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AN ANALYTICAL MODEL FOR SIMULATING THE CRITICAL BEHAVIOR OF ELASTOMERIC SEISMIC ISOLATION BEARINGS
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.
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
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
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
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
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
(b) Bearing cross section (taken from
(a) Illustration of isolated structure
Warn and Whittaker 2006)
Fig. 1-1 Seismic isolation
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 3DBasis (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.
- b. c.
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.
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:
- Evaluate the simulation capability of existing semi-empirical bearing models of elastomeric bearings using experimental data from past studies.
- 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.
- Develop an improved, analytical bearing model that requires only material and physical parameters based on the knowledge gained from the sensitivity studies.
- Employ the improved analytical bearing model for numerical earthquake simulation of an isolation system composed of elastomeric bearings.
- Evaluate the models using existing experimental data from past studies and data generated from physical earthquake simulation testing performed as part of this study.
- Expand the parameter set beyond that available from existing experimental data using threedimensional 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 semiempirical 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.
AN ANALYTICAL MODEL FOR SIMULATING THE CRITICAL BEHAVIOR OF ELASTOMERIC SEISMIC ISOLATION BEARINGS
