MECHANISTIC MODEL FOR LEAD RUBBER BEARINGS

MECHANISTIC MODEL FOR LEAD RUBBER BEARINGS

ABSTRACT

 

Seismic isolation is a technique used to shift the fundamental period of a structure to a long range period which reduces the forces a structure attracts during a seismic event. Two, widely used bearings in seismic base isolation of structures are elastomeric and lead rubber bearings. A typical elastomeric bearing consists of a number of layers of rubber alternated with steel shims bonded between two rubber layers. The addition of a lead core inserted in a central mandrel hole results in a lead rubber bearing (LRB). The lead enhances the bearings energy dissipating in an earthquake event. When elastomeric or LRBs are simultaneously subjected to vertical compressive load and increasing lateral displacement, the shear force equilibrium path can exhibit a critical point, beyond which the bearing exhibits negative stiffness. Semi-empirical models to simulate this behavior for elastomeric and LRB have been developed in the past. These models rely on experimentally calibrated parameters, making them impractical for design. Recently, a particular mechanics based model, developed for an elastomeric bearing only, approaches the modelling using vertical springs and a simple bi-linear constitutive relationship to represent the rotational behavior of an elastomeric bearing. Overarching goal of the present study is to build on the mechanics based elastomeric model to develop a LRB model. The elastomeric model is modified to include hysteretic behavior of LRB and also uses a Newton type numerical solution technique to solve for response of the bearing. The model, proposed in this study, utilizing vertical springs approach has shown to be capable of simulating the strength,

stiffness and hysteretic behavior of LRB well.

 

 

 

 

 

 

TABLE OF CONTENTS

Acknowledgements……………………………………………………………………….. vi

Chapter 1 INTRODUCTION …………………………………………………………………………… 01

1.1 General ……………………………………………………………………………………………… 01

1.2 Problem Statement ……………………………………………………………………………… 02 …

2.5 Experimental response……………………………………………………………………………… 10

2.6 Summary………………………………………………………………………………………………… 11

Chapter 3 MECHANISTIC MODEL FOR ELASTOMERIC BEARING…………………… 12

3.1 Introduction…………………………………………………………………………………………….. 12

3.2 Equilibrium and Compatibility equations……………………………………………………. 14

3.3 Newton-Raphson Method…………………………………………………………………………. 16

3.4 Moment-rotation relationship……………………………………………………………………. 22

Chapter 4 MECHANISTIC MODEL FOR LEAD RUBBER BEARING…………………… 28

4.1 Introduction…………………………………………………………………………………………….. 28

4.2 Lead Model…………………………………………………………………………………………….. 30

4.3 Equilibrium and Compatibility equations……………………………………………………. 32

4.4 Newton-Raphson method………………………………………………………………………….. 35

Chapter 5 MODEL RESULTS AND EVALUATION)…………………………………………….. 37

5.1 General…………………………………………………………………………………………………… 37

5.2 Elastomeric bearing…………………………………………………………………………………. 37

5.2.1 Experimental response…………………………………………………………………….. 37

5.2.2 Model Simulation…………………………………………………………………………… 38

5.3 LRB……………………………………………………………………………………………………….. 40

5.3.1 Experimental response…………………………………………………………………….. 40

5.3.2 Model simulation for monotonic loading…………………………………………… 41

5.3.3 Model simulation for cyclic loading………………………………………………….. 42

5.4 Comparison of results………………………………………………………………………………. 43

5.5 Discussion………………………………………………………………………………………………. 47

Chapter 6 SUMMARY AND CONCLUSIONS………………………………………………………. 49

6.1 Summary………………………………………………………………………………………………… 49

6.2 Conclusions…………………………………………………………………………………………….. 50

6.3 Recommendations for future research………………………………………………………… 51

References………………………………………………………………………………………………………….. 52

Appendix……………………………………………………………………………………………………………. 54

A.1 Elastomeric bearing…………………………………………………………………………………. 54

A.2 Lead Rubber Bearing………………………………………………………………………………. 59

 

1.3 Research Objective …………………………………………………………………………….. 04

1.4 Tasks ………………………………………………………………………………………………… 04

1.5 Scope of Research ………………………………………………………………………………. 04

1.6 Organization of thesis …………………………………………………………………………. 05

Chapter 2 BACKGROUND …………………………………………………………………………….. 06

2.1 Introduction ……………………………………………………………………………………….. 06

2.2 Two-spring model ………………………………………………………………………………. 07

2.3 Semi-empirical models ……………………………………………………………………….. 08

2.4 Elastomeric Mechanistic Model …………………………………………………………… 09

           

 

 

Chapter 1 INTRODUCTION

1.1 General

Base isolation is used in seismic resistance of structures by introducing flexibility to a structural system and, hence, shifting the structure’s fundamental natural period to a long period range, e.g., 3 to 5 seconds. This is typically achieved by placing isolators at the base of a structure. During an earthquake event this period shift translates into reduced floor acceleration and reduced inter-story drift demands on the superstructure compared to a conventional (non-isolated) structural system. For a design level earthquake event this means that the structure effectively remains elastic. The elastic behavior of the structure also means no ductile deformations or large deformation/displacements which in turn reduces possibility of structural and non-structural damage. This reduced demand in displacements, however, comes at the condition that the high displacement demand at the structure isolator interface can be taken care of. The advantage of the structure remaining elastic for a design level event makes base isolation one of the most efficient seismic resistance techniques.

Base isolation is done by means of bearings. Two most commonly used base isolation bearings are: a) friction pendulum and b) lead rubber. The focus of this research is on lead rubber bearing (LRB). An elastomeric bearing is low damping rubber bearing which consists of alternating layers of rubber and steel shims bonded together. Fig. 1-1 shows a photograph of a section cut, longitudinally, through the height of the bearing exposing the steel shims and rubber layers. The photograph also shows a hole in the middle of the bearing. If a lead core is plugs this hole it is a LRB it is an elastomeric bearing.

 

 

Fig. 1-1. Photograph of an elastomeric bearing (Han and Warn 2014)

Numerous mathematical models (refer to Chapter 2 for details) have been proposed to simulate behavior of elastomeric and LRB subjected to simultaneous lateral displacement and vertical compressive load, which is a typical state of the bearing under a structure in an earthquake event. LRB models, so developed, are semi-empirical in nature and also lump the plasticity or the hysteretic element in an element that both translates and rotates, however, this does not accurately reflect the kinematics of lead rubber bearings in base isolated structures. Hence, due to these limitations, a mechanistic model development for lead rubber bearings is undertaken in the present study.

1.2 Problem Statement

The typical condition of an elastomeric bearing being subjected to vertical compressive load and increasing lateral displacement simultaneously leads to a state where the shear force can pass through a critical point (Fig. 1-2). The bearing, beyond this critical displacement and critical lateral force exhibits, negative tangential horizontal stiffness and a condition of unstable equilibrium. This behavior can have important implications on the behavior of a building as well the stability of a bearing. The current study focuses on simulating this critical response of a LRB using a mechanics based model.

 

Fig.1-2 Critical lateral force response of a bearing

Experiments have demonstrated this bearing response (Sanchez et al. 2014). Semiempirical bearing models (Nagarajaiah and Ferrell 1999; Izuka 2000; Yamamoto et al. 2009; Kikuchi et al. 2010) have been shown to simulate the influence of compressive vertical load and lateral displacement on the lateral force response with reasonable accuracy. This approach gives good agreement with the experiment but is not of much use to a design engineer as it involves performing experiments on the bearing in order to calibrate empirical parameters that will enable design and analysis of a bearing.

A recent model proposed by Han and Warn (2014) to simulate the behavior of elastomeric bearings, which utilizes a set of parallel vertical springs and a bilinear constitutive relationship to represent the moment rotational relationship of elastomeric bearings, is adopted and further extended to simulate the behavior of lead-rubber bearings.

1.3 Research Objective

The primary objective of this study is to develop a model that will be able to simulate the lateral force-displacement behavior of LRBs. The approach adopted to achieve this objective is to utilize a mechanics based model proposed by Han and Warn (2014).

1.4 Tasks

1. The model proposed by Han and Warn (2014) employed with some

modifications is re-derived and solved using a Newton type numerical scheme to obtain the equilibrium path solution.

2. A Bouc-wen hysteretic element is added to the Han and Warn bearing model to simulate the energy dissipated by the lead-core.
3. Simulations are performed with the re-derived elastomeric bearing model and new lead-rubber bearing model to generate data that is compared with experimental data from Sanchez et al. (2013) to assess the capabilities of the two models for replicating the monotonic force-displacement response under different vertical force levels.
4. Simulations and results are presented for cyclic loading on LRB.
   • Scope of Research

The scope of present research is limited to the bearing models being subjected to shear force, lateral displacement and moments. Boundary condition at the top of bearing is free to translate horizontally but restrained against rotation. Simulations of bearing behavior for cyclic and monotonic lateral displacement conditions on one type of elastomeric and LRB each (refer to section 5.1 for details of bearings) are performed.

• Organization of thesis

This report contains six Chapters. A review and description of past models developed for bearings is presented in Chapter 2. Following which the development of Han and Warn (2014) elastomeric bearing model with some modifications is presented in Chapter 3. Chapter 4 contains the addition of Bouc-Wen element to the elastomeric bearing model and details the complete development of the LRB model. Results of lateral force response simulations of the bearing models (elastomeric and LRB) are evaluated against experimental response in Chapter 5. Chapter 6 presents the main conclusions from this study and also provides recommendations for future work.  A list of references and an

appendix is also included.

 

MECHANISTIC MODEL FOR LEAD RUBBER BEARINGS

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