. 4-

16. For low vertical load level, i.e. ranging 30-40% of Pcro, the maximum relative error is approximately

20% or less for all bearing shape factors. As the ratio of P/Pcro increases, the maximum relative error

increases. For P/Pcro = 70% and 80%, the relative error is significantly larger but less than 50% for all

models. Figure 4-17 presented the relative error for peak shear force and each P/Pcro ratio. The results

presented in Fig. 4-17 are similar to Fig. 4-16, in that as P/Pcro increases the relative error increases.

54

Fig. 4-16 Further evaluation result for analytical models in critical displacement

4.6 Discussion

A mechanistic bearing model for simulating the critical behavior of elastomeric bearing is proposed

in this chapter. The analytical model simulates the critical behavior reasonable well and with reasonable

accuracy by comparison to experimental data from previous studies especially when considering the

model does not rely upon empirical parameters that must be experimentally calibrated. Due to the limited

amount of experimental data, 3D FE modeling was employed to further identify the fidelity of the

analytical model in an expanded parameter set. Comparing to the result from highly detailed FE models,

the analytical model shows better agreement in both peak shear force and critical displacement at low

P/Pcro level than the high P/Pcro level. As the ratio of P/Pcro increases, the maximum difference between

FE model result and analytical model simulation result increases as well.

0 10 20 30 400

10%

20%

30%

40%

50%P/P

cro = 30%

Rel

ativ

e er

ror

0 10 20 30 400

10%

20%

30%

40%

50%P/P

cro = 40%

0 10 20 30 400

10%

20%

30%

40%

50%P/P

cro = 50%

Rel

ativ

e er

ror

0 10 20 30 400

10%

20%

30%

40%

50%P/P

cro = 60%

0 10 20 30 400

10%

20%

30%

40%

50%P/P

cro = 70%

Shape factor: S

Rel

ativ

e er

ror

0 10 20 30 400

10%

20%

30%

40%

50%P/P

cro = 80%

Shape factor: S

55

The Method of Morris sensitivity analysis technique is used to define the parameter set for the

parametric FE study and to quantify the sensitivity of the FE model output to variations in the model

parameters with the goal of elucidating the underlying mechanism controlling the critical behavior. From

the results of the study, the number of layers, N, the individual rubber layer thickness tr, and the outer

bearing diameter, Do, are the three most important parameters for the output, εcr, over different vertical

load level. This result supports the findings from the global sensitivity analysis presented in Chapter 3, in

that, the critical behavior, specifically critical displacement, is controlled by the rotational behavior of the

bearing.

Fig. 4-17 Further evaluation result for analytical models in peak shear force

0 10 20 30 400

10%

20%

30%

40%

50%P/P

cro = 30%

Rel

ativ

e er

ror

0 10 20 30 400

10%

20%

30%

40%

50%P/P

cro = 40%

0 10 20 30 400

10%

20%

30%

40%

50%P/P

cro = 50%

Rel

ativ

e er

ror

0 10 20 30 400

10%

20%

30%

40%

50%P/P

cro = 60%

0 10 20 30 400

10%

20%

30%

40%

50%P/P

cro = 70%

Shape factor: S

Rel

ativ

e er

ror

0 10 20 30 400

10%

20%

30%

40%

50%P/P

cro = 80%

Shape factor: S

56

Chapter 5

Numerical and experimental earthquake simulation of an isolation system composed of elastomeric bearings

5.1 General

Numerical earthquake simulation is an important tool for the design of seismically isolated

structures. Widely used numerical earthquake simulation models (Nagarajaiah et al. 1991, Ryan et al.

2005) assume a constant, positive, second-slope stiffness, for unidirectional excitation that does not

account for the reduction in the tangential horizontal stiffness with increasing lateral displacement

adequately. Semi-empirical bearing models (Iizuka 2000, Yamamoto et al. 2009, Kikuchi et al. 2010) that

are able to simulate the influence of vertical compressive force on the bearing shear force response,

require numerous empirical parameters that must be calibrated from experimental testing. An analytical

bearing model for simulating the critical behavior of elastomeric bearing is proposed in Chapter 4. The

analytical model is employed into a numerical earthquake simulation model in this chapter for simulating

the earthquake response of individual bearing and isolation system.

5.2 Scope

In this chapter, a dynamic model of an isolation system composed of elastomeric bearings is

developed to extend the analytical bearing model proposed in Chapter 4 for application to earthquake

simulation. The dynamic model is used to simulate experimental earthquake simulation testing of two

isolation systems composed of elastomeric bearings, one system tested as part of this study and the other

as part of a previous study (Sanchez et al. 2013). The results of the numerical simulation are compared to

those obtained from the physical earthquake simulation tests to evaluate the dynamic model.

5.3 Dynamic model

The focus here is to develop a dynamic model that can be used to simulate physical earthquake

simulation studies to evaluate the capabilities of the model. To be consistent with the setup of the physical

57

earthquake simulation testing, the model (Fig. 5-1) assumes a rigid mass super-structure supported by

four bearing elements, two on each side, with three degrees-of-freedom, specifically, horizontal

translation, vertical translation and rotation. A more detailed description of the formulation of the

analytical dynamic model follows.

Fig. 5-1 Illustration of the dynamic analytical model: (a) undeformed; and (b) deformed configurations

An illustration of the dynamic model of the isolation system composed of elastomeric bearings is

presented in Fig. 5-1. The super-structure consists of a rigid mass and an upside down rigid tee. The rigid

mass has translational mass, m, and mass moment of inertia I. The distance between the center of mass

and base of the rigid tee is denoted as hc. The distance between the left and right bearings is denoted as b.

m, I

hc

x

z

b

φ

φg

ug

Reference

point

um

(φ+φg)h

c

Rigid

vg

PL

b.

a.

Fix

Miy

Fiz

FDzL

FL

FDxL

FR

FDxR

PR

FDzR

vm

Bearing model

*

*

Mass

58

The bearings response is simulated using an extension of the analytical bearing model proposed in

Chapter 4. The dynamic model is two-dimensional and has three degrees-of-freedom (DOF) specifically,

lateral displacement, um, vertical displacement, vm, and rotation, , associated with the intersection of the

segments of the rigid tee. The response of two bearing elements on the left and the two on the right are

summed for the purpose of dynamic equilibrium of the isolated model. Three components of ground

motion, specifically, horizontal, ug, vertical, vg, and rotational, g are considered for the formulation.

5.3.1 Formulation of the equations of motion

The equations of motion are derived from equilibrium of the system in the deformed configuration,

illustrated in Fig. 5-1b, and D’Alembert’s principle. From equilibrium of the rigid mass and tee the

following equations are derived:

0 : 0x ix DxL DxR L RF F F F F F (5-1)

* *0 : 0z iz DzL DzR L RF F F F P P (5-2)

* *0 : 02

y ix c iy R DzR L DzL

bM F h M P F P F (5-3)

where Fix, Fiz, and, Miy are the inertial forces and moment on the mass due to the ground motion defined

in terms of the ground motion components in Eqs. (5-4) through (5-6).

ix m g c gF m u u mh (5-4)

iz m gF m v v (5-5)

2iy c gM I mh (5-6)

The damping forces for the left bearing elements FDxL, FDzL; and the right bearing elements FDxR, FDzR are

defined according to:

FDxL c

xLu

m (5-7)

2DzL zL m

bF c v

(5-8)

59

DxR xR mF c u (5-9)

2DzR zR m

bF c v

(5-10)

where cxL and czL are the sum of the horizontal and vertical viscous damping coefficients of the two left

bearing elements, respectively; cxR and czR are the sum of the horizontal and vertical viscous damping

coefficients of the two right bearing elements, respectively. The dynamic vertical force, PL* and shear

force, FL, response of the two left bearing elements are determined by the analytical bearing model.

Similarly the response of the two right bearing elements, PR*, and FR, are determined by the analytical

bearing model. Substituting Eqs. (5-4) through (5-10) into the equilibrium Eqs. (5-1) through (5-3) results

in the following dynamic equations of motion.

m c hL hR m L R g g cm u h c c u F F m u h (5-11)

* *m vL vR m L R gmv c c v P P mv (5-12)

2 2 * * 212 2

2 2c m c vL vR L R c g c g

bmh u I mh c c b P P mh u I mh (5-13)

Assuming the static weight of the superstructure is distributed evenly to each bearing, the damping

coefficients for the individual bearing in x and z directions cx and cz are calculated using:

24x nx

mc

(5-14)

24z nz

mc

(5-15)

where β is the damping ratio of each individual bearing which is calculated from test result; ωnx, ωnz is

natural frequency of the isolation system in x and z direction, respectively.

5.3.2 Analytical bearing model

Analytical bearing model proposed in previous chapter is used to simulate the nonlinear bearing

behavior in the dynamic analytical model for the isolation system. Illustrations of the analytical bearing

60

model are presented in Fig. 5-2a. These illustrations show the model in both the undeformed and

deformed configurations where the deformed configuration is divided into static deformation, due to static

weight of the superstructure and kinematic deformations due to the simultaneous lateral force, F, and

vertical force, P.

Fig. 5-2 Undeformed and deformed configurations for analytical model for elastomeric bearing

The dynamic equilibrium equations are derived with respect to the statically deformed configuration.

However, equilibrium of the bearing model and the corresponding shear force response are a function of

the total vertical force on the bearing. Therefore, the total vertical force in each bearing, P, is the sum of

the static weight on each bearing, W, and the dynamic vertical force, P* applied on each bearing. The

dynamic vertical forces in each bearing are calculated by using the displacement response from the

dynamic model in Eq. (5-16).

*

*

for the left bearings2

for the right bearings2

L v m

R v m

P k v b

P k v b

(5-16)

In Eq. (5-16), kv is the instantaneous vertical stiffness of the individual bearing, which is calculated using

(Warn et al. 2007):

hshear

spring

vertical

springs

u

v

s

θ

x’

y’

rigid

roller

vst

W

Fvk

P

undeformedstatically

deformation

dynamic

deformation

61

2

2 2

12 1 21

1

vov

o

kk

u

a D

(5-17)

where a is the ratio of the inner to outer bearing diameters; kvo is the initial vertical stiffness of the bearing

which is calculated using:

c bvo

r

E Ak

T (5-18)

where Tr, Ab and Ec are previously defined.

Similarly with vertical forces, the total vertical deformation, v, is the sum of the static, vst, and

kinematic, vk, components according to Eq. (5-19).

st kv v v (5-19)

The vertical deformation due to the static weight is:

stvo

Wv

k (5-20)

Considering equilibrium and compatibility of the deformed bearing model are derived in Eqs. (4-4)

through (4-7). Equations (4-4) through (4-7) are differentiated with respect, s, θ, and P to obtain the

system of equations presented in Eq. (5-21) and solved using the incremental analysis procedure

described in Iizuka (2000).

1sin cos sin 1 0

1 0cos sin cos 0 0

0 0

0

cos sin 0 cos 0 sin

i i i i

i i i ii

si ii i s

i iii ii

sii i i i i i

i

st

h

h

uMF h

s

s

sv v M

P uP

F dQP F

ds

v

P

(5-21)

The differential quantities for step i, (∂Ms /∂θ)i and (∂Ms /∂P)i are estimated by a backward finite

difference approximation using response quantities from step i-1 and i according to Eqs. (5-22) and (5-23).

( 1)

( 1)

s si is

i i i

M MM

(5-22)

62

( 1)

( 1)

s si is

i i i

M MM

P P P

(5-23)

The number of parallel springs, 12, is determined from a convergence study of the analytical bearing

model that is described in previous Chapter 4.

5.3.3 Stepwise solution procedure

Equations (5-11) through (5-13) are solved using Newmark’s average acceleration method

(Newmark 1959) and a stepwise process. To initiate the solution procedure, Equations (5-22) to (5-25) are

used for the first time step assuming linear spring properties to avoid an ill-conditioned problem [6]. At

the first time step, the following procedure is executed:

1. Newmark’s method is used to solve Eqs. (5-11) through (5-13) for 1um, 1vm, and, 1;

2. Quantities: 1um, 1vm, and, 1 are passed to the bearing model for each bearing element;

3. Equation (5-16) is used to calculate 1P* that is added to the static weight W to determine 1P;

4. Equations (5-22) through (5-25) are used to calculate 1θ 1s, 1v and 1F;

5. 1F and 1P are passed to Eqs. (5-11) through (5-13).

Beyond the first time step, Newmark’s method is used to solve Eqs. (5-11) through (5-13) for the

displacement response at step i+1, iteration 1, 11

i mu , 11

i mv , and 11

i . Bearing forces PL*, PR

* and FL, FR

are calculated using the analytical bearing model and a modified Newton-Raphson iteration scheme. The

following procedure is used:

1. For iteration j, displacement response quantities, 1ij

mu , 1ij

mv , and 1ij , are passed to

bearing model for each bearing element;

2. Equation (5-16) is used to calculate i+1P* that is added to the static weight W to determine

i+1P;

3. Equation (5-29) is used to calculate incremental response quantities i(Δθ) i(Δs), i(Δv) and

i(ΔF) of each bearing element;

63

4. Quantities: i(Δθ) i(Δs), i(Δv) and i(ΔF)are added to the value from step i, iθ is, iv and iF to

obtain the bearing response at step i+1, i+1θ, i+1s, i+1v and i+1F;

5. Quantities: i+1F and i+1P are passed to Eqs. (5-11) through (5-13);

6. The modified Newton-Raphson method is used to solve Eqs. (5-11) through (5-13) for

incremental displacement response for iteration j, 1i mju , 1i m

jv , and 1ij ;

7. Add 1i mju , 1i m

jv , and 1ij to the value from iteration j, 1i

jmu , 1i

jmv , and 1i

j to

obtain 11

ij

mu , 1

1i

jmv , and 1

1i

j ;

8. Check whether the tolerance criteria of norm displacement presented in Equation 32 is met;

if the criteria is met, quantities, 11

ij

mu , 1

1i

jmv , and 1

1i

j are the displacement response for

time step i+1, i+1um, i+1vm, i+1; if not repeat 1-7 until the criteria is met.

2 2 2

61 1 11 1 1

1 1 1

10j j j

i m i m i

i m i m i

u v

u v

(5-24)

5.4 Physical earthquake simulation testing

To evaluate the dynamic analytical model, physical earthquake simulation testing of an isolation

system composed of four elastomeric bearings was performed using the George E. Brown Jr., Network

for Earthquake Engineering Simulation (NEES) shake tables at the University at Buffalo. In this section, a

summary of the test setup, instrumentation, test bearings and testing program is presented.

5.4.1Test setup and instrumentation

Figure 5-3 presents an illustration and photograph of the experimental setup. The setup consisted of

five concrete blocks for payload, a steel frame, four elastomeric bearings, four 5-channel reaction load

cells bolted to the earthquake simulator platform and instrumentation. Four concrete trapezoidal blocks

each weighing 8.8 kN and one rectangular concrete block weighing 89 kN were used for artificial mass.

The dimensions of the rectangular concrete block are 3657 mm in length (East-West), 1829 mm wide and

610 mm in height. Each trapezoidal block has an estimated mass moment of inertia of 5.0×108 kg·mm4

64

and a center of mass estimated to be 103 mm from the bottom of the block. Two 152 mm by 152 mm

square lengths of timber were placed between the trapezoidal and rectangular concrete blocks for ease of

assembling the setup. The total payload on the bearings, including concrete blocks, steel frame and

ancillary hardware was 135 kN.

(a) Illustration

(b) Photo

Fig. 5-3 Dynamic shake table test setup

65

The shear force, vertical force and moment at each bearing location were measured using 5-channel

reaction load cells annotated in Figure 5-3b. Absolute displacements were measured using seventeen

spring potentiometers attached to a reference frame and connected to either the earthquake simulator

platform or isolated specimen. The relative horizontal displacement response across each individual

bearing and at the center of mass was obtained by subtracting absolute displacement data from the

appropriate string potentiometer. In total, thirty three accelerometers were installed on the shake table

platform and the isolated structure to measure the input motion and response of the isolated mass to

provide redundancy for the determination of the base shear. An illustration of the plan view of the

experimental setup showing the position of each bearing identification number is presented in Fig. 5-4.

Fig. 5-4 Plan view of bearing setup

5.4.2 Test bearing specimen

The isolation system consisted of a set of four annular low-damping natural rubber bearings. Low-

damping natural rubber bearings where chosen over the more commonly used lead-rubber bearings to

eliminate the added complexity of the lead-core that exhibits, for example, thermo-mechanical behavior

(Kalpakidis and Constantinou 2009a, b). The bearings were proportioned assuming a prototype horizontal

period of TP = 2.5 s and a target dead load compressive pressure of pP = 3.58 MPa. For shake table testing

a time scaling factor of 2 was specified resulting in a length scale factor of 4 according to the

requirements of dynamic similitude and a target model period, TM = 1.25 s. A model target compressive

pressure pM = 3.58 MPa (pressure scale factor equals 1) was used to determine the bonded rubber area

#2

N

E

#1

#3 #4

x

y2438 mm

2438 mm

66

having an outer bonded rubber diameter of 114 mm, and inner bonded rubber diameter of 19 mm. An

assumed effective shear modulus of 0.62 MPa at 100% shear strain resulted in a total rubber thickness, Tr,

of 64 mm that was divided into 20 individual rubber layers each 3.2 mm thick and 19 steel shim plates

each 1.9 mm thick as shown in Figure 5-5. A summary of the model bearing details is presented in Table

5-1.

Fig. 5-5 Details for test bearings

Table 5-1 Details of experimentally tested bearings

Description Symbol Unit Value

Outer diameter Do mm 114 Inner diameter Di mm 19

Single rubber layer thickness tr mm 3.2 Single shim layer thickness ts mm 1.9

Number of rubber layers N – 20 Shape factor S – 7.5

5.4.3 Test program

The test program consisted of dynamic characterization tests using sinusoidal base excitation and

earthquake simulation tests using recorded earthquake ground motions that were scaled both in time and

amplitude. A series of 2 Hz sinusoidal base excitation with peak acceleration ranging from 0.05 to 0.3 g

were performed to establish the shear force-lateral displacement response of the bearings from which the

67

effective shear modulus at 25% shear strain and the effective damping of the bearings, , were determined.

The effective damping is calculated using Eq. (5-25).

2max2

D

h

E

k u

(5-25)

where kh is the effective horizontal stiffness, ED is the energy dissipated per cycle, umax is the peak lateral

displacement. The damping factor calculated using Eq. (5-25) assumes the energy dissipated does not

depend on temperature. This assumption is supported by the findings of Kalpakidis and Constantinou

(2009b) that showed the measured changes of a lead-rubber bearing’s properties were due entirely to

changes attributed to the thermal dependency of the lead’s strength. From the result of characterization

test, the damping ratio of each bearing changes within 10% when the maximum strain level changes from

25% to 120%. Therefore, the strain dependencies are neglected in the damping ratio calculation as well.

Table 5-2 presents a summary of the Go and values determined for the individual bearings. The peak

lateral displacement was limited to 75 mm, corresponding to 120% shear strain during the

characterization tests. It should be noted the stiffness of Bearing 4, and therefore Go, is significantly larger

than for the other three bearings resulting in a small amount of torsion during uni-directional shaking.

Table 5-2 Result from dynamic characterization test

Quantity Unit Bearing 1 Bearing 2 Bearing 3 Bearing 4

Go MPa 0.57 0.55 0.49 0.81 % 4.9 5.0 7.2 5.0

The ERZ-NS component of ground motion recorded at the 95 Erzincan station during the 1992

Erzincan, Turkey earthquake was selected and used for the earthquake simulation tests. The recorded

ground motion was obtained from the Pacific Earthquake Engineering Research Center (PEER 2012)

database. A summary of the test program is presented in Table 5-3. For the earthquake simulation testing,

the isolation system was subjected to a single horizontal component of ground motion that was repeated

with three different intensities, i.e. 50%, 95% and 75%. After each earthquake simulation, white noise

excitation with peak acceleration of 0.05g, i.e. low amplitude frequency rich motion, was used to re-

68

center the bearings to eliminate any small residual displacements prior to the subsequent test. The ground

motions were time-scaled, i.e. compressed, to 0.5 of the length of the original record prior to conducting

the earthquake simulation tests to be consistent with the time scaling factor of 2 assumed when

proportioning the bearings. The motion at the center of the shake table platform was recorded and used as

input for the numerical simulation of the analytical dynamic model. During the excitation at 95% intensity,

a significant torsional response was detected in the isolation system. Therefore, testing proceeded at a

lower intensity, i.e. 75% in the direction of y-axis, which resulted in failure of two of the bearings and the

conclusion of the testing program.

Table 5-3 Program for earthquake simulation test Test No. Ground motion Component Direction Intensity

1 Erzincan ERZ-NS y-y 50% 2 Erzincan ERZ-NS y-y 95% 3 Erzincan ERZ-NS x-x 75%

5.5 Evaluation the dynamic response of bearing model

In this section, the results of the numerical simulations are compared to the results of the

experimental shake table tests to evaluate the dynamic analytical model for simulating the earthquake

response of an isolation system composed of elastomeric bearings. The material properties, i.e. the initial

shear modulus and horizontal effective damping of each individual bearing obtained from the

characterization test (presented in Table 5-2) were used as the material parameters for the numerical

simulations. The vertical effective damping of each individual bearing is assumed to be equal to the

horizontal effective damping of the bearing.

5.5.1 Comparison between numerical and experimental simulation

Figure 5-6 presents a comparison of the lateral displacement histories of the center of mass

obtained from numerical simulation, denoted Model, and experimental shake table testing, denoted

Experimental, for Test 1, that was performed with the ground motion at 50% intensity. The relative error

between the model and experimental for the peak lateral displacement is 6.0%. However the model over-

69

predicts the post-peak displacement response. Figure 5-7 presents a comparison of the shear force

histories for each individual bearing. For the analytical model the shear force in the individual bearing is

the sum of the lateral force, F, from the bearing model and the damping force, FDx. The relative error

between the model and experimental peak shear force ranges from 4.7% to 11.6% for all the bearings.

While the model estimates the maximum shear force response well, it again over-predicts the post-peak

shear force response with a phase-shift as the response decays. Figure 5-8 presents a comparison of the

vertical force history for each bearing from Test 1. The vertical force is the sum of the static weight on the

bearing, W, the dynamic vertical force, P*, and the damping force, FDz. The relative error in the peak shear

force between the model and experimental responses ranges from 1.8% to 5.6% for all bearings.

Fig. 5-6 Comparison of lateral displacement history at the center of mass for the Erzincan ground

motion at 50% intensity

0 5 10 15 20−120

−60

0

60

120

Time (s)

Late

ral d

ispl

acem

ent (

mm

)

Experimental

Model

70

Fig. 5-7 Comparison of shear force history in the individual bearings for the Erzincan ground


Fig. 5-8 Comparison of vertical force history in the individual bearings for the Erzincan ground


A comparison of the individual bearing and total isolation system shear force-lateral displacement

responses from the model with experimental responses for Test 1 are presented in Fig. 5-9. Neither

experimental test results nor model simulation result show a reduction in the tangential horizontal

−10

−5

0

5

10

#1−NE

ExperimentalModel

She

ar fo

rce

(kN

)

#2−NW

0 5 10 15 20−10

−5

0

5

10

#3−SW

Time (s)

She

ar fo

rce

(kN

)

0 5 10 15 20

Time (s)

#4−SE

20

30

40

50

#1−NE

Experimental

Model

Ver

tical

forc

e (k

N)

#2−NW

0 5 10 15 2020

30

40

50

Time (s)

Ver

tical

forc

e (k

N)

#3−SW

0 5 10 15 20

Time (s)

#4−SE

71

stiffness for any of the individual bearing and, as a result, the isolation system. The comparison presented

in Fig. 5-9 shows good agreement between the experimental and model for the effective stiffness of the

individual bearings. Furthermore, the relative error in cumulative energy dissipated between the model

and experimental results range from 11.1% to 14.6% for all the bearings and total system.

Fig. 5-9 Comparison of individual and total shear force - lateral displacement response for

Erzincan ground motion at 50% intensity

−12

−8−4

048

12#1−NE Experimental

She

ar fo

rce

(kN

)

−12

−8−4

048

12Model

−12

−8−4

048

12#2−NW

She

ar fo

rce

(kN

)

−12

−8−4

048

12

−12

−8−4

048

12#3−SW


She

ar fo

rce

(kN

)

−12

−8−4

048

12


−12

−8−4

048

12#4−SE

She

ar fo

rce

(kN

)

−12

−8−4

048

12

Lateral displacement (in)

−200 −100 0 100 200−30−20−10

0102030


She

ar fo

rce

(kN

)

Total

−200 −100 0 100 200−30−20−10

0102030


72

Figure 5-10 presents a comparison of the lateral displacement histories of the center of mass for Test

2, performed with the ground motion at 95% intensity. The relative error in lateral displacement between

the model and experimental result is 7.0%. Better agreement for the post-peak response is observed for

Test 2 than for Test 1 (Figure 5-6). Figure 5-11 presents a comparison of the shear force histories in each

individual bearing for Test 2. The relative error between the model and experimental peak shear force

response ranges from 0.3% to 10.9%. The model response appears to agree well with the experimental

shear force response in each individual bearing until the lateral displacement response is sufficiently small

where a phase shift is observed. Figure 5-12 presents a comparison of the vertical force history in the

individual bearings for Test 2. The relative error in the peak shear force response between the model and

experimental results ranges from 1.4% to 15.8% for all bearings. For all response histories from Test 2,

the model simulates the experimental response reasonably well until the response becomes small, after

approximately 10 s, where the model over-estimates the experimental response with a phase-shift.

A comparison of the shear force-lateral displacement responses from the model and experimental

results for Test 2 is presented in Fig. 5-13. From Fig. 5-13, it the model is able to simulate the stiffness

reduction in Bearings 1, 2 and 3 as observed in the experimental results. The relative error in cumulative

energy dissipated between the model and experimental results ranges from 4.8% to 15.7% for all bearings.

However, the model is not simulating the increased area of the hysteresis loop at the peak lateral

displacement when the tangential horizontal stiffness is negative as observed in the experimental response.

The relative error between various experimental and model simulation response quantities for Tests 1 and

2 are summarized presented in Table 5-4.

73

Fig. 5-10 Comparison of lateral displacement history at the center of mass for the Erzincan ground


Fig. 5-11 Comparison of shear force history in the individual bearings for the Erzincan ground


0 5 10 15 20−120

−60

0

60

120

Time (s)

Late

ral d

ispl

acem

ent (

mm

)

Experimental

Model

−10

−5

0

5

10

#1−NE

ExperimentalModel

She

ar fo

rce

(kN

)

#2−NW

0 5 10 15 20−10

−5

0

5

10

#3−SW

Time (s)

She

ar fo

rce

(kN

)

0 5 10 15 20

Time (s)

#4−SE

74

Fig. 5-12 Comparison of total vertical force history in the individual bearings for the Erzincan

ground motion at 95% intensity

20

30

40

50

#1−NE

Experimental

Model

Ver

tical

forc

e (k

N)

#2−NW

0 5 10 15 2020

30

40

50

Time (s)

Ver

tical

forc

e (k

N)

#3−SW

0 5 10 15 20

Time (s)

#4−SE

75

Fig. 5-13 Comparison of individual and shear force - lateral displacement response for the 95%

intensity test.

−12

−8−4

048


She

ar fo

rce

(kN

)

−12−8−4

048

12Model

−12

−8−4

048

12#2−NW

She

ar fo

rce

(kN

)

−12

−8−4

048

12

−12

−8−4

048

12#3−SW


She

ar fo

rce

(kN

)

−12

−8−4

048

12


−12

−8−4

048

12#4−SE

She

ar fo

rce

(kN

)

−12

−8−4

048

12


−200 −100 0 100 200−30−20−10

0102030


She

ar fo

rce

(kN

)

Total

−200 −100 0 100 200−30−20−10

0102030


76

Table 5-4 Relative error of model simulation result comparing to experimental test result

Response quantity Intensity of

Ground motion Bearing 1 Bearing 2 Bearing 3 Bearing 4

Maximum lateral displacement

50% 6.0% 6.0% 6.0% 6.0%

95% 7.0% 7.0% 7.0% 7.0%

Maximum shear force 50% 11.6% 10.8% 4.7% 5.2%

95% 10.9% 5.6% 0.3% 9.9%

Maximum vertical force

50% 5.6% 5.1% 1.8% 1.8%

95% 15.8% 8.3% 1.4% 9.5%

Dissipated energy 50% 14.6% 14.4% 11.1% 11.2%

95% 15.7% 14.1% 7.4% 4.8%

Following Test 2, Test 3 was performed with the ground motion intensity reduced to 75% and

applied in the transverse y-direction. During Test 3, two bearings failed possibly due to damaged

sustained during the previous test. While only qualitative, Figure 5-14 presents a comparison of the shear

force-lateral displacement responses from Test 3 with the response obtained from numerical simulation

with the Erzincan ground motion amplitude scaled to 105% intensity. For the numerical simulation the

intensity of the ground motion was increased until a solution could not be obtained because the global

base shear of the isolation system exhibited negative tangential stiffness precluding a numerical solution.

While this comparison can only be considered qualitative, it does illustrate the analytical bearing model is

able to simulate bearing response into a regime where the tangential horizontal stiffness is negative that

might have caused Bearings 1 and 2 to become unstable.

77


test resulting in failure of bearings

5.5.2 Simulation of experimental tests by Sanchez et al. and comparison of results

Additional data from shake table tests performed by Sanchez et al. (2013) with an isolation system

composed of elastomeric bearings was obtained from the NEES Project Warehouse to further evaluate the

dynamic analytical model because of the premature failure of Bearings 1 and 2. Details of the Sanchez et

al. tests can be found in Sanchez et al. (2013). However, the experimental setup used by Sanchez et al.

−12

−8−4

048


She

ar fo

rce

(kN

)

−12−8−4

048

12Model

−12

−8−4

048

12#2−NW

She

ar fo

rce

(kN

)

−12

−8−4

048

12

−12

−8−4

048

12#3−SW


She

ar fo

rce

(kN

)

−12

−8−4

048

12


−12

−8−4

048

12#4−SE

She

ar fo

rce

(kN

)

−12

−8−4

048

12


−200 −100 0 100 200−30−20−10

0102030


She

ar fo

rce

(kN

)

Total

−200 −100 0 100 200−30−20−10

0102030


78

was similar to that shown in Fig. 5-3 with one notable exception being the use of steel mass plates

resulting in a larger payload and a lower center of mass. The larger payload allowed Sanchez et al. to test

bearings with a larger outer bonded diameter, i.e. 165mm. The parameters of the dynamic analytical

model, e.g. Go, cxL, czL, cxR, czR among others were determined from the experimental data and the physical

dimensions of the test setup described in (Warn et al. 2007) just as just as for the prior comparison. Figure

5-15 presents a comparison of the shear force – lateral displacement response of each bearing and the

total system from the model simulation with the experimental data with the ERZ-NS component scaled to

150% intensity. The bearing numbering scheme used by Sanchez et al. differs from that used for the

shake table testing in this study (see Fig. 5-4). The numbering scheme of Sanchez et al. was adopted for

the presentation of the numerical simulation results herein. From the comparison presented in Fig. 5-15,

the results from the model simulation agree well with the general trend and behavior observed from the

experimental data for the individual bearings and the total system response. The relative error between the

peak lateral displacement from the model and experimental data is 14.5%. The relative error between the

peak shear force response from model and experimental simulations ranges from 6.2 to 12.8%. From Fig.

5-15, the dynamic analytical model is capable of simulating the reduction in horizontal tangential stiffness

to the point where the individual bearing shear force passes through zero and negates in sign as observed

in the response of Bearing 3. One notable shortcoming of the analytical model is the underestimation of

the energy dissipated especially at the peak displacements where the tangential horizontal stiffness is

negative.

79


simulations with the Erzincan ground motion scaled to 150% intensity.

5.6 Further demonstration of the analytical model

The previous comparisons illustrated the dynamic analytical model is able to simulate the earthquake

response of individual elastomeric bearings and the total system response with reasonable accuracy

including bearing response in a regime where the horizontal tangential stiffness is negative include shear

force response passing through zero and changing sign at the peak displacement excursion. To assess the

−20

−10

0

10

20#1−NW Experimental

She

ar F

orce

(kN

)

−20

−10

0

10

20Model

−20

−10

0

10

20#2−NE

She

ar F

orce

(kN

)

−20

−10

0

10

20

−20

−10

0

10

20#3−SW


She

ar F

orce

(kN

)

−20

−10

0

10

20


−20

−10

0

10

20#4−SE

She

ar F

orce

(kN

)

−20

−10

0

10

20


−200 −100 0 100 200−50

−25

0

25

50


She

ar fo

rce

(kN

)

Total

−200 −100 0 100 200−50

−25

0

25

50


80

models capability to simulate changing vertical load demands, the Sanchez et al. simulation was repeated

by increasing the height of the center of mass, hc, from 375 mm (as for comparison in Fig. 5-15) to 1500

mm. By only changing hc, only the overturning moment and resulting vertical loads on the individual

bearings are altered.

Figure 5-16 presents the results of numerical simulations of the Sanchez et al. tests for the cases with

hc = 375 mm (original) and hc = 1500 mm. Individual bearing and total responses for the hc = 375 mm

case are shown in the left column and those for the hc = 1500 mm case in the right column. In each plot a

dashed reference line is included to illustrate the difference in maximum shear force response from the

original hc = 375 mm case. Increasing hc resulted in only a modest increase in the maximum lateral

displacement of the system from 173 mm to 175 mm. For Bearings 1 and 3, both on the west side,

increasing hc results in a decrease in peak shear force at approximately -100 mm of displacement due to

the decreased compressive load as a result of the increased overturning moment. Specifically, the

maximum shear force of Bearing 1 is 12 kN for the model of hc = 375 mm in comparison to 10.9 kN for

the model of hc = 1500 mm, a 10% reduction. For Bearings 2 and 4, on the east side, the maximum shear

forces increased as hc increased. Specifically, the maximum shear force in Bearing 2 increased from 10.2

kN to 11.2 kN as hc increased from 350 mm to 1500mm. The increase in shear force response of the east

bearings (2 and 4) does not equal the decrease in the west bearings (1 and 3) as indicated by the slight

decrease in the maximum total shear force response of the isolation system from 42.9 kN to 41.7 kN. The

results presented in Fig. 5-16 show the model is able to account for the change in vertical load on the

bearings due to the changed overturning moment caused by increasing the height of the center of mass

though for this example the changes were modest because the height could not be increased beyond 1500

mm. The comparison does, however, illustrate how the analytical bearing model could potentially

simulate the shear force response of a bearing at the perimeter of a mid-rise, i.e. 10-15 story, or under

braced frames in base-isolated building due to overturning effects.

81

Fig. 5-16 Comparison of model shear force - lateral displacement response from numerical

simulation of Sanchez et al. tests with Erzincan ground motion scaled to 150% intensity and varying hc

−20

−10

0

10

20#1−NW

hc = 375mm

She

ar F

orce

(kN

)

hc = 1500mm

−20

−10

0

10

20#2−NE

She

ar F

orce

(kN

)

−20

−10

0

10

20#3−SW

She

ar F

orce

(kN

)

−20

−10

0

10

20#4−SE

She

ar F

orce

(kN

)


−200 −100 0 100 200−50

−25

0

25

50


She

ar fo

rce

(kN

) Total

−200 −100 0 100 200


82

5.7 Discussion

A dynamic analytical model to simulate the earthquake response of an isolation system composed of

elastomeric bearings has been presented. The dynamic analytical model consists of a rigid mass and rigid

super-structure supported by four bearing elements. An analytical bearing model developed in previous

chapter was extended to account for varying vertical force and was used to simulate the response of the

individual elastomeric bearing elements. Experimental shake table testing of an isolation system

composed of four elastomeric bearings was conducted to collect data to evaluate the dynamic analytical

model. The analytical dynamic model was developed to simulate the behavior of elastomeric bearings

into a regime where the horizontal tangential stiffness becomes negative resulting and even reversal of the

sign of the shear force at the peak lateral displacement as observed in the experimental data from previous

tests (Sanchez et al. 2013).

From the comparisons of the experimental and numerical results, the analytical dynamic model

presented here was shown to simulate the individual bearing and system response reasonably well given

the model requires only material and physical parameters. Further evaluation using the results of Sanchez

et al. (2013) suggested the dynamic analytical model is capable of simulating the experimentally observed

phenomena such as negative tangential stiffness and reversing of the shear force at the peak lateral

displacement. However, the comparison between the numerical simulation and the experimental

simulation of Sanchez et al. revealed the model does not adequately simulate the energy dissipated, area

within the force-displacement loop, particularly at the peak displacement excursion when the tangential

horizontal stiffness is negative. Furthermore, the vertical deformation in the vertical springs due to a

change in vertical load is neglected in the model. This component of deformation was neglected in an

effort maintain a level of model simplicity. However, the implication of this simplification on the fidelity

of the model simulation needs to be evaluated.

83

Chapter 6

Conclusion

6.1 Summary

The overarching objectives of this dissertation are to develop a fundamental understanding of the

mechanism(s) leading to the reduction in tangential horizontal stiffness observed in elastomeric bearings

with increased lateral displacement and to develop an analytical model for simulating the behavior.

The capabilities of two semi-empirical models (Nagarajaiah and Ferrell 1999, Iizuka 2000) for

simulating critical behavior of elastomeric bearings at various lateral displacements were evaluated using

data from past experimental studies (Buckle et al. 2002; Weisman and Warn 2012; Sanchez et al. 2013).

Critical displacement obtained from simulations with the Iizuka model compared well with experimental

data both in terms of the general trends and the relative error. A global variance-based sensitivity analysis

was performed on the Iizuka model whereby the sensitivity of the critical displacement, ucr, simulation to

variability in four independent material and model parameters, specifically, G, r, s1 and s2 was determined.

The results of sensitivity analysis demonstrated the critical displacement ucr is most sensitive to the

rotational spring parameter, r, at large shear strain level.

Based on this finding, an analytical bearing model was developed that uses a number of parallel

vertical springs instead of one rotational spring to simulate the rotational behavior requiring parameters

related only to the physical and material properties of the bearing. The analytical model was shown to

simulate the critical behavior with reasonable accuracy by comparison to experimental data from previous

studies especially when considering the model does not rely upon empirical parameters that must be

experimentally calibrated. Three-dimensional (3D) finite element (FE) model was further employed into a

one-at-a-time sensitivity technique, i.e. the Method of Morris, to support the findings that the rotational

behavior is controlling the critical behavior of elastomeric bearings. Furthermore, by comparing to the

result from 3D FE models, the analytical bearing model shows similar fidelity with the highly detailed 3D

FE model.

84

A dynamic model of an isolation system composed of elastomeric bearings is developed to extend

the analytical bearing model for application to numerical earthquake simulation and to evaluate the

capability of the model for simulating the earthquake response of elastomeric bearings. From the

comparisons between the results from the experimental and numerical simulations, the bearing model

developed in this study was shown to simulate the individual bearing and system response reasonably

well given the model contains only material and physical parameters. However, evaluation of the model

with data from Sanchez et al. revealed the model does not adequately simulate the dissipated energy, i.e.

the area within the force-displacement loop, specifically at the peak displacement excursion where the

tangential horizontal stiffness is negative.

6.2 Specific conclusions

The specific outcomes from this dissertation are listed below:

1. Based on the findings from global sensitivity analysis, the parameter controlling the

nonlinear rotational behavior of an individual bearing is the key to simulating the critical

behavior of elastomeric bearing at large lateral displacements.

2. The parallel vertical spring approach for representing the rotational behavior is reasonable

and providing similar fidelity for simulating critical behavior to models utilizing a

concentrated rotational spring and semi-empirical formulations.

3. The analytical bearing model developed in this study has similar fidelity by comparison to

detailed 3D FE modeling.

4. The analytical model is able to generally simulate the earthquake response of individual

bearings with reasonable accuracy accounting for the effects of vertical load fluctuation due

to overturning on the shear force response.

6.3 Significance of study

The findings of this dissertation research, for the first time, identified the rotational behavior is the

mechanism that controls the critical behavior, i.e. stability, of elastomeric bearings at large lateral

85

displacement. An improved, more practical, analytical bearing model that does not rely upon empirical

parameters is developed for simulating the critical behavior with reasonable accuracy. The analytical

bearing model can be implemented into numerical earthquake simulation software, such as OpenSees, to

simulate the earthquake response of structures isolated with elastomeric bearing. The ability to simulate

individual bearing response beyond the stability limit into a regime where the tangential horizontal

stiffness is negative will allow design professionals and academics to assess the global stability of the

isolation system instead of just individual bearing stability.

6.4 Recommendations for future research

While the evaluation of the proposed analytical bearing model showed that the model is capable of

simulating the monotonic and earthquake response of elastomeric bearing, a few shortcomings and

opportunities to expand the model were identified.

From the evaluation of the numerical earthquake simulation the model underestimated the area

within the loop when the bearing displacement exceeded the stability limit. While mostly speculation at

this point, one hypothesis is that this underestimation is due to the assumed bilinear elastic constitutive

relationship for the vertical spring elements does not fully represent the behavior of rubber in tension, for

example Mullins’ effect (Mullin 1966). The dissipated energy of an individual bearing beyond the

stability limit might be better simulated using a plastic constitutive relationship of vertical springs.

Lead-rubber bearings are widely used for seismic isolation in the United States and around the world.

It is suggested that the analytical bearing model developed in this study could be expanded to simulate the

lead-rubber bearings by adding a plastic element to represent the contribution of the lead core behavior

into the model.

The dynamic model for the isolation structure composed of elastomeric bearings assumes that the

vertical deformation in the vertical springs due to a change in vertical load is relative small and can be

neglected. However, the implication of this simplification on the fidelity of the model simulation needs to

be validated in the future. Furthermore, the bearing model in the dynamic model requires an input of

86

vertical force, P, and lateral displacement, u, of each bearing, so that the compatibility of vertical

displacement over each bearing is not imposed. This shortcoming can be improved by using input of the

displacement responses, i.e. u, and, v, instead.

87

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Appendix A

Instrumentation used in dynamic test

The instrumentations used in dynamic shake table test are listed in this appendix. Elevation and side

view of the instrumentation layout is also provided.

Table A.1 List of instrumentations

Channel Instrument Notation Measurement Location Axis

1 LC1N Load cell force B1 Z

2 LC1Sx Load cell force B1 X

3 LC1Sy Load cell force B1 Y

4 LC1Mx Load cell moment B1 X

5 LC1My Load cell moment B1 Y
















21 ACC1x Accelerometer acceleration B1 bottom end-plate X

22 ACC2y Accelerometer acceleration B1 bottom end-plate Y

23 ACC3z Accelerometer acceleration B1 bottom end-plate Z









91

Table A.1 List of instrumentations (Cont’d)

Channel Instrument Notation Measurement Location Axis


33 ACC13x Accelerometer acceleration B1 top end-plate X

34 ACC14y Accelerometer acceleration B1top end-plate Y

35 ACC15z Accelerometer acceleration B1 top end-plate Z


37 ACC17y Accelerometer acceleration B2 top end-plate Y








45 ACC25x Accelerometer acceleration center of table X

46 ACC26y Accelerometer acceleration center of table Y

47 ACC27z Accelerometer acceleration center of table Z

48 ACC28x Accelerometer acceleration center of steel plate X

49 ACC29y Accelerometer acceleration center of steel plate Y

50 ACC30z Accelerometer acceleration center of steel plate Z

51 ACC31x Accelerometer acceleration center of top concrete block X

52 ACC32y Accelerometer acceleration center of top concrete block Y

53 ACC33z Accelerometer acceleration center of top concrete block Z

54 STP1x String Pot. displacement B1 bottom end-plate X

55 STP2x String Pot. displacement B1 top end-plate X

56 STP3x String Pot. displacement B4 bottom end-plate X

57 STP4x String Pot. displacement B4 top end-plate X

58 STP5y String Pot. displacement B1 bottom end-plate Y

59 STP6y String Pot. displacement B1 top end-plate Y

60 STP7y String Pot. displacement B2 bottom end-plate Y

61 STP8y String Pot. displacement B2 top end-plate Y

62 STP9x String Pot. displacement table under B1 X

63 STP10x String Pot. displacement table under B4 X

64 STP11y String Pot. displacement table under B1 Y

65 STP12x String Pot. displacement surface of concrete block X

66 STP13y String Pot. displacement surface of concrete block Y

67 STP14z String Pot. displacement B1 Z




92

Fig. A-1 Instrumentation layout

93

Appendix B

Additional test specimen

Another set of bearings (S=3.75) were tested on the shake table in this dissertation. The details of

S=3.75 bearings is shown in this appendix.

Fig. B-1 Bearing Detail for S=3.75 Bearing

94

Appendix C

Earthquake simulation testing program

Test program for earthquake simulation in this study is listed in this appendix. A total 59 tests were

applied on the two isolation systems. Ground motions from three earthquakes are used in the experimental

testing, i.e. Erzincan (ERZ), Takatori (TAK) and Rinaldi (RRS).

Table C.1 Test program for S=7.5 bearings

Test NO. Test name Excitation Amplification

x y z x y z

1 erz1h1 ERZ-NS 0.25 2 erz1h2 ERZ-NS 0.5 3 erz1h3 ERZ-NS 0.65 4 erz1h4 ERZ-NS 0.75 5 erz1h5 ERZ-NS 0.85 6 erz1h6 ERZ-NS 0.9 7 tak1h1 TAK090 0.25 8 tak1h2 TAK090 0.4 9 rrs1h1 RRS228 0.15 10 rrs1h2 RRS228 0.35 11 rrs1h3 RRS228 0.5 12 erz1h7 ERZ-NS 0.955 13 rrs1h4 RRS228 0.55 14 rrs1h5 RRS228 0.6 15 rrs1h6 RRS228 0.65 16 erz1v1 ERZ-NS ERZ-UP 0.5 0.5 17 erz1v2 ERZ-NS ERZ-UP 0.75 0.7518 erz1v3 ERZ-NS ERZ-UP 0.85 0.8519 erz1v4 ERZ-NS ERZ-UP 0.9 0.9 20 erz1v5 ERZ-NS ERZ-UP 0.95 0.9521 erz1v6 ERZ-NS ERZ-UP 0.75 0.7522 erz1h2r ERZ-NS 0.5 23 erz1v7 ERZ-NS 0.5 24 erz1h4r ERZ-NS 0.75

95

Table C-2 Test program for S=3.75 bearings

Test NO. Test name Concrete

block number Excitation Amplification x y z x y z

1 s7erz1h1 7 ERZ-NS 0.2 2 s7erz1h2 7 ERZ-NS 0.6 3 s7erz1h3 7 ERZ-NS 0.7 4 s7erz1h4 7 ERZ-NS 0.6 5 s5erz1h1 5 ERZ-NS 0.4 6 s5erz1h2 5 ERZ-NS 0.6 7 s5erz1h3 5 ERZ-NS 0.8 8 s5erz1h4 5 ERZ-NS 0.9 9 s5erz1h5 5 ERZ-NS 1.0

10 s6erz1h1 6 ERZ-NS 0.4 11 s6erz1h2 6 ERZ-NS 0.6 12 s6erz1h3 6 ERZ-NS 0.6 13 s6erz1h4 6 ERZ-NS 0.8 14 s6erz1h5 6 ERZ-NS 1.0 15 s6erz1h6 6 ERZ-NS 1.2 16 s6erz1h7 6 ERZ-NS 1.4 17 s6erz1h8 6 ERZ-NS 1.0 18 s6erz1h9 6 ERZ-NS 1.5 19 s6erz1h10 6 ERZ-NS 1.6 20 s6erz1h11 6 ERZ-NS 1.7 21 s6rrs1h1 6 ERZ-NS 0.2 22 s6rrs1h2 6 ERZ-NS 0.6 23 s6rrs1h3 6 ERZ-NS 0.8 24 s6rrs1h4 6 ERZ-NS 0.9 25 s6rrs1h5 6 ERZ-NS 1.0 26 s6erz1h12 6 ERZ-NS 1.8 27 s6erz1h13 6 ERZ-NS 1.9 28 s8erz1h1 8 ERZ-NS 0.25 29 s8erz1h2 8 ERZ-NS 0.6 30 s8erz1h3 8 ERZ-NS 0.8 31 s8erz1h4 8 ERZ-NS 1.0 32 s8erz1h5 8 ERZ-NS 1.1 33 s8erz1h6 8 ERZ-NS 1.2

96

Appendix D

Parameter set used in method of Morris

Fifty-four FE models are used in the sensitivity analysis and evaluation of the analytical model.

Detail of the parameter set is listed in Table D-1.

Table D-1 Parameters used in sensitivity analysis for FE models

NO. N tr ts tc Do Di G K

1 30 0.25 0.09 0.85 17.5 1.3 95 125 2 30 0.25 0.17 0.85 17.5 1.3 95 125 3 14 0.25 0.17 0.85 17.5 1.3 95 125 4 14 0.65 0.17 0.85 17.5 1.3 95 125 5 14 0.65 0.17 2.05 17.5 1.3 95 125 6 14 0.65 0.17 2.05 17.5 0.5 95 125 7 14 0.65 0.17 2.05 37.5 0.5 95 125 8 14 0.65 0.17 2.05 37.5 0.5 135 125 9 14 0.65 0.17 2.05 37.5 0.5 135 325

10 10 0.25 0.13 1.45 32.5 0.9 65 425 11 10 0.25 0.13 1.45 32.5 0.9 65 225 12 10 0.25 0.13 1.45 12.5 0.9 65 225 13 26 0.25 0.13 1.45 12.5 0.9 65 225 14 26 0.25 0.21 1.45 12.5 0.9 65 225 15 26 0.25 0.21 2.65 12.5 0.9 65 225 16 26 0.25 0.21 2.65 12.5 0.9 105 225 17 26 0.25 0.21 2.65 12.5 1.7 105 225 18 26 0.65 0.21 2.65 12.5 1.7 105 225 19 14 0.55 0.07 2.35 27.5 1.5 95 375 20 14 0.95 0.07 2.35 27.5 1.5 95 375 21 14 0.95 0.07 2.35 27.5 0.7 95 375 22 14 0.95 0.07 2.35 27.5 0.7 95 175 23 30 0.95 0.07 2.35 27.5 0.7 95 175 24 30 0.95 0.07 2.35 27.5 0.7 135 175 25 30 0.95 0.15 2.35 27.5 0.7 135 175 26 30 0.95 0.15 1.15 27.5 0.7 135 175 27 30 0.95 0.15 1.15 47.5 0.7 135 175 28 14 0.45 0.13 2.35 32.5 1.3 95 425 29 14 0.45 0.13 2.35 12.5 1.3 95 425 30 14 0.45 0.13 2.35 12.5 1.3 95 225 31 14 0.45 0.13 1.15 12.5 1.3 95 225 32 14 0.45 0.13 1.15 12.5 1.3 135 225 33 14 0.45 0.21 1.15 12.5 1.3 135 225

97

Table D-1 Parameters used in sensitivity analysis for FE models (Cont’d)

NO. N tr ts tc Do Di G K

34 14 0.85 0.21 1.15 12.5 1.3 135 225 35 30 0.85 0.21 1.15 12.5 1.3 135 225 36 30 0.85 0.21 1.15 12.5 0.5 135 225 37 14 0.45 0.11 1.75 37.5 1.5 135 225 38 14 0.45 0.11 1.75 17.5 1.5 135 225 39 14 0.45 0.11 1.75 17.5 1.5 95 225 40 14 0.85 0.11 1.75 17.5 1.5 95 225 41 14 0.85 0.11 0.55 17.5 1.5 95 225 42 14 0.85 0.11 0.55 17.5 1.5 95 425 43 14 0.85 0.11 0.55 17.5 0.7 95 425 44 30 0.85 0.11 0.55 17.5 0.7 95 425 45 30 0.85 0.19 0.55 17.5 0.7 95 425 46 34 0.95 0.17 1.15 42.5 1.7 105 175 47 34 0.95 0.17 1.15 42.5 1.7 105 375 48 34 0.95 0.17 1.15 42.5 0.9 105 375 49 34 0.95 0.17 1.15 22.5 0.9 105 375 50 34 0.95 0.17 2.35 22.5 0.9 105 375 51 18 0.95 0.17 2.35 22.5 0.9 105 375 52 18 0.55 0.17 2.35 22.5 0.9 105 375 53 18 0.55 0.09 2.35 22.5 0.9 105 375 54 18 0.55 0.09 2.35 22.5 0.9 65 375

XING HAN

Education

B.S. Civil Engineering (Honors School), July 2008

Harbin Institute of Technology, Harbin, China

M.S. Structural Engineering, July 2010

Harbin Institute of Technology, Harbin, China

Ph.D. Civil Engineering, Expected December 2013

The Penn State University, University Park, PA

Research experience

2011-present Graduate Research Assistant, Department of Civil and Environmental Engineering, The

Penn State University

2008-2010 Graduate Research Assistant, School of Civil Engineering, Harbin Institute of

Technology

2006-2008 Research Assistant, School of Civil Engineering, Harbin Institute of Technology

Journal publication

1. Han, X., Keller, C. A., Warn, G. P., and Wagener, T. (2013). “Identification of the controlling

mechanism for predicting critical loads in elastomeric bearings.” J. Struct. Eng.,

10.1061/(ASCE)ST.1943-541X.0000811.

2. Han, X., Warn, G. P. “A mechanistic model to simulate the critical displacements of elastomeric

bearings.” Journal of Structural Engineering (submitted for review).

3. Han, X., Warn, G. P. “Earthquake simulation of an isolation system composed of elastomeric

bearings considering the influence of vertical load.” Earthquake Engineering and Structural

Dynamics (submitted for review).

an analytical model for simulating the critical …

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