Full-Scale Testing, Modelling and Analysis of Light-Frame

Full-Scale Testing, Modelling and Analysis

of Light-Frame Structures Under

Lateral Loading

Phillip J. Paevere

A thesis submitted in total fulfilment

of the requirements of the degree of Doctor of Philosophy

February 2002

Department of Civil and Environmental Engineering

The University of Melbourne

Abstract

i

Abstract

ii

Abstract

The differing needs and expectations of building owners, users and society are driving

a change towards a technology-intensive, performance-based approach to the design

and evaluation of light-frame structures. A critical underlying assumption of the

performance-based philosophy is that performance can be predicted with reasonable

accuracy and consistency. Development of improved performance prediction

technologies, for light-frame structures, requires a detailed understanding of the

structural behaviour of light-frame buildings, as well as the environmental loadings to

which they are subjected during their lifetime. Full-scale structural testing in the

laboratory, combined with analytical modelling, are essential in obtaining this

understanding.

This thesis presents the results of experimental and analytical investigations into the

performance of light-frame structures under lateral loading. The specific objectives of

this research are to: 1) develop simple, experimentally validated numerical models of

light-frame structures, which can be used to predict their performance under lateral

loads, particularly seismic loads; and 2) collect experimental data suitable for

validation of detailed finite-element models of light-frame structures.

To meet these objectives, a series of full-scale experiments were conducted on a

North American style single-storey L-shaped timber-frame house. In these

experiments, the distribution of the reaction forces underneath the walls, and the

displaced shape of the house were measured in detail under static and static-cyclic

lateral loading. The natural frequencies of the house were also determined using

dynamic impact tests.

A range of analytical models were developed and validated against the experimental

results. All of the analytical models incorporate pinching, degrading hysteretic

elements, which can accurately simulate the force-displacement, and energy

dissipating characteristics of light-frame structural components under cyclic lateral

loading. System identification methodologies were developed to determine the

Abstract

iii

hysteresis parameters from the experimental results, and to facilitate linking of the

different models. The analytical models were then used to conduct deterministic and

stochastic response analyses of light-frame structures under earthquake loading, and

to examine the response sensitivity to the model type, and the assumed structural

period and damping.

The experimental results have provided the most detailed picture of the reaction

forces beneath a non-symmetrical light-frame structure, under lateral loading, ever

recorded. They have shown that there is potential for significant sharing and

redistribution of applied lateral load, between the external shear-walls of a light-frame

house, through the roof and ceiling diaphragm, under both elastic and inelastic

response conditions. The most commonly used design techniques for lateral load

distribution in light-frame structures do not allow for such sharing of load between the

walls.

The analytical modelling results have shown that relatively simple modelling

strategies can be used to simulate the lateral load-sharing characteristics that were

observed in the experiments with reasonable accuracy. The importance of the

interaction between the shear-walls, and the effect of this interaction on seismic

performance prediction was also highlighted by the modelling work. The results

showed that the single-storey test house is highly unlikely to collapse under

earthquake loading due to direct shaking, even for a large event, but could sustain

significant damage. It was demonstrated that the inherent uncertainties due to the

random nature of seismic excitation, and the assumptions used in the modelling

process, have a significant effect on the predicted performance of light-frame

structures.

Statement

iv

Statement

This thesis comprises only the author’s original work, except where due

acknowledgement is made in the text. To the best of the author’s knowledge, this

thesis contains no material previously published or written by another person, except

where due reference is made in the text, or indicated in the Preface of this thesis. This

thesis contains no material which has been accepted for the award of any other degree

or diploma in any university or other institution. This thesis is less than 100,000

words in length, exclusive of figures, tables, references and appendices.

Phillip Paevere

September 2001

Acknowledgements

v

Acknowledgements

It is a great pleasure to acknowledge the valuable contributions of the many people

who have helped to make this project possible. Firstly, thankyou to my supervisors,

Dr. Greg Foliente (CSIRO), and Associate Professor Nick Haritos (The University of

Melbourne) for their guidance, patience and encouragement, which has been

invaluable.

Thank you also, to the following people for their assistance, advice and support

relating to the experimental work: Jay Crandell of the NAHB Research Center, for

providing the test house details and advice on various aspects of the experimental

program; Dr. Emad Gad from The University of Melbourne for his assistance in the

dynamic house testing; Lyndon Macindoe and Rod Banks of CSIRO, for the design of

the experimental instrumentation systems; Craig Seath of CSIRO for his work in

building the house; and the CSIRO workshop team for their expert assistance in

constructing various laboratory equipment.

I would also like to express my appreciation to the CSIRO, the NAHB Research

Center, and The University of Melbourne for their financial and in-kind support for

this project.

Finally, I would like to thank my wife Bridget, and our three children, Patrick, Lydia

and Jasper for the joy and inspiration they have provided during the course of my PhD

studies.

Table of Contents

vi

Table of Contents

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

Statement....................................................................................iv

Acknowledgements.....................................................................v

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

List of Tables ........................................................................... xii

List of Figures ..........................................................................xiv

Notation .....................................................................................xx

Abbreviations..........................................................................xxv

Preface ....................................................................................xxvi

CHAPTER 1 Introduction and Overview ..............................1

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

1.2 A Framework for Combined Experimental and Analytical

Analysis of Light-Frame Structures .......................................................... 2

1.3 A Framework for Improving Design Procedures..................................... 5

1.4 Performance-Based Design and Evaluation ............................................. 6

1.5 Overview of Common Design Procedures used for Lateral Load

Distribution .................................................................................................. 7

1.6 Summary of Research Needs and Opportunities ................................... 10

1.7 Project Objectives and Scope ................................................................... 11

1.7.1 Overall Objectives of Research Program................................................ 11

1.7.2 Specific Objectives and Scope of the Work Presented ........................... 12

1.8 The CUREE Caltech Woodframe Project .............................................. 13

Table of Contents

vii

1.9 Thesis Overview......................................................................................... 14

1.9.1 Structure of Literature Review................................................................ 14

1.9.2 Structure of Thesis .................................................................................. 15

CHAPTER 2 Experiment Description and Results.............17

2.1 Introduction ............................................................................................... 17

2.2 Overview of Light-Frame Testing............................................................ 19

2.2.1 Introduction ............................................................................................. 19

2.2.2 General Behaviour of Light-Frame Systems and Components............... 20

2.2.3 Whole Building Testing of Light-Frame Structures ............................... 22

2.2.4 Summary of Research Needs and Opportunities – Light-Frame

Testing..................................................................................................... 26

2.3 Experiment Description............................................................................ 32

2.3.1 Background ............................................................................................. 32

2.3.2 Testing Program ...................................................................................... 33

2.3.3 Description of the Test House................................................................. 34

2.3.4 Load and Reaction Measurement............................................................ 44

2.3.5 Displacement Measurement .................................................................... 47

2.3.6 Data Management ................................................................................... 47

2.3.7 Documentation ........................................................................................ 51

2.4 Elastic Testing............................................................................................ 51

2.4.1 Introduction ............................................................................................. 51

2.4.2 Test Description ...................................................................................... 51

2.4.3 Elastic Testing Results ............................................................................ 52

2.5 Dynamic Impact Testing........................................................................... 67

2.5.1 Experiment Description........................................................................... 67

2.5.2 Experiment Results and Discussion ........................................................ 68

2.5.3 Comments on Experiment....................................................................... 70

2.6 Destructive Testing.................................................................................... 75

2.6.1 Introduction ............................................................................................. 75

2.6.2 Loading Mechanism................................................................................ 75

2.6.3 Global Hysteresis Response .................................................................... 78

Table of Contents

viii

2.6.4 Individual Wall Hysteresis Responses .................................................... 80

2.6.5 Ceiling Diaphragm Hysteresis Responses............................................... 87

2.6.6 Displaced Shapes..................................................................................... 88

2.6.7 Load Distribution .................................................................................... 89

2.6.8 Damage Status......................................................................................... 90

2.6.9 Comments on Experiment..................................................................... 107

2.7 Conclusions .............................................................................................. 108

2.7.1 Elastic Testing ....................................................................................... 109

2.7.2 Dynamic Impact Testing ....................................................................... 110

2.7.3 Destructive Testing ............................................................................... 111

2.7.4 Recommendations for Further Research ............................................... 112

CHAPTER 3 Hysteresis Modelling .....................................115

3.1 Introduction ............................................................................................. 115

3.2 Overview of Hysteresis Modelling ......................................................... 116

3.2.1 Introduction ........................................................................................... 116

3.2.2 Phenomenological Hysteresis Models .................................................. 116

3.2.3 Phenomenological Versus Mechanics-Based Hysteresis Models......... 127

3.2.4 Sensitivity of Response to Hysteresis Modelling.................................. 131

3.3 Overview of System Identification......................................................... 136

3.3.1 Introduction ........................................................................................... 136

3.3.2 System Identification Methodology...................................................... 138

3.4 Differential Hysteresis Model Formulation .......................................... 139

3.4.1 Introduction ........................................................................................... 139

3.4.2 Model Formulation................................................................................ 140

3.5 System Identification of Hysteresis Parameters ................................... 143

3.5.1 Introduction ........................................................................................... 143

3.5.2 System Identification for a Range of Different Systems....................... 144

3.5.3 Parallel System Identification ............................................................... 146

3.5.4 Identification of Hysteresis Parameters for Whole-Building Test

Data ....................................................................................................... 148

3.6 Summary and Conclusions..................................................................... 167

Table of Contents

ix

CHAPTER 4 Structural Modelling.....................................169

4.1 Introduction ............................................................................................. 169

4.2 Overview of Seismic Response Analysis Techniques and

Structural Modelling of Light-Frame Structures ................................ 171

4.2.1 Background ........................................................................................... 171

4.2.2 Static Analysis....................................................................................... 173

4.2.3 Pushover Analysis ................................................................................. 174

4.2.4 Time-History Analysis .......................................................................... 175

4.2.5 Response Spectrum Analysis ................................................................ 177

4.2.6 Monte Carlo Simulation........................................................................ 179

4.2.7 Random Vibration Analysis and Equivalent Linearisation................... 180

4.2.8 Whole-Building Models of Light-Frame Structures. ............................ 183

4.2.9 Scope of Structural Modelling and Seismic Response Analysis in

Current Work......................................................................................... 187

4.3 SDOF model............................................................................................. 189

4.3.1 Background ........................................................................................... 189

4.3.2 Formulation ........................................................................................... 189

4.3.3 Equivalent Linearisation of SDOF model............................................. 190

4.3.4 Extension of SDOF Model to MDOF Systems..................................... 191

4.4 Hysteretic Shear-Building Model .......................................................... 191

4.4.1 Background ........................................................................................... 191

4.4.2 Matrix Formulation ............................................................................... 192

4.3.4 State Vector Formulation ...................................................................... 196

4.4.4 Equivalent Linearisation of Hysteretic Shear-Building Model............. 197

4.5 Hysteretic Shear-Wall Model ................................................................. 202

4.5.1 Background ........................................................................................... 202

4.5.2 Formulation ........................................................................................... 203

4.6 Finite Element Model.............................................................................. 205

4.7 Hybrid Response Analysis ...................................................................... 208

4.7.1 Background ........................................................................................... 208

4.7.2 Formulation ........................................................................................... 209


Table of Contents

x

CHAPTER 5 Seismic Response Analysis ...........................213

5.1 Introduction ............................................................................................. 213

5.2 Ground Motions ...................................................................................... 214

5.2.1 Introduction ........................................................................................... 214

5.2.2 CUREE Ground Motions ...................................................................... 214

5.2.3 SAC Suite of Ground Motions.............................................................. 215

5.3 Deterministic Seismic Response Analyses Using SDOF and Shear-

Building Models....................................................................................... 221

5.3.1 Introduction ........................................................................................... 221

5.3.2 Response Analysis of Test House Using SDOF Model........................ 221

5.3.3 Response of Three-Storey Building Using Shear-Building Model....... 243

5.4 Stochastic Response Analyses Using Equivalent Linearisation .......... 249

5.4.1 Introduction ........................................................................................... 249

5.4.2 Single-Storey Timber-Frame House ..................................................... 250

5.4.3 Three-Storey Timber-Frame Building .................................................. 255

5.5 Seismic Response Analyses using Hysteretic Shear-Wall Model........ 259

5.5.1 Introduction ........................................................................................... 259

5.5.2 Shear-Wall Model Details..................................................................... 260

5.5.3 Comparison Between Shear-Wall Model and Experimental

Responses .............................................................................................. 264

5.5.4 Comparison Between Shear-Wall Model and SDOF Model

Responses .............................................................................................. 266

5.5.5 Comparison Between Shear-Wall Model Response Under Uni-

Directional and Bi-Directional Excitations ........................................... 270

5.5.6 Analysis of Seismic Demands on Individual Walls Under Bi-

Directional Earthquakes ........................................................................ 270


5.6.1 Sensitivity Study of Single-Storey House Using SDOF Model............ 278

5.6.2 Response Analysis of Example Three-Storey Building Using Shear-

Building Model ..................................................................................... 279

5.6.3 Stochastic Response Analysis Using Equivalent Linearisation ............ 280

5.6.4 Response Analysis of Test House Using Shear-Wall Model................ 281

Table of Contents

xi

CHAPTER 6 Summary, Conclusions and

Recommendations ..................................................................285

6.1 Key Findings ............................................................................................ 285

6.2 Detailed Summary and Conclusions...................................................... 287

6.2.1 Introduction ........................................................................................... 287

6.2.2 Full-Scale Experiments on L-Shaped Test-House ................................ 287

6.2.3 Hysteresis Modelling and System Identification .................................. 289

6.2.4 Structural Modelling ............................................................................. 290

6.2.5 Seismic Response Analysis................................................................... 291

6.3 Recommendations for Further Research .............................................. 295

CHAPTER 7 References ......................................................299

APPENDIX A Summary of Full-Scale Elastic Testing

Results .....................................................................................315

APPENDIX B Summary of Full-Scale Destructive

Testing Results........................................................................331

APPENDIX C Generalised Reduced Gradient

Algorithm ................................................................................361

APPENDIX D Equivalent Linearisation Coefficients.......363

APPENDIX E Publications Arising From Research.........367

List of Tables

xii

List of Tables

Table 2.1 – Summary of full-scale experiments on timber structures (Fischer et

al. 2001)............................................................................................................... 28

Table 2.2 – Summary of full-scale experiments on light-frame structures. ................ 29

Table 2.3 – Summary of test house construction and materials.................................. 37

Table 2.4 – Summary of elastic testing program. ....................................................... 56

Table 2.5 – Results summary for elastic tests. ............................................................ 64

Table 2.6 – Load-sharing in elastic tests under single point load. ............................. 67

Table 2.7 – Initial in-plane stiffness and capacity characteristics of whole house,

and separate wall systems. .................................................................................. 85

Table 2.8 – Results summary for selected displacement cycles of destructive test. ... 86

Table 2.9 – Damage observations at different stages during the destructive test. .... 103

Table 2.10 – Structural damage status of structural sub-systems after final load

cycle of destructive test ..................................................................................... 104

Table 3.1 – Description of system properties and hysteresis model parameters...... 142

Table 3.2 – Fitted hysteresis parameters for various pinching, degrading

structural systems. ............................................................................................. 151

Table 3.3 – Fitted hysteresis parameters, using parallel system identification for

two different shear-wall systems....................................................................... 156

Table 3.4 – Fitted hysteresis parameters for L-shaped test-house and its

individual wall sub-systems. ............................................................................. 159

Table 4.1 Analysis capabilities for different modelling strategies........................... 188

Table 5.1 – CUREE ground motions: Set of 20 ordinary ground motions with

10% probability of exceedance in 50 years....................................................... 216

Table 5.2 – CUREE ground motions: Set of 6 near-fault ground motions with

2% probability of exceedance in 50 years......................................................... 217

Table 5.3 – SAC ground motions for Los Angeles with 10% probability of

exceedance in 50 years. ..................................................................................... 218

List of Tables

xiii





Table 5.6 – Statistics of displacement demand predictions from SDOF model of

test house, under SAC and CUREE ground motions. ....................................... 226

Table 5.7 – Median displacement demands for SDOF models with different

periods and strengths under SAC and CUREE ground motions....................... 238

Table 5.8 – 90th percentile displacement demands for SDOF models with

different periods and strengths under SAC and CUREE ground motions. ....... 239

Table 5.9 – Statistics of inter-storey displacement demand predictions from

shear-building model of example three-storey timber building, under SAC

and CUREE ground motions............................................................................. 245

Table 5.10 – Natural frequencies and associated Rayleigh damping coefficients

for hysteretic shear-wall model. ........................................................................ 262

List of Figures

xiv

List of Figures

Figure 1.1 – Framework for experimental and modelling studies (Foliente 1997a)......3

Figure 1.2 – Framework for development of improved design procedures

(Foliente, 1998) ......................................................................................................5

Figure 2.1 – Typical two-storey light-frame construction (NAHBRC, 2000). ............30

Figure 2.2 – Typical hysteresis data from tests of components in light-frame

construction. .........................................................................................................31

Figure 2.3 – Details of test house. ................................................................................38

Figure 2.4 – Photographs of test house during construction. .......................................43

Figure 2.5 – Metal straps embedded in the top plate of walls W1-W4........................44

Figure 2.6 – Photographs of load cell system. .............................................................46

Figure 2.7 – Photographs of displacement measurement system.................................48

Figure 2.8 – Ceiling level displacement gauge locations.............................................49

Figure 2.9 – Loading mechanisms used in elastic testing. ...........................................50

Figure 2.10 – Horizontal and vertical distribution of self-weight for test house. ........57

Figure 2.11 – Displaced shape and reaction forces (N) for elastic test 3. ....................58

Figure 2.12 – Displaced shape and reaction forces (N) for elastic test 5. ...................60

Figure 2.13 – Displaced shape and reaction forces (N) for elastic test 6. ...................62

Figure 2.14 – Results summary for elastic test 12. .....................................................66

Figure 2.15 – Equipment used in dynamic impact tests...............................................71

Figure 2.16 – Excitation and accelerometer locations for dynamic impact testing. ....71

Figure 2.17 – Example acceleration time histories (three repeats) and power

spectra from non-destructive dynamic impact tests. ............................................72

Figure 2.18 – Normalised sum of the power spectra from dynamic impact tests. .......74

Figure 2.19 – Photographs of load application system for destructive test..................77

Figure 2.20 – Displacement-based loading protocol used in the destructive test. .......78

Figure 2.21 – Global hysteresis response of whole house in the North-South

direction................................................................................................................81

Figure 2.22 – Backbone of global hysteresis response of whole house in the

North-South direction...........................................................................................81

List of Figures

xv

Figure 2.23 – Hysteresis response of wall systems in North-South and East-West

directions. .............................................................................................................82


directions, plotted on same scale..........................................................................83

Figure 2.25 – Comparison of wall W3 and W4 hysteresis responses for initial

load cycles of destructive test...............................................................................84

Figure 2.26 – Diagram showing rigid-body rotation, and racking distortion of

section of roof and ceiling diaphragm over W2, W3 and W4, for selected

displacement levels. .............................................................................................91

Figure 2.27 – Approximate hysteretic behaviour of roof and ceiling diaphragm

over walls W2, W3 and W4. ................................................................................91

Figure 2.28 – Displaced shape of house perimeter and undeformed edge at

different stages of destructive test (loading as shown in diagram at bottom). .....92

Figure 2.29 – Distribution and magnitude of X-direction reaction forces (kN) at

different stages of destructive test. .......................................................................94

Figure 2.30 – Distribution and magnitude of Y-direction reaction forces (kN) at


Figure 2.31 – Distribution and magnitude of Z-direction reaction forces (kN) at


Figure 2.32 – Percentage of X-direction reaction taken by each wall sub-system

during destructive test. .......................................................................................100

Figure 2.33 – Percentage of X-direction reaction taken by cross-walls during

destructive test....................................................................................................101

Figure 2.34 – Maximum uplift force in each wall during destructive test. ................102

Figure 2.35 – Photos of damaged house after destructive test. ..................................105

Figure 3.1 – Examples of piece-wise linear hysteresis models (Loh and Ho,

1990)...................................................................................................................119

Figure 3.2 – Comparison of idealised pinched hysteretic system with energy

equivalent and displacement equivalent elasto-plastic systems. ........................120

Figure 3.3 – Model by Stewart (1987). ......................................................................121

Figure 3.4 – PWL model from Sivaselvan and Reinhorn (1999)...............................121

Figure 3.5 – Model by Dolan (1989)..........................................................................123

Figure 3.6 – Model by Kasal and Xu (1997)..............................................................123

Figure 3.7 – Model by Mostaghel (1999). .................................................................124

List of Figures

xvi

Figure 3.8 – Model by Deam (2000)..........................................................................124

Figure 3.9 – Possible hysteresis shapes of the basic Bouc-Wen model for n=1

(Baber, 1980)......................................................................................................126

Figure 3.10 – Diagram showing hybrid approach to response analysis using both

mechanics-based and phenomenological models (Foliente et al., 1998b). ........130

Figure 3.11 – Hysteresis pinching and degradation effects on reliability

estimation. ..........................................................................................................134

Figure 3.12 – SDOF hysteretic structural system. .....................................................141

Figure 3.13 – Experimental and fitted hysteresis for a timber framed shear-wall

without blocking [experimental data from Karacabeyli and Ceccotti (1998)]...152

Figure 3.14 – Experimental and fitted hysteresis for a light-gauge steel-framed

house with plasterboard lining [experimental data from Gad (1997)]. ..............153

Figure 3.15 – Experimental and fitted hysteresis for a pre-cast concrete wall to

slab connection [experimental data from Robinson et al. (1999)]. ....................154

Figure 3.16 – Experimental and fitted hysteresis for a one-room Japanese-style

post and beam house – [experimental data from Watanabe et al. (1998)]. ........155

Figure 3.17 – Parallel system identification example 1 ............................................157

Figure 3.18 – Parallel system identification example 2 .............................................158

Figure 3.19 – Experimental and fitted hysteresis of test house..................................160

Figure 3.20 – Experimental and fitted hysteresis of wall W1 from test house. .........161






Figure 4.1 – Spectral densities of wind and earthquake loads, compared with

natural frequencies of common structures (Ferry-Borges and Castanheta,

1971)...................................................................................................................172

Figure 4.2 – Capacity Spectrum method (Chopra and Goel, 1999) ...........................176

Figure 4.3 – Response Spectrum method (Chopra, 1995). ........................................178

Figure 4.4 – Equivalent Linearisation method ...........................................................182

Figure 4.5 – Shear-building model.............................................................................193

Figure 4.6 – Shear-wall model ...................................................................................204

Figure 4.7 – FE model of house .................................................................................207

List of Figures

xvii

Figure 4.8 – Hybrid Response Analysis (Kasal et al., 1999) .....................................211

Figure 5.1 – Details of SDOF model used for single-storey test house sensitivity

study. ..................................................................................................................222

Figure 5.2 – Displacement demands for SDOF model (T=0.129 sec) for different

levels of assumed equivalent viscous damping, under 10/50 and 2/50

CUREE and SAC earthquakes. ..........................................................................224

Figure 5.3 – Displacement demands from SDOF model (T=0.147 sec) for

different levels of assumed equivalent viscous damping, under 10/50 and

2/50 CUREE and SAC earthquakes. ..................................................................225

Figure 5.4 – Displacement demand predictions from SDOF model (with 2%

damping) under 10/50 and 2/50 CUREE and SAC ground motions.................230





Figure 5.7 – Example hysteretic responses of SDOF model under selected

ground motions...................................................................................................233

Figure 5.8 – Comparison of median displacement demands for different assumed

equivalent viscous damping levels.....................................................................234

Figure 5.9 – Comparison of 90th percentile displacement demands for different

assumed equivalent viscous damping levels. .....................................................236

Figure 5.10 – Comparison of median displacement demands for SAC and

CUREE ground motions.....................................................................................241

Figure 5.11 – Details of shear-building model for example three-storey building. ...245

Figure 5.12 – Inter-storey displacement demand predictions for example three-

storey building under 10/50 and 2/50 CUREE and SAC earthquakes. ..............246

Figure 5.13 – Hysteretic responses of example three-storey building under

nr94rrs ground motion........................................................................................247

Figure 5.14 – Comparison of maximum drift ratio predictions for SDOF and

shear-building models, for example three-storey building under 10/50 and

2/50 SAC and CUREE ground motions.............................................................248

Figure 5.15 – Comparison of response statistics calculated using Monte-Carlo

simulation, and Equivalent Linearisation under stationary white noise

excitation (max = 0.5g) for SDOF model of test house. ...................................252

List of Figures

xviii

Figure 5.16 – Example of hysteretic response of SDOF model under stationary

white noise excitation (max = 0.5g). ..................................................................254

Figure 5.17 – Comparison of peak and standard deviation of SDOF model

responses under first 20 seconds of CUREE ground motions. ..........................254

Figure 5.18 – Comparison of response statistics calculated using Monte-Carlo

Simulation, and Equivalent Linearisation under stationary white noise

excitation (max = 0.5g) for example three-storey building................................256

Figure 5.19 – Example of hysteretic response of shear-building model under

stationary white noise excitation (max = 0.5g). .................................................258

Figure 5.20 – Details of shear-wall model for single-storey test house. ....................263

Figure 5.21 – Comparison of model prediction and experimental results for

distribution of in-plane reaction forces in walls W1-W4 under static-cyclic

loading................................................................................................................266

Figure 5.22 – Comparison between displacement demand prediction from shear-

wall model and SDOF model under 10/50 and 2/50 CUREE and SAC

ground motions...................................................................................................268

Figure 5.23 – Comparison of averaged response from walls W1 to W4 in shear-

wall model with response from SDOF model, for selected ground motions. ....269

Figure 5.24 – Comparison between displacement demand predictions, averaged

for walls W1 to W4, under bi-directional and uni-directional SAC ground

motions. ..............................................................................................................271

Figure 5.25 – Displacement demand for individual walls, under bi-directional

SAC ground motions..........................................................................................274

Figure 5.26 – Predicted in-plane hysteresis responses of individual walls under

bi-directional la32 ground motion......................................................................275

Figure 5.27 – Distribution of load to individual walls under la32 ground motion. ...276

Figure 5.28 – Median of displacement demands from shear-wall model, for

individual walls, under bi-directional SAC ground motions..............................277

Figure 5.29 – 90th percentile of displacement demands from shear-wall model,

for individual walls, under bi-directional SAC ground motions........................277

List of Figures

xix

Notation

xx

Notation

The following variables and symbols are used throughout thesis.

A = parameter that regulates ultimate hysteretic restoring force

1a = Rayleigh damping coefficient

2a = Rayleigh damping coefficient

B = matrix of the expected values of the products of the forcing

functions and the response vectors

C = the damping matrix of a MDOF system

Ci = expectations required to compute 3eC

3eC = linearisation coefficient of 2y

c = viscous damping coefficient

ci = viscous damping coefficient of ith mass

E(•) = expected value

error = error between experimentally and model determined hysteresis

used in system identification

erf(•) = error function

erfc(•) = complementary error function

exp(•) = exponential function

uE = sub-matrix of covariance matrix, S

uE� = sub-matrix of covariance matrix, S

zE = sub-matrix of covariance matrix, S

uzE = sub-matrix of covariance matrix, S

uzE�

= sub-matrix of covariance matrix, S

uuE� = sub-matrix of covariance matrix, S

Eexp = experimentally determined energy dissipation in system

identification error function

Notation

xxi

Emod = model calculated energy dissipation in system identification

error function

F = fundamental frequency of vibration (Hz)

f(t) = mass-normalised forcing function

F(t) = forcing function

Fu = maximum load

f = forcing function vector

F = a general vector of time-varying actions

Fexp = experimentally determined restoring force in system


Fmod = hysteresis model calculated restoring force in system


G = matrix that contains the coefficients of Y

g = acceleration due to gravity (9.8 m/s2)

Hα = matrix that contains the hysteretic elements

ihα = hysteretic coefficient of ith element

h(z) = pinching function

ˆ{ }I = influence vector

I = identity matrix

IGL(•) = generalised Gauss-Laguerre quadrature

Isn = standard sin integral

Isum = standard summation

k = stiffness

Kα = linear part of the stiffness matrix

3eK = linearisation coefficient of 3y

Ki = expectations required to compute 3eK

ikα = linear spring coefficient of ith element

m = mass

mi = mass of ith element

M = the mass matrix of a MDOF system

n = parameter which controls the 'sharpness' of yield

Notation

xxii

q = pinching parameter that controls the percentage of ultimate

restoring force zu where pinching occurs

Qi = total restoring force of the ith mass

p = pinching parameter that controls the initial drop in slope

R = a general differential operator

[ ( ), ( ); ]R u t z t t = non-damping restoring force of non-linear system

r = total number of lumped masses in MDOF model

S = zero-mean time lag covariance matrix

S� = time derivative of S

0S = power spectral density of white noise excitation (m2/s)

sgn(•) = signum function

T = fundamental period of vibration (seconds)

t, to, tf = time

u = displacement

u� = velocity

u�� = acceleration

iu = relative displacement of ith mass

iu� = relative velocity of ith mass

iu�� = relative acceleration of ith mass

U = a general vector of system response

We = weighting factor for energy error in system identification error

function

Wf = weighting factor for force error in system identification error

function

{ }X = vector of global displacements

{ }X� = vector of global velocities

{ }X�� = vector of global accelerations

{ }Y = vector of global response

{ }Y� = time derivative of{ }Y

ix = displacement of ith mass with respect to ground

Notation

xxiii

ix� = velocity of ith mass with respect to ground

ix�� = acceleration of ith mass with respect to ground

gx�� = ground acceleration

Aix�� = absolute acceleration

1,iy = iu – ith displacement element of vector {Y}

2,iy = iu� - ith velocity element of vector {Y}

3,iy = iz - ith hysteretic displacement element of vector {Y}

1,iy� = time derivative of 1,iy



{Z} = vector of the hysteretic components of the displacements

z = hysteretic component of the displacement

z� = time derivative of z

zi = hysteretic displacement of ith mass

iz� = time derivative of zi

zu = ultimate value of hysteretic displacement

α = ratio of linear to non-linear contribution to restoring force

β = hysteresis shape parameter

γ = hysteresis shape parameter

( )Γ • = Gamma function

i∆ = constants for expected values needed by 3eC and 3eK

ijδ = Kronecker delta

νδ = strength degradation parameter

ηδ = stiffness degradation parameter

δψ = pinching parameter that controls the rate of change of ζ 2

ε = calculated energy dissipation

1ζ = controls the severity of pinching

2ζ = controls the rate of pinching

ζ s = parameter that indicates degree of pinching

Notation

xxiv

η = stiffness degradation

oθ = integration limit of snI

λ = pinching parameter that controls the rate of change of ζ 2 as ζ1

changes

εµ = mean ofε

1ξµ = mean of 1ξ

2ξµ = mean of 2ξ

ηµ = mean ofη

νµ = mean ofν

1* 2*,µ µ = linearisation coefficient constants

ν = strength degradation

ξ ,ξo = viscous damping ratio

23ρ = correlation coefficient of 2y ( or u� ) and 3y (or z)

1* 2*,σ σ = linearisation coefficient constants

1,uσ σ = standard deviation of u ( or 1y )

2,uσ σ�

= standard deviation of u� ( or 2y )

3,zσ σ = standard deviation of z ( or 3y )

ψ o = pinching parameter that contributes to the amount of pinching

ω 0 = natural frequency of linear system = k m/

Abbreviations

xxv

Abbreviations

The following abbreviations are used throughout this thesis.

CSIRO = Commonwealth Scientific and Industrial Research Organisation

CUREE = Consortium of Universities for Research in Earthquake Engineering

DE = distributed element

DOF = degree-of-freedom

EQL = Equivalent Linearisation

FE = Finite Element

GRG = Generalised Reduced Gradient

MCS = Monte-Carlo simulation

MDOF = multi-degree-of-freedom

NAHB = National Association of Home Builders

ODE = ordinary differential equation

PWL = piece-wise linear

PWNL = piece-wise non-linear

RVA = Random Vibration Analysis

SDOF = single-degree-of-freedom

US = United States

USA = United States of America

Preface

xxvi

Preface

Three refereed conference papers have been produced throughout the course of this

research and are reproduced in the Appendix. One journal paper is currently under

review. These publications are listed below.

Chapter 3 has in part been presented in the following:

• Paevere, P. J. and G. C. Foliente. 1999. "Hysteretic Pinching and Degradation

Effects on Dynamic Response and Reliability." Pp. 771-79 in Proceedings of

the Eighth International Conference on the Application of Statistics and

Probability. Sydney, 12-15 December. Ed. R. E. Melchers and M. G. Stewart.

A.A. Balkema, Rotterdam.

Chapter 4 has in part been presented in the following:

• Paevere, P. J., G. C. Foliente, and N. H. Haritos. 1998. "On Finding an

Optimum MDOF Inelastic System Model for Dynamic Reliability Analysis."

Pp. 215-222 in Proceedings of the Australasian Conference on Structural

Optimisation. Sydney, Australia, February 11-13. Ed. G. P. Steven, O. M.

Querin, H. Guan, and Xie Y. M. Oxbridge Press, Victoria, Australia.

• Paevere, P. J., N. H. Haritos, and G. C. Foliente. 1998. "A Hysteretic MDOF

Model for Dynamic Analysis of Offshore Towers." Pp. 513-17 in Proceedings

of the Eighth International Offshore and Polar Engineering Conference.

Montreal, Canada, May 24-29.

Currently under review:

• Paevere, P. J., G. C. Foliente, and B. Kasal. "Load Distribution and Load-

Sharing Mechanisms in a One-story Woodframe Building.", Paper submitted

for publication in the Journal of Structural Engineering, ASCE. First

Submitted, September 2001.

Preface

xxvii

Chapter 1 – Introduction and Overview

1

CHAPTER 1

Introduction and Overview

1.1 Background

Many people in the world live and work in light-frame buildings. They rely on these

structures for protection against natural disasters such as tropical cyclones and

earthquakes, and have an inherent expectation of their safety. Although light-frame

buildings have anecdotally performed well in natural disasters, many have also

suffered extensive damage, causing financial ruin and social upheaval. In the 1994

Northridge Earthquake, for example, twenty four deaths, and financial losses of more

than US$20 billion occurred as a result of damage in light-frame construction. This

represented most of the fatalities, and more than half of the total damage bill (Office

of Emergency Services, 1995; Kircher et al., 1997).

Until quite recently, modern technology has played only a small role in the design and

construction of light-frame buildings, particularly those made from timber. The

construction techniques have been developed over long periods of time, based mainly

on tradition and experience. In other, relatively modern forms of construction, such

as pre-stressed concrete, technology has been applied extensively, and forms the basis

of the design philosophy and methodology.

Inevitably, societal expectations will dictate that the design and evaluation of light-

frame structures also incorporates state-of-the-art technology, to ensure adequate and

consistent structural performance in natural disasters, and so that structural

performance can be targeted to suit differing user needs and expectations. Such

technology is a necessary prerequisite for a ‘performance-based’ approach to

development of new building products and systems, and will lead to enhanced

innovation and trade in the light-frame construction industry.


2

Development of improved performance prediction technologies requires a detailed

understanding of the structural behaviour of light-frame buildings, as well as the

environmental loadings to which they are subjected during their lifetime. Structural

testing in the laboratory, field measurements and disaster surveys combined with

analytical modelling are essential in obtaining this understanding.

1.2 A Framework for Combined Experimental and

Analytical Analysis of Light-Frame Structures

A light-frame building is an assemblage of several components or sub-assemblies

with repetitive members such as walls, floors and roof systems connected by inter-

component connections such as bolts, metal straps or proprietary connectors forming

a three-dimensional highly indeterminate structural system. Because very little is

known about how the load is shared and distributed in such a complex structural

system, gross simplifying assumptions must be made in structural evaluation and

design. This can result in either over- or under-strength elements being present in a

structure, resulting in either over-conservative (and therefore uneconomical) or less

safe structures.

Figure 1.1 shows a framework for analytical and experimental studies of structural

performance of light-frame buildings. Most analytical and experimental work found in

the literature focuses on understanding the response behaviour and mechanisms at the

sub-system level, such as shear-walls – Level 1 in Figure 1.1 (Foliente and Zacher

1994). Recommendations arising from studies at this level typically ignore system

effects, and the fact that the actual forces that the sub-system experiences, depends

primarily on the geometric and structural characteristics of the whole building – Level

2 in Figure 1.1. For example, Level 1 tests do not always take into account the effect

of boundary conditions on the results. Boughton’s (1988) summary of tests on full-

scale houses (Level 2) and isolated wall components (Level 1) demonstrated that

boundary conditions in the wall test ‘can influence the stiffness, ultimate load and

failure characteristics of components under test’.


3

Whole building

Subassemblies(e.g., walls, floors)

Elements(e.g., steel, timber, panel

sheathing)

Inter-componentconnections

Simple joints andfasteners

Level 1 Relations

Level 2 Relations

Figure 1.1 – Framework for experimental and modelling studies (Foliente 1997a).

He also concluded that the commonly accepted assumptions regarding load-sharing

and ductility were invalid, and in some cases un-conservative, but that structural

redundancies which were usually ignored, unintentionally compensated for this.

Full-scale whole house testing is needed to properly understand the system behaviour

and to prove the validity of isolated component test results and their interpretations

(Boughton 1988; Foliente and Zacher 1994).

Analytical modelling strategies which are consistent with the framework in Figure 1.1

are also needed to extend the usefulness of the experimental data, and predict

structural behaviour of the whole building, its subassemblies and element-level

components under real environmental loadings such as tropical cyclones and

earthquakes. Ideally, the analytical models should incorporate the physical

relationships between individual element and whole-building response and vice-versa.

To achieve this, a ‘hybrid’ modelling approach, which uses a range of linked

analytical models, is preferable to a single ‘monolithic’ model.

A detailed ‘monolithic’ modelling approach may be suitable for analysing whole-

building and corresponding component response under simple deterministic loads, but

may not be appropriate when complex random loads such earthquakes need to be

considered. Building responses under these types of loads are inherently non-linear

and random in nature, and are best quantified using Monte-Carlo simulation (MCS),


4

or similar techniques. These methods are extremely computationally intensive and are

not suitable for ‘monolithic’ structural models with a large number of degrees-of-

freedom (DOF).

Under a ‘hybrid’ strategy, the worst-case output from a simple (i.e. few DOF) global

response prediction model, based on many repeated analyses, can be used as the input

to a more detailed model which can then determine individual element responses.

Conversely, the output from a detailed element-level model, can be used to determine

the properties of a simple global model (in a way analogous to physical testing),

which can then be used to derive global response statistics, based on many repeated

analyses.

The hybrid modelling approach enables response predictions to be made which

include system effects, linkages between element-level and global response, and

consideration of the variability in environmental loading. The link between these

models is critical, so that they can be used in tandem to span across analysis domains

(i.e. global to local, simple to complex, deterministic to probabilistic). This linkage is

best facilitated using system identification, where the model parameters for the simple

model are determined in a systematic fashion, from the output of the complex model,

and vice versa.

Because analytical models are generally much simpler than the real-world systems

they represent, experimental testing is, in turn, needed in the development, refinement

and validation of the analytical models. Different types of models require different

types and densities of response data for the purposes of validation. Unfortunately,

there have been relatively few experimental studies on whole-building response which

have provided the right kind and/or amount of data needed to fully develop and

validate models which cater for both element-level and global response. Some

advances towards this end have been made in studies conducted by Phillips et al.

(1993), Kasal et al. (1994), Gad (1997) and Fischer et al. (2001). These are reviewed

in Chapters 2 and 4.


5

1.3 A Framework for Improving Design Procedures

Analytical and experimental studies of the structural performance of light-frame

buildings are crucial in the development of design procedures. Figure 1.2 shows an

ideal way of developing improved design methods, both for engineered construction

and conventional, non-engineered (or deemed-to-comply) construction. Improvements

are made based not only on calibration of design methods to field observations and

historical performance (e.g., Crandell and McKee 2000), but also on improved

building performance models and state-of-the-art analysis techniques (right-most

block in Figure 1.2).

“Conventional” Construction Engineered Construction Analysis Based on “FirstPrinciples”

• Prescriptive or deemed tocomply provisions

• For simple building types andshapes

• Simplified guidelines

• Span tables and charts

• Diagrams and figures of requiredconstruction details

• Code-specified loads andmaterial properties

• Engineering calculations(equations, tables and diagrams)

• Element-by-element design

• Historical performance

• Loads: realistic representation

• Structure type: element/joint,subsystem, whole building

• Analytical models: lumped mass,frame or component, FiniteElement

• Analysis: static, dynamic,reliability

C U R R E N T A P P R O A C H I D E A L A P P R O A C H

Figure 1.2 – Framework for development of improved design procedures (Foliente, 1998).

Currently, light-frame timber buildings in the USA may be non-engineered

(commonly called ‘conventional’ construction), fully engineered or mixed (i.e.,

combined conventional and engineered construction) (Cobeen 1997; Foliente 1998).

For the most part, conventional construction provisions have little or no direct

relations with engineered design provisions. Combined conventional and engineered

construction, which is increasingly practiced in California and other US states, results

in significant variations in design practice even in the same locality (Cobeen 1997).

In Australia (which has a similar style of light-frame timber construction to the USA),

deemed-to-comply provisions for light-frame timber construction are given in a set of

span tables and supporting specifications for various members of the house. Used


6

widely by builders throughout Australia, including regions with high winds, these

provisions have been developed based on accepted engineering design standards

(MacKenzie 2000) – the centre and left blocks in the diagram in Figure 1.2 show this

development process. This is a rational approach, and has also been applied (although

to a lesser extent, compared to the Australian practice) in the development of the

Wood Frame Construction Manual for one- and two-family dwellings in high wind

areas in the south-eastern USA (AFPA 1995).

The ‘ideal approach’ in Figure 1.2 provides for faster and cheaper development of

innovative building products by minimising testing requirements and field trials, and

also provides the opportunity to increase trade between regions with different building

practices, by demonstrating equivalent performance of regionally differing

construction systems. This approach is consistent with the shift towards technology-

intensive ‘performance-based’ design procedures, which are gaining wide acceptance.

1.4 Performance-Based Design and Evaluation

During recent years, there has been a shift towards ‘performance-based’ design and

assessment of engineering structures such as buildings and bridges. Currently, most

of the design and evaluation of these structures is based on prescriptive engineering

design methods (i.e. centre block of Figure 1.2), which are contained in building

codes and standards. The basic idea behind the performance approach is that a

structure is assessed in terms of compliance with specific performance criteria, rather

than compliance with a set of generalised prescriptive design rules. The advantage of

the performance-based approach is that the designer is freed from the constraints of

overly-generalised prescriptive rules, allowing for more innovative design solutions.

Foliente (2000), proposes that ‘the performance approach is, in essence, the practice

of thinking and working in terms of ends rather than means’.

One of the key drivers behind the move towards performance-based design of

buildings, is the increasingly diverse needs of building owners, users and society. The

‘one size fits all’ approach, which underpins most prescriptive methods, is not

appropriate in a modern and increasingly sophisticated world. In the context of the


7

design and evaluation light-frame structures, a more flexible performance-based

approach seems ideal, since the majority of light-frame structures are residential

dwellings. The residential sector of the construction industry has by far the largest

diversity of owner/occupier expectations and requirements, and would therefore

benefit most through its implementation into practice. The performance approach is

consistent with the ‘ideal approach’ shown in the framework for development of

improved design procedures in Figure 1.2. However, the critical underlying

assumption behind the performance-based philosophy, is that performance can be

predicted with consistency and accuracy – this is a key challenge which needs to be

addressed before performance-based design can be truly beneficial.

Another challenge in implementing true performance-based design in practice, is the

development of probabilistic analysis techniques and performance criteria.

Environmental loadings on structures are random in nature, and hence they are best

described in probabilistic terms. Structural response to these loads is also random,

and as such, any performance criteria relating to the response levels, are also best

described probabilistically.

State-of-the-art technologies, including whole-structure testing and modelling, and

probabilistic analysis techniques must be utilised to understand the behaviour of light-

frame structures, if accurate performance prediction under natural disaster loading is

to be achieved. The development of experimentally validated analytical models of

light-frame structures is essential in working towards this goal.

1.5 Overview of Common Design Procedures used

for Lateral Load Distribution

Design of light-frame structures to resist wind and earthquake loads involves the

design of the structure’s lateral force resisting system. This usually consists of a

system of shear walls and diaphragms which are connected together to transfer the

loads to the building’s foundations. The design procedure can be summarised as

follows (NAHBRC, 2000):


8

1) Determine building geometry and layout of walls, floors and roof.

2) Calculate an equivalent wind or earthquake lateral design load.

3) Distribute the lateral design load to the shear-walls via the floor and roof

systems.

4) Determine the shear-wall and diaphragm requirements to resist the calculated

proportion of the applied load (from step 3).

5) Determine, hold-down and inter-component connection, and special detailing

requirements.

Because light-frame structures are highly indeterminate three-dimensional systems,

crude simplifying assumptions are usually made in determining the distribution of the

applied lateral loads (i.e. step 3 above). This is problematic, because the design of

individual sub-systems and components are highly dependent on the accuracy and

reliability of the lateral force distribution method used. This step in the analysis and

design process is critical, because if one chooses an inappropriate method of

distributing these loads, any subsequent effort spent on detailed engineering design of

individual components may be wasted. Thus it is possible to have an ‘engineered’

house that is potentially less safe than the one that has not been ‘engineered’, if the

load distribution is not estimated correctly.

The three most commonly used lateral force distribution methods are: 1) Tributary

Area; 2) Total Shear; and 3) Relative Stiffness. These are discussed in detail in the

Residential Structural Design Guide: 2000 Edition (NAHBRC, 2000) and are

described briefly in the following.

Tributary Area Method

The Tributary Area method is the most commonly used lateral force distribution

method, and is based on an assumption that the horizontal diaphragms are flexible

compared to the walls. Under this assumption, the lateral forces are distributed in

proportion to the tributary area or mass associated with the shear walls, rather than

their stiffness. The method is analogous to a series of flexible beams on rigid

supports, with varying line loads. It has been shown that this approach can lead to

inaccurate results for certain plan configurations (Kasal and Leichti 1992) and can


9

lead to both conservative and non-conservative results. This model does not consider

the effect of stiffness on load distribution and is based on assumptions, which are

rarely met in practice.

Total Shear Method

This method is the second most popular, and requires minimal calculations. The

method dictates that the total shear resistance of all the walls in a given storey must

add up to the total applied shear resulting from the design load. The distribution of the

shear resistance is left up to the judgement of the designer. The distribution should be

determined such that a desirable response (such as minimum torsion under earthquake

loading) will result. The reliability of this method depends totally on the judgement

of the designer, and could result in poor performance if the load distribution is not

estimated with reasonable accuracy.

Relative Stiffness Method

This approach is the converse of the tributary area approach. It is based on an

assumption that the horizontal diaphragms are rigid compared to the walls. Under this

assumption, the lateral forces are distributed to the walls in proportion to their

stiffness rather than their associated tributary area. If the rotation of the building is

considered, then the method is analogous to a rigid beam on elastic springs, and some

re-distribution of the applied load, due to the torsional response can be

accommodated. This method is conceptually more precise than the other two, and is

the only method which considers torsion and the associated re-distribution of the

loads within the system, but still represents a very rough approximation to the actual

structural behaviour.

Comments

These three methods are completely different in their underlying philosophy, and as

such, can produce quite different results for identical buildings under identical loads.

If performance-based methods are to be successfully applied in the design and

assessment of light-frame structures, more accurate and reliable lateral force


10

distribution methods need to be developed, based on a detailed understanding of the

structural behaviour. This understanding is best facilitated through the development

of experimentally validated analytical models of light-frame structures.

1.6 Summary of Research Needs and Opportunities

The regulatory shift towards technology-intensive performance-based design

procedures, and the differing needs and expectations of building owners, users and

society, will inevitably drive technological advances in the design and evaluation of

light-frame structures. Development of reasonably accurate performance prediction

technologies requires a detailed understanding of the structural behaviour of light-

frame buildings, which can only be achieved through combined structural testing in

the laboratory and analytical modelling in both the deterministic and probabilistic

domains.

Key research areas which are critical to understanding the behaviour of light-frame

systems and the development of better performance prediction capabilities include:

• Development of a range of analytical models for light-frame structures, which

consider system effects, the load paths within the structure, the links between

component and global responses and the inherent variability in environmental

loads.

• Experimental validation of analytical models through whole-building tests and

tests on components and sub-assemblies. Distribution of applied lateral load

within full-scale structures needs to be measured, so that analytical models

which are capable of predicting the load distribution can be validated.

• Linking of analytical models using system identification, so that local to

global, simple to complex, and deterministic to probabilistic analysis domains

can be crossed.


11

1.7 Project Objectives and Scope

1.7.1 Overall Objectives of Research Program

The long-term vision for this project is to make light-frame structures more affordable

and safer for people than they currently are through appropriate efficiency in the

structural system. It has been highlighted that a path towards this goal is the

development of tools and procedures that can be used to establish the structural

performance with reasonable accuracy, and the performance criteria for light-frame

buildings, and to optimise their construction in some manner, (i.e. find the most

economical design for a given performance level). Following this path, the objective

of this project is to develop experimentally validated numerical models of light-frame

structures, which are capable of accurately predicting structural performance, and can

be used to improve current design procedures.

It is anticipated that the project findings will contribute to the basic understanding of

structural behaviour of light-frame buildings under lateral loads, and to the

development of safer and more affordable housing in the future.

To meet the project objectives, a research plan has been formulated in four phases,

with each phase having its own specific aims.

1) Full-Scale Testing

In the first phase of the project, the aim is to conduct tests on a full-scale L-shaped

timber-frame house, and to measure the load distribution and deflected shape in fine

detail. The experiments are designed and conducted for the purposes of validating a

range of numerical models (see Phase 3), and also to examine the load-sharing and

system effects under a range of elastic and inelastic response conditions.

2) Component and sub-assembly testing

The aim in this phase of the project is to determine the characteristics of the

subassemblies, connections and components used in the test house, to facilitate full

validation of a finite-element (FE) model of the house (See Phase 3), and to compare


12

the behaviour of subassemblies when isolated to when they are part of the full

structure.

3) Structural Modelling and Validation

In this phase of the project, the aim is to develop a range of numerical models and

analysis techniques, which can be used to predict the performance of light-frame

structures under seismic loading. The models will be validated against the

experimentally determined responses from Phase 1. Three types of models will be

developed to cover a range of analysis capabilities

• non-linear FE model

• hysteretic shear-wall model

• hysteretic single-degree-of-freedom (SDOF) and shear-building models

System identification techniques will also be developed to systematically determine

model parameters from experimental data, and to facilitate linking of the models, so

that different analysis domains can be crossed.

4) Performance Analysis

The aim in this phase is to examine the performance of typical light-frame structures,

including the house tested in Phase 1, under earthquake loading using deterministic

and stochastic response analyses.

1.7.2 Specific Objectives and Scope of the Work Presented

The work presented in this thesis does not cover every aspect of the overall research

program described in section 1.7.1. The development and validation of the FE model

of the house (part of Phase 3), and the testing of the individual components and sub-

assemblies (all of Phase 2) are the subjects of separate research projects, and are not

presented here. It is important that they are included in the overview and description

of the testing program to give the reader an appreciation of the wider scope of the

project goals.

The specific aims of this PhD project are: 1) to develop simple, experimentally

validated numerical models of light-frame structures, which can be used to predict


13

their performance under seismic loading; and 2) collect experimental data suitable for

validation of more complex non-linear finite-element models of light-frame

structures.

The scope of the current work includes:

• full-scale testing of the house (Phase 1)

• development of hysteretic SDOF, shear-building and shear-wall models with

stochastic response analysis capability (Phase 3)

• development of system identification techniques to facilitate hybrid modelling,

and determination of model parameters from experimental data (Phase 3); and

• seismic performance prediction using hysteretic SDOF, shear-building and

shear-wall models (Phase 4)

1.8 The CUREE Caltech Woodframe Project

In response to the extensive damage caused to light-frame timber construction in the

1994 Northridge Earthquake, an extensive research project was initiated in the USA to

improve the seismic resistance of light-frame timber buildings. The project is being

conducted by the Consortium of Universities for Research in Earthquake Engineering

(CUREE), and is referred to as the CUREE-Caltech Woodframe project. An overview

can be found in Hall (2000). The scope of the CUREE project is broad, and includes

field investigations, full-scale house testing, component testing, analytical modelling

and development of design procedures. This project was still ongoing at the time of

publication of this thesis. Because of the currency of the project, and its direct

relevance to the work presented here, it warrants special mention so as to outline the

key similarities and differences.

The overall goals of the CUREE project, and the goals outlined for this project in

section 1.7, are quite similar, in that both projects aim to improve the performance of

light-frame structures in the long term. The work presented here is focussed only on

laboratory testing and analytical modelling at present.


14

The full-scale testing conducted so far in the CUREE project involved shake-table

testing of a two-storey house. The main objective in this test was to determine the

dynamic characteristics and the seismic performance of the test structure under

various levels of seismic shaking and for different structural configurations. Full

details of the CUREE shake table tests can be found in Fischer et al. (2001). The full-

scale testing in the work for this thesis involves static and static-cyclic testing of a

non-symmetrical single-storey house. The main objective in the testing presented

here was to determine the complete distribution of laterally applied load throughout

the structure in detail (not included in the CUREE test) and to obtain data for the

validation of numerical models, which can predict the load-path, including a FE

model.

The analytical modelling tasks in the CUREE project also have similar goals to those

stated for this project. Modelling of system behaviour is emphasised, based on data

from both system and component tests. The emphasis in the analytical modelling

work presented here in this thesis is also on system behaviour, but perhaps leans more

towards modelling in the probabilistic domain, hybrid strategies and system

identification.

The issues that need to be addressed before reliable performance prediction tools for

light-frame structures become a reality, are numerous and diverse. One of the major

issues is that these structures are complex in nature and come in a huge variety of

configurations. Even a large initiative like the CUREE project can only begin to

address these issues. The experimental and analytical work presented in this thesis

will complement and extend the knowledge gained from the ongoing work of the

CUREE-Caltech Woodframe project, and vice-versa.

1.9 Thesis Overview

1.9.1 Structure of Literature Review

Literature relevant to the project background and the overall project objectives is

reviewed in Chapter 1. Although each of the individual topics that are covered in this


15

thesis are inter-related, they are generally treated as separate topics in the literature,

and hence, are reviewed separately at the beginning of the appropriate chapter.

1.9.2 Structure of Thesis

In Chapter 1, background information, and justification for the research project is

presented. Some key research needs are identified, and overall and specific project

aims are identified. Literature relevant to the overall project objectives is reviewed.

In Chapter 2, the general behaviour of light-frame systems, and full-scale testing of

light-frame structures are reviewed. The experiments conducted on a full-scale L-

shaped house are outlined in detail, including a description of the loading and

response measurement systems. The experimental results are presented and

discussed, and some conclusions based purely on the experiments are drawn.

In Chapter 3, modelling of hysteretic behaviour, and determination of hysteresis

model parameters using system identification are reviewed in the context of light-

frame structures. A differential model of hysteresis appropriate for use in the

structural analysis of light-frame systems is presented. The parameters of the

differential hysteresis model are determined from the experimental results using non-

linear gradient-based system identification. A ‘parallel’ system identification

technique is demonstrated, where a single set of hysteresis parameters is fitted to

multiple data sets simultaneously.

In Chapter 4, seismic response analysis techniques are reviewed. Hysteretic SDOF,

shear-building, and shear-wall models, for the seismic analysis of light-frame

structures are formulated using differential hysteresis elements. An Equivalent

Linearisation (EQL) scheme formulated for determining response statistics of a SDOF

model under random loads, by Foliente et al. (1996), is extended to a multi-degree-of-

freedom (MDOF) shear-building model. FE, and hybrid modelling strategies, for

seismic response analysis are outlined.


16

In Chapter 5, the results of the analytical modelling of the seismic response of the L-

shaped test house and an example three-storey light-frame building are presented.

Two different suites of ground motions which are used in the analyses are discussed

and summarised, then response and sensitivity studies of single- and three-storey

light-frame structures are presented. EQL is used to determine the response statistics

of the single- and three-storey light-frame buildings under white noise excitations, and

the results are compared with those from MCS. Comparisons are made between

predictions from different modelling techniques, and finally the seismic demands on

individual walls of the test house are predicted in terms of response statistics.

Chapter 6 summarises the key conclusions from the research and provides a summary

of the work presented in the thesis.

Appendix A contains an extended summary of the full-scale elastic testing results.

Appendix B contains an extended summary of the destructive testing results.

Appendix C contains a summary of the Reduced-Gradient system identification

algorithm.

Appendix D contains the equations for the linearisation coefficients, which are used in

the statistical equivalent linearisation schemes for the shear-building and SDOF

models.

Appendix E contains the peer-reviewed conference papers produced throughout the

course of the research.

Chapter 2 – Experiment Description and Results

17

CHAPTER 2

Experiment Description and Results

2.1 Introduction

Structural testing in the laboratory is essential for understanding the response of

structural components and systems under different loading conditions. Physical

experiments allow detailed monitoring of the structural behaviour and response under

controlled conditions, and provide information that can be used directly to improve

new and existing products and design tools and procedures. Importantly, laboratory

testing is also crucial in the development, calibration and validation of analytical

models of structures.

This chapter addresses the experimental work conducted in this project. It begins

with a discussion and review on testing of light-frame structures and then outlines and

describes the physical experiments conducted on a full-scale North American style

one-storey L-shaped timber-frame house. A description of the loading and response

measurement systems used in the experiments is given, and the results of the

experiments are summarised and presented. Some conclusions are drawn based purely

on the results of the experiments.

The experimental program for the entire project is broken into five stages:

1. Elastic testing – presented in section 2.4

2. Dynamic impact vibration testing – presented in section 2.5

3. Destructive testing – presented in section 2.6

4. Isolated wall testing – not covered in this thesis

5. Component testing – not covered in this thesis


18

In the ‘elastic’ tests, the distribution of the reaction forces underneath the walls of the

house, and a variety of displaced shapes, under very small lateral loads were

measured. Inelastic behaviour in these tests, and any associated damage to the house,

was minimised by limiting displacements to around 1mm. In the vibration tests, the

fundamental racking mode frequencies of the structure were determined using impact

vibration measurements. In the destructive testing, the distribution of the reaction

forces underneath the walls, and the displaced shape of the house, were measured

under static-cyclic lateral loading up to +/- 120mm.

A major objective of the experiments described here was to collect data for calibration

and validation of a suite of numerical models of the structure, comprising:

• a detailed FE model

• a hysteretic shear-wall model

• lumped mass models (SDOF and shear-building)

Each of these models requires a different density of response data for the purposes of

validation. Hence, the experiment and the instrumentation were designed such that

both global and also very localised response data could be gathered or derived. In the

elastic and destructive tests, 250 separate channels of load and displacement data were

measured providing a detailed picture of the distribution of forces throughout the

structure and the deflected shape of the house during the tests.

The other major objective of the whole-structure testing, was to completely measure

the displaced shape and distribution of the applied loads throughout the structure

under elastic and inelastic response conditions, to examine the amount of load-

sharing, and the load-redistribution mechanisms.

Full details of the test house are given in section 2.3.3


19

2.2 Overview of Light-Frame Testing

2.2.1 Introduction

Studies of structural damage in light-frame structures after natural disasters have been

a primary means of validating and improving design methods. This has proven to be a

very slow and inconsistent process for optimising safety, and generally lacks the

quantitative aspect needed to improve engineering tools. Furthermore, data collected

from these efforts do not provide information on processes leading to failure.

Structural testing in the laboratory or the field is useful because it allows close

monitoring of the structural behaviour and response under controlled loading

conditions, and provides information that can be used directly to improve new and

existing products and design tools and procedures.

Analytical models for analysis of light-frame structural systems are also needed to

extend the usefulness of the experimental data, predict structural behaviour under

specified natural hazard loads and conduct parametric response studies. These models

are useful in research and can assist in the development and/or calibration of code

requirements and design procedures. As highlighted in Chapter 1, models are

generally much simpler than the real-world systems they represent, and experimental

testing is needed for their development, refinement and validation. The integration of

laboratory testing and analytical modelling studies of light-frame buildings is crucial

to provide the necessary technical understanding for the development of rational

design tools and procedures which will ensure the levels of safety and economy which

are demanded by building occupants and society.

Although the inter-dependence of testing and modelling is vitally important, it is

convenient to review them as separate topics, since they are mostly treated as such in

the literature. In the following sections, the general behaviour of light-frame systems,

and full-scale experimentation of light-frame construction is reviewed. Analytical

modelling of the behaviour of light-frame construction is reviewed separately in

Chapters 3 and 4.


20

2.2.2 General Behaviour of Light-Frame Systems and Components

Light-frame structures are assembled from components or sub-assemblies with

repetitive members such as walls, floors and roof systems connected by inter-

component connections such as nails, bolts and metal plate connectors. One common

configuration of a light-frame structure is shown in Figure 2.1. Resistance to lateral

loads is commonly provided by shear-wall or diagonal bracing systems, connected

through horizontal diaphragms such as the roof and floors. These elements can be

connected together in an almost infinite variety of configurations. The detail of the

various connectors and fasteners, and the panelling and materials vary from region to

region, and vary even within the same locality (Cobeen, 1997). Foliente and Zacher

(1994), Barton (1997) and Gad (1997) have reviewed the literature on experimental

testing of light-frame timber and steel structural components. Much of this work has

focussed on the behaviour under reverse cyclic loading, due to the complicated

characteristics of the load-displacement behaviour (or hysteresis) of such systems.

The majority of the available experimental cyclic test data are based on tests of

connections and sub-assemblies such as shear-walls and diaphragms.

A common observation from the sub-assembly tests is that the hysteretic behaviour of

a sub-system is governed by the hysteretic characteristics of its primary connection

(Dowrick 1986). For example, Stewart (1987) and Dolan (1989) reported that the

behaviour of shear-walls is dominated by the nailed sheathing connection. It is

therefore crucially important to characterise the hysteretic behaviour of the primary

connections and/or sub-assemblies in order to characterise the overall behaviour of

light-frame structural systems.

Figure 2.2 shows typical hysteresis data from cyclic tests of connections and

subassemblies in light-frame construction. Figures 2.2 (a), (b) and (c) show the

hysteretic behaviour of a plywood sheathed nailed joint, a light-gauge steel stud wall

with plasterboard lining and a plywood sheathed shear-wall, respectively. Several

characteristic features of the cyclic response of these systems can be noted (Foliente

1995):


21

• a non-linear, inelastic load-displacement relationship with no distinct yield

point

• progressive loss of stiffness with each loading cycle (stiffness degradation)

• degradation of strength when cyclically loaded to the same displacement level

(strength degradation)

• pinched hysteresis loops (i.e. the hysteresis loops appear to have been

‘pinched’ together about the origin).

The observed non-linearity, lack of distinct yield point, and the strength degradation

of the hysteresis of these systems, stems from the combination of material properties

of the individual components that make up the system such as the nails, screws, plate

connectors, timber, steel etc. The observed degradation in stiffness and the pinching

of the hysteresis loops, are a result of the behaviour of the dowel-type fasteners which

connect the elements of the light-frame structure together. These connections loosen

due to the distortion of the fastener holes, and become slack under cyclic loading

resulting in slippage under reversal of the applied load. As the amplitude of the

applied cyclic load or displacement increases, the amount of slippage also increases.

Initial slackness, due to shrinkage or clearances at fastener holes has also been

observed in some tests (Dean et al. 1989).

The response of light-frame systems to cyclic loading is also load-rate dependent (for

timber) and exhibits memory, so that the response of the system at any given point in

time depends on the past force-displacement history (Whale, 1998). Memory in the

hysteresis and load-rate-dependence cause difficulties when determining how to apply

loads and forces in physical experiments on light-frame components. The order and

number of inelastic excursions as well as the rate of load or displacement application

in a cyclic test can significantly alter the behaviour. Several testing protocols have

been proposed and are currently used (ISO 1999, Foliente et al. 1998a, Shepherd

1996). Recently, as part of the CUREE-Caltech Woodframe Project (Hall, 2000),

physical testing protocols have been specifically developed to represent the seismic

demands imposed by earthquakes on timber buildings (Krawinkler et al., 2000).


22

The complex nature of the hysteresis which is typical for light-frame construction,

makes it difficult to analyse or predict the behaviour, particularly under dynamic

loads. In a dynamic analysis, the response of the structure is sensitive to the natural

frequency, which is a function of the structural mass and the stiffness, and on the

strength capacity (Paevere and Foliente, 1999). As can be seen in Figure 2.2, the

stiffness and strength capacity of light-frame components varies considerably,

depending on the previous response. This means that in order to accurately predict

the dynamic behaviour using a numerical model, it must be able to track and vary the

strength and stiffness appropriately in the hysteresis (Chopra and Kan 1973; Iwan and

Gates, 1979). In Chapter 3, numerical modelling of hysteretic behaviour is reviewed

in detail and a differential hysteresis model is presented which can accurately

characterise the complex hysteretic behaviour of light-frame structures and

components.

2.2.3 Whole Building Testing of Light-Frame Structures

Testing of isolated components and sub-assemblies used in light-frame construction,

as described in the previous section, provides an indication of the load and

displacement capacity of the test component. Isolated component testing can also be

used to approximately derive overall structural behaviour when used in conjunction

with analytical models. The results of these tests are limited however, because

boundary conditions, loading mechanisms, and system effects may not be able to be

simulated properly in an isolated component experiment. Assumptions about the

load path(s) which are dominant within the structure must be made, yet load-sharing

and distribution in light-frame construction are not well understood (Boughton, 1988).

These effects can only be studied through full-scale whole-system testing, combined

with analytical modelling.

Testing of full-scale light-frame structures dates back to 1957 in Canada, where one

of the first reported experiments was conducted by Dorey and Schriever (1957), on a

single-storey timber-frame house. Since then the bulk of the tests on light-frame

structures have been conducted in the USA and Australia during the last three

decades. A number of full-scale static and dynamic tests of Japanese-style post-and-


23

beam heavy timber-frame houses have also been conducted in Japan, but these are not

considered to be ‘light-frame’ structures. An extensive review of the Japanese studies

is given in Fischer et al. (2001) and a summary of this review is reproduced in Table

2.1.

In the following, the static and dynamic testing of full-scale light-frame structures is

reviewed. A summary of the experimental studies reviewed herein is given in Table

2.2 Analytical modelling of the whole-building behaviour of light-frame structures is

reviewed in Chapter 4.

Static Testing

Since Dorey and Schriever (1957), static testing of full-scale light-frame structures

has been conducted by Tuomi and McCutcheon (1974), Reardon (1986, 1990),

Stewart et al. (1988), Reardon and Mahendran (1988), Phillips (1990), Reardon and

Henderson (1996), and Richins et al. (2000) amongst others. Many of the studies

have pointed towards the importance of the interactions between the sub-assemblies

of the light-frame building and the load-sharing mechanisms. The effect of the

boundary conditions and the loading mechanisms on the behaviour of individual

components in the whole building are also highlighted in some of the studies.

Tuomi and McCutcheon (1974) conducted static tests on a full-scale single-storey

house. They investigated the racking resistance contributions provided at various

stages of construction, by adding them incrementally. They also examined the racking

resistance under combined wind and snow loading, and the deflection and cracking

behaviour around door and window openings.

Stewart et al. (1988) tested two manufactured homes under simulated wind-loading.

This study was focussed on the effect of transverse walls on racking resistance, and

the interaction between the roof diaphragm and the shear-walls. They concluded that

the roof system was far stiffer than the shear-walls, such that the system could be

appropriately modelled as a stiff beam on elastic foundations.


24

Richins et al. (2000) also conducted a series of structural tests on a typical

manufactured home under simulated design-level wind loads. In this test, distributed

loads were applied using a pressurised air bag on the side of the house, and

concentrated loads were applied at the ceiling level. They measured the global

displacements of the house and reactions in the tie-down straps and concluded that the

racking and slip displacements were small under design-level wind loads.

Phillips (1990) and Phillips et al. (1993) recorded the reaction forces beneath

individual walls under static-cyclic loading, to address the problem of load-sharing

and load-redistribution within a non-linear system. The main conclusions from this

work were: 1) the wooden roof behaved as a rigid diaphragm and contributed

significantly to lateral load-sharing among the shear-walls; 2) up to 20% of the lateral

load was carried by the transverse walls at small load levels; 3) the transverse walls

did not contribute to load-sharing; and 4) the stiffness contributions provided by

individual layers of sheathing were directly additive. These experiments offered

important insight into load-sharing in a full-scale building with regular geometry.

Properties of individual members, connections and sub-assemblies of the

experimental house were established prior to the entire building test. This makes the

test suitable for validation of analytical models (Kasal et al., 1994).

At the James Cook Cyclone Structural Testing Station in Australia, different

configurations of light-frame houses under static loads, representative of high wind

loading, have been tested. The testing has included single-storey timber-frame brick-

veneer houses (Reardon, 1986; Reardon and Mahendran, 1988), a two-storey split-

level timber-frame brick-veneer house (Reardon and Henderson, 1996) and a single-

storey light gauge steel-framed brick-veneer house (Reardon, 1990). The main focus

of all the studies was to examine the response to a design level wind load, including

uplift. Wind-tunnel testing was also used to determine appropriate load distributions

used in the experiments. These studies highlighted the importance of the component

interactions, and the effect of the boundary conditions and non-structural components,

but are only applicable for the small displacement levels induced by wind-loading

conditions simulated in the experiments. Some of these tests also showed that

commonly accepted assumptions regarding load-sharing and ductility were invalid,


25

and in some cases un-conservative, but that structural redundancies which were

usually ignored, unintentionally compensated for this.

Dynamic Testing

Dynamic testing of full-size light-frame structures has been far less common than

static testing, presumably due to the high costs involved. Tests have been conducted

by Suzuki et al. (1996), Gad et al. (1998), Gad (1997) and most recently by Beck et al.

(2001) and Fischer et al. (2001). A summary of these studies is given in Table 2.2.

A number of full-scale dynamic tests of Japanese-style post-and-beam heavy timber-

frame houses have also been conducted in Japan. Tanaka et al. (1998), Kohara and

Miyazawa (1998), Ohashi et al. (1998), Yamaguchi and Minowa (1998) and Seo at al.

(1999) have all conducted dynamic tests, but these are not reviewed here as they are

not ‘light-frame’ construction. These studies are reviewed extensively in Fischer et

al. (2001) and a summary of this review is reproduced in Table 2.1.

Suzuki et al. (1996) and Gad et al. (1998) used an impact hammer to estimate the

natural frequencies and mode shapes of three different wooden houses and cold-

formed steel-framed houses, respectively. None of the reported experiments contained

instrumentation to measure forces between components, making it very difficult to

determine the load distribution within the structure from these results alone. Some

indication of load paths, under elastic response conditions can be derived from the

measured mode shapes. In more recent experiments of this type, Beck et al. (2001)

conducted forced and ambient vibration testing of low-rise light-frame buildings,

combined with analysis of recorded earthquake responses, to determine the natural

period and damping vales. They used their results to determine a more accurate

formula for estimating the fundamental period of timber structures for use in

earthquake design procedures.

Gad (1997) tested a one-room (2.3m x 2.4m) brick-veneer house with a light gauge

steel frame and plasterboard lining. These tests were mainly focussed on the

contribution of the non-structural components to the seismic response, particularly the


26

plasterboard lining. Racking tests, swept frequency dynamic tests and shake-table

tests were performed incrementally as the non-structural components were added to

the frame. Critical components such as brick-ties and plasterboard fasteners were also

tested. In accordance with previous whole structure tests, Gad also found that

boundary conditions imposed on a wall by the surrounding structure have a significant

effect on the lateral load resisting capacity. Analytical modelling based on the

experiments was also undertaken (see Chapter 4).

Most Recently, Fischer et al. (2001) have conducted an extensive program of shake-

table tests at the University of California, San Diego, on a two-storey timber frame

house as part of the CUREE-Caltech Woodframe project. Broad details of this project

are given in Chapter 1. The structural components used in the house were full-scale

although the plan dimensions (4.9m x 6.1m) were smaller than a typical house due to

the size of the shake-table. Their objective was to determine seismic performance

under different levels of seismic shaking and for different structural configurations.

Their results showed that a fully engineered timber-frame house has better seismic

performance than a conventionally constructed house. The tests were conducted

incrementally to assess the contributions of different elements to the response and it

was found that the non-structural wall finishes considerably stiffened the structure and

reduced the response level. Individual material and connection properties were

obtained for the components used in the house, and frequency and damping evaluation

tests of the whole structure were conducted. Significant analytical modelling studies

have also been undertaken (see Chapter 4), based on the experimental results.

2.2.4 Summary of Research Needs and Opportunities – Light-Frame Testing

1. The properties and characteristics of the inter- and intra-component connections

govern the overall behaviour of light-frame structures. Understanding the

hysteretic behaviour of these connections through physical testing and accurate

analytical modelling is therefore essential in order to predict performance of light-

frame structures.


27

2. Simulation of the boundary conditions and loading mechanisms provided by the

surrounding structure are essential when testing light-frame connections and sub-

assemblies such as walls. This understanding can only be achieved through

integrated whole-building and component testing and analytical modelling of the

components and the whole structure.

3. Previous whole-building tests have been mainly focussed on failure observations

and determination of global behaviour. Even though in some cases a lot of

information was collected, these test results alone have limited engineering use.

Experimentally validated mathematical models for the analysis of light-frame

structural systems are needed to extend the usefulness of experimental data and to

perform parametric and sensitivity studies.

4. No detailed conclusions about load path within the structure [with the exception of

Phillips’ (1990) test on a house with regular plan and elevation] can be drawn.

Load-sharing and load-distribution mechanisms need further study, especially on a

house with irregular plan and elevation.

5. Little is known about specific properties of materials, connections and sub-

assemblies used to construct the full-building experimental models. This makes it

difficult to use the information for validation of analytical models. Most testing in

the past was planned and conducted without consideration of analytical modelling

requirements.

6. Full-scale response data of light-frame buildings under cyclic loading need to be

obtained (from laboratory tests and field measurements of instrumented

buildings), compared with component test results and used in validation and

refinement of analytical models.


28

Table 2.1 – Summary of full-scale experiments on timber structures (Fischer et al. 2001).


29

Table 2.2 – Summary of full-scale experiments on light-frame structures.

Reference Test Specimen Loading Focus of Research

Tuomi and McCutcheon

(1974)

Single-storey timber-

frame with plywood

bracing

Earthquake

lateral static-cyclic

Contributions of different structural

elements to racking resistance

Reardon

(1986)


frame with brick-

veneer

Wind

lateral and uplift point

loads

Response under cyclonic design

wind loads

Reardon and

Mahendran

(1988)


frame with brick-

veneer

Wind


loads

Response under design wind loads

Stewart et al.

(1988)

Single-storey

manufactured

Wind

lateral point loading

and pressurised air

bag

Racking resistance provided by

transverse walls and interaction of

roof-diaphragm and walls

Phillips

(1990)


frame with plywood

bracing

Earthquake

lateral load-controlled

static-cyclic loading

Load-sharing, effect of transverse

walls and stiffness contributions of

sheathing layers

Reardon

(1990)

Single-storey light-

gauge steel-framed

with

brick-veneer

Wind


loads

Response under cyclonic design

wind loads

Reardon and

Henderson

(1996)

Two-storey split-level

brick veneer

Wind


loads

Performance of different structural

elements, and response levels under

design wind loads

Williams et al

(2000)

Single-storey

manufactured house

Wind

lateral point loading

and pressurised air

bag

Response under design wind loads

Suzuki et al.

(1996)

Two-storey light-frame

timber houses

Impact

impact hammer and

electric vibrator

Measure natural frequencies and

mode shapes

Gad et al.

(1998)

Light-gauge steel-

framed houses

Impact

impact hammer


mode shapes

Gad

(1998)

One-room light-gauge

steel-framed house

Earthquake

lateral static-cyclic,

swept frequency and

ground motion using

shake-table

Contribution of plasterboard lining to

earthquake resistance

Beck et al.

(2001)

low-rise timber

buildings

ambient vibration

and centrifugal shaker


damping

Fischer et al.

(2001)

Two-storey timber-

frame house with OSB

sheathing

Earthquake

swept frequency and

ground motion using

shake-table

Examine seismic performance under

different levels of seismic shaking for

various building configurations


30

Figure 2.1 – Typical two-storey light-frame construction (NAHBRC, 2000).


31

(a) plywood sheathed nailed joint on wooden stud (Foliente, 1995)

-40

-30

-20

-10

0

10

20

30

-30 -20 -10 0 10 20 30

Displacement (mm)

Load

(kN)

(b) light-gauge steel stud wall with plasterboard sheathing (Gad, 1999)

-20

-10

0

10

20

-200 -100 0 100 200

Displacement (mm) (c) plywood sheathed shear-wall (Kawai, 1998)

Figure 2.2 – Typical hysteresis data from tests of components in light-frame

construction.


32

2.3 Experiment Description

2.3.1 Background

In order to address the research needs outlined in the previous section and in Chapter

1, a program of experiments, on a full-scale L-shaped timber-frame house and its

individual components and sub-assemblies was designed. A major objective of the

experiments was to collect data for calibrating and validating a suite of numerical

models of the structure, comprising:

• a detailed FE model

• a hysteretic shear-wall model

• lumped mass models (SDOF and shear-building)

The other major objective of the whole-structure testing, was to completely measure

the displaced shape and distribution of the applied loads throughout the structure

under elastic and inelastic response conditions, to examine the amount of load-

sharing, and the load-distribution mechanisms.

All testing was carried out at the Commonwealth Scientific & Industrial Research

Organisation (CSIRO) division of Building, Construction and Engineering at their

full-scale testing facility in Melbourne, Australia. This facility is capable of testing

full-scale structures (up to three storeys high) to destruction under static or static-

cyclic loading. The facility has a ceiling height of 12 metres and a reaction wall with

a point load capacity of 100 kN. The main feature of the laboratory is the strong

floor, which has an area of 370 square metres and is fully accessible from underneath,

via the basement. The floor can withstand distributed loading of 48 kPa and point

loading of 500 kN with negligible deflection.

The design and layout of the test house, were chosen to represent a typical ‘stick

frame’ or ‘stud-frame’ single-storey house in North America. The specifications were

jointly developed by the National Association of Home Builders (NAHB) Research

Centre in the USA and researchers at CSIRO and North Carolina State University.

Slight modifications were made to the construction materials used (Australian


33

equivalents to US materials were used) and the floor plan (the size of the rooms were

adjusted slightly to match the strong floor anchorage layout in the CSIRO full-scale

testing facility).

The house consisted of the ‘structural’ elements only (i.e. no finishes such as doors,

windows, trims or cladding were used). The experiment was designed in this way so

that it best satisfied the implicit assumptions which are the basis of the FE modelling

strategy (as outlined in Chapter 4), and so that the behaviour of the main structural

elements would be the basis for any conclusions (see section 2.9). Although finishes

were not included, the house shell was constructed such that it could be clad

externally with vinyl or timber siding on the walls and vinyl, slate or asphalt tiles on

the roof.

2.3.2 Testing Program

The overall experimental testing program comprises five stages:

1 Elastic tests – small point loads were applied laterally, at the ceiling or ridge

level, at various locations in the house to determine the extent and the

distribution of the load within the structure. To avoid irrecoverable damage, the

total house displacement in these tests was targeted to be less than ±1.0mm.

2 Non-destructive dynamic tests – the natural frequencies of the house in the

main racking modes, were obtained by measuring the acceleration response

from dynamic impact tests using an instrumented hammer.

3 Destructive test – a lateral cyclic load was applied statically (in line with the

long direction) until failure, cyclic displacements up to +/- 120mm were

applied.


34

4 Isolated wall tests – in a separate research project, all of the walls in the house

will be tested individually (to destruction) as isolated or free-standing walls.

5 Coupon tests – in a separate research project, the materials and typical details

used in the house will also be tested; these tests will include:

• materials tests

o bending properties of sheathing panels

o modulus of elasticity of all framing materials

• sheathing joints

o plywood to stud

o gypsum to stud

• inter-component connections

o wall to wall (tee junction and corner junction)

o ceiling to wall

o roof truss to wall

Note that the experiments in stages 4 and 5, are the subjects of separate research

projects, and are not presented here. Their purpose is to facilitate full validation of a

three-dimensional non-linear FE model, and to compare the behaviour of

subassemblies when isolated to their behaviour when acting as part of the full

structure. They are included in the description of the testing program to give the

reader an appreciation of the wider goals that the work in this thesis forms a part of.

2.3.3 Description of the Test House

For the purposes of the experiments described here, the house was assumed to be built

on a concrete slab foundation and to consist of the ‘structural’ elements only, and the

plasterboard lining. The following non-structural elements were not installed in the

test house:

• tapes and joint compound in the interior joints between intersecting walls

• tapes and joint compound or cornices between the wall and ceiling lining

• windows and doors and their frames


35

• interior and exterior door and window trims

• exterior wall and roof cladding

The house walls were constructed from Australian Radiata Pine with an average

density (at 12% moisture content) of 550 kg/m3 (Southern Yellow Pine could be

considered the US equivalent). Wall frames were assembled from 90x35mm studs at

400mm centres, without blocking, double 90x35mm top plate and a 90x45mm bottom

plate. Studs were end-nailed to top and bottom plates using 3.05x75mm machine-

driven nails (two per end). Hold-down restraints (i.e. on end studs) were not used in

any of the walls. The bottom plate of all walls was anchored to a grid of load cells

using 12.5mm bolts and plate washers at approximately one-metre spacings.

The internal lining was a 13mm gypsum board (plaster board), laid horizontally, and

attached by self-drilling screws at 300mm spacings. The gypsum lining was not

connected with tape and joint compound in the corners between the ceiling and the

roof panels, or between intersecting walls. However, in-plane joints in the gypsum

lining within the wall and ceiling panels were joined with tape and joint compound.

Wall bracing was provided by 9.5mm external plywood sheathing, attached with

2.87x50mm machine driven nails at 150/300mm spacings (i.e. 150mm on perimeter,

300mm internally). The roof was constructed of pre-fabricated trusses (pine) laid out

without blocking on 600mm spacings, sheathed with 12.5mm plywood and fastened

using 2.87 x 50mm machine driven nails at 150/300mm spacings.

A summary of the house construction and materials is given in Table 2.3. Plan

dimensions of the test house, the coordinate system and the wall notation used are

shown in Figure 2.3 (a), and elevations are shown in Figure 2.3 (b). Overall details of

the house, including a summary of the framing and sheathing are depicted in Figures

2.3 (c) and (d). A summary of the individual wall dimensions and sheathing layout are

shown in Figure 2.3 (e). Photographs of the test house at different stages of

construction are given in Figure 2.4.


36

The four main shear-resisting walls in the house in the North-South direction, walls

W1-W4, were carefully chosen to represent a range of different but typical wall

configurations used in a timber-frame house. Wall W1 represents the front of a

garage, and is a short load-bearing wall with a very large opening, Wall W2 is in

effect a medium length load-bearing wall with a small door opening and two small

window openings. Wall W3 is a long non-load bearing wall, clad with plasterboard on

both sides, with only one narrow doorway opening in the middle. Wall W4 is a long

load bearing wall with both small and large openings. The perimeter walls are

connected to the roof and ceiling system with ‘triple-grip’ type connecters, which join

the top plate of the wall to the bottom chord of the roof trusses. Note that wall W3 is

not structurally connected to the roof and ceiling system, as the roof trusses are

designed to span between the perimeter walls. However the roof trusses are installed

with some light metal brackets (with slotted nail holes) which connect the bottom

chord to the top plate of wall W3 to ensure that the trusses remain ‘in-plane’.

Each of the four main walls was constructed with a 3mm steel strap embedded in the

double 90x35mm top plate, as shown in Figure 2.5 (a). The connection between the

steel strap and the loading mechanism, through a rod-end is shown in Figure 2.5 (b).

The additional strength provided by the steel strap may have had a minor effect on the

behaviour of the top plate during the destructive test. The strap was considered

necessary however, to provide a loading mechanism which would allow applied loads

to be distributed along the length of the walls in both the ‘pull’ and ‘push’ directions,

and to avoid localised failure mechanisms from occurring at the loading points, which

would be likely under the severe loads and displacements which were to be imposed

during the destructive test. Such localised failure mechanisms would render the

destructive experiment meaningless.

The non-symmetrical L-shaped plan of the house was chosen for two reasons: 1) to

extend and complement previous work done by Phillips (1990) on a symmetrical plan

house with different wall types; and 2) to ensure some torsional component to the

response, and hence full engagement of all of the structural sub-systems including the

roof. This is important for obtaining an understanding of the load paths within the

building.


37

Table 2.3 – Summary of test house construction and materials.

Wall Framing: Studs: 90x35mm MGP 10 spaced @ 400mm centres without blocking Lower Plate: 90 x 45mm MGP 10 Top Plate: double 90 x 35mm MGP 10 Header Plates: 190x90 (short span); 290x90 (long span) Lower Plate Anchorage: 12.7mm bolts with plate washers spaced @ approx. 1000mm MGP 10 Characteristic Properties: Average density (at 12% moisture content): 550 kg/m3

Bending (90 x 35mm):16 MPa, Bending (90 x 45mm) 19 MPa Tension Parallel to Grain: 8.9 MPa, Shear: 5.0 MPa Compression parallel to grain: 24 MPa

E: 10 000 MPa (short duration), G: 670 MPa (short duration) Roof Framing: Trusses: Pre-fabricated ‘gang-nailed’ pine trusses spaced @ 600mm centres; no blocking; Truss Connectors: Connected to top plate with pryda ‘triple-grip’ plate connectors Truss Top and Bottom Chords: 90x35mm MGP10 Truss Web Members: 70x35mm MGP10 Plywood Bracing: Walls: 2400 x 1200 x 9.5mm F11 Bracing Ply (Laid Vertically) Roof: 2400 x 1200 x 12.5mm F11 Ply (Laid Horizontally) Complies with AS 2269-1994, Plywood - Structural F11 Plywood Characteristic Properties:

Bending: 11.0 MPa, Tension: 6.6 MPa, Shear 1.80 MPa E:10500 MPa, G: 525 Mpa Gypsum Board Lining: 1200 x 2400 x 13mm Gypsum Board (Laid Horizontally ) Complies with AS 2588-1998, Gypsum Board Gypsum characteristic Properties:

Breaking force perpendicular to wrapped edge: 490 N Breaking force parallel to wrapped edge: 200 N

Minimum nail pull resistance: 270 N Fixing Nails: For use in framing All connections use 3.05 x 75mm machine driven (Senco nail gun) Stud to top and bottom plate: end-nailed x 2 Stud to stud: face nailed at 600mm spacings Sheathing Nails: For use on plywood 2.87 x 50mm machine driven (Senco nail gun) Spaced @ 150mm on perimeter, 300mm internally Sheathing Screws: For use on Gypsum wallboard ‘6 gauge’ x 30mm Needle-point Type 1 Gypsum board Screws (Self-Driven) Spaced @ 300mm on studs and truss bottom chords


38

6.2 3.0

5.1

6.15.1

6.1 W1

W2 W3 W4

W5

W6W7

W8

W9

N

S

EW

N

S

EW

2.0

destructive testloading points

‘push’

‘pull’

X

YZ

X

YZ

(a) Floor plan, coordinate system and wall numbering notation (dimensions in m)

Figure 2.3 – Details of test house.


39

East Elevation

West Elevation

North Elevation

SouthElevation

2.44.0

9.2

11.2

2.4

4.0

Pitch = 22o

Pitch = 28o

(b) Elevations (dimensions in m)

Figure 2.3 (cont’d) – Details of test house.


40

12.5mm ply roof

Pre-Fabricated Pine Roof-Trusses @ 600mm centres

13mm gypsum lining

90x35mm studs @ 450mm centres

Load-cell grid:60 load cells @ 1m centres

9.5mm ply bracing

90x35mm cords70x35mm web

N EW S

N EW S

ZYXZYX

(c) Schematic drawing of test house



41

90x35 studs @ 450 centres(no blocking)

90x45 bottom plate

double 90x35top plate

290x90 header

290x90 garage beam

plywood box beam over garage door

290x90 header

190x90 headers

190x90 header

N E

W S

N E

W S

(d) Wall framing only (dimensions in mm)



42

(e) Wall dimensions (m) and sheathing


gypsumply none

• All dimensions in metres• All walls 2.4m high• All doorways 2.0m high unless indicated• Small windows 1.0m wide x1.2m high• Large window 2.4m wide x 1.2m high• All walls gypsum sheathed

on side not shown

gypsumply none gypsumply none

• All dimensions in metres• All walls 2.4m high• All doorways 2.0m high unless indicated• Small windows 1.0m wide x1.2m high• Large window 2.4m wide x 1.2m high• All walls gypsum sheathed

on side not shown

W4

11.2

1.21.22.4 1.2 2.8

W3

11.2

6.01.24.0

W2

5.1

0.25 0.51.15

0.850.351.0 1.0

W7&6

6.0

2.0 0.4 1.2 2.4

W9

9.2

1.2 1.2 1.23.2 2.4

W8

3.0

0.5 1.1 1.4

W5

7.2

1.2 1.24.8

W1

6.1

0.6 0.64.9

2.1

1.21.2


43

(a) Wall framing only – near completion

(b) Plywood-sheathed walls and roof trusses

(c) Completed house without gypsum board lining

Figure 2.4 – Photographs of test house during construction.


44

(a) Diagram indicating location of embedded steel straps and loading points

(b) Close-up view of load cell connection to embedded strap via

universal joint

Figure 2.5 – Metal straps embedded in the top plate of walls W1-W4.

2.3.4 Load and Reaction Measurement

Reaction forces were measured underneath the bottom plate of the house throughout

the static and destructive experiments, through a grid of 60 load-cell units spaced

approximately one-metre apart. The load cell units were held in place and attached to

the strong floor of the laboratory via a heavy metal frame. The entire grid of load

cells supporting the house is shown in Figure 2.6 (a). The configuration of the three

cells in a typical unit, and the connection to the heavy metal frame are shown in

Figure 2.6 (b). The load cell units were connected to the bottom plate via a steel

channel as shown in Figures 2.6 (c). The pin connection between the steel channel and

the load cell prevents moments being applied to the load cell in the direction of

loading. The entire load cell system had a stiffness of 350 kN/mm in the X direction,

325 kN/mm in the Y direction and 465 kN/mm in the Z direction. This means that the

load cells were around ten times stiffer than the initial stiffness of the whole house in

the direction of loading (see section 2.6). Loads in the actuators (i.e. the applied

loads) were also measured using calibrated load cells, as shown in Figure 2.5 (c).

Each load-cell unit, as shown in Figure 2.6 (b) is manufactured from three identical

shear-beam type load cells, which are connected together and oriented such that each


45

individual load cell measures the load in one principal direction [i.e. X, Y or Z as

shown in Figures 2.6 (b) and 2.3 (a)]. Equilibrium checks were performed at various

stages during their installation, and also during the experiments on the house by

comparing with actuator forces (see results in Appendix A and B), where the load cell

units were generally accurate to within ± 2% of the actuator loads. The load-cell units

were placed as close as possible to the edges of openings where the overturning forces

would be a maximum.

The steel channels which were used to connect the load cell units to the wall bottom

plates were intended to simulate the concrete slab foundation of the house. The

channels were continuous underneath the walls, but not across any of the door

openings or wall intersections. The use of the channels would have had some effect on

the distribution of load to individual load cells, particularly in the vertical direction

due to the redistribution of overturning actions, but it was the only viable option for

measuring the reaction forces under the house in such detail and in three directions.

This was not considered to be a major limitation however, since one of the major

objectives of the experiment was to gather data for calibration of a FE model of the

house. For FE model calibration, the load measurement system can be included in the

model by supporting the model of the house on an equivalent structure. Once the

model is calibrated, the effect of the reaction force measurement system on the

behaviour of the house can be examined and quantified.


46

12

11

10

7

8

9

13 14 15 16 17

33605958

57

56

55

54

53

1

52

51

50

494847

18 19 20

21

22

23

24

25

26

27

28

29

30

61 62 63

3132

42

4

41

4039 38 37 36 35 34

43 44 45 46

W1

W9

W7

W2

W6

W3

W8

W4

W5

2

(a) Grid of load cells supporting entire house

X cellX cellX cellX cell

Y cellY cellY cellY cell

Z cellZ cellZ cellZ cell

X

Y

Z

(b) Typical load cell units

(c) Load cell units and pin connections to supporting channel

Figure 2.6 – Photographs of load cell system.


47

2.3.5 Displacement Measurement

Displacements were measured throughout the static and destructive experiments in

three principal directions. The displacements were measured at the intersection of all

walls at both the top plate level, and also at the extreme corners of the roof. Wall W4

was more extensively instrumented than the other walls during the destructive test,

with diagonal displacements measured across all of the openings. This was done to

provide detailed calibration data for the FE model of the house. The bottom plate

displacements were measured to monitor any slippage or lifting of the bottom plate

from the load cell grid.

Four types of digital displacement gauges were used to measure displacements:

• 300mm digital callipers for large in-plane (X) displacements at the top plate level

• 150mm digital callipers for large out-of-plane (Y) and vertical (Z) displacements

• 25mm digital gauges for all bottom plate displacements and accurate measurement

of small displacements at selected top plate locations

• 450mm analogue potentiometer type gauges for all roof measurement and for

openings on wall W4

•

Photographs of the displacement measurement system are given in Figure 2.7 and the

position of the ceiling-level displacement gauges during the static and destructive

testing is presented in Figure 2.8.

2.3.6 Data Management

All of the load and displacement data generated in the static and destructive tests was

logged on an IBM Compatible PC. Altogether, 250 channels of data were recorded

during the experiments to completely capture the load distribution and displacement

response of the structure. Readings were taken once every two seconds, which was

sufficiently fast, given the slow rates of loading (see section 2.6). All of the load cells

and analogue displacement gauges were connected together by four separate RS485

lines which were then read into the serial port of the PC via RS232 converters.


48

(a) Measurement of bottom

plate displacement

(b) Measurement of large in-plane wall

displacements at top plate

(c) Displacement gauge support frames for

in-plane and out-of-plane top plate

displacements

(d) Attachment of displacement gauge

supports to house support frame

(e) Displacement gauge on roof above W2

(f) Displacement gauges on W4 openings

Figure 2.7 – Photographs of displacement measurement system.


49

W1

W2 W3 W4

W5

W6

W7

W8

W9

X and Y: in-plane wall gauges

X and Y: Roof mounted gauges

Z: uplift gauges

Key:

X

YZ

X and Y: in-plane wall gauges

X and Y: Roof mounted gauges

Z: uplift gauges

Key:

X

YZ

X

YZ

Figure 2.8 – Ceiling level displacement gauge locations.


50

(a) Threaded rod and turnbuckle at walls W1-W4

(b) Cable and turnbuckle at South end of ridge

(c) Portable car jack

Figure 2.9 – Loading mechanisms used in elastic testing.


51

The digital displacement gauges were multiplexed onto eight additional RS232 serial

ports on the computer. The entire instrumentation system, data visualisation and

storage was managed by a Microsoft EXCEL based application running under the

Windows NT4.0 operating system. The actuators used in the destructive test (see

section 2.6) were controlled by a separate PC with specialised controller software.

2.3.7 Documentation

In addition to the instrumentation described above, three fixed video cameras, a

roaming video camera and stills cameras were used to capture images of the damage

at different locations throughout the house during the destructive experiment.

Engineers were observing the damage states throughout the house, and describing it in

front of the video cameras.

2.4 Elastic Testing

2.4.1 Introduction

The primary aim in the elastic testing phase of the experimental program was to

determine the distribution of reaction forces underneath the house for a range of

elastic loading configurations. The measured data is to be used for validation of the

FE model under each of the different loading configurations to ensure that it correctly

predicts different load paths under elastic response conditions. In addition to this, the

measurements are to be used to better understand the system effects and load

distribution in light-frame structures.

2.4.2 Test Description

In this series of tests, small point loads, all less than approximately 10 kN, were

applied at the ceiling level at various places throughout the house and in various

combinations. Loads were also applied at the southern end of the ridge. The complete


52

distribution of the reaction forces underneath the bottom plate, and the displaced

shape of the house were measured in each test. The tests are referred to as ‘elastic’

tests since they were designed to limit the displacements to less than around 1mm, so

as not to induce any significant damage or permanent deformation. However, even

under these small displacement levels, the response is not purely ‘elastic’.

A summary of the tests conducted is given in Table 2.4, and in Appendix A. The

loads were applied manually using three different methods, depending on where the

load was to be applied:

• For loads applied at the top plate of walls W1, W2, W3 and W4, in the North-

South direction, a threaded rod with turnbuckle and 100 kN load cell assembly

was used. The rod was connected to the metal strap embedded in the top plate of

walls W1-W4 (as shown in Figure 2.5) at one end, and to a fixed reaction frame at

the other. Loads could be applied in both the ‘push’ and ‘pull’ directions at these

locations (directions as shown in Figure 2.3 (a)).

• For loads applied at the ridge level, a cable and turnbuckle with a 10 kN load

cell was used. The cable was connected to the ridge of the house via a long piece

of timber, nailed through the plywood sheathing into the top chord of the roof

trusses at the ridge. The other end of the cable was connected to the reaction wall

of the laboratory. Loads could only be applied in the ‘pull’ direction at the ridge.

• For all other locations, the loads were applied using a portable scissor-type car

jack and a one 10 kN load cell. The car jack applied load to the house via a thick

metal bearing plate (to avoid crushing) and a portable reaction frame, which was

bolted to the floor. Loads could only be applied in a pushing manner using this

method

Photographs of the three different loading systems are given in Figure 2.9.

2.4.3 Elastic Testing Results

The first set of measurements in the elastic testing (test 1) were taken to capture the

distribution of the vertical reactions of the house under its own self-weight, without

any applied loading. The results are shown in Figure 2.10 (a). Some of the load cells


53

are showing negative reactions, indicating that the bottom plate is lifting in that

region. This is most likely due to stresses caused when constructing the house and

bolting it to the support channel since checks of the load cells before and after testing,

without the structure present, indicated all cells to be working properly. All load cells

were zeroed before any of the static or destructive tests were conducted, to eliminate

any unrelieved stresses. The total sum of the vertical reaction forces shown in Figure

2.10 (a) is 50.775 kN, hence the mass of the house is 50.775 / 9.8 = 5.18 tonne. This

is consistent with the estimated mass, based on the nominal mass of the construction

materials used in the house. The approximate vertical distribution of the self-weight

is shown in Figure 2.10 (b). The estimated contributions from the roof, ceiling and

wall systems are given, with approximate vertical locations. The location of the

vertical centre of mass is also shown, and is located approximately at the ceiling level

of the house.

After the self-weight had been measured, the load cells were zeroed and the series of

elastic tests outlined in Table 2.4 were conducted. Note that these results do not

include gravity forces since the load-cells were all zeroed after the weight had been

measured. The results for tests 3, 5 and 6 are presented graphically in Figures 2.11,

2.12 and 2.13, respectively. Results for the entire series of tests are summarised in

Table 2.5 and in Appendix A. An example of the results summary, as given in the

appendices is shown in Figure 2.14, for elastic test 12 (see Appendix A for notes on

this type of figure). These load distribution measurements provide the most detailed

picture of the reaction forces beneath a non-symmetrical light-frame structure, under

lateral loading, ever recorded.

Figure 2.11 indicates that significant load-sharing occurs between the main shear-

resisting walls in the house. It can be seen from the distribution of the in-plane (X

direction) reaction forces that a significant amount (50%) of the applied load is

resisted by the non-loaded walls. The vertical reactions (Z direction) are also

distributed to the non-loaded external walls. The applied load is distributed to the

non-loaded external walls mainly through the roof and ceiling diaphragm, and

partially through the transverse-walls, since the X-direction reactions in the

transverse-walls are very small in comparison to the in-plane walls. The applied load

is also distributed to the internal wall W3. This wall is not rigidly connected to the


54

roof and ceiling diaphragm and hence, the load is distributed partially via the

transverse-walls and partially via the non-structural connection between the top plate

and the roof trusses. The in-plane reactions in the external cross-walls (Y direction),

indicate the torsional nature of the response.

Similar behaviour is also true for elastic test 5, shown in Figure 2.12, where the load

is applied to wall W4 only, and elastic test 6, shown in Figure 2.13, where the load is

applied to wall W8 only. This behaviour is generally true for all of the elastic tests. In

Table 2.6, the amount of load resisted by the loaded wall (for tests where only a single

point load is applied) is compared to the amount of load resisted by the rest of the

structure. This indicates that between 19% and 78% of the applied load can be shared

by other parts of the structure under these conditions. As for the examples shown in

Figures 2.11 to 2.13, the applied load in all of these tests is generally distributed to the

non-loaded external walls through the roof and ceiling diaphragm. The amount of

load-sharing depends on the stiffness of the loaded wall, the connection to the roof

and ceiling diaphragm, and the configuration of the surrounding structure. As should

be expected under ‘elastic’ response, the amount of load-sharing is related to the

location of the loaded wall in the structure and the relative stiffness of the loaded wall

compared to the surrounding structure. Generally, the stiffer walls resist a larger

proportion of the applied load. In the test house, wall W3 is the stiffest wall, followed

by wall W4. A summary of the wall stiffness and capacity characteristics is given in

Table 2.7 (see section 2.6).

The amount of load-sharing observed in these elastic experiments, is a lower bound

estimate. This is because the applied loads are very small and hence all of the

structural sub-systems may not be fully engaged. It is shown in section 2.6 on

destructive testing, that the load-sharing increases significantly once the house is

pushed into the inelastic range.

In elastic test 12, which is summarised in Figure 2.14, all four North-South walls (i.e.

walls W1 to W4) were pushed simultaneously until their displacement reached around

1.1mm. Under this loading, the total sum of the X-direction reaction forces (i.e.

applied loads) was 27.5 kN, and no structural damage or permanent deformation was

apparent – all of the house walls essentially responded elastically. A hurricane-level


55

design wind load for the test house is around 30 kN, based on the wind load

provisions as specified in the Residential Structural Design Guide: 2000 Edition

(NAHBRC, 2000) for a 120-mph zone with an exposure ‘B’, which is equivalent to a

suburban exposure condition based on ASCE 7-98 (ASCE, 2000). Given that the

applied load was more than 90% of this design wind-load, and the response of the

structure was essentially elastic, the results of this test indicate that the test-house

should easily withstand this design wind load with minimal or no structural damage.

It is important to highlight that this conclusion is based purely on the lateral wind load

to the walls and does not consider uplift pressure on the roof. It should be noted that

test 12 was not intended to be a ‘wind-load’ test as such, but rather another load case

for model validation. The uniform displaced shape, and the associated load

distribution, are unlikely to be representative of the displacements and load

distribution under a real wind load. It should also be noted that no conclusions about

non-structural damage can be drawn, since the test house did not have trims, paint,

doors, windows etc, installed.


56

Table 2.4 – Summary of elastic testing program.

Test Description* Applied Load (N)

1 Dead Load Test gravity only

2 Load W1 to 0.3t 2780 (W1)

3 Load W2 to 0.5t 4790 (W2)

4 Load W3 to 0.5t 4920 (W3)

5 Load W4 to 0.5t 4920 (W4)

6 Load W8 to 0.5t 5183 (W8)

7 Load W8 to 0.5t, W2 to 0.5t 5134 (W8), -5070 (W2)

8 Load W5 to 0.5t 5160 (W5)

9 Load W5 to 0.5t, W4 to 0.8t 5170 (W5), -8040 (W4)

10 Load W2 & W5 to –0.5t, 0.3t -5120 (W2), 3184 (W5)

11 Load between W2 & W3 to 0.8t 6795 (btw W2, W3)

12 Push W1, W2, W3, & W4 1.1mm

1096 (W1), 5431 (W2), 15000 (W3), 6500 (W4)

13 Load roof ridge (5 deg. west) -5088 (ridge @ 5 deg. west)

14 Load roof ridge (10deg. west) -5208 (ridge @ 10 deg. west)

15 Load roof ridge (20 deg. east) -2797 (ridge @ 20 deg. west)

* t = tonne


57

722

1115

-1040

1962

1765

2505

1932

235

991473

14581351

1373

-73

907719

-673

900722

1328

1983

153

1423

289361

616525

800

-430

674399

1364

291

822

1472

1299

569 1431

2509

2277

1617

1607

2519

1031

976

2321

415

168334

1484

-1

792

90

-3

-1

Total Self Weight = 50.775 kN

(a) Measured horizontal distribution of vertical reactions under self-weight (in N).

Roof = 22.3 kN

Ceiling = 7.5 kN

Walls = 21 kN

0.8 m

1.0 m

1.4 m

Vertical Centre of Mass

0.8 m

2.35 m

(b) Approximate vertical distribution of self-weight (in kN).

Figure 2.10 – Horizontal and vertical distribution of self-weight for test house.


58

48

117

9177-1

16

87

20

-7

719

14

399393341

322333285

229

16 -26

86

2

6874

73

8480

8769

83

8890

75

85 7

-43

7

98105

103110

106104

9168

9180

13

17

27

21

4

Figure 2.11 – Displaced shape and reaction forces (N) for elastic test 3.

X Reactions

Displaced Shape

4790 N

Y

XZ


59

18-5

19-26

-149

-132

-105-117

-79

-74

40

5562

-24

59-28

-6-2

-5

46

160 117122

104

134

71

1-1

0-4

-4-3

-13

0-27 -82

-40-33

-256

6-4

12

-2-14

0

2-9

-69

-87

-19

-21-10

93

7495

-98

441

-476

-38

121 54 -10

40

74

112

-194

-15

175121

-520

494

1491

734

-223

61

-466

-328

-220

-111-111-85

-47

-7

70

111-55

-54

-25

4

42

69

110 13

-12

-45-52

-249

8244

-96

-49

245

-93

289

-206

212219

83

-29

-68-10

-270

-91-22

Figure 2.11(cont’d) -Displaced shape and reaction forces (N) for elastic test 3.

Z Reactions

Y Reactions

Y

XZ


60

1028

18103

33

-2

1

35

-22

33

7472

6461

665843

41

5-1

6

7988

87

9891

10684

106

115118

106

117

14

4

21

347330294

332328299

282208

306269

1-6

48

012

8

-158


Displaced Shape

X Reactions

4920 N

Y

XZ


61

40

-315

69

72

7861

8483

12

1718 -10

-1-4

-1-1

-212

-50-55

-53 -80 -101

-50

1-1

-1

-2-2

-20

01

14

68

-8

-13-8

-31

17-10

1

71-29

6

1

24

7784

-15

-184

-103-90

-78

-17

106

-75

95125 55

459

57

70

-44

-829

19

-192

5620

2

149-52

-2

-173

-92

-66

-47-73-110

-68

-18

78

128-83

-74

-31

-5

40

75

112103

-47-34

-19

-362

473

187

-231

-106

751

-320

668

-674

169191198

-112-93

-37

-420-231-150

Figure 2.12 (cont’d) -Displaced shape and reaction forces (N) for elastic test 5.

Z Reactions

Y Reactions

Y

XZ


62

20

47

-8-7

4

63 -4

-6

-5

-6-7

-3-2

9

1588

11

20

8

2 -8

-2

3-1-8

-10-11

-10-12

-15-25

-8

-7-8

-8-9

1

2

-13

12

19

202115

189

20

1915

-4

-8

-28

9

-2

02

1


Displaced Shape

X Reactions

5183 N

Y

XZ


63

37-8

-730

149141

134132

141135

146

242 226158

12-1

10

-1

529

8984

82

91

114

112

0-3

1

44

8946-1-6

1

36

118

257

199

130

91

-1-4

-3

33

16960

-5

20

113141

518549

71

90

7785

303191

63

161159 4348 -18

-4-58

22845

-180-131

1385711

0-34

-60-176

127 31-1

-14-3914

3-2

-4-6

-83

11775

478

-8-30

26

-185

166327

-165

-32

-12-55

-41

71

-623

-452

-182

-138-1269

-342

1025

-225

-175-3116

Figure 2.13 (cont’d) – Displaced shape and reaction forces (N) for elastic test 6.

Z Reactions

Y Reactions

Y

XZ


64

Table 2.5 – Results summary for elastic tests.

Test Measurement W1 W2 W3 W4 W5 W678 W9

in-plane displacement (mm) 1.03 0.28 0.03 0.06 0.02 0.00 0.13

X reaction (N): sum = 2861 610 1046 674 461 19 23 29

Y reaction:(N): sum = -45 35 29 -7 -7 624 487 -1206 2

Uplift:max (N) 861 174 73 159 170 266 10


X reaction (N): sum = 4676 333 2303 957 955 12 64 52

Y reaction:(N): sum = -15 6 29 -28 -14 900 -15 -893 3

Uplift:max (N) 476 520 85 249 466 194 10


X reaction (N): sum = 4886 53 305 3751 490 86 56 144

Y reaction:(N): sum = 80 7 3 -21 1 80 -29 39 4

Uplift:max (N) 73 88 375 107 170 153 0


X reaction (N): sum = 4835 67 438 1195 2995 16 73 51

Y reaction:(N): sum = 45 16 -6 4 -15 -610 -20 676 5

Uplift:max (N) 75 192 110 674 420 112 0


X reaction (N): sum = 117 52 78 -132 169 -2 -34 -13

Y reaction:(N): sum = 5004 52 45 219 278 711 2496 1203 6

Uplift:max (N) 0 176 83 623 175 342 126


X reaction (N): sum = -4957 -312 -2313 -1287 -838 -34 -93 -81

Y reaction:(N): sum = 5033 49 10 241 304 -268 2590 2107 7

Uplift:max (N) 246 773 156 683 0 372 348


X reaction (N): sum = 29 90 468 72 -587 -8 -9 3

Y reaction:(N): sum = 3795 21 61 137 50 2784 731 12 8

Uplift:max (N) 77 41 77 408 339 106 32


65

Table 2.5 (cont’d) – Results summary for elastic tests.

Test Measurement W1 W2 W3 W4 W5 W678 W9


X reaction (N): sum = -7857 -39 -318 -1317 -5994 -12 -122 -55

Y reaction:(N): sum = 4961 -7 94 168 75 4801 819 -990 9

Uplift:max (N) 122 209 141 1417 149 107 417


X reaction (N): sum = -5189 -284 -2061 -1505 -1157 -50 -74 -58

Y reaction:(N): sum = 3212 20 -7 66 62 1668 411 993 10

Uplift:max (N) 185 747 165 339 62 202 223


X reaction (N): sum = 6286 321 1922 2041 1597 207 89 109

Y reaction:(N): sum = -229 15 12 -28 -18 369 125 -705 11

Uplift:max (N) 437 524 169 342 528 163 0


X reaction (N): sum = 27515 803 4337 13951 7255 200 316 652

Y reaction:(N): sum = 234 62 44 -103 -41 726 42 -496 12

Uplift:max (N) 1091 1135 1563 1566 1286 618 0


X reaction (N): sum = -4859 -228 -1260 -1137 -2102 -23 -71 -39

Y reaction:(N): sum = 193 -6 -4 14 20 -37 137 68 13

Uplift:max (N) 249 489 146 614 0 123 272


X reaction (N): sum = -4977 -204 -1058 -1738 -1839 -19 -69 -48

Y reaction:(N): sum = 656 -6 3 22 20 386 182 48 14

Uplift:max (N) 246 415 202 522 0 102 246


X reaction (N): sum = 2450 -138 -670 -872 -695 -3 -31 -41

Y reaction:(N): sum = -792 -9 -14 -5 -3 -643 -191 74 15

Uplift:max (N) 185 220 103 210 0 30 112


66

Top plate displaced shape (plan) X Reactions

W1 W2 W3 W4 W5 W8 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totapplied load (N) 1096 5430 15000 6500 0 0 0 sum 803 4337 13951 7255 200 316 652 27515

displ (mm) -1.18 -1.17 -1.1 -1.11 0.03 - 0.07

Z reactions Y Reactions

load (N) W1 W2 W3 W4 W5 W678 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totmax 1155 1382 2016 1808 0 269 955 downforce sum 62 44 -103 -41 726 42 -496 234min -1091 -1135 -1563 -1566 -1286 -618 0 uplift

Figure 2.14 – Results summary for elastic test 12.


67

Table 2.6 – Load-sharing in elastic tests under single point load.

Test Loaded Wall

Loading Direction

% Resisted by Loaded Wall

% Resisted by Other Walls Combined

2 W1 N-S 22% 78% 3 W2 N-S 50% 50% 4 W3 N-S 81% 19% 5 W4 N-S 63% 37% 6 W8 E-W 57% 43% 8 W5 E-W 78% 22%

2.5 Dynamic Impact Testing

2.5.1 Experiment Description

In any dynamic modelling of structural behaviour, it is crucially important to correctly

characterise the fundamental frequencies of vibration of the structure, if accurate

response predictions are to be obtained. In this phase of the testing, the natural

racking-mode frequencies of the test house in the North-South direction and the East-

West direction were obtained by measuring the acceleration response from dynamic

impact tests. The North-South direction dynamic tests were performed both before

and after the elastic testing was conducted, to examine whether a noticeable change in

dynamic characteristics had occurred as a result of any damage caused to the house by

the elastic tests.

In the dynamic tests, the house was instrumented with five accelerometers and was

excited by hitting it with a rubber-tipped, instrumented impact hammer at the top-

plate level. Photographs of the equipment used are given in Figure 2.15. The structure

was excited at three different locations, along the top plate, to ensure that the

fundamental racking displacement mode was revealed in each direction. To ensure

repeatability, the test was repeated six times at each excitation location. The

excitation and accelerometer locations used in the vibration tests are shown in Figure

2.16.


68

2.5.2 Experiment Results and Discussion

During the experiments, the acceleration time history responses were recorded at each

accelerometer, for the three different excitation locations. The power spectrum of each

response, for each excitation location, was then calculated to give an indication of the

frequency content of the response. Each power spectrum was then examined for the

lowest natural frequency, with the lowest common frequency for each direction

representing the first mode response of the entire structure in that direction. The basic

assumption behind this method is that if the fundamental frequency is being excited in

the structure, it should be apparent in the responses from all locations for a given

direction. It is also assumed that the house is responding as a SDOF system, with the

mass concentrated at the ceiling level (see section 2.5.3 about test limitations).

Examples of the acceleration time histories and power spectra for three repeats of

selected tests, conducted before the elastic testing, are shown in Figure 2.17. These

plots show a consistent value for the fundamental frequency of 13.6 Hz (T=0.075 sec)

in the North-South direction and 14.8 Hz (T=0.068 sec) in the East-West direction.

The resulting power spectra from all of the experiments were combined, and the

normalised sum of the spectra is shown in Figure 2.18 (a). This plot shows the

fundamental frequencies in each direction more prominently. It can be concluded

from these tests that the house has a higher initial stiffness in the East-West direction

than the North-South direction. This is not obvious from the layout of the house,

since it is shorter in the East-West direction, however there are very few openings in

the East-West oriented walls (W5-W9), and as a result of this, their combined

stiffness could be higher than the North-South Walls (W1-W4) which have many

openings.

In Figure 2.18 (b), the North-South direction normalised spectra, from before and

after the elastic testing are compared. This shows that the frequency has shifted

slightly from 13.6Hz to 13.3 Hz during the elastic testing, and that the area under the

main spectral peak has increased slightly. The slightly wider spectral peak and the

small frequency shift indicate that some change has occurred in the dynamic

characteristics, inferring that some damage has occurred as a result of the elastic


69

testing. However, the change is only very small, and may not be significant, since the

potential level of error in this type of experiment is quite high (see section 2.5.3).

The fundamental frequency of the test house in the North-South direction can also be

derived using the results of other experiments. The initial lateral stiffness of the house,

measured at the ceiling level during the initial load-cycles of the destructive testing, in

the North-South direction, is 36 kN/mm (see section 2.6). The measured mass is 5.18

tonne, and can be assumed to be lumped at the ceiling level (see Figure 2.10).

Therefore the fundamental frequency can then be calculated approximately as

follows:

360.00518 13.3

2 2

kmFπ π

= = = Hz (2.1)

Which agrees with the value of 13.3 Hz calculated using the vibration measurements

taken after the elastic testing. This agreement is quite fortuitous given the error

associated with this method (see section 2.5.3), but does indicate that the test house

may be reasonably approximated as a SDOF system with all of the mass (above the

floor) concentrated at the ceiling level. As shown in Figure 2.10 (b), the mass which

participates in the dynamic response of the test house is distributed above and below

the ceiling level. For a real house, the vertical mass distribution, and the height of the

centre of mass will depend on the type of cladding (i.e. timber, stucco, vinyl) and roof

tiles (i.e. slate, vinyl, asphalt) which are used, and may not necessarily be located at

ceiling level. Dynamic response sensitivity to the assumed mass is addressed in

Chapter 5.

Assuming that the East-West fundamental frequency is the measured value of 14.8 Hz

(T=0.068 sec), the initial stiffness can be calculated as 44.8 kN/mm as follows:

2 2(2 ) 0.00518(29.6 )k m Fπ π= = = 44.8 kN/mm (2.2)

It is not possible to compare this result against the results of the elastic or destructive

testing, since the responses in the East-West direction were not suitable for


70

determining all of the individual wall stiffness values. This is because the loading in

the destructive experiments was applied in the North-South direction only.

2.5.3 Comments on Experiment

The results of these tests, when considered in isolation, must be used with caution,

since it may not be appropriate to use them for predicting either the static or dynamic

behaviour of the house. The natural frequencies measured by this method are unlikely

to be representative of the dynamic response of the house under earthquake loading.

This is because the vibration amplitudes induced in the structure by these tests are

very small, because the energy imparted onto the structure by the hammer is also

small. Hence, the measured vibration frequency is based on the initial tangent

stiffness of the house. As described in section 2.2, the load-displacement curve for a

timber structure is non-linear from the origin, hence the stiffness changes

significantly, depending on the value of load or displacement. For this structure, the

initial stiffness in the North-South direction (37 kN/mm) is approximately double that

calculated at a displacement of 1.0mm (17.5 kN/mm). This means that the measured

vibration frequency, is highly dependent on the amplitude of vibration induced in the

test, and hence, the use of a different excitation system (such as a larger hammer or a

vibrator), may result in a significantly different observed frequency of vibration. The

dependency of the natural frequency on the magnitude of the displacement response

(as well as the response history) poses a challenge when modelling and analysing the

dynamic behaviour of this type of structure. These challenges are outlined in more

detail and addressed in Chapters 3 and 4.

As indicated earlier, another limitation of these tests, is that the results are based on an

‘unfinished’ house which does not have, roof-tiles, cladding, trims, doors or windows,

and hence the mass, and hence the vertical and horizontal distribution of the mass in

the test-house, may be significantly different than for a ‘real’ house. Also, the

stiffness of a ‘real’ house may be higher due to the non-structural finishes, especially

if an exterior finish similar to ‘stucco’ is used.


71

(a) Accelerometer attached at ceiling

level of wall W3

(b) Instrumented hammer used to excite

the structure

Figure 2.15 – Equipment used in dynamic impact tests.

N

S

EW

Loc 2Loc 1Loc 3

a1 a2 a3

a4 a5

Loc 2Loc 1

Loc 3

a1

a2

a3

a4

a5

NS Test EW Test

Figure 2.16 – Excitation and accelerometer locations for dynamic impact testing.


72

Acceleration (m/s2) Spectral density

-1

-0.5

0

0.5

1

1.5 2.5 3.5

-1

-0.5

0

0.5

1

1.5 2.5 3.5

-1

-0.5

0

0.5

1

1.5 2.5 3.5

0.00E+001.00E-062.00E-063.00E-064.00E-065.00E-066.00E-067.00E-06

0 10 20 30 40 50 60 70

0.00E+00

5.00E-06

1.00E-05

1.50E-05

2.00E-05

0 10 20 30 40 50 60 70

0.00E+00

5.00E-06

1.00E-05

1.50E-05

2.00E-05

0 10 20 30 40 50 60 70

Time (sec) Frequency (Hz)

(a) Accelerometer a3, North-South test configuration, excitation LOC 1

Figure 2.17 – Example acceleration time histories (three repeats) and power spectra from non-destructive dynamic impact tests.


73

Acceleration (m/s2) Spectral density

-1

-0.5

0

0.5

1

0.5 1.5 2.5

-1

-0.5

0

0.5

1

1 2 3

-1

-0.5

0

0.5

1

1 2 3

0.00E+001.00E-062.00E-063.00E-064.00E-065.00E-066.00E-06

0 10 20 30 40 50 60 70

0.00E+001.00E-062.00E-063.00E-064.00E-065.00E-066.00E-06

0 10 20 30 40 50 60 70

0.00E+001.00E-062.00E-063.00E-064.00E-065.00E-066.00E-06

0 10 20 30 40 50 60 70

Time (sec) Frequency (Hz)

(b) Accelerometer a5, East-West test configuration, excitation LOC 1

Figure 2.17 (cont’d) – Example acceleration time histories (three repeats) and power spectra from non-destructive dynamic impact tests.


74

0

0.2

0.4

0.6

0.8

1

5 10 15 20 25 30

Frequency (Hz)

NormalizedSpectral Density

14.8 Hz East-West

13.6 HzNorth -South

(a) Comparison of the NS and EW direction spectra before elastic testing was

conducted

0

0.5

1

5 10 15 20 25 30

Frequency (Hz)

NormalizedSpectral Density

13.6 HzBefore Elastic

Testing

13.3 HzAfter Elastic

Testing

(b) Comparison of the NS direction spectra before and after elastic testing was

conducted

Figure 2.18 – Normalised sum of the power spectra from dynamic impact tests.


75

2.6 Destructive Testing

2.6.1 Introduction

In the destructive testing, the displaced shape and the distribution of forces throughout

the structure under reverse-cyclic lateral loading were measured in detail. The primary

objectives of the destructive test were to: 1) obtain response data for numerical

models which incorporate the whole house behaviour, such as the shear-wall model

outlined in Chapter 4, and a detailed FE model which incorporates three-dimensional

system behaviour; and 2) to help understand whole-building response, system effects

and load distribution in light-frame structures.

2.6.2 Loading Mechanism

The main objectives in this phase of the testing are concerned with whole-building

behaviour and the load-sharing and system effects. Hence, an important consideration

in designing the loading, was to maximise the three-dimensional or torsional nature of

the response during the test, and ensure full engagement of all of the structural sub-

systems, including the roof and ceiling diaphragm. It was therefore decided to load

one side of the house only, with identical static-cyclic displacements applied to walls

W3 and W4, at the top plate level, under displacement control. Although this is

unrealistic in terms of any natural environmental loading, it was desirable for the

purposes of validating and calibrating the numerical models. If the models are to be

used to predict load-sharing and system effects, then they must be validated against

experiments which have exhibited such behaviours. Once the models are validated,

they can then be used to investigate house behaviour under more realistic

environmental loading scenarios. Model validation under the experimental loading

conditions, and subsequent analysis of behaviour under earthquake loading is

demonstrated for a ‘shear-wall’ model of the house in Chapter 5.

Simultaneous loading of all four North-South shear-walls was considered (i.e. walls

W1-W4), but was decided against because the load-sharing and system effects would

have been minimal and difficult to quantify with all walls loaded under displacement


76

control. Displacement control was used in the destructive experiment because of the

difficulties associated with load-controlled destruction of the house. This would have

required the design of a complex multi-actuator hydraulic control system, or an

impractically large spreader beam structure.

Three different loading rates were used in the destructive test, for different levels of

applied displacement:

• For displacements up to 10mm a rate of 2mm/min

• For displacements between 11-50mm a rate of 15mm/min

• For displacements between 51-120mm, a rate of 25 mm/min

These load rates were chosen to eliminate any dynamic effects in the response, and so

that an appropriate density of data could be obtained over one working day. The

displacements were applied using two motorised screw jacks, one of 200 kN capacity

at wall W4 and one of 100 kN capacity at wall W3. Photographs of the loading

system used in the destructive testing are given in Figure 2.19.

The loading protocol used in the experiment is shown in Figure 2.20. As was

highlighted earlier, the aim in determining the loading protocol, was to induce the

type(s) of structural response in the experiment, that are the most appropriate for

validating and calibrating the numerical models. In this work, the models will be

unique in that they will incorporate a range of phenomena, including 3D system

effects (especially in the FE model), and cyclic degradation in the walls (in the shear-

wall model) and inter-component connections, therefore the loading regime should

enable these phenomena to be observed in the experiment, by using reverse-cyclic

loading.

A reverse-cyclic loading protocol is in fact often used to simulate the way a test

specimen may respond under an earthquake load. Under earthquake loading, many

reversals of the load direction may occur. This is particularly important for a light-

frame structure, given that this type of construction is prone to cyclic degradation of

strength and stiffness, and pinching of the hysteresis (see section 2.2). Therefore

cycling of the load is vital when validating any numerical models which are to be used


77

for predicting response to earthquake loading. In addition to this, the irregular floor

plan will mean that the response under ‘push’ and ‘pull’ direction loading may be

different, hence reversal of the loading direction is also important.

It was therefore decided to use the reverse-cyclic protocol shown in Figure 2.20,

which is conceptually based on the draft ISO standard for joints (ISO, 2000; Foliente

et al., 1998a). This is a displacement-based protocol. Recently, cyclic loading

protocols designed specifically for timber structures have been developed by

Krawinkler et al. (2000) as part of the CUREE project (Hall, 2000). These were not

available when the experiments were conducted.

(a) Motorised screw-jack, push-rod and load cell assembly attached to walls W3 and W4

(b) Motorised screw jack and support

(c) House ready for destructive testing

Figure 2.19 – Photographs of load application system for destructive test.


78

-120

-80

-40

0

40

80

120

0 1 2 3 4Time (hrs)

Applied Displ (mm)

Figure 2.20 – Displacement-based loading protocol used in the destructive test.

2.6.3 Global Hysteresis Response

In Figure 2.21, the approximate global hysteresis response of the whole building, in

the direction of loading (North-South) is presented. The value on the ‘Load’ axis was

calculated by summing all of the reaction forces in the North-South direction in the

load-cells under the bottom plates, at a given point in time, and then repeating this for

all points in time during the experiment. These values are the same as the values of

the total applied load in the actuators. The value on the ‘Displacement’ axis is the

approximate centroidal displacement of the house in the direction of loading. The

global hysteresis response in Figure 2.21 contains no information about the response

of the house in the East-West direction. It will be shown later that the response in this

direction is minimal.

The global hysteresis response is characterised by ‘pinching’ at the origin, and

degradation of the strength and stiffness under cyclic loading. As described in section


79

2.2, This type of behaviour is to be expected from a light-frame system connected

with dowel-type fasteners such as nails and screws.

The initial stiffness of the whole house, is approximately 36 kN/mm, calculated at a

displacement of 0.2mm. The secant stiffness at 1.0mm displacement is 22.5 kN/mm

(see Table 2.7), indicating that the load-displacement curve is highly non-linear from

the beginning. This is also highlighted by the load-displacement envelope of the

global hysteresis, which is shown in Figure 2.22. The total capacity of the house, in

the direction of loading, is around 100 kN with maximum capacity being reached at

about 30mm displacement. The house is still resisting 80 kN at 80mm displacement

and 70 kN at 110mm displacement, indicating this type of construction is highly

ductile.

Given the severe loads and displacements applied to the building in this experiment,

and the global response of the structure, this type of single-storey building is well

placed to resist any lateral design-level load without danger of total collapse. However

this exact response cannot necessarily be extrapolated to other situations based on the

results of this experiment alone, though similar behaviour under similar conditions

can be expected.

Although total collapse under earthquake loading appears unlikely for the type of

single-storey building tested, the level of damage incurred is an important

consideration. More than half of the total cost of property damage from the 1994

Northridge earthquake was due to damage in timber-frame buildings (Kircher et al.,

1997) [note that around 95% of buildings in Los Angeles County are constructed

from timber]. The damage levels observed during the destructive test, for different

displacement levels, and the damage status of the house is discussed in section 2.6.8.

The results show that the damage on the plasterboard wall (W3) is more severe than

for the plywood-braced walls (W1,W2,W4) for a given displacement response. Also,

the damage levels incurred under a real earthquake may be very different in different

parts of the structure, depending on the layout and configuration of the walls. It is

therefore important to examine the results in more detail than the global response

provides, at the sub-system level. In the following section, the response of the

individual walls is examined.


80

2.6.4 Individual Wall Hysteresis Responses

In Figure 2.23, the in-plane hysteresis response of each of the different walls of the

house are shown separately. The out-of-plane contributions of the cross-walls (i.e.

walls W5 to W9) to the North-South direction response are also shown, lumped into a

single sub-system, as they are effectively acting together, and are negligible in

magnitude when plotted separately. The in-plane contributions to the East-West

direction response of walls W6, W7 and W8 are also lumped into a single sub-system,

as they are effectively acting as a single unit in the East-West direction due to the

geometry of the wall layout, and are small in magnitude when considered separately.

Note that the out-of-plane contribution of walls W1 to W4, to the total East-West

response is negligible and is not plotted. In Figure 2.24, the same data is plotted, with

all the plots on the same scale, to give an indication of the relative response of each

different sub-system. The in-plane initial stiffness and capacity characteristics for

each of the wall sub-systems is given in Table 2.7. Stiffness values for the lumped

wall system W678 were estimated using the values from other walls, and the

perforated shear wall method (NAHBRC, 2000), as they could not be reliably

determined from the experimental data, due to the very small in-plane deformation of

wall W678.

Except for the doubly-sided gypsum clad wall W3, the hysteretic characteristics of all

of the walls are quite similar, and exhibit similar behaviour to the global hysteresis

shown in Figure 2.21. Wall W3 is initially stiffer than the other walls, but is more

brittle and loses capacity at a much faster rate during the test. A comparison of the

initial cycles from the hysteresis responses of walls W3 and W4 is given in Figure

2.25. This shows that the onset of inelastic behaviour for wall W3 occurs at around

5mm displacement, compared to around 10mm displacement for wall W4.

Under current practice, the testing and analysis of residential shear-wall structures are

based mainly on the results of isolated wall testing. The results presented here, go a

step further than this, to provide an insight into how the shear-walls behave when they

are part of a whole structure.


81

-120

-80

-40

0

40

80

120

-120 -80 -40 0 40 80 120

Displ (mm)

Load (kN)

Figure 2.21 – Global hysteresis response of whole house in the North-South

direction.

-120

-80

-40

0

40

80

120

-120 -80 -40 0 40 80 120

Displ (mm)

Load (kN)

Figure 2.22 – Backbone of global hysteresis response of whole house in the North-South direction.


82

Load (kN)

-10

-5

0

5

10

-80 -40 0 40 80-40

-20

0

20

40

-80 -40 0 40 80

-50

-25

0

25

50

-120 -80 -40 0 40 80 120-50

-25

0

25

50

-120 -80 -40 0 40 80 120

-20

-10

0

10

20

-120 -80 -40 0 40 80 120-20

-10

0

10

20

-20 -10 0 10 20

-6-4-2024

-20 -10 0 10 20-20

-10

0

10

20

-20 -10 0 10 20 Displacement (mm)


directions.

W1 N-S

W3 N-S W4 N-S

W2 N-S

W5-9 N-S out-of-plane

W5 E-W

W6-8 E-W W9 E-W


83

Load (kN)

-50

-25

0

25

50

-120 -80 -40 0 40 80 120-50

-25

0

25

50

-120 -80 -40 0 40 80 120

-50

-25

0

25

50

-120 -80 -40 0 40 80 120-50

-25

0

25

50

-120 -80 -40 0 40 80 120

-50

-25

0

25

50

-120 -80 -40 0 40 80 120-50

-25

0

25

50

-120 -80 -40 0 40 80 120

-50

-25

0

25

50

-120 -80 -40 0 40 80 120-50

-25

0

25

50

-120 -80 -40 0 40 80 120 Displacement (mm)


directions, plotted on same scale.

W1 N-S

W3 N-S W4 N-S

W2 N-S

W5-9 N-S out-of-plane

W5 E-W

W6-8 E-W W9 E-W


84

-50

-25

0

25

50

-20 -10 0 10 20Displ (mm)

Load (kN)

W3

-50

-25

0

25

50

-20 -10 0 10 20Displ (mm)

Load (kN)

W4

Figure 2.25 – Comparison of wall W3 and W4 hysteresis responses for initial load cycles of destructive test.


85

One of the big problems in isolated wall testing, is defining the boundary conditions,

especially the amount of uplift restraint to provide, so as to simulate the restraint

conditions provided by the surrounding structure, in a real house. The type and level

of uplift restraint used in an isolated test can dramatically affect the load and

displacement capacity which is determined. In the future it is planned to test the main

shear-wall walls in the house as isolated walls, and then compare the isolated wall and

the whole system responses. The individual wall response data obtained from the

whole-house and isolated wall testing can then be used to refine existing isolated wall

testing techniques. This is beyond the scope of the current work. In this work, in

Chapter 5, the global and sub-system experimental data presented here is used in the

development and validation of lumped-mass and shear-wall models for prediction of

global and sub-system responses in light-frame buildings under earthquake loading.

Table 2.7 – Initial in-plane stiffness and capacity characteristics of whole house, and separate wall systems.

Structural system and Orientation House W1* W2* W3 W4 W5* W6-8* W9*

Characteristic

N-S N-S N-S N-S N-S E-W E-W E-W

Initial Tangent Stiffness

(kN/mm) 36 0.67 4 21 9 12 12# 16

Stiffness at 1.0mm displacement

(kN/mm) 22.5 0.55 3.5 11.5 6 8 9# 11

Maximum Load

(kN) 100 12* 30* 33 50 14* 5* 18*

Displacement at Maximum Load

(mm) 32 58* 62* 10 35 18* 4* 11*

Maximum Displacement

(mm) 105 58* 62* 116 117 18* 4* 11*

Load at Maximum Displacement

(kN) 70 12* 30* 14.5 15.4 14* 5* 18*

* Maximum capacity may not have been reached # Estimated value


86

Table 2.8 – Results summary for selected displacement cycles of destructive test.

Applied Displ. (mm)

Measurement W1 W2 W3 W4 W5 W678 W9

in-plane displacement (mm) 2.75 2.86 6.51 7.31 0.81 - 0.68

X reaction (kN): sum = 74.3 1.50 7.84 32.78 28.97 0.31 1.06 1.84

Y reaction:(kN): sum = -1.0 0.36 0.01 0.01 -0.11 -4.96 -0.49 4.20 10

Uplift:max (kN) 2.03 3.16 3.11 7.16 3.54 1.15 0.13


X reaction (kN): sum = 101.4 2.76 13.54 35.52 42.70 0.43 1.92 4.59

Y reaction:(kN): sum = -1.5 0.49 -0.04 0.07 0.01 -8.23 -1.07 7.27 20

Uplift:max (kN) 4.10 5.44 4.16 9.24 4.94 1.88 0.37


X reaction (kN): sum = 112.8 4.98 20.20 26.63 51.29 0.85 2.46 6.39

Y reaction:(kN): sum = -2.2 0.70 -0.02 -0.09 -0.35 -12.11 -1.68 11.33 40

Uplift:max (kN) 7.04 9.16 3.52 12.56 7.03 3.57 1.23


X reaction (kN): sum = 102.6 6.29 23.83 19.16 42.70 0.79 2.77 7.10

Y reaction:(kN): sum = -2.7 0.88 0.17 0.02 -0.49 -14.30 -2.56 13.61 60

Uplift:max (kN) 8.04 9.37 2.98 10.98 8.25 5.32 2.14


X reaction (kN): sum = 91.6 6.74 26.20 17.56 34.55 0.74 2.67 3.14

Y reaction:(kN): sum = -2.9 0.90 0.17 0.02 -0.42 -15.68 -2.94 15.10 80

Uplift:max (kN) 7.44 9.95 2.82 9.55 9.05 6.38 2.89


X reaction (kN): sum = 81.2 7.90 30.26 13.86 21.47 0.87 2.61 4.23

Y reaction:(kN): sum = -3.44 0.97 0.02 -0.39 -0.49 -18.24 -2.76 17.46 120

Uplift:max (kN) 6.86 8.40 2.58 6.66 10.73 6.68 3.89


87

2.6.5 Ceiling Diaphragm Hysteresis Responses

It is evident from the hysteresis plots of the individual walls in Figure 2.23. that the

displacements of the non-loaded walls W1 and W2 are generally around half of the

value for the loaded walls, W3 and W4. The displacement difference is due to two

factors: 1) the rigid-body rotation of the house under the applied loading; and 2) the

in-plane shear distortion (or racking) of the roof and ceiling diaphragm. The rigid-

body rotations and racking type distortions of the roof and ceiling diaphragm, at

different stages during the destructive test are shown diagrammatically in Figure 2.26.

Only the main section of the roof covering walls W2, W3 and W4 is shown.

Because the displacements of the roof system relative to the laboratory strong-floor

were measured at the corners of the house during the test, the rigid-body rotation and

the racking displacements of the roof and ceiling diaphragm can be separated, and an

approximate hysteresis behaviour for the roof system can be derived. The

approximate in-plane racking hysteresis of the roof and ceiling system is given in

Figure 2.27. The data on the displacement axis of Figure 2.27 was derived by

calculating the in-plane shear distortion, in the North-South direction, of the main

section of the roof and ceiling diaphragm, between walls W2 and W4. The values

were derived from the roof-mounted displacement gauges. Note that only the section

of roof covering walls W2, W3 and W4 was considered, and that some of the

displacements required for the calculations were estimated from other nearby gauges.

The data on the load axis is the measured restoring force under wall W2. This can be

assumed to be the approximate ‘load’ if the roof and ceiling diaphragm is assumed to

be the main path for transferring the applied loads from the Eastern (i.e. loaded) side

of the house, over to the western (i.e. non-loaded) side of the house. The only other

path by which load can be distributed to walls W1 and W2 is via the cross walls, and

it is shown in Figure 2.23, that the cross-wall responses are small compared with the

walls in-line with the loading. So if the contribution of the cross-walls to the load-

sharing is ignored, then the only path for the applied load to be distributed to wall W2

is the roof system, and as such the reaction under wall W2 can be considered as the

approximate in-plane shear reaction force for the roof and ceiling diaphragm over

walls W2, W3 and W4.


88

The hysteresis plot in Figure 2.27 indicates that this main section the roof and ceiling

diaphragm is flexible, but is relatively rigid compared to the walls, since the

maximum shear distortion is only around 10-15mm, whereas the maximum

displacement of wall W2 is around 60mm. Figure 2.27 also indicates slight inelastic

behaviour of the roof system under the applied loading. This is consistent with the

damage observations outlined in section 2.6.8, which indicated that there were signs

of very slight working of the fasteners holding the plywood and gypsum on to the roof

trusses on the Eastern side of the house.

2.6.6 Displaced Shapes

The displaced shapes of the house, at various stages during the test, are shown in

Figure 2.28. The undeformed edges are also shown for reference. Only the perimeter

walls are shown, and the roof has been removed for clarity. These plots are snapshots

of the displaced shape, taken at the time of peak displacement on selected cycles of

displacement in the ‘push’ direction. Note that the displacements of walls W3 and

W4 are equal, since the actuators are attached to these two walls and are operated in

displacement control mode. It should also be noted that wall W3 is not structurally

connected to the roof and ceiling diaphragm, so the displaced shape shown

corresponds to the displacements at the top plate level of the walls, and not the

displaced shape of the ceiling and roof diaphragm.

It is clear from Figure 2.28 that the major component of the displacement is in the

direction of the loading. Very little transverse or vertical displacement is apparent. At

the maximum applied displacement of 120mm, the maximum displacement transverse

to the loading direction is 19mm, which is 16% of the in-line displacement. The

maximum vertical displacement, due to stud uplift is around 10mm, which is 8% of

the in-line displacement.


89

2.6.7 Load Distribution

The distribution of the reaction forces underneath the bottom plate of all the walls in

the house, at various stages during the test, for the X, Y and Z directions are shown in

Figures 2.29, 2.30 and 2.31, respectively. Actual values of the reactions at each load

cell are given on the figures as well. These diagrams are graphical snapshots of the

reaction force data, taken at the time of peak displacement in the ‘push’ direction, and

correspond to the same points in the test as the displaced shapes in Figure 2.28. A

summary of the individual wall displacements and the reaction forces, for the same

selected cycles of displacement, is given in Table 2.8, and a summary of all reaction

and displacement data, over the entire destructive test, is given in Appendix B.

Figure 2.29 shows that as the applied load on walls W3 and W4 is increased, it is

increasingly resisted by walls W1 and W2 in the X direction (in-plane), even though

there is no directly applied load on these walls. Figure 2.31 shows a similar behaviour

for the Z-direction reactions. Figure 2.29 also shows that the X-direction load in the

cross walls are very small compared to the in-plane loads. This implies that the load

is transferred into these walls mainly via the roof system. This load-sharing becomes

more pronounced as the applied displacement is increased, and the load-carrying

capacity of walls W3 and W4 is reduced as they are loaded into the inelastic range.

Figures 2.29 and 2.31 show that the distribution of the loads throughout the structure

change significantly during the experiment. In Figure 2.30 the torsional nature of the

response is indicated by the presence of in-plane loads in the end walls. The

magnitudes of these loads are relatively small but are certainly not negligible.

In Figure 2.32, the percentage of the total in-plane load, which is taken by each wall

sub-system at the maximum point in the loading cycle, is plotted for both the ‘push’

and ‘pull’ directions. Here it is highlighted that the distribution of the load throughout

the structure, changes significantly during the experiment, the amount of load resisted

by each sub-system depending on the level of applied displacement. Initially the

stiffest wall (W3) resists the majority of the load, but as the displacement increases

and its response becomes inelastic, the more ductile wall W4 takes over. Walls W1,

W2 and the cross walls also take more load as the applied displacement is increased.


90

It should be highlighted that wall W3 is not directly connected to the roof and ceiling

diaphragm, and was loaded such that it would have the same displacement as wall W4

in the destructive experiment. Because of this, no conclusions can be made about the

load-sharing and redistribution between wall W4 and the internal wall W3.

In Figure 2.33, the percentage of in-plane load which is taken by the out-of-plane

walls is plotted as a function of the applied displacement level. Initially, only 4% of

the total load is taken by the cross-walls, but this increases to a maximum of 12% for

very large displacements.

In Figure 2.34, the maximum uplift force, which is measured under each wall sub-

system is plotted for both the ‘push’ and ‘pull’ directions. Here it is highlighted that

the maximum wall uplift forces throughout the structure also change during the

experiment depending on the level of applied displacement. The re-distribution of the

vertical forces follows a similar trend to the in-plane forces, whereby the forces are

increasingly transferred to the non-loaded walls (W1 and W2) as the loaded walls

(W3 and W4) lose capacity. This is primarily because the vertical reactions result

from the in-plane actions, and are hence dependent on the same lateral load-sharing

mechanisms.

2.6.8 Damage Status

The damage status of the various parts of the house at different levels of displacement,

are presented in Table 2.9. Damage levels are described as either minor, moderate or

severe, and these levels are used as the damage indicators in the analytical modelling

in Chapter 5. The damage status of the different structural sub-systems in the house,

after the final load cycle, is presented in Table 2.10. Photographs of the damage

incurred, at the end of the test, are shown in Figure 2.35. The damage to walls W3

and W4 was quite severe, with the gypsum board sustaining most of the damage. It

should be noted that the apparent damage for a finished house, with trims, and door

and window frames may be more severe for a given displacement, although the

capacity and stiffness may be significantly increased.


91

10mm120mm 100 80 60 40 20

W2 Reaction Force

W4

W9

W5E

W

SN

E

W

SN

Applied Load

Figure 2.26 – Diagram showing rigid-body rotation, and racking distortion of section of roof and ceiling diaphragm over W2, W3 and W4, for selected displacement levels.

-40

-30

-20

-10

0

10

20

30

40

-15 -10 -5 0 5 10 15

Displ (mm)

Load (kN)

Figure 2.27 – Approximate hysteretic behaviour of roof and ceiling diaphragm over

walls W2, W3 and W4.


92

10mm

20mm

40mm

Figure 2.28 – Displaced shape of house perimeter and undeformed edge at different stages of destructive test (loading as shown in diagram at bottom).


93

60mm

80mm

120mm

Figure 2.28 (cont’d) – Displaced shape of house perimeter and undeformed edge at different stages of destructive test (loading as shown in diagram at bottom).


94

0.20.6

0.50.20.10.1

0.0

0.2

1.0

0.0

0.0

1.0

1.31.4

1.71.90.40.1

0.0

0.00.0

0.2

2.1

2.82.0

2.4

3.12.5

2.32.2

2.8

3.63.2

3.9

0.6

0.3

0.02.7

3.62.5

3.83.03.0

3.12.2

2.62.4

0.4

0.5

0.0

0.1

10mm

0.3

1.7

1.60.30.1

0.10.1

0.5

3.7

0.0

0.0

2.7

3.4

3.53.3

3.5

1.10.3

0.1

0.10.0

0.4

2.2

2.71.9

2.4

2.82.1

2.12.0

2.4

3.12.6

3.41.8

0.5

0.0

4.5

5.44.3

6.35.14.6

6.64.2

4.43.4

0.10.0

0.4

0.01.3

0.0

0.2

20mm

0.3

1.7

1.60.30.1

0.10.1

0.5

3.7

0.0

0.0

2.7

3.4

3.53.3

3.5

1.10.3

0.1

0.10.0

0.4

2.2

2.71.9

2.4

2.82.1

2.12.0

2.4

3.12.6

3.41.8

0.5

0.0

4.5

5.44.3

6.35.14.6

6.64.2

4.43.4

0.10.0

0.4

0.01.3

0.0

0.2

40mm

Figure 2.29 – Distribution and magnitude of X-direction reaction forces (kN) at different stages of destructive test.

Y

X Z


95

0.6

2.6

2.7

0.50.2

0.1

0.5

3.6

0.00.1

0.1 0.0

4.0

4.2

4.64.2

4.8

1.40.6

0.00.3

0.5

1.71.3

1.3

1.71.60.7

1.51.3

1.4

2.21.7

2.7 1.8

0.6

0.0

3.1

4.43.3

5.34.33.5

7.4

4.6

3.73.1

1.0

0.5

0.21.2

0.1

0.4

60mm

0.6

2.9

2.7

0.50.3

0.1

0.2

0.9

0.00.1

0.1 0.0

4.3

4.6

4.94.6

5.4

1.60.8

0.0

0.4

0.4

1.71.2

1.2

1.71.40.4

1.51.2

1.4

2.11.6

2.30.6

0.52.5

3.62.3

4.13.22.5

6.5

3.8

3.42.7

1.3

0.5

0.21.1

0.3

0.4

80mm

0.9

3.4

2.9

0.70.4

0.10.0

0.1

1.0

0.20.1

0.15.3

5.1

5.55.2

6.3

1.71.1

0.8

0.4

1.20.7

0.6

1.30.9

0.3

1.40.7

1.2

1.91.4

2.20.3

0.5

0.00.1 1.2

2.40.8

2.51.51.6

5.22.9

2.01.32.4

0.30.41.0

0.3

0.6

120mm

Figure 2.29 (cont’d) – Distribution and magnitude of X-direction reaction forces (kN) at different stages of destructive test.


96

0.1-0.0

0.00.20.8

0.9 -0.0-0.0

0.7

1.0

0.30.4

0.5 0.0

0.1-0.1

-0.0-0.0

0.0

0.00.0

-0.1-0.3 -0.8

-0.6

-0.3-0.1

0.1-0.1

0.00.0

0.0-0.0

-0.0-0.0

0.1-0.1

0.0-0.1

-0.4-0.3

-0.3-0.4

0.2-0.0

-0.0-0.0

0.0

-0.1

0.2

-0.1

0.20.40.4

-0.4-0.3

-0.0

-0.8-0.5

-1.5

10mm

0.2-0.1

0.00.30.8

1.3 -0.0

0.4

0.9

1.5

0.1

0.80.8 -0.0

0.1-0.2

0.0-0.1

0.10.10.0

-0.4 -0.7 -1.2

-0.9

-0.6-0.0

0.2-0.1

0.00.0

0.0-0.0

-0.0

0.00.1

-0.1

0.0

0.4

-0.6-0.3

-0.4

-0.7

0.3-0.1

-0.0

0.10.1

-0.1

0.4

-0.2

0.41.0

1.0

-0.6-0.6

-0.3

-1.3-1.2

-1.9

20mm

0.3-0.1

0.10.50.8

1.5-0.0

1.01.0

2.0

0.4

0.80.9

-0.1

0.2-0.2

0.0-0.2

0.0

0.10.0

-0.9 -1.5-1.8

-1.2

-0.7-0.0

0.1-0.1

0.00.0

-0.0-0.0

0.10.0

0.1-0.1

-0.2

1.0

-0.4-0.5

-0.7

-1.3

0.4-0.2

0.0-0.1

0.3-0.2

0.6

-0.3

0.4

1.92.0

-0.6

-0.8-0.7

-1.8-1.9

-2.4

40mm

Figure 2.30 – Distribution and magnitude of Y-direction reaction forces (kN) at different stages of destructive test.

Y

X Z


97

0.3-0.2

0.10.61.1

1.7-0.0

1.41.1

2.4

0.50.7

1.0-0.1

0.4-0.1

-0.0-0.3

-0.0

0.1-0.0

-1.3 -2.0-2.0

-1.2-0.9

0.00.1

-0.1

0.1-0.1

-0.0-0.0

0.10.1

0.0-0.1

-0.1

1.2

-0.6-0.7

-0.9

-1.4

0.4-0.1

0.0-0.2

0.3-0.3

0.6

-0.3

0.4

2.42.3

-0.6

-0.9-0.9

-2.0-2.2

-2.7

60mm

0.3-0.2

0.10.71.3

1.8-0.0

1.71.4

2.6

0.60.5

1.0-0.2

0.40.0

-0.0-0.3

-0.0

0.1-0.0

-1.5 -2.3-2.3

-1.2-1.4

-0.00.1

-0.1

0.1-0.0

-0.1-0.0

0.20.1

0.0-0.1

-0.1

1.4

-0.7

-0.8-0.9

-1.1

0.3-0.1

-0.0-0.2

0.3-0.3

0.5

-0.3

0.5

2.62.4

-0.5

-0.9-1.1

-1.8

-2.4

-2.9

80mm

0.2-0.2

0.1

0.81.7

2.1

-0.0

2.11.7

2.9

0.6

1.01.5

-0.2

0.00.2

0.1-0.5

0.1

0.10.1

-1.8 -2.8-2.6

-1.3 -1.8

0.10.1

-0.1

0.1-0.0

-0.3

-0.1

0.10.1

0.0-0.1

-0.2

1.6

-1.0

-1.3-1.2

-1.6

0.2-0.0

-0.0

0.10.0

0.2

0.4

-0.2

0.4

2.82.6

-0.3

-0.7

-1.4

-2.1

-2.8

-3.1

120mm

Figure 2.30 (cont’d) – Distribution and magnitude of Y-direction reaction forces (kN) at different stages of destructive test.


98

-0.4

2.5

-2.0

1.32.1 0.8

0.3-0.1

0.1

1.5

-1.0

-0.30.7 0.3

-3.2

1.20.2

-0.0

3.0-1.3

0.2

-2.5

-1.3-0.9

-0.3 -1.5-3.1-3.0

-0.6

3.1

5.2

-2.3

-2.0

-0.9

-0.6

0.9

2.5

3.6 1.3

-1.1-0.5

-0.6

-4.5

4.92.9-2.0

-2.7

7.8

-4.8

7.9

-7.2

2.22.62.5

-1.1-1.1

-1.0

-3.5-2.3-2.4

10mm

-1.7

4.7

-4.1

2.74.0 1.3

0.6-0.2

-0.4

1.4

-1.4-0.8

1.3 0.7

-5.4

2.10.5-0.0

5.1

-2.5

0.6

-4.9

-2.1

-1.3

0.0 -1.4-3.8

-4.2

-0.7

3.8

6.4

-2.9

-2.2

-0.9

-0.8

1.0

2.8

4.1 1.8

-1.0-0.7

-0.7

-7.7

8.95.7

-3.5-6.1

12.7

-8.7

13.5

-9.2

2.23.5

3.7

-1.8-1.9

-1.1

-4.9-4.0

-3.3

20mm

-3.5

9.0

-7.0

4.96.7

2.1 1.0 -0.6-1.2

1.0

-3.6

-1.4

2.4 1.3

-9.2

3.80.6-0.0

7.6

-4.2

1.3

-7.0

-3.5

-1.9

1.0 -0.4-1.8-3.5

-0.4

2.7

4.2

-3.2

-1.6

-0.9-1.2

1.1

2.4

3.0 2.2

-0.4-0.7

-0.9

-10.2

14.8

7.4

-6.6-10.3

17.9

-12.6

19.4

-6.9

-0.6

3.34.4

-2.7-2.6-0.9

-5.3-5.4-4.5

40mm

Figure 2.31 – Distribution and magnitude of Z-direction reaction forces (kN) at different stages of destructive test.

Y

X Z


99

-5.1

12.6

-8.0

6.2

8.4

2.3 1.0 -1.2-2.1

0.4

-5.3

-2.0

2.9 1.6

-9.4

4.60.3

-0.0

9.0

-5.5

2.1

-8.3

-5.6

-2.9

2.1 0.6-1.5-3.0

-0.3

2.0

3.1-1.8

-0.9

-0.9-1.4

0.8

1.81.5

2.7

0.2

-0.9-1.2

-6.0

16.2

5.3

-5.9-9.9

14.4

-11.0

19.5

-5.0

-1.1

1.54.5

-3.5-3.4-0.9

-4.3

-5.6-4.8

60mm

-6.2

14.6

-7.4

6.8

8.8

2.0 1.0 -1.6 -2.9

0.9

-6.4

-2.2

3.0 1.9

-9.9

4.70.4

-0.0

9.7

-6.3

2.7

-9.1

-6.6

-3.5

3.4 1.0-1.2-2.8

-0.8

1.9

3.1-1.0

-0.1-0.8

-1.6

0.5

1.3-0.4

3.0

0.7

-0.8-1.4 0.5

12.3

1.8

-4.1-8.6

12.7

-9.5

17.1

-4.1

-1.50.6

4.5

-3.5-3.4-0.5

-2.7

-5.8-5.5

80mm

-4.9

16.5

-6.9

8.69.3

1.31.1

-2.1 -3.9

-0.2

-6.7

-5.6

1.52.1

-8.4

4.4-0.0

-0.0

10.2

-5.5

3.7

-10.7

-9.7

-5.0

6.1

1.4-1.0-2.6

-0.9

1.0

2.3-1.1

0.3-1.8-2.4

0.1

2.2-0.1

3.8

2.7

-0.8

-2.2

7.7

10.2

0.1-0.8-3.8

4.5

-6.7

13.4

-2.8

-2.2-1.5

4.1

-3.1-2.40.3

-2.3

-6.7-6.0

120mm

Figure 2.31 (cont’d) – Distribution and magnitude of Z-direction reaction forces (kN) at different stages of destructive test.


100

0%

10%

20%

30%

40%

50%

60%

0 20 40 60 80 100 120

Applied Displacement (mm)

% o

f X R

eact

ion

W1 W2 W3 W4 X-Walls

(a) Push direction

0%

10%

20%

30%

40%

50%

60%

0 20 40 60 80 100 120


% o

f X R

eact

ion

W1 W2 W3 W4 X-Walls

(b) Pull direction

Figure 2.32 – Percentage of X-direction reaction taken by each wall sub-system during destructive test.


101

0%

4%

8%

12%

16%

0 20 40 60 80 100 120


% o

f X

Rea

ctio

n in

Cro

ss-W

alls

push direction

pull direction

Figure 2.33 – Percentage of X-direction reaction taken by cross-walls during destructive test.


102

0

2

4

6

8

10

12

14

0 20 40 60 80 100 120

Applied Dsiplacement (mm)

Max

imum

Upl

ift F

orce

(kN)

W1 W2 W3 W4 W5 W678 W9

(a) Push direction

0

2

4

6

8

10

12

14

0 20 40 60 80 100 120

Applied Dsiplacement (mm)

Max

imum

Upl

ift F

orce

(kN)

W1 W2 W3 W4 W5 W678 W9

(b) Pull direction

Figure 2.34 – Maximum uplift force in each wall during destructive test.


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Table 2.9 – Damage observations at different stages during the destructive test.

Displacement Damage Observations

5 mm First significant inelastic behaviour on plasterboard wall

No visible signs of fastener or sheathing damage

10 mm First significant inelastic behaviour on plywood braced walls

First visible signs of plasterboard cracking around corners of openings

Peak load for plasterboard wall

20 mm Minor cracking of plasterboard at corners of openings

First visible signs of plasterboard screw tearing and pulling

35 mm Moderate plasterboard cracking at corners of openings

First visible signs of plywood nail tearing and pulling

Peak load for plywood braced walls

50 mm Moderate to Severe plasterboard cracking at corners of openings

Moderate tear-out and pull-through of plasterboard screws

Minor tear-out and pull-through of corner nails on plywood bracing panels

Minor separation of sheathing materials from frame

Minor stud bending and uplift

75 mm Severe plasterboard cracking at corners of openings

Severe tear-out and pull-through of plasterboard screws

Moderate tear-out and pull-through of corner nails on plywood bracing

panels

Moderate separation of sheathing materials from frame

Moderate stud bending and uplift

+100 mm Severe cracking of plasterboard

Severe pull-through and tear-out of plasterboard screws

Severe tear-out of corner nails on plywood bracing panels

Severe separation of all sheathing materials from frames

Severe stud bending and uplift


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Table 2.10 – Structural damage status of structural sub-systems after final load cycle of destructive test

System Damage Status after final load cycle

W1 Moderate cracking of gypsum board at corners of opening; no visible damage to

plywood sheathing; still holding maximum load.

W2 Severe cracking of gypsum around openings; some splitting of plywood sheathing

and nail tear-out at corners; slight separation of plywood sheathing from studs; still

holding near to maximum load

W3 Severely damaged gypsum both sides especially at intersections with cross and

end walls; complete loss of capacity with large sections of gypsum board

completely separated from studs with screws pulled through the gypsum and/or

broken; moderate cracking of gypsum around the door opening; visible studs are

noticeably bent under large loadings; wall has broken through the end-walls at

both ends; capacity reduced to less than 50% of maximum.

W4 Severe damage to gypsum around all openings, some pull-through of gypsum

screws and separation of gypsum from studs; severe tear-out and withdrawal of

plywood nails, many plywood sheets completely separated from studs; severe

stud uplift and bending at wall-ends and openings; capacity reduced to 50% of

maximum.

W5 Wall W3 has broken through W5 at the intersection of the two walls, and the studs

have separated at this location, otherwise intact with very minor signs of damage

at perimeter fastener locations.

W6 Intact and apparently undamaged.

W7 Intact and apparently undamaged.

W8 Small section of wall has distorted severely due to movement of wall W3.

W9 Wall W3 has broken through W9 at the intersection of these two walls, and the

studs have separated at this location, otherwise intact with very minor signs of

damage at perimeter fastener locations.

Roof &

Ceiling

Some perimeter screws on ceiling gypsum board showing slight signs of wear and

pull-through, damage to gypsum limited to sections near loading points, otherwise

completely intact; nails for plywood roof sheathing show minor signs of tilting and

withdrawal on roof above wall W4 although plywood sheets generally undamaged

and intact; some minor nail withdrawal on truss clips (i.e. plate connectors

between roof trusses and top plate), and some permanent deformation of truss

clips; roof trusses appear undamaged and intact, roof system as a whole still

transferring load from W3 & W4 to W2 & W1 at maximum capacity.


105

Stud uplift in W4 door opening

Severe crack in W4 window opening

Almost undamaged truss-clips above W2

Moderate cracking on gypsum board, W1 garage door opening

Wide view of damage on W2

Severe cracking of gypsum board in W2 window opening

Figure 2.35 – Photos of damaged house after destructive test.


106

Nail distortion on roof sheathing

Gypsum board broken separated from studs at north end of W3

Broken studs at North end of W3 have pushed through W5

Gypsum board broken separated from studs at south end of W3

Figure 2.35 (cont’d) – Photos of damaged house after destructive test.


107

Separation of Plywood and gypsum from W4

Severe distortion of studs on W4 door opening

Figure 2.35 (cont’d) – Photos of damaged house after destructive test.

2.6.9 Comments on Experiment

The experiments and results described herein are based on an unfinished house, which

consisted of the structural elements only, and the gypsum board lining. An

unfinished house was chosen as the basis for the experiments, because the main aims

of the work were to examine load-sharing between the structural elements, and to

collect data for validation of analytical models, which do not consider any non-

structural elements. Also, there is a large variability in the type, robustness and

durability of the non-structural components which could be used, and their presence

may introduce too many unknowns into the experiments, making it difficult to

interpret the data and draw meaningful conclusions. The limitation of this approach is

that the inclusion of the non-structural elements in the experiment (i.e., tapes in the

corners of the interior panel materials, cornices or similar ‘non-structural’ links

between roof and wall panels, windows, doors, exterior wall finish) may alter the load

paths at small displacements. For example, taping and plastering of the joint between


108

wall W3 and the ceiling diaphragm may establish a more rigid connection between the

roof and ceiling diaphragm and the wall, which will then become more prominent in

resisting applied lateral loads. The presence of the non-structural finishes may also

increase the strength and stiffness of the house, and also increase the apparent damage

status. This could be addressed by testing a finished house to compare and enhance

the results presented here, and so that the results could be compared against field

observations of damage on similar houses in past earthquakes . It would also be

desirable to conduct similar experiments on light-frame houses with a range of

different configurations than the house described herein, to provide additional data for

model validation.

Another limitation stems from the loading regime used in the destructive experiment.

This limitation applies to almost all structural testing, in that the interaction between

the applied load in the laboratory and the specimen, may not accurately represent the

‘real’ in-service load-structure interaction. In the destructive test, identical

displacements were applied at the top plate of walls W3 and W4, to ensure that all

structural sub-systems and load-sharing mechanisms were engaged. This meant that

no conclusions could be made about the load-sharing and redistribution between wall

W4 and the internal wall W3. This also restricted the structure from rotating in the

area between the two load points and hence induced some unrealistic local stresses on

wall W5.

2.7 Conclusions

This chapter has presented a review of the general behaviour of light-frame systems,

and full-scale testing of light-frame structures. The results of structural testing

conducted on a full-scale L-shaped timber-frame house are summarised and

presented. Three series of tests were conducted:

• elastic testing under a variety of small point loads

• vibration-based dynamic impact testing

• destructive testing under reverse static-cyclic lateral loading


109

Data gathered from the experiments is to be used in validating different types of

numerical models which represent the structural behaviour of the house. These

include a FE model (not in this Thesis), and hysteretic shear-wall, SDOF and shear-

building models (see Chapter 5).

On the basis of the experimental results presented in this chapter, the following

conclusions can be drawn.

2.7.1 Elastic Testing

Small point loads, less than 10 kN, were applied to the walls of the house at the

ceiling level, in different configurations, and the displaced shape and the distribution

of the reaction forces under elastic response was measured in detail.

• Significant load-sharing occurs between the external shear-resisting walls in

the house under elastic response conditions. When a concentrated load is

applied to a single wall, the load is distributed to the non-loaded external walls

mainly through the roof and ceiling diaphragm. Between 19% and 78% of the

applied load can be shared by the rest of the structure, depending on the

location of the wall in the structure, the connection to the roof and ceiling

diaphragm and the relative stiffness of the loaded wall compared to the

surrounding structure.

• The amount of load-sharing observed in the elastic experiments, is a lower

bound estimate, since not all of the structural sub-systems are engaged at small

displacement levels. The load-sharing and redistribution increases once the

house is pushed into the inelastic range.

• A uniform displacement field was applied to all four North-South walls (i.e.

walls W1 to W4) until the applied load reached 90% of a hurricane level

design wind load. Under this loading, the response of the structure was

essentially elastic with no damage observed. The results of this test indicated


110

that the type of house tested should easily withstand this design wind load

with minimal or no structural damage. It is important to highlight that this

conclusion is based purely on the lateral wind load to the walls and does not

consider uplift pressure on the roof.

2.7.2 Dynamic Impact Testing

The natural racking-mode frequencies of the house in the North-South and East-West

directions were obtained by measuring the acceleration response from dynamic

impact tests. The experiments were conducted before and after the elastic tests to

check for signs of damage.

• The natural frequencies determined from the dynamic tests in the North-South

direction were 13.6 Hz (T=0.074 sec) before the elastic testing, and 13.3Hz

(T=0.075 sec) after the elastic testing, with a slightly wider spectral peak

observed in the latter tests. This indicated that a slight change had occurred in

the dynamic characteristics, inferring that some slight damage had resulted

from the elastic testing. However, the change is only very small, and could be

within the error bound of the experiment.

• The natural frequency in the East-West direction was 14.8 Hz (T=0.068 sec)

measured before elastic testing

• The calculated natural frequency in the North-South direction, derived using

the stiffness from the initial load-cycles of the destructive testing, and the

measured mass of the structure is 13.3 Hz, which is in agreement with the

value determined from the vibration tests. This indicates that a SDOF model,

with all mass lumped at the ceiling level, may be appropriate for global

response analysis of the test house.

• The natural frequencies calculated using dynamic impact tests must be used

with caution, since the measured vibration frequency, is highly dependent on


111

amplitude of vibration induced in the test, and hence depends on the excitation

level used.

2.7.3 Destructive Testing

Identical static-cyclic displacements of up to +/-120mm were applied at the ceiling

level, on one side of the house, in the North-South direction. The displaced shape and

the distribution of reaction forces throughout the structure under the inelastic response

were measured, and the following observations were made:

• The global hysteresis response of the house was characterised by ductile

behaviour, ‘pinching’ at the origin, and degradation of the strength and

stiffness under cyclic loading. The hysteresis characteristics of the individual

walls were similar to each other, and to the global response, except for the

internal wall W3, which was more brittle.

• The roof and ceiling diaphragm in the house provided a robust load-

distribution path from the loaded to the non-loaded walls which are

structurally connected to the diaphragm. It behaved as a flexible diaphragm

under the applied loading, but was relatively rigid compared to the walls.

• The load-displacement relationship of the whole house, and of the individual

components was non-linear from the origin. The initial stiffness of the whole

house, measured at a displacement of 0.2mm, was 36 kN/mm, whereas the

stiffness measured at 1mm displacement was 22.5 kN/mm.

• The total strength capacity of the house, in the direction of loading, was

around 100 kN with maximum capacity being reached at about 30mm

displacement. Beyond the maximum strength capacity, the house resisted 80

kN at 80mm displacement and 70 kN at 110mm displacement, indicating this

type of construction is highly ductile and not prone to sudden collapse.


112

• As the applied load was increased on walls W3 and W4, the percentage of the

load resisted by walls W1 and W2 also increased, even though no load was

applied to these walls. The applied load was re-distributed into these walls

mainly via the roof and ceiling diaphragm. The distribution of the load

throughout the structure, changed significantly during the experiment, with the

amount of load resisted by each sub-system depending on the level of applied

displacement and the structural integrity of the other subsystems. This is due

to the highly ductile nature of the structure.

• For small loads, 4% of the total applied load was taken by the out-of-plane

walls. At peak load, this increased to 9%, and then further to a maximum of

12% beyond the peak load.

2.7.4 Recommendations for Further Research

• In the experiments presented here, all the observations are based on an

unfinished house, which consists of the structural elements only. Inclusion of

the non-structural elements in the experiments may alter the load paths at

small displacements. The presence of the non-structural finishes may also

increase the strength and stiffness of the house, and also increase the apparent

damage status. This could be addressed by testing a finished house to

compare and enhance the results presented here, and so that the results could

be compared against field observations. It would also be desirable to conduct

similar experiments on light-frame houses with a range of different

configurations than the house described herein, to provide additional data for

model validation.

• The results of this experiment give an insight into how the shear-walls behave

when they are part of a whole structure. Since the current practice for shear-

wall design and analysis is based on isolated wall testing, it is recommended to

test the main shear-walls of the test-house as isolated walls, in order to

examine the link between the isolated wall and the whole system responses.


113

• It is also recommend that a three-dimensional FE model be fully validated

against the experimental results presented herein and also for another house

with a very different configuration than the current test house so that the

experimentally validated model can be used to conduct sensitivity studies for a

wide range of practical house configurations. This is needed to provide general

recommendations for lateral force distribution in light-frame buildings.

• Once the FE model has been validated it should be used as a tool in the

development of practical and reasonably accurate design methods and

recommendations for lateral force distribution in light-frame structures.


114

Chapter 3 – Hysteresis Modelling

115

CHAPTER 3

Hysteresis Modelling

3.1 Introduction

As was indicated in the overview of light-frame structural behaviour in Chapter 2,

analytical modelling of the behaviour of light-frame structures under severe cyclic

loading requires an accurate mathematical representation of the force-displacement

relationship of the critical structural components. This chapter addresses the

modelling of the force-displacement relationship, or hysteresis, for light-frame

structures. It begins with a review and discussion on hysteresis modelling and system

identification techniques, which can be used to determine hysteresis model

parameters. Next, a hysteresis model which is suitable for modelling the behaviour of

light-frame structures under seismic loading is described. The model is based on the

Bouc-Wen differential model of hysteresis. The Generalised Reduced Gradient

(GRG) method of system identification is then used to determine the parameters for

the modified Bouc-Wen hysteresis model, for a range of different experimental data

sets, which are taken from the literature. A parallel system identification approach is

outlined and illustrated through two examples. In these examples a single set of

hysteresis model parameters is fitted to two different experimental data sets

simultaneously. The two different data sets are derived from identical test specimens

under different applied loading. Finally, the modified Bouc-Wen hysteresis

parameters are determined for the test house using the experimental results presented

in Chapter 2. The GRG method is used to determine hysteresis parameters for the

whole-building response, and for the individual wall sub-systems. Three different

structural models which incorporate modified Bouc-Wen differential hysteretic

elements are formulated in Chapter 4. The models are used for prediction of global

and individual wall responses in Chapter 5.


116

3.2 Overview of Hysteresis Modelling

3.2.1 Introduction

Typical hysteresis data obtained from cyclic tests of light-frame structures, as shown

in Figure 2.2, exhibit a non-linear load-displacement relationship with no distinct

yield point, cyclic degradation of strength and stiffness, and pinching about the origin

In addition to this, the behaviour exhibits memory, such that the resisting force

depends not only on the instantaneous displacement, but on the past displacement

history as well (see section 2.2.2).

Since the hysteretic characteristics of the inter- and intra-component connections

govern the global hysteretic behaviour of light-frame structures under cyclic loading

(see section 2.2.2) it is therefore desirable to characterise this behaviour as accurately

as practically possible in any analytical model used for the prediction of structural

behaviour under earthquake loads.

Because of the complex nature of the behaviour, and the difficulties involved in

simulating such behaviour, many simplified hysteresis models have been developed

which ignore some (or even most) of the features observed in experiments. Generally,

it is the very simplistic models which are used in earthquake engineering practice.

Researchers have also developed very complex hysteresis models which attempt to

represent hysteretic behaviour as accurately as possible. These models have been

primarily used for research purposes and are not routinely used for seismic analysis.

Some of the different hysteresis modelling approaches, and the more commonly used

hysteresis models are described and reviewed in the following section.

3.2.2 Phenomenological Hysteresis Models

Countless hysteresis models have been developed for the purposes of structural

analysis. This is mainly because of the many different material and structural

configurations which are used in the built environment, which have had models

developed specifically for them, often with many variations. There are two main types

of hysteresis model: 1) phenomenological; and 2) mechanics-based [e.g. reinforced


117

concrete fibre models (Fardis, 1991)]. There are far too many models to review

comprehensively in the work herein, and hence, a selection of models, often used for

modelling the behaviour of light-frame structures are reviewed. Most of the available

models are phenomenological, and are discussed in the following. Mechanics-based

models are discussed and compared to phenomenological models in section 3.2.3.

Piece-wise linear (PWL) models of hysteretic behaviour have been developed

primarily for seismic analysis of concrete and steel structures. Some examples of

PWL models are shown in Figure 3.1. The elasto-plastic and bi-linear models, shown

in Figure 3.1 (a) and 3.1 (b) are the simplest approximations to inelastic behaviour,

and have historically been the most widely used approaches in hysteresis modelling,

particularly for concrete and steel structures. The advantage of these approaches is the

simplicity and hence ease of implementation. The limitation, when using these models

to predict the behaviour of light-frame structures, is that they are perhaps too

simplistic to accurately represent the real behaviour – they do not take account of the

cyclic degradation in strength and stiffness and pinching of the hysteresis which is

observed in laboratory experiments.

The effect of including or excluding these experimentally observed phenomena in

predictive structural modelling is disputed. The conclusions of the previous research

undertaken to determine the effect of hysteretic modelling on response prediction are

inconclusive and conflicting (see section 3.2.4). The physical effect of these

phenomena is, however, more intuitive.

The effect of stiffness degradation can significantly alter the dynamic response of a

structure under seismic loading, since any change in stiffness causes a corresponding

shift in the natural frequency of the structure. If the shift in the natural frequency is

towards the dominant frequency in the earthquake excitation, then the structure’s

response can be amplified dramatically due to resonance effects (see section 4.2.1).

Strength degradation can also be physically important, since many structural collapses

are caused by progressive weakening of the structure resulting from the violent load-

reversals experienced under earthquake loading. The presence of pinching in a

structural response can dramatically alter its energy absorption characteristics. The

energy dissipated by a structure is given by the area enclosed by the hysteresis trace,


118

and hence a pinched hysteresis response, for a given deflection level, will have vastly

different energy absorbing capacity to a non-pinched response. Conversely, the

displacement response for a given energy dissipation, can be quite different for a

pinched and a non-pinched system. This concept is illustrated diagrammatically in

Figure 3.2. Strength and stiffness degradation also have an effect on the shape of the

trace, by causing it to change over time. This in turn affects the energy dissipation

characteristics.

Although the elasto-plastic and bi-linear models, in their simplest form, have no way

of incorporating the important behavioural characteristics observed in the laboratory

and the field (such as cyclic degradation and pinching), many other PWL hysteresis

models have been developed, which address these limitations to varying degrees. The

modified Clough model and the Q-hysteresis model shown in Figure 3.1 (c) and 3.1

(d) incorporate stiffness degradation. The Takeda model and the Slip model shown

in Fig 3.1 (e) and 3.1 (f) incorporate stiffness degradation and pinching effects

commonly observed in reinforced concrete structures.

The PWL model shown in Figure 3.3, was developed by Stewart (1987) for nailed

sheathing to timber connections, and includes pinching and stiffness degradation, as

well as an option to include initial slackness in the first cycle of loading. Many other

PWL models have been developed for the many different connection configurations

used in timber construction. These are reviewed in Foliente (1993).

Sivaselvan and Reinhorn (1999) developed a complex but general PWL hysteresis

model which incorporates strength and stiffness degradation and pinching. It is an

extension of Park’s three parameter model (Park et al., 1987). Models such as this are

much more powerful in their predictive capability and more general in their

application, but are very complex due to the ‘rule-based’ definition of the behaviour.

The complexity is illustrated by the branch numbering convention for the model

which is shown in Figure 3.4.


119

Figure 3.1 – Examples of piece-wise linear hysteresis models (Loh and Ho, 1990).


120

Idealised pinched hysteresis

load

displacement

D

Energy = E

elasto-plastic:energy equivalent

D/2Energy = E

elasto-plastic:displacement

equivalent

D

Energy = 2E

FuStiffness = k

k k

Fu Fu

Figure 3.2 – Comparison of idealised pinched hysteretic system with energy equivalent and displacement equivalent elasto-plastic systems.


121

Figure 3.3 – Model by Stewart (1987).

Figure 3.4 – PWL model from Sivaselvan and Reinhorn (1999).


122

Another approach to hysteresis modelling is the piece-wise non-linear (PWNL)

approach, where the hysteresis response is defined by a series of non-linear segments.

Dolan’s (1989) hysteresis model for nailed sheathing connections, shown in Figure

3.5, divides the hysteresis trace into four segments, and four boundary conditions,

with each segment described by an exponential equation. Kasal and Xu (1997)

developed a 20 parameter hysteresis model with full pinching degrading functionality,

where the loading and unloading paths are defined by several segments described by

exponential functions. Continuity requirements can be imposed at the transitions

from segment to segment, or released, to model discontinuities. This model is shown

in Figure 3.6.

A more general family of hysteresis models has been developed, and is referred to as

the distributed element (DE) family of models. This type of model was proposed by

Biot (1958) and Iwan (1966), and is based on a series of parallel spring, damper and

friction elements which when combined are capable of modelling complex hysteretic

behaviour. This approach to modelling has attracted recent attention from Mostaghel

(1999) and Deam (2000) who have both developed DE-based hysteresis models with

full pinching, degrading functionality which can accurately approximate real

hysteretic behaviour as observed in experiments on light-frame structures. These

models are described diagrammatically in Figures 3.7 and 3.8. The advantage of the

DE hysteresis models is that they can be defined analytically through mathematical

equations, rather than through a complicated set of rules, and can approach smooth

behaviour when many parallel elements are included. DE models can also be

formulated such that the parameters all have a direct physical interpretation, like a

mechanics-based model (see section 3.2.3).

Although the DE and PWNL hysteresis models are mathematically neater than the

PWL rule-based models, they are still quite complex in their formulation. A more

mathematically elegant approach to hysteresis modelling, in the opinion of the author,

is the differential approach to hysteresis modelling, where the hysteresis trace is

defined by a differential equation.


123

Figure 3.5 – Model by Dolan (1989).

Figure 3.6 – Model by Kasal and Xu (1997).


124

Figure 3.7 – Model by Mostaghel (1999).

Figure 3.8 – Model by Deam (2000).


125

A differential hysteresis model was initially proposed and formulated by Bouc (1967)

and has been the subject of many revisions, modifications and applications. Wen

(1976;1980), Baber (1980), Baber and Wen (1981), Baber and Noori (1985; 1986),

Casciati (1989), Capecchi (1991), Foliente (1993; 1995), Madan et al. (1997),

Sivaselvan and Reinhorn (2000) and Wang and Wen (2000) have all modified the

model in some way. Many others have applied the model in their research. It is clear

from the sheer volume of literature associated with the differential model, that

researchers have been attracted by its mathematical elegance and simplicity.

Baber (1980) and Maldonado et al. (1987) examined in detail the properties of the

Bouc-Wen model. The derivation of the model from the theory of visco-plasticity and

its resemblance to the endochronic constitutive theory are discussed in Mettupalayam

and Reinhorn (2000) and Sivaselvan and Reinhorn (1999). A limitation of the early

form of the differential model was first reported by Maldonado et al. (1987) and

Casciati (1989). They observed that the model would not form a closed loop during

partial loading-unloading cycles and would effectively soften during reloading

without stress reversal. Thyagarajan (1990) also observed this limitation. He

compared the response of DE models to the Bouc-Wen differential model under

different loading regimes and showed that the Bouc-Wen model can drift

unrealistically along the displacement axis under these circumstances. This limitation

has relatively minor consequences in the case of seismic analysis however because

seismic excitations tend to induce large stress reversals which are relatively few in

number. In this case the artificial drifting effect is only minor.

Wen’s (1976;1980) modification to Bouc’s original model is probably the most

significant, as he generalised the model so that it could be used for random vibration

analysis (RVA) through Equivalent Linearisation (EQL). In EQL, the non-linear

differential system is replaced by a ‘statistically equivalent’ linear system, for which

the techniques of RVA can be used to calculate structural response to random

excitations in terms of statistical parameters rather than deterministic values. The

differential model is ideal for use in EQL because of its mathematical simplicity. The

methodology of the EQL for the differential model of hysteresis is outlined in more

detail in Chapter 4.


126

Figure 3.9 – Possible hysteresis shapes of the basic Bouc-Wen model for n=1 (Baber, 1980).


127

Baber and Wen (1981) enhanced the differential model to include stiffness and

strength degradation and extended it to MDOF systems. Baber and Noori (1985;

1986), and Foliente (1993) added pinching capability whilst maintaining its

tractability for EQL, but for SDOF systems only.

In the following chapters of this thesis, Foliente’s (1993) pinching degrading

modification of Bouc’s original model is used extensively. The hysteretic shapes that

the modified Bouc-Wen model is capable of generating are shown in Figure 3.9 and

demonstrated in section 3.5. The model parameters are described in Table 3.1. The

model is formally outlined in section 3.4, and implemented in two different MDOF

systems in Chapter 4 – a hysteretic shear-building model, and a hysteretic shear-wall

model. An EQL scheme for the shear-building model is also formulated and

implemented. A system identification method is developed to determine the

parameters of the modified Bouc-Wen model, based on experimental data, in section

3.5.

3.2.3 Phenomenological Versus Mechanics-Based Hysteresis Models

As described earlier, hysteresis models can be classified as either phenomenological

or mechanics-based. The underlying philosophy of the phenomenological approach to

hysteresis modelling is that the experimentally observed behaviour of the physical

system can be represented by some form of mathematical function (or set of rules),

without having regard to the physical and mechanical processes which cause that

behaviour. Phenomenological hysteresis models at the member level can be easily

assembled into a complete structural system. The alternative to this, is the mechanics-

based approach. Mechanics based-models are based on the individual component

behaviour, and interactions between components in the system [e.g. reinforced

concrete fibre model by Fardis (1991)]. They provide better insight into how material

properties affect the response, and allow prediction of cyclic response of a system

based only on the properties of its individual elements.


128

Most of the hysteresis models outlined in section 3.2.2 are phenomenological in

philosophy. The DE family of models are the only possible exception, as they can be

derived from either a phenomenological or mechanics-based philosophy. Some of the

parameters of some of the phenomenological models do not have any physical

significance, although physical interpretations of their effect on model response can

be made. The advantage of the phenomenological approach to modelling is that in

theory, it allows great flexibility. A large range of systems can be represented by a

single model type, and very complex behaviour can be represented succinctly, and in

a mathematical form which is advantageous for higher-order analyses, such as in the

case of the differential model, which is tractable in approximate RVA using EQL (see

section 4.2.7). The disadvantage to this approach is that the model parameters can

often, in practice, only be obtained from experimental data, using some form of

system identification technique (see section 3.5). Each new structural or material

configuration to be modelled requires new experimental data. A solution to eliminate

this limitation is proposed later in this section, and is outlined diagrammatically in

Figure 3.10. Another limitation of the phenomenological approach to hysteresis

modelling, is that when the model parameters are based purely on an experiment of a

specific structural configuration, the parameters determined for one set of data, under

a single input function, may not necessarily predict the response under a different

input function. A method for minimising this problem, through ‘parallel system

identification’, is discussed in section 3.5.

Under the mechanics-based philosophy, physical processes and interactions between

conceptualised components of a structural system are modelled using the basic

principles of mechanics. For example, a nailed timber joint can be represented by the

combined mechanical behaviour of the fastener and of the timber members. Under

this approach, experimental data is not necessarily required for identification of model

parameters (although it is desirable), as the material properties of the individual

elements define the system. The material properties are already available for common

materials, based on previous experiments. However, if new materials are involved,

then parameters must be determined from new experiments. A mechanics-based

model for fasteners in timber structures has been developed by Foschi (2000). In this

mechanics-based model, the connecter (i.e. nail) is modelled as an elasto-plastic beam

in a non-linear medium (i.e. timber) which only acts in compression, allowing the


129

formation of a gap between the nail and the timber. The model automatically adapts

to the input , and develops pinching and degradation as the nail-timber gap increases.

A big advantage of mechanics-based modelling is that the model parameters all have a

direct physical meaning, and hence the values for the parameters are more easily

estimated when experimental data is not available. This approach is often preferred by

practitioners and researchers due to its compatibility with their understanding of the

way that physical and mechanical systems behave. However, the nature of the

inelastic behaviour of complex configurations may not always be fully understood.

A major disadvantage with the mechanics-based approach is the lack of flexibility –

i.e. every different mechanical or structural configuration requires a different ‘model’

to represent its behaviour, and every different failure mechanism associated with each

configuration must be specifically accounted for and included in the model. If a

particular failure mechanism is not considered, then there is no way that the model

can predict it. Mechanics-based models can also be mathematically messy, and are

generally not tractable in higher-order analyses. Models of large systems can become

extremely computationally demanding.

Phenomenological and mechanics-based models need not be used exclusively; they

can in fact be used in tandem via system identification, as is shown in the example in

Figure 3.10. In this scheme, mechanics-based hysteresis loops for a joint are

synthetically generated, eliminating the need for configuration-based experimental

data. System identification can then be used to directly obtain the parameters of a

phenomenological model, based on the output from the mechanics-based model. This

approach is only effective when the mechanics-based model gives accurate prediction

of the joint response. If this is not the case, then testing should be performed, the

results from which should then be used to directly obtain the parameters of a

phenomenological model using system identification.


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Mechanics-BasedModel Input

* Member

* Fastener

Cyclic or Arbitrary Load

Mechanics-BasedModel Response

System ID

Phenomenological Model

Incorporate

Structural Model

Load Historyu(t) -Displacementv(t) -Velocitya(t) -Acceleratione(t) -Energy::

System Response

Frame

Lumped Mass

Finite Element

-200

-100

0

100

200

0 4 8 12 16

-30

-15

0

15

30

0 30 60 90

Series1

Figure 3.10 – Diagram showing hybrid approach to response analysis using both

mechanics-based and phenomenological models (Foliente et al., 1998b).


131

3.2.4 Sensitivity of Response to Hysteresis Modelling

The previous section described a selection of the numerous hysteresis models that

have been developed, and highlighted the varying analytical complexity and

predictive capability of different approaches. Simple hysteresis models are desirable

for many reasons, including computational efficiency and ease of implementation, and

hence hysteretic pinching and degradation are often ignored in the hysteresis

modelling for structural analysis under seismic loads. This leads to some obvious

questions: Is this practice acceptable? Does the simplification produce conservative or

non-conservative results? Many researchers have attempted to answer these

questions, but their findings have been inconsistent.

Moss and Carr (1987) studied the dynamic response of several low-rise timber

structures, under different earthquake excitations, and varying approximations in

modelling the hysteresis. They concluded that variation in the hysteresis loop shape

does not have a major influence on the displacement demand under earthquake

excitation, although some of the results do not appear to support that conclusion.

Stewart (1987) also concluded that pinching, stiffness degradation, and hysteretic

damping have little influence on the predicted dynamic response, in terms of

displacement demand.

In contrast to the above findings, Aschheim et al. (1998) have shown that the peak

roof displacements of a reinforced concrete moment-resistant frame building that have

been computed using bilinear elements that lack cyclic stiffness degradation were less

than the actual displacements recorded during the 1994 Northridge earthquake. Other

models with stiffness degradation have been shown to better model the displacement

response and apparent elongation of period of this building (Moehle et al., 1997).

These corroborate the earlier finding by Chopra and Kan (1973), who investigated

how the ductility requirements for multi-storey, MDOF systems with stiffness

degrading behaviour compare with those with ordinary bilinear properties only.

Chopra and Kan also concluded that: 1) The differences in ductility requirements due

to stiffness degradation are generally smaller than those associated with probabilistic


132

variability of ground motions; and 2) Stiffness degradation leads to increased ductility

requirements for stiff buildings (i.e. period T=0.5 sec or lower) but has little effect on

ductility requirements for flexible buildings (i.e. T=2.0 sec or higher). This is

significant in the context of light-frame construction, since most light-frame structures

have periods within the range T=0.1 to T=1.0 seconds (Foliente and Zacher, 1994),

where the sensitivity to ground motion is the greatest.

Iwan and Gates (1979) studied the effects of deterioration, stiffness degradation and

crack-slip (i.e. pinching) on ductility demands and effective linear system parameters

(i.e. period and damping). They obtained and compared the maximum response of six

types of hysteretic systems – ranging from the simple bilinear to the stiffness-

degrading and pinching type – for an ensemble of twelve earthquakes. They found

that at low to moderate levels of excitation, pinching could reduce the system

response, but under strong excitations pinching effects are much less apparent. The

primary effect of deterioration and stiffness degradation is to increase the effective

period of the system.

More recently, Aschheim and Black (1999) studied the effects of prior earthquake

damage on response of SDOF systems that exhibit degradation and pinching.

Computed peak displacement responses were compared between an initially-

undamaged oscillator and a previously-damaged oscillator but not between oscillators

with different hysteresis models.

The commentary to the 1997 US National Earthquake Hazard Reduction Program

Recommended Provisions recognises the effects of pinching on energy dissipation

capacity of the system and hints that it may affect the assignment of response

modification coefficient used for design (Building Seismic Safety Council 1997).

Paevere and Foliente (1999) compared the effect of nine different hysteresis

modelling approaches, on predicted response and reliability under a range of

earthquake excitations, for one, two and three-storey Japanese style timber houses.

They showed that the assumed hysteretic degradation and pinching in a SDOF model

can have a significant effect on the prediction of peak responses and reliability. The

nine different types of hysteresis model used in this study are shown in Figure 3.11


133

(a). Each model type was subjected to a suite of artificially generated ground

motions, with variable peak ground accelerations, and a measure of the reliability was

calculated for each model type. The results are shown in Figure 3.11 (b). The

calculated reliability indices (β) given on the vertical axes, are defined as the value of

the standard normal variate, of the probability of exceeding the corresponding peak

displacement value given on the horizontal axes. Results are shown for system

frequencies of 5.5, 3.9 and 2.6 Hz, which are representative of one, two and three

storey houses, respectively. The plots shows that the hysteresis model used can have a

significant effect on the value of the reliability prediction obtained.

All of the studies reviewed in this section have provided useful insights, but have

adopted differing methodologies and used different response quantities for the basis of

the conclusions. Hence there is no general agreement or recommendations regarding

the effects of hysteretic pinching and degradation on system response. Some studies

have highlighted circumstances where the hysteresis assumptions can have a

significant impact on response predictions. Comprehensive studies need to be

undertaken to determine when it is appropriate to use simplified hysteresis models,

and when it is advisable to use a more complex model. It should always be understood

however, that the hysteresis modelling approach should be appropriate to the

objectives of the analysis.


134

Hysteresis Cases Case Hysteresis

Type Strength Degradation

Stiffness Degradation

Pinching

1 Bouc-Wen

Yes* Yes* Yes*

2 PWL Yes Yes Yes 3 PWL Yes Yes No 4 PWL No Yes Yes 5 PWL No Yes No 6 PWL Yes No No 7 PWL Yes No Yes 8 PWL No No Yes 9 PWL No No No

* Fitted to experimental data

-50

-25

0

25

50

-30 -20 -10 0 10 20 30Displ (cm)

Ft (kN) ExperimentFitted BWBNF (Case1)

Ft(kN)

-50

-25

0

25

50

-10 -5 0 5 10-50

-25

0

25

50

-10 -5 0 5 10

-50

-25

0

25

50

-10 -5 0 5 10-50

-25

0

25

50

-10 -5 0 5 10

-50

-25

0

25

50

-10 -5 0 5 10-50

-25

0

25

50

-10 -5 0 5 10

-50

-25

0

25

50

-10 -5 0 5 10-50

-25

0

25

50

-10 -5 0 5 10

Case 2 Case 3

Case 4 Case 5

Case 6 Case 7

Case 8 Case 9

Displ(cm)

(a) Hysteresis modelling cases 1-9 used to calculate reliability estimates under Tokyo ground motions (Paevere and Foliente,1999)

Figure 3.11 – Hysteresis pinching and degradation effects on reliability estimation.


135

-1.0

0.0

1.0

2.0

3.0

0 5 10 15 20 25 30

123456789

ββββ

2.6 Hz

-1.0

0.0

1.0

2.0

3.0

0 5 10 15 20 25 30

123456789

ββββ

3.9 Hz

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 5 10 15 20 25 30

123456789

peak displ (cm)

ββββ

5.5 Hz

(b) Reliability estimates (β) under Tokyo ground motions for hysteresis modelling cases 1-9 (Paevere and Foliente,1999).

Figure 3.11 (cont’d) – Hysteresis pinching and degradation effects on reliability estimation.


136

3.3 Overview of System Identification

3.3.1 Introduction

A simple mathematical expression, that describes the behaviour of a structural system

under a time-varying load can be written in the following synthetic form:

ˆ ˆRU F= (3.1)

where R is a differential operator, U is the vector of the system response and F is

the vector of the time-varying actions. Two types of analysis are relevant in seismic

performance studies of light-frame structures:

1) The operation performed by the operator R is known and the actions defined by the

vector F are also known. This is a response analysis problem and requires the

solution of the response, described by vector U . This type of analysis is addressed in

Chapters 4 and 5 of this thesis.

2) When the vectors F and U are known, the problem is reduced to identifying the

operator R. This is known as a system identification problem and forms the basis of

the topic discussed in the following review.

There are basically two steps in constructing a model of the constitutive or hysteretic

relations of systems under cyclic loads from measured data: (a) select a family of

candidate models that represents the general behaviour of the physical system, and (b)

choose a particular member of this family that best describes the observed data.

Hysteresis model selection or development is typically in mind in step (a) and

parameter estimation or system identification in step (b). Once the hysteresis relations

of the materials, elements or systems have been identified/selected, these elements are

then assembled to model the structural system.

As outlined in section 3.2, many hysteresis models are already available for light-

frame structures, and a preferred model for use in this work, based on the Bouc-Wen

differential model, is outlined in section 3.4. However the need to obtain accurate and

reliable model input values has always been identified as a critical research need


137

(Foliente 1997a, 1997b). The most commonly used method for evaluation of

hysteresis model parameters from experimental data, is trial-and-error, often using a

visual goodness-of-fit criterion. This is tedious and inconsistent. A systematic and

consistent method of parameter estimation is offered by system identification

procedures.

System identification is the process for constructing a mathematical description or

model of a physical system when both the input to the system and the corresponding

output are known. In structural engineering, the input is usually an excitation and the

output is the displacement, velocity, or acceleration response of the structure to this

excitation. Alternatively, as is often the case in seismic structural testing, the input

can be a displacement pattern and the output the corresponding pattern of resisting

force. The particular model obtained from the identification process should produce a

response that closely matches the system output, given the same input.

System identification methods may be generally classified as parametric or non-

parametric (Imai et al., 1989), or, in terms of sampled data or target quantities to be

estimated, as time-domain or frequency-domain. Parametric identification involves

iteratively modifying the parameters so as to minimise some objective function and is

independent of the model formulation, while non-parametric identification determines

the transfer function of the model in terms of its analytical representation and solves

for parameters directly. In the time domain methods, parameters are determined from

observational data, which are sampled in time, whereas in frequency domain methods,

damping ratios and modal response quantities are identified using frequency domain

measurements. In seismic response analysis of light-frame structures, the techniques

for parametric, time-domain identification are of most relevance because of the time-

dependent nature of load actions on a structure and the typical non-linear behaviour of

the response. Frequency domain methods inherently involve averaging of temporal

information, and hence discard some of the temporal detail. For light-frame systems,

in which the parameters can be expected to degrade over time, given the

experimentally observed characteristics, this averaging is not acceptable, and hence

only time domain methods are considered herein.


138

System identification is relevant to structural analysis problems for two major

reasons: (a) for obtaining a system model from laboratory or field data for predicting

the structural response of similar real-world systems under adverse environmental

loadings, and (b) for estimating the existing conditions of structures for the

assessment of damage and deterioration. The latter may be used to obtain an

‘improved’ or ‘updated’ mathematical model that better represents the characteristics

of the existing structure.

3.3.2 System Identification Methodology

The methodology of all parametric system identification algorithms is given by the

following three steps:

1) Determine the form of the model and the system parameters

2) Select a criterion function (e.g., ‘goodness-of-fit’) to compare the model response

to the actual response, when both the model and the actual system are subjected to the

same input, and

3) Select an algorithm for systematic modification of the parameters to minimise

discrepancies between the model and the actual system responses.

Many algorithms have been developed for the system identification process (i.e. step

3, above) such as, Simplex Method, Gauss-Newton method, Kalman Filtering,

Genetic Algorithm, Simulated Annealing and Gradient based methods. The theory,

and descriptions of the algorithms, are widely available in textbooks on optimisation

and operations research.

In one of the few studies on system identification of hysteresis parameters, in a

structural engineering context, Kunnath et al. (1997), used a Gauss-Newton approach

to determine the hysteresis parameters for reinforced concrete sub-assemblages. The

hysteresis model used in their study was based on the Bouc-Wen model described

earlier. Zhang et al. (2001) also determined Bouc-Wen hysteresis parameters for

structural systems, using different identification techniques, but these results were

based on simulated (i.e. model-generated) input data rather than real experimentally

determined data.


139

In this work, in section 3.5, the GRG method is used to identify the hysteresis

parameters of the modified Bouc-Wen model using a range of different input-output

data sets derived from physical experiments on light-frame structures. The GRG

method is an extension of the Wolfe algorithm (Charnes and Cooper, 1957) which can

accommodate both a non-linear objective function and also non-linear constraints on

the parameters. In essence, the method employs linear, or linearised constraints,

defines a new set of variables which are normal to some of the constraints, and then

transforms the gradient to this new basis (Himmelblau, 1972). The method is very

commonly used as it has been implemented into the ‘Microsoft Excel’ spreadsheet

software. Full details of the GRG method are given in Lasdon et al. (1978), and a

summary of the algorithm is reproduced in Appendix C.

3.4 Differential Hysteresis Model Formulation

3.4.1 Introduction

The following section outlines the differential hysteresis model which is used in the

structural modelling and performance analysis sections of this thesis. In Chapter 4, the

hysteresis model is implemented in to three different structural model representations:

• Hysteretic SDOF model (see sections 4.3 and 3.4)

• Hysteretic shear-building model (see section 4.4)

• Hysteretic shear-wall model (see section 4.5)

The hysteresis model is based on a version of the Bouc-Wen differential model which

was enhanced by Baber and Wen (1981), Baber and Noori (1985; 1986) and Foliente

(1993; 1995) to incorporate strength and stiffness degradation and pinching

functionality. These enhanced features make the model suitable for the analysis of

light-frame systems under seismic loads. In addition to this, the model is also tractable

in RVA through EQL, whereby response statistics under random excitations can be

determined directly, without the need to use computationally expensive simulation

techniques (described in section 4.2). A complete description and formulation of the

hysteresis model, including the theoretical basis, is given in Foliente (1993). A


140

summary of the essential equations and parameters of the model, and demonstration

of its capabilities are given in the following section.

3.4.2 Model Formulation

Consider a hysteretic structural system as shown in Figure 3.12. Assuming a single-

degree-of-freedom only, and a forcing function F(t), the equation of motion may be

generally written as:

mu cu R u t z t t F t�� [ ( ), ( ); ] ( )+ + = (3.2)

where u is the displacement, and the dots represent derivatives with respect to time.

The restoring forces acting on the mass m are broken into discrete components. The

inertial restoring force is given by mu�� . The damping restoring force is given by cu�

and is assumed linear. The non-damping restoring force is given by R u t z t t[ ( ), ( ); ] and

consists of a linear and a hysteretic (non-linear) component. The linear component is

given by αku and the hysteretic component is given by ( )1−α kz , where α is a

weighting constant representing the relative participations of the linear and non-linear

terms. If the equation of motion is mass normalised, the following standard form is

obtained.

�� ( ) ( )u u u z f t+ + + − =2 10 0 02

02ξ ω αω α ω (3.3)

where f(t) is the mass-normalised forcing function (i.e. acceleration); all other

undefined symbols are described in Table 3.1 The hysteretic displacement z is a

function of the time history of u. It is related to u by the following first-order non-

linear differential equation.


141

1( )

( )n nAu u z z u z

z h zν β γ

η

− − + =

� � �� (3.4)

where:

h z z u qz /u( ) . exp sgn �= − − −LNM

OQP10 1

222ζ ζb gc h (3.5)

ζ ε ζ ε1 10( ) [ . exp( )]= − −s p (3.6)

ζ ε ψ δ ε λ ζψ2 1( ) ( )( )= + +o (3.7)

ν ε δ εν( ) .= +10 (3.8)

η ε δ εη( ) .= +10 (3.9)

ε = z zudttt f

�0

(3.10)

1/

( )

n

uAZ

ν β γ = +

(3.11)

inertiaforcingfunction

F(t)m

linear

nonlinear(hysteretic)

viscousdamping

Figure 3.12 – SDOF hysteretic structural system.


142

Table 3.1 – Description of system properties and hysteresis model parameters.

Parameter / Property

Description

ω 0 natural frequency of the linear system = k m/

ξo damping ratio of the linear system

k Initial stiffness of the linear system

Fu maximum load

A parameter that regulates ultimate hysteretic restoring force

α ratio of linear to non-linear contribution to restoring force

β, γ hysteresis shape parameters (see Figure 3.9)

n parameter which controls the 'sharpness' of yield

ε calculated energy dissipation

δ δν η, strength, stiffness degradation parameters

( ) ( ),ν ε η ε strength and stiffness degradation functions

h(z) pinching function; if h(z) = 1.0, model does not pinch.

( )1ζ ε function that controls the severity of pinching

( )2ζ ε function that controls the rate of pinching

ζ s parameter that indicates degree of pinching

λ pinching parameter that controls the rate of change of ζ 2 as ζ1 changes

q pinching parameter that controls the percentage of ultimate restoring force

zu where pinching occurs

p pinching parameter that controls the initial drop in slope

ψ o pinching parameter that contributes to the amount of pinching

δψ pinching parameter that controls the rate of change of ζ 2


143

The model is capable of simulating a range of softening and hardening behaviour,

depending on the values of the main shape parameters β and γ , the range of basic

hysteresis shapes which are possible for various values of β and γ are shown in Figure

3.9. The cyclic degradation of strength and stiffness are linear functions of the

energy dissipation (Eqs. 3.8 and 3.9) and are controlled by the parameters νδ and ηδ .

Hysteretic pinching is also a function of energy dissipation (Eqs. 3.6 and 3.7) and is

controlled by the parameters ζ s , λ , q, p, ψ o and δψ .

By adjusting the parameters of the model, which are described in Table 3.1, a wide

range of different hysteretic behaviour(s) can be simulated. Some of these are

demonstrated in section 3.5. Although the model is powerful in its capability to

simulate complex behaviour, the relatively large number of model parameters means

that trial-and-error or eyeball evaluation of the parameters for a specific experimental

data set is tedious and inconsistent. A systematic and consistent method of parameter

estimation is needed. This is offered by system identification techniques, which are

demonstrated in the following section.

3.5 System Identification of Hysteresis Parameters

3.5.1 Introduction

In the following section, the GRG method of system identification is used to

determine the parameters for the modified Bouc-Wen model of hysteresis, as outlined

in the previous section, for a range of different experimental data sets. A parallel

system identification philosophy is also outlined and illustrated through two

examples. In these examples a single set of hysteresis model parameters is fitted to

two different experimental data sets simultaneously. The two different data sets are

derived from identical test specimens under different applied loading. Finally, the

modified Bouc-Wen hysteresis parameters are determined for the full-scale

experiment described in Chapter 2. The GRG method is used to determine the

parameters for the whole-building response, as well as individual wall sub-systems.


144

The hysteresis model which is used in this thesis (the modified Bouc-Wen model) and

the system identification algorithm employed herein (GRG method) need to be

validated for a range of test data, especially those obtained from evaluation of light-

frame subassemblies, and full-scale timber buildings. The following sections

demonstrate the versatility and applicability of the modified differential hysteresis

model, when the model parameters are identified using the GRG system identification

algorithm.

3.5.2 System Identification for a Range of Different Systems

To demonstrate the capability of the modified Bouc-Wen model to accurately track

the force-displacement level for a range of systems at all displacement levels, the

hysteresis parameters for four different structural systems were determined using the

GRG method, based on experimental data.

The four different structural systems considered are:

• A plywood sheathed shear-wall under pseudo-dynamic earthquake loading

(Karacabeyli and Ceccotti, 1998)

• A plasterboard lined light gauge steel frame under static-cyclic loading (Gad,

1997)

• A pre-cast concrete wall to slab connection under static-cyclic loading

(Robinson et al., 1999)

• A one-room Japanese post and beam house under pseudo-dynamic earthquake

loading (Watanabe et al., 1998)

Each of the four systems are modelled using a displacement controlled approach.

Under this approach, the input to the system is assumed to be the displacement

pattern, and the output of the system is the restoring force. This is how most static-

cyclic tests for seismic resistance are conducted. The parameters of the modified

Bouc-Wen model for each of the systems were then determined using the GRG

method, and the experimentally determined force-displacement data.


145

The objective function to be minimised was the error between the experimental and

model data. This error was based on the difference between the experimentally and

model determined restoring force and energy dissipation as follows

( ) ( )2 2

exp mod exp modf eerror W F F W E E = − + − ∑ ∑ (3.12)

where:

Fexp = experimentally determined restoring force (i.e. load)

Fmod = model calculated restoring force

Wf = weighting factor for force error

Eexp = experimentally determined energy dissipation

Emod = model calculated energy dissipation

We = weighting factor for energy error

The weighting factors We and Wf are used to control the importance of the force or

the energy component of the error in the system identification procedure.

Theoretically, if the force component of the error is zero, then the error in the energy

dissipation will also be zero because the hysteresis traces will match identically. In

practice, the error in the force calculation is never zero, and hence there are a set of

solutions which will give a similar level of error. By optimising for both force and

energy simultaneously, the set of parameters which also has the closest shape (and

hence energy dissipation) to the experimental data is computed. This is desirable

from a modelling point of view, as it may be important in some types of analysis to be

able to accurately predict the energy dissipation as well as the level of force or

displacement under seismic loading. In all of the system identification presented in

this thesis, the values of We and Wf were chosen such that the error contributions from

‘energy’ and ‘force’ were equal in magnitude. Each time the identification procedure

converged to a solution, We and Wf were adjusted to give equal ‘force’ and energy’

error components, and the procedure was re-run, until no further convergence could

be obtained.

A summary of the four experiments and the identified model parameters is given in

Table 3.2. Plots comparing the experimental- and the model-calculated responses, in


146

terms of the hysteresis response as well as energy dissipation are given in Figures 3.13

to 3.16. It is clear from these plots that the model is very good in simulating the

force-displacement response of the systems shown, even under extreme levels of

applied displacement. The energy dissipation can also be simulated very accurately.

Each of these four systems exhibit the kind of non-linear, pinching and cyclically

degrading behaviour which is typical of light-frame structures, and also other systems

(such as the reinforced concrete connection in Figure 3.15).

It should be noted, that in the displacement-controlled approach adopted here, no

dynamic response is considered, and hence the mass and damping parameters do not

form part of the identification problem. The hysteresis model used in the

identification procedure, calculates the restoring force as a function of the

displacement only, and hence only the parameters which affect the shape of the

hysteresis are determined. This is appropriate for identifying parameters from

displacement-controlled physical testing (such as static-cyclic testing) where the

‘input’ is the displacement pattern and the ‘output’ is the restoring force. It is

assumed in this work that this approach is also appropriate for pseudo-dynamic test

data. The procedure can be just as easily used to determine the parameters of the

system based on dynamic test data (i.e. from shake-table) as well. For this case, the

‘input’ would be the excitation acceleration and the ‘output’ would be the

displacement, velocity or acceleration response, which must be calculated using a

dynamic model.

3.5.3 Parallel System Identification

A possible limitation of the phenomenological approach to hysteresis modelling,

where the parameters are based purely on experimental data, is that the model

parameters determined for one set of data, under a single input function, may not

necessarily predict the response under a different input function. This can be a

problem because the whole purpose of having a model, is to use it for predicting

responses to unknown future excitations.


147

In order to determine the parameters of a phenomenological model, which is valid for

an infinite set of input functions (i.e. excitations), it is necessary to derive a single set

of model parameters, based on an infinite number of experiments where each

experiment uses an identical specimen and a different excitation. This is clearly not

possible. However, it is possible to derive the model parameters based on a finite

number of experiments which are conducted under a finite and realistic range of

excitations. In order for this to be possible, the system identification for each

different excitation data set must be conducted simultaneously or in parallel, and there

must be a single set of parameters in existence, which can adequately model the

behaviour of the system under all the excitations.

In this work, identical types of shear-wall construction, which have been physically

tested under two different excitations, are used to demonstrate this concept of parallel

system identification and show that it is feasible to derive a single set of satisfactory

model parameters for a system under at least two different excitations.

In the first example, a timber-frame shear-wall which has been tested ‘pseudo-

dynamically’ under a displacement response time history, obtained from dynamic

analysis (Karacabeyli and Ceccotti, 1998) using two different scalars of the

Northridge earthquake is fitted using the parallel system identification. The unique

set of parameters for this system is given in Table 3.3, and the comparison between

the experiment and model data is shown in Figure 3.17. The fit between the model

and experiment is excellent in terms of both the hysteresis trace and the energy

dissipation. In this example, the two different excitations used are similar in content

but different in magnitude.

In the second example a different configuration of timber shear-wall, which has been

tested under static-cyclic and then pseudo-dynamic earthquake loading (Kawai 1998)

is fitted using the parallel system identification. These two excitations are quite

different in content and in magnitude and are therefore more of a challenge to fit in

parallel than the data in the previous example. The unique set of parameters for this

system is also given in Table 3.3, and the comparison between the experiment and

model data is shown in Figure 3.18. Note that the values for stiffness and ultimate

load are different for the two different systems. This is because the static cyclic test


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was conducted on a single wall panel, and the pseudo-dynamic test was conducted on

a double wall panel. The construction type is identical, and it was therefore assumed

that all parameters except the stiffness and ultimate load could be considered

identical, and the system identification was constrained as such. The fit between the

model and experiment is still good in terms of both the hysteresis trace and the energy

dissipation, but is not as good as in the first example, or in the single dataset examples

shown earlier.

Figures 3.17 and 3.18 show that it is possible to obtain a satisfactory

phenomenological model, defined by a single set of parameters, which can simulate

the response of a system to very different excitations obtained from experiments.

This means that the ‘hybrid’ analysis technique shown in Figure 3.10 can also be used

in a similar fashion. A mechanics-based model (instead of an experiment) could be

used to generate the response under a range of different excitations. The parallel

system identification, can then be used, in principle, to compute a single

phenomenological model which satisfactorily simulates all the responses. The

feasibility of this technique, and the implications for analytical modelling and

performance prediction warrant further study. It may well be the case that models

based on multiple data sets with different excitations (analogous to ground motion

variability), are preferable to models based on experimental data containing multiple

repeats under the same excitation (analogous to material or system variability). If this

were the case then there may be implications in the development of future state-of-

the-art analytical modelling and laboratory testing strategies.

3.5.4 Identification of Hysteresis Parameters for Whole-Building Test Data

The previous sections have demonstrated the capabilities of the modified Bouc-Wen

differential model of hysteresis in simulating the experimentally observed behaviour

of light-frame structural systems taken from the literature. In the following section,

modified Bouc-Wen model parameters are identified for the full-scale destructive

experiment described in Chapter 2. Firstly, the parameters are determined for the

whole house response. These parameters are used later, in Chapter 5, in a SDOF

model of the house, for deterministic and stochastic seismic response analyses. The


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model parameters of the individual walls are also determined (where possible). These

parameters are used later in a hysteretic shear-wall model of the house, which

includes elements to represent the characteristics of the individual walls. This model

is developed and presented in Chapter 4, and is used in seismic response analysis in

Chapter 5.

As for the previous examples, the model parameters for the whole house and each of

the individual wall sub-systems were determined using the GRG method. As before,

the model was set with displacement as the input and restoring force as the output,

and as such the mass and damping have no influence. This is appropriate for this

static-cyclic displacement controlled experiment. The parameters were determined by

minimising the error between the experimentally derived and model determined

restoring forces and energy dissipation (simultaneously), with We and Wf determined

for equal ‘force’ and ‘energy’ error contribution.

A summary of the identified parameters for the whole house and individual walls is

given in Table 3.4. Plots comparing the experimental and the model calculated

response for the whole house and individual walls are given in Figures 3.19 to 3.25.

These plots demonstrate a very good fit for all the walls in terms of both hysteresis

trace and energy dissipation, however wall W4 is not as good as the others in the final

load cycles. Wall W4 is of similar construction to wall W2 (i.e. plywood and

plasterboard sheathed) but is pushed to much higher displacements than wall W2 in

the experiment. It seems that at very large displacements, the cyclic degradation

cannot be modelled as a linear function of energy dissipation – which is the

assumption in the differential hysteresis model. However at smaller displacement

levels, the degradation of the plywood/plasterboard sheathed walls can be adequately

modelled as a linear function of energy dissipation (i..e. for walls W1, W2, W5, W9)

in the differential hysteresis model. For wall W3, which is sheathed with plasterboard

on both sides, it appears that the degradation can be modelled as a linear function of

energy dissipation at all displacement levels considered.

It should be noted that the wall stiffness values given in Table 3.4 are quite different

from those derived from the initial cycles of the destructive testing, given in Table

2.7. This is because the stiffness values in Table 2.7 are based on the initial tangent


150

stiffness of the walls (values in first row), and on the secant stiffness at 1.0mm

displacement (values in second row), as measured in the destructive experiment. The

values in Table 3.4 however, are derived from the system identification process as

outlined above. They are calculated such that the best-fit hysteresis trace is obtained

over the entire response, not just at the origin, and have little to do with the initial or

secant (1.0mm) stiffness. This is because the hysteresis is non-linear from the origin.

The identified wall and house stiffness values, in Table 3.4, are appropriate for use in

seismic response modelling, as they enable accurate predictions well into the inelastic

range.

Note also that the degradation and pinching parameters, given in Table 3.4, are

identical for walls W1 and W2. Only the parameters which affect the stiffness and

ultimate strength are different (i.e. k an γ). These walls are of similar construction

and exhibit very similar hysteretic characteristics (except for the strength and

stiffness), so this is reasonable. The system identification for these walls was carried

out in parallel, with the problem constrained such that all the pinching and

degradation parameters were the same for both walls. The identical situation applies

to walls W5 and W9. The pinching and degradation characteristics of these two walls

are very similar – but the stiffness and strength are different, so the identification was

constrained appropriately. Parameters for wall W6 to W8 (i.e. walls W6, W7 and W8

grouped into a single wall), could not be fitted based on the experimental data,

because the response under the applied loading was too small. The pinching and

degradation parameters for wall W6 to 8 were assumed to be the same as for walls

W5 and W9, and the stiffness and strength parameters were estimated, using the

results from other walls, and the perforated shear-wall method (NAHBRC, 2000). It

should also be noted that in the experiment, walls W1, W2, W5, W6-8, and W9 had

no load applied directly, and as a result, experienced much smaller displacement

levels than the loaded walls (W3 and W4). Under these less severe displacements, the

degradation and pinching are not as pronounced, and as a result, the parameters which

control pinching and degradation do not affect the fit to the test data, as significantly

as for walls W3 and W4. The parameters assigned to all the walls, are technically

only valid for the range of displacements which they were subjected to in the

experiment, however, for the purposes of the modelling in Chapter 5, they are


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assumed to be valid for all displacement levels considered. Ideally, all of the

individual walls would have different parameters, because they have different

configurations and hence different behaviour would be expected, except for perhaps

the two end walls W5 and W9 which are almost identical except for their length.

However it was not possible to configure a single experiment of this type, such that all

the walls would all be subjected to large enough displacements to completely induce

the degrading pinching behaviour that would be preferable for the parameter fitting.

Table 3.2 – Fitted hysteresis parameters for various pinching, degrading structural systems.

System Timber-frame (4x2) plywood-sheathed

shear-wall without blocking

Light gauge steel frame house with

plasterboard lining

Pre-cast concrete panel to concrete slab connection

Japanese style timber framed post and beam house

Loading Pseudo-Dynamic VanCouver-0.57g Static-Cyclic Static-Cyclic Pseudo-Dynamic

Kobe-0.8g

Experiment Reference

Karacabeyli and Ceccotti (1998)

Gad (1997)

Robinson et al. (1999)

Watanabe et al. (1998)

k (kN/cm) 25.0 8.0 40.0 23.2

Fu (kN) 50.0 35.0 50.0 50.0

n 1 1 1 1

A 1.6 0.74 0.6 1

α

0.00100 0.00001 0.00001 0.00001

δν

0.0653 0.0000 0.0069 0.0040

δη

0.0727 0.0129 0.0766 0.5078

ζs 0.682 0.827 0.858 0.958

q 0.00357 0.03177 0.00000 0.00000

p 0.713 0.118 1.501 12.061

ψ

0.3513 0.8744 0.1982 0.0689

δψ

0.0032 0.0021 0.0000 0.0003

λ

1.147 2.000 1.269 0.318

β

0.809 0.714 0.599 2.000

γ -0.602 -0.745 -0.443 -1.735


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Figure 3.13 – Experimental and fitted hysteresis for a timber framed shear-wall without blocking [experimental data from Karacabeyli and Ceccotti (1998)].


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Figure 3.14 – Experimental and fitted hysteresis for a light-gauge steel-framed house with plasterboard lining [experimental data from Gad (1997)].


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Figure 3.15 – Experimental and fitted hysteresis for a pre-cast concrete wall to slab connection [experimental data from Robinson et al. (1999)].


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Figure 3.16 – Experimental and fitted hysteresis for a one-room Japanese-style post and beam house – [experimental data from Watanabe et al. (1998)].


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Table 3.3 – Fitted hysteresis parameters, using parallel system identification for two different shear-wall systems.

System Single skin timber

shear-wall post and beam construction

Double skin timber shear-wall post and beam construction

Timber-frame (4x2) plywood sheathed shear-wall

with blocking

Loading Static-Cyclic Pseudo-Dynamic Kobe-0.8g

Pseudo-Dynamic Northridge-0.23g

Pseudo-Dynamic Northridge-0.38g

Experiment Reference Kawai (1998) Karacabeyli and Ceccotti (1998)

k (kN/cm) 5.0 10.0 25.0

Fu (kN) 10.0 17.0 50.0

n 1 1 1

A 1 1 1

α

0.00001 0.00001 0.00001

δν

0.0100 0.0100 0.0072

δη

0.0275 0.0275 0.0005

ζo 0.945 0.945 0.952

q 0.0815 0.0815 0.0834

p 2.000 2.000 0.504

ψ

0.1320 0.1320 0.0016

δψ

0.0050 0.0050 0.0036

λ

0.398 0.398 10.000

β

0.200 0.200 0.915

γ 0.303 0.153 -0.415


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Figure 3.17 – Parallel system identification example 1 Experimental and fitted hysteresis responses for timber framed shear-wall under

0.23g Northridge earthquake and 0.38g Northridge Earthquake [experimental data from Karacabeyli and Ceccotti (1995)].


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Figure 3.18 – Parallel system identification example 2 Experimental and fitted hysteresis responses for Japanese style timber-frame shear-

wall under static-cyclic loading (single wall) and 0.8g Kobe earthquake (double-wall) [experimental data from Kawai (1998)].


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Table 3.4 – Fitted hysteresis parameters for L-shaped test-house and its individual wall sub-systems.

System House (N-S) W1 W2 W3 W4 W5 W6-8 W9

k (kN/cm) 119.0 4.0 20.0 60.0 35.0 40.0 40.0 45.0

Fu (kN) 106.0 12.0 30.0 35.0 55.0 30.0 30.0 40.0

n 1 1 1 1 1 1 1 1

A 1 1 1 1 1 1 1 1

α

0.00001 0.07533 0.07533 0.00010 0.00001 0.00001 0.00001 0.00001

δν

0.0164 0.0107 0.0107 0.1500 0.0150 0.0500 0.0500 0.0500

δη

0.0373 0.0130 0.0130 0.0500 0.0108 0.0500 0.0500 0.0500

ζs 0.951 0.900 0.900 0.980 0.960 0.950 0.950 0.950

q 0.00076 0.03413 0.03413 0.00152 1E-06 1E-06 1E-06 1E-06

p 3.478 3.672 3.672 6.374 2.783 2.428 2.428 2.428

ψ

0.0574 0.0261 0.0261 0.0805 0.0497 0.1160 0.1160 0.1160

δψ

0.0107 0.0267 0.0267 0.0875 0.0182 0.0001 0.0001 0.0001

λ

1.798 0.712 0.712 0.466 1E-06 0.838 0.838 0.838

β

0.219 0.500 0.500 0.258 0.200 1.690 1.690 1.690

γ 0.904 -0.167 0.167 1.456 0.436 -0.356 -0.190 -0.565


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Figure 3.19 – Experimental and fitted hysteresis of test house.


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Figure 3.20 – Experimental and fitted hysteresis of wall W1 from test house.


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3.6 Summary and Conclusions

This chapter presented a review of hysteresis modelling and system identification

techniques, and then determined hysteresis model parameters for a range of different

systems taken from the literature, and for global and individual wall responses of the

test-house described in Chapter 2. A ‘parallel’ system identification procedure was

also demonstrated. The key points and findings are summarised as follows.

• The literature on hysteresis modelling indicates that there is no general

agreement regarding the effects of hysteretic pinching and degradation on

predicted system response. Some studies have highlighted circumstances

where the hysteresis assumptions can have a significant impact on response

predictions. Comprehensive studies need to be undertaken to determine when

it is appropriate to use simplified hysteresis models, and when it is advisable

to use a more complex model.

• A formulation for a modified Bouc-Wen differential hysteresis model was

presented based on work by Foliente (1993; 1995). This model has pinching

and strength and stiffness degradation functions to accurately simulate the

behaviour of light-frame structures under seismic loading. The pinching and

degradation in the model are controlled by energy dissipation.

• The GRG method of system identification was used to determine the

parameters for the modified Bouc-Wen hysteresis model, for four different

experimental data sets, taken from the literature. These included timber,

concrete and light-gauge steel systems. The model-computed hysteresis

closely matched the experimental data in all cases, in terms of both hysteresis

shape, and energy dissipation. The examples demonstrated that system

identification provides a systematic and consistent method of estimating

hysteresis model parameters from real test data. The examples also

demonstrated the capabilities of the modified Bouc-Wen hysteresis model in

simulating real test data which exhibits highly pinched and degrading

behaviour.


168

• A ‘parallel’ system identification approach was outlined and illustrated

through two examples. It was demonstrated that a single set of hysteresis

model parameters can be satisfactorily fitted to two different experimental data

sets simultaneously, where the two different data sets are derived from

identical test specimens under different applied loading. The first example

involved a timber-frame shear-wall which was tested pseudo-dynamically

under two different scalars of the Northridge earthquake. The second example

involved a Japanese-style timber shear-wall, which was tested under static-

cyclic and then pseudo-dynamic earthquake loading The feasibility of this

technique, and the implications for analytical modelling and performance

prediction warrant further study, as this finding may have implications for

future analytical modelling and laboratory testing strategies.

• Finally, the modified Bouc-Wen hysteresis parameters were determined for

the test house using the experimental results presented in Chapter 2. The GRG

method was used to determine hysteresis parameters for the whole-building

response, and for the individual wall sub-systems. The model-computed

hysteresis closely matched the experimental data in all cases, in terms of both

hysteresis shape, and energy dissipation.

Chapter 4 – Structural Modelling

169

CHAPTER 4

Structural Modelling

4.1 Introduction

In the previous chapter, the modelling of the force-displacement relationship, or

hysteresis, for light-frame structures was addressed. This chapter extends from this to

examine analytical modelling of the seismic response of the complete light-frame

structure. Analytical modelling is very important for gaining understanding of

structural performance during earthquakes. Physical experiments, such as those

described in Chapter 2, provide valuable information about the behaviour of specific

structural components or full-scale configurations under specific loading scenarios,

but modelling is required to extend this knowledge so that it can be used to predict

responses of different configurations to future unknown events. Various modelling

strategies can be used to predict structural responses under specified earthquakes, or

to a spectrum of likely earthquakes. Such models can be used for parametric studies

to determine how various structural or earthquake characteristics contribute to the

overall performance, and for the development and calibration of codes of practice to

be used by designers. Modelling is also important to help understand data gathered

during damage surveys after seismic events, and data from controlled physical

experiments. The models can then be used to direct future experimental testing and

damage surveys in the right direction.

The long-term vision for this project is to make light-frame structures more affordable

and safer for people than they currently are through appropriate efficiency in the

structural system. Necessary to this goal is the development of modelling tools and

procedures that can be used to establish the structural performance, and the

performance criteria for light-frame buildings. These can then be used to optimise

their construction in some manner, i.e., find the most economical or environmentally-


170

friendly design for a given performance level. Accurate performance prediction for

light-frame buildings under natural disaster loading is important to provide the

consistent and reasonable levels of safety expected by the building occupants, as well

as the affordability and efficiency demanded by society. This technology is also

needed to facilitate innovation in the light-frame construction industry, by allowing a

more flexible performance-based approach to the development of new building

products and systems. The critical underlying assumption behind the performance-

based approach, is that performance can be predicted with consistency and reasonable

accuracy.

To help address these issues, this project aims to develop a range of experimentally

validated numerical models of the simple L-shaped one-storey house presented in

Chapter 2, and predict the performance of this structure under earthquake loading,

both deterministically and in terms of response statistics.

In this chapter, a range of different modelling approaches and strategies, covering a

range of modelling sophistication, are examined and presented. All the structural

models that have been developed use modified Bouc-Wen hysteretic elements, to

incorporate the non-linear structural behaviour which inevitably occurs when a light-

frame structure is subject to an extreme seismic event. The modified Bouc-Wen

hysteretic element is described and formulated in Chapter 3.

This chapter begins with an overview of common seismic response analysis

techniques, and a review of structural modelling of light-frame structures. Next,

SDOF and shear-building models are discussed in relation to modelling light-frame

structural response to seismic loads. A hysteretic shear-building model, for

deterministic response analysis of multi-storey light-frame structures is then

presented, and an EQL scheme, for direct stochastic response analysis is formulated

for the hysteretic shear-building. Next, a ‘shear-wall’ model for light-frame structures

is presented, in which each wall is represented by a single hysteretic element, with the

walls connected together at the ceiling level by a rigid frame. A strategy for

modelling light-frame structures using sophisticated FE models is discussed and

finally, a hybrid approach to response analysis is presented, using both complex and

simple structural models, linked via system identification. The hybrid approach


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facilitates the transition between global and local response predictions and between

the stochastic and deterministic domains.

4.2 Overview of Seismic Response Analysis

Techniques and Structural Modelling of Light-

Frame Structures

4.2.1 Background

Modelling of structural response to seismic loading involves the discretization of time

and space into a computable formulation. Structures are spatially continuous and

often complex in their form. Although continuous representations of very simple

structures are mathematically possible, complex systems are most easily and

practically analysed when discretized spatially in some manner. Three methods of

spatial discretization used in structural modelling are: 1) lumped-mass procedure; 2)

the method of generalised displacements; and 3) FE method; specific details of these

methods can be found in various textbooks (Clough and Penzien, 1993; Chopra 1995).

Seismic loading, and structural response to such loads is dynamic in nature, in the

sense that it is time-varying. This is important because most low rise light-frame

structures have a fundamental mode of vibration within the range of T=0.1 to T=1.0

seconds (Beck 2001, Foliente and Zacher 1994), which is similar to the range of

frequencies dominant in the spectral content of typical earthquakes. This is shown

graphically in Figure 4.1 where spectral densities of wind and earthquake loads are

compared with natural frequencies of common structures. This indicates that the

structural response of a light-frame structure could be amplified by resonance effects.

Because of this fact, seismic response analysis of light-frame structures must consider

the dynamic nature of the loads in some manner, and this usually involves some

means of temporal discretization. In contrast to this, the dynamic characteristics of

the load and response are not so important when considering wind loading, because


172

the spectral content of the wind is dominated by much lower frequencies than the

fundamental structural response and hence the response is essentially static.

Another important aspect in the seismic response analysis of light-frame structures is

the hysteresis relationship used to model the non-linear force-deformation behaviour.

In Chapters 2 and 3 it was highlighted that the components of light-frame structures

suffer degradation of strength and stiffness, under cyclic loading. This can result in

the dynamic characteristics of the structure (i.e. the natural frequency) changing

during an earthquake, which may shift the structural response towards resonance and

hence amplify the response. Seismic response analysis should therefore include

consideration of the cyclic degradation of the system properties, which may result

from the violent load reversals experienced by a structure during an earthquake. The

dynamic behaviour of timber structures is further complicated by the dependence of

the material properties to the rate and the duration of loading.

Figure 4.1 – Spectral densities of wind and earthquake loads, compared with natural frequencies of common structures (Ferry-Borges and Castanheta, 1971).

One of the biggest challenges in seismic response analysis, is dealing with the

uncertainties involved, and the associated variability in the response. There is

uncertainty due to the model representation of the real system, due to discretization,


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parameter estimation and the assumed physical behaviour inherent in the model,

uncertainty due to the variability in the structural properties, and most importantly,

uncertainty due to the variability in the assumed excitation. Although significant in

magnitude, the uncertainty due to the modelling assumptions and the structural

properties is relatively much smaller than the uncertainty associated with the

earthquake loading used in the analysis. A structure that has been analysed based on

only one or two earthquake records may behave very differently when analysed under

an earthquake with different characteristics. Deterministic modelling cannot account

for these uncertainties, yet it is the treatment of these uncertainties, and particularly

uncertainty in the loading used in the analysis, which is probably the most important

aspect of seismic response analysis. Non-deterministic or stochastic methodologies

are essential in order to develop useful performance prediction tools for light-frame

buildings under natural disaster loading.

There are six main methods of seismic response analysis, which address the issues

raised above to varying degrees using different methodologies. The methods are :

• Static analysis

• Non-linear pushover analysis

• Time-history analysis

• Response- spectrum analysis

• Monte-Carlo simulation

• Random Vibration Analysis

Each of these methods is described briefly in the following sections, followed by a

review of the structural modelling of light-frame structures.

4.2.2 Static Analysis

Static analysis is the simplest method for analysing structural response to an

earthquake. It is the most commonly used method in design, since it forms the basis

of most seismic design codes. In this method, the structure is designed to resist a

static lateral force, which is assumed to be equivalent to the lateral force exerted by an

earthquake. The magnitude and distribution of the equivalent static lateral forces are


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determined using an estimate of the fundamental period of the building and the mass

of the structure. The method assumes that the structural response is such that its

maximum deflected shape under earthquake loading is similar to the fundamental

mode shape, and that the mode shape may be approximated by a simple equation. In

effect the building is assumed to behave as a SDOF system. Various versions of this

procedure are used in different design codes around the world, but they are all based

on these same underlying assumptions. This approximate method is far too simplistic

to realistically predict structural response and as a result of this, has been constantly

refined in an attempt to address its limitations. The advantage of the method is of

course its simplicity, which is the main reason for its wide acceptance. The method

may be appropriate for simple regular buildings. For more complex buildings, it is

most useful at the preliminary analysis and design stage, before more sophisticated

analyses are conducted.

4.2.3 Pushover Analysis

Non-linear pushover analysis techniques are becoming increasingly popular in

earthquake response analysis in earthquake research and design. In this method an

increasing monotonic load is applied to a model of a structure which incorporates

non-linear elements. The structural model will typically contain linear beam and

column elements, with plastic hinges at pre-defined locations. The sequence of

formation of the plastic hinges and the critical global collapse mechanism are tracked,

and an estimate of the global lateral load-displacement relationship for the structure

can be obtained. The advantages and disadvantages of pushover analysis were

extensively examined by Krawinkler and Seneviratna (1998). This study

recommended these methods be used with care since they cannot represent the

dynamic nature of seismic structural response with a high degree of accuracy.

An extension to this method, originally developed by Freeman (1978) and elaborated

by Fajfar (1999), is known as the Capacity Spectrum Method and is described

graphically in Figure 4.2. In this method, the load-deformation relationship for the

structure is obtained from a non-linear pushover analysis and then plotted together

with the displacement demand spectrum, after conversion of the variables into


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compatible forms. The displacement demand spectrum is derived from a response

spectrum (see section 4.2.5) for an earthquake with characteristics appropriate for the

building site. The point where the capacity diagram intersects the design spectrum is

known as the ‘demand point’, and is an estimate of the maximum displacement which

will be experienced by the structure under an earthquake with the characteristics

represented in the demand spectrum. Krawinkler (1995) pointed out that the inherent

natural frequency associated with this demand point may have little to do with the

dynamic response of the inelastic system. In essence, the limitations of this method

are that it is primarily based on a static analysis, although dynamic considerations are

included in the derivation of the site response spectrum, in a simplified form.

4.2.4 Time-History Analysis

In time-history analysis, the response of the structure under a specified acceleration

time-history is estimated, based on the equations of motion. Closed form solutions can

be obtained when the excitation can be described analytically but the responses are

typically computed using numerical time-stepping methods such as the Newmark-β

method (Newmark, 1959) or the Wilson-θ method (Wilson et al., 1973). These

methods are based on dynamic equilibrium using the equations of motion, where the

excitation and response are discretized in time. The complete time-history of the

response to a given excitation is calculated, and hence evaluations of peak responses

and crossings of various thresholds can be obtained. In this approach, the structural

model is formulated in terms of the equations of motion, so unlike response spectrum

or pushover methods, the dynamics of the problem are inherently considered. Non-

linearities including cyclic degradation and pinching can be included in the model,

and their effect on the dynamic response is therefore also included in the analysis.

Non-linear time-history analysis is the most accurate method for predicting seismic

responses, but for complex structures it can be very numerically intensive. The major

limitation of this method, as with all deterministic methods, is that the calculated

response is entirely dependent on the prescribed input function, which is usually

unknown. This problem can be addressed to some extent by the use of simulation

techniques (see section 4.2.6) but intense numerical computation is required.


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Figure 4.2 – Capacity Spectrum method (Chopra and Goel, 1999).

Time-history analysis has been used widely by researchers, and has provided the

underlying basis and understanding used to develop and refine many of the simplified

analysis techniques. It is currently becoming more widely used in general earthquake

engineering practice as commercial analysis software packages now routinely include

time-history analysis functionality. Implementation of non-linear time-history

analysis functionality into commercial software is less common, but is increasing with

the increasing interest in, and availability of, various types of hysteretic elements

which can simulate non-linear behaviour.


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4.2.5 Response Spectrum Analysis

In the response spectrum method, the maximum response of a structure to an

earthquake design spectrum is calculated directly, using an estimate of the structure’s

fundamental period and damping ratio. A response spectrum is obtained by plotting

the peak responses to an earthquake for different combinations of natural frequency

and damping ratio. The values of the peak responses are obtained from time-history

analyses of a SDOF oscillator. A response spectrum for the El-Centro ground motion

is shown in Figure 4.3 (a). To create a more practical spectrum for use by designers,

the process is then repeated for many different earthquake records, and the results are

averaged and smoothed into a design spectrum as shown in Figure 4.3 (b). The peak

pseudo-acceleration, pseudo-velocity or displacement demand, corresponding to the

estimated natural period of the building is taken directly from the design spectrum,

and can be compared with the structure’s capacity to resist that demand level. The

essence of this method, is that the dynamic analysis has already been performed for

the full range of possible structures, and a range of different earthquakes, using very

simple models, and then presented in a convenient format for use by practitioners.

The method can be extended to MDOF systems using modal combination methods

such as square-root-sum-of-squares or complete-quadratic-combination, although

these methods have no theoretical basis. Inelastic response can also be included

through the use of an inelastic design spectrum, as is also shown in Figure 4.3 (b). The

inelastic spectrum is a function of the assumed structural ductility, as well as the

natural period and damping. The inelastic response spectrum can be obtained from

time-history analysis of inelastic oscillators, or can be approximated based on the

elastic spectrum and a correction factor (Mahin and Bertero, 1981), although the

relationship between the elastic and inelastic spectrum is not clear. Full details of the

basic Response Spectrum Method can be found in most structural dynamics textbooks

(Clough and Penzien, 1993; Chopra 1995). The advantage of this method is that it is

as simple to use as a static analysis, but takes into account the basic vibrational

properties of the structure, such as natural period and damping, and also the dynamic

characteristics of the ground motion, which are used to derive the spectrum.


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(a) Elastic response spectrum

(b) Elastic and inelastic design Spectrum

Figure 4.3 – Response Spectrum method (Chopra, 1995).


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The main disadvantages stem from the fact that the design spectra are derived using

SDOF dynamic models, which often are too simple to accurately represent the

dynamic characteristics of real structures. The validity and accuracy of the use of

modal superposition techniques and inelastic design spectra (based on elastic values)

are also questionable.

4.2.6 Monte Carlo Simulation

A limitation of all of the above methods is that they are deterministic in nature,

whereas seismic response behaviour is essentially a non-deterministic or stochastic

phenomena. A structure that has been analysed based on only one or two earthquake

records may behave very differently when analysed under an earthquake with

different characteristics. This poses a problem because the purpose of seismic

response analysis is to predict responses to future unknown excitations. The

variability which is inherent due to the unknown nature of this excitation, is the

biggest contributor to the overall uncertainty of seismic response predictions. Other

sources of uncertainty stem from the modelling assumptions and discretization, and

the randomness of the material properties which determine the structural

characteristics. Sources of uncertainty in the analysis of structural response are

discussed in Melchers (1999).

Monte Carlo Simulation (MCS) is one method for dealing with these uncertainties,

and is based on repeated deterministic analyses, such as time-history analyses. MCS

techniques emerged in the 1960’s and 1970’s following the advent of digital

computers, which made it possible and practical to apply the technique. The basic

concepts of MCS are given in many texts, and are described in the context of

structural dynamics in a pioneering paper by Shinozuka (1972).

In MCS, the sources of uncertainty in the analysis are identified and quantified in

terms of statistical distributions of the model parameters or classes of excitations.

Parameter values and excitations are then sampled at random and deterministic

analyses are performed repeatedly. The output of these analyses can then be collated

and expressed as distributions of response quantities, such as peak displacement,


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rather than as single deterministic values. Statistical distributions can then be fitted to

the responses so that the response quantities can be expressed in probabilistic terms.

Extreme values of a given response level, can then be associated with a given

probability level. MCS can be used to estimate the probabilities of exceedance of

different limiting states of structural response, or to study the contributions of various

sources of uncertainty to the overall variability of the response. The technique can be

applied to any system, so long as a deterministic model of the system is available.

The accuracy of the results of MCS depend on the accuracy of the model, the accurate

quantification of the statistical distributions associated with the model parameters, the

number of simulations performed, the generation of truly random numbers, and

importantly, in the accuracy in characterising the excitations. Unfortunately, real

ground motion records for particular sites are very scarce, and hence MCS techniques

usually depend on digitally simulated records, or on records from locations which are

judged to be appropriate. So practically speaking, MCS is only as good as the

simulated excitations which are used. MCS is rarely used by practitioners because it

is extremely computationally intensive, especially for complex structures. However

various importance sampling techniques, such as ‘Latin-Hypercube’ and ‘Russian

Roulette and Splitting’, have been developed to reduce the computational intensity.

These techniques are examined in Pradlwarter and Schueller (1999). As computer

power increases, and demand for more accurate analysis grows, MCS or similar

techniques are likely to become more commonly used in the future.

4.2.7 Random Vibration Analysis and Equivalent Linearisation

Random Vibration Analysis (RVA) was developed in the 1950’s in response to the

emergence of jet propelled aircraft. According to Roberts and Spanos (1990),

engineers discovered that they could not analyse complex phenomena, such as the

spatial and temporal variation of in-flight pressure on aircraft panels, using traditional

deterministic techniques. A new probabilistic approach was required which described

the excitations and responses in terms of statistical parameters rather than

deterministic values. The computers of the day were not capable of performing


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significant simulation-based analyses (such as MCS), and so simplified stochastic

response analysis techniques were developed to calculate response statistics.

In RVA, excitations are modelled as stochastic processes, such as filtered white

noises or Poisson processes, and response statistics, such as the standard deviation of

the response, are solved for directly, without the need for simulation. Extreme values

of the response are related to the standard deviation (this is demonstrated in section

5.4), and can be calculated using various methods such as those developed by

Davenport (1964), Shinozuka et al. (1968) or Michaelov et al. (2001). One very

rough rule of thumb is to take the maximum value of the response as three times the

standard deviation. The basic concepts and methods of RVA are given in a number of

texts (eg, Yang 1986; Soong and Grigoriu 1993; Roberts and Spanos 1990). In the

RVA technique, the stochastic characteristics of the response can be directly related to

the stochastic characteristics of the excitation through the systems frequency response

function, provided the system is linear. However, for non-linear systems, exact

solutions are very difficult, and only exist for a very small class of systems. Because

of this limitation, approximate solutions have been developed. Approximate

solutions include Markov methods, perturbation and functional series methods,

moment closure techniques and EQL. These and other techniques are reviewed in

Spanos and Lutes (1986).

For seismic response analysis, which can be strongly non-linear, the EQL technique

has been widely used (Branstetter et al. 1988, Roberts and Spanos 1990) because it

gives reasonably good results (accurate to within 10-15%), even for strongly non-

linear systems, and can be implemented easily for MDOF systems (Cho, 2000). The

basic concept of EQL is outlined in Figure 4.4., where it is shown that a non-linear

system is replaced by a ‘statistically equivalent’ linear system. The ‘statistically

equivalent’ stiffness and damping parameters of a linear system are typically derived

by minimising the mean square error between the responses of the linear and non-

linear systems under random excitation. If the excitation for each of the systems is

assumed to be a random process, then the response difference, or ‘error’ between the

linear and non-linear system is also a random process. This error can be calculated

and minimised using stochastic formulations of the equations of motion.


182

nonlinearsystem

C K Z

mF(t)

nonlinearsystem

C K Z

mF(t)

‘statistically equivalent’linear system

Ce Ke

mF(t)

‘statistically equivalent’linear system

Ce Ke

mF(t)

Random VibrationAnalysis

Random VibrationAnalysis

EquivalentLinearisation

EquivalentLinearisation

µµµµ

Response Statistics

σσσσ

µµµµ

Response Statistics

σσσσ

Figure 4.4 – Equivalent Linearisation method.


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The resulting ‘statistically equivalent’ linear system is then used for RVA, and the

response statistics of this system are computed. Full details of the general procedure

of EQL are given in textbooks on random vibration such as Soong and Grigoriu

(1993), Roberts and Spanos (1990) and Cho (2000).

The EQL technique is an approximate alternative to MCS for calculating response

statistics of a non-linear structure under random excitation. However MCS is

becoming more and more practicable, due to the rapid and continuing increase in

computational power. Even if MCS becomes widely used in practice in the future,

simplified techniques for stochastic response analysis will still be required for

analysing complex structures with many degrees of freedom, and for preliminary

assessment of structural response statistics in lieu of simulation.

4.2.8 Whole-Building Models of Light-Frame Structures.

In the preceding sections, different seismic response analysis techniques and

methodologies have been outlined. Each of these methodologies requires some kind

of structural model to predict response to a prescribed input. The structural model is

defined here as the discretized mathematical representation of the mass, damping,

stiffness and hysteretic characteristics of the structure, used to numerically calculate

responses to a given loading.

The complexity of a structural model should be chosen according to the objectives of

the analysis, and the computational requirements of the response analysis technique

which is used. The choice of model complexity is important in the seismic response

analysis of light-frame structures, because real light-frame structures are highly

complex, highly redundant systems, which can be difficult to discretize or simplify

meaningfully. A range of whole-building models with varying degrees of complexity

and analysis capabilities, have been developed for light-frame structures. Models for

static analysis (predominantly) have been developed in studies by Gupta and Kuo

(1987), Yoon (1991), Schmidt and Moody (1989), Kasal (1992), Gad (1997), He et

al. (2001), Koerner et al. (2000) and Andreasson (2000). Models for dynamic


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analysis have been developed by Gad (1997), Ceccotti et al. (2000) and Fischer et al.

(2001). These models are reviewed in the following.

Static Analysis Models

The models selected for review below, have been developed predominantly for static

analysis, and are only suitable for seismic response analysis using either static or

pushover techniques (see sections 4.2.2 and 4.2.3).

Gupta and Kuo (1987) and Kuo (1989) developed a whole-building model to analyse

the house tested by Tuomi and McCutcheon (1974) (see section 2.2.3). The house

model was constructed from simple elastic shear-wall elements which included uplift

and racking deformations. They were able to predict the experimental results with

reasonable accuracy but were not capable of predicting non-linear behaviour. Yoon

(1991), extended this model to include non-linear behaviour, and also analysed the

house tested by Tuomi and McCutcheon (1974).

Schmidt and Moody (1989) developed a three-dimensional model assembled from

simplified shear-wall representations, which incorporated non-linear fasteners,

connected together through rigid roof and floor diaphragms. The model was used to

predict the behaviour of the house tested by Tuomi and McCutcheon (1974) (see

section 2.2.3).

Kasal (1992) developed a three-dimensional FE model based on the ANSYS FE

software. The three-dimensional model was used to evaluate load-sharing in a full

structure, based on the full-scale experiments on a simple box-style house by Phillips

(1990). In a similar fashion, the experiments described in Chapter 2 of this thesis are

to be used to validate the FE models by Kasal (1992) and study the load-sharing, but

for a ‘real’ house with irregular plan layout and different wall configurations. More

details of this FE model are given in section 4.6.

Gad (1997) developed FE models, similar to those developed by Kasal (1992) for a

one-roomed brick-veneer house that he had tested (see section 2.2.3). The FE models

were formulated with static analysis in mind, and were validated using the results of


185

racking tests. The models were then used to study the sensitivity of the racking

response to different boundary conditions and structural configurations.

Koerner et al. (2000) developed a three-dimensional FE model of the manufactured

house tested by Richins et al. (2000) under simulated wind loads. Full composite

behaviour of the timber-frame and the OSB sheathing panels was assumed and hence

all walls and diaphragms were modelled using orthotropic shell elements. Inelastic

‘link elements’ were used to model the connections between the sub-assemblies. The

model was verified using the results of static experiments conducted on the house (see

section 2.2.3) and was used to demonstrate the importance of boundary conditions on

the predicted global displacements. Due to the complexity of this type of model, it is

best suited to static analyses, or limited dynamic analyses.

He et al. (2001) developed a non-linear FE model for light-frame timber buildings,

which implemented a mechanics-based hysteresis element (Foschi, 2000) to represent

individual sheathing to frame connections. The model uses the basic material

properties, and the load-deformation characteristics of the connectors as the input

parameters. The model was verified using the results of individual wall-tests and then

used to examine the three-dimensional response of eccentric structures. This model is

capable of predicting individual nail response within a three-dimensional building,

and is therefore quite complex, and is at the moment, most appropriate for static

analyses only.

Andreasson (2000) developed FE models of multi-storey timber-frame buildings

using the ANSYS FE software. Linear isotropic beam and shell elements were used

to model the framing and sheathing elements, respectively, and non-linear-elastic

springs were used to model the connections within and between diaphragms. The

models were validated and calibrated based on a program of racking tests on different

configurations of shear-walls, and small specimen connection tests. The calibrated FE

model was then used to investigate the interaction between the various structural sub-

assemblies and the load-sharing under static loading, and it was shown that load

distribution is mainly influenced by the inter-component connections. No cyclic or

dynamic loading effects were considered in this study. The FE model did not include


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elements which incorporate inelastic hysteretic behaviour, and are therefore only

suitable for monotonic static analyses of light-frame structures.

Dynamic Analysis Models

The models reviewed below have been developed predominantly for seismic response

analyses using time-history analysis techniques (see section 4.2.4). The models by

Gad (1997) and Fischer et al. (2001) may also be appropriate for limited simulation

studies (see section 4.2.6), due to their relative simplicity.

The model developed by Gad (1997), is a simplified model for dynamic analysis of a

single-room brick-veneer house using RUAMOKO (Carr, 1998). The model used the

Stewart model of hysteresis (Stewart, 1987) to represent the in-plane hysteretic

behaviour of the in-plane walls. The mass associated with the bricks in the out-of-

plane walls was also included in the model, as it significantly affects the dynamic

response in a brick-veneer structure. The model was verified using the results of

shake-table and swept-frequency dynamic tests (Gad, 1997). The model was used for

sensitivity analyses, and to determine the extra loading which is imposed on the

structural frame during an earthquake, by the out-of plane brick-veneer walls.

Ceccotti et al. (2000) developed a more detailed whole-building model for dynamic

analysis, using the DRAIN-3D software. In their approach, individual walls of the

light-frame structure are represented by equivalent frames with semi-rigid joints. The

frames are then connected together with braced diaphragms with equivalent

characteristics to the floor system. A ‘Florence’ pinching hysteresis model (Ceccotti

and Vignoli, 1989) is used for the semi-rigid connections in the equivalent frames.

The hysteresis parameters for the equivalent frames were determined from individual

wall tests. The model was used to perform time-history response analyses of a multi-

storey timber frame building under various historical earthquakes, and to determine

the effect of diaphragm flexibility on seismic performance.

A simpler whole-building model for dynamic analysis has been developed by Fischer

et al. (2001), as part of the CUREE-Caltech Woodframe project (Hall, 2000). In their

‘pancake’ model, which is implemented using RUAMOKO (Carr, 1998), each of the


187

walls of a two-storey box-type timber frame structure are represented by zero-length

hysteretic springs which use the Stewart (1987) model of hysteresis. The contribution

of the out-of-plane walls is ignored. The roof diaphragm is modelled using plate

elements, and the floor diaphragm is modelled using a combination of plate and beam

elements. The model was used to predict the results of shake-table tests conducted on

a full-scale two-storey timber-frame house (Fischer et al., 2001) (see section 2.2.3).

Six other dynamic analysis models, of varying complexity, were also developed to

predict the results of the testing by Fischer et al. (2001) as part of an international

benchmarking exercise, sponsored by the CUREE project. The objective of the

exercise was to assess the state-of-the-art numerical analytical models in blind

predicting the inelastic seismic response, measured during shake-table tests. Details of

the results of the models and the outcomes can be found in the proceedings of a

workshop on the exercise (CUREE, 2001). Most of the participants used some form

of hybrid modelling approach (see section 4.7), where detailed models were used to

characterise individual wall behaviour, and simpler models used to predict dynamic

response. The model presented by Ceccotti et al. (2000), which is described above,

was most successful in predicting the dynamic response.

4.2.9 Scope of Structural Modelling and Seismic Response Analysis in

Current Work

The previous sections have provided an overview of a variety of structural modelling

and seismic response analysis techniques, which range from the conceptually and

computationally simple to the conceptually complex and computationally intensive.

Each model and method has advantages and disadvantages and the choice of method

of-course depends on the analysis objectives (Seible and Kingsley 1991; Foliente

1997a). In the following sections of this chapter, different structural models,

appropriate for the analysis of the dynamic seismic response (deterministic and

stochastic) of light-frame structures, are discussed, developed and presented. Five

types of models are developed or examined:

1. SDOF model – section 4.3

2. Shear-Building model – section 4.4


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3. Shear-Wall Model – section 4.5

4. Finite-Element Model – section 4.6

5. Hybrid Approach – section 4.7

The analysis capabilities of these modelling strategies are summarised in Table 4.1.

The shear-building, and shear-wall models were developed by the author, specifically

for this work, as was the stochastic EQL scheme for the shear-building model. The

models incorporate the modified Bouc-Wen differential model of hysteresis

(described in Chapter 3) to account for inelastic structural behaviour and are used for

seismic response analysis in Chapter 5. This includes analysis of the test house

described in Chapter 2, and an example three-storey building. The FE and hybrid

modelling strategies are introduced, so as to put some of the broader goals of this

project into context, but they are not used in this work.

Table 4.1 Analysis capabilities for different modelling strategies

Model Input/Output Resolution Analysis Capabilities

SDOF /

Shear-Building - global only

- static

- dynamic

- probabilistic

Shear-Wall - global

- sub-assembly level

- static

- dynamic

- limited probabilistic

Finite-Element

- global

- wall level

- panel / Member level

- connector level

- static

- limited dynamic

- limited probabilistic

Hybrid

- global

- wall level

- panel / Member level

- connector level

- static

- dynamic

- probabilistic


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4.3 SDOF model

4.3.1 Background

A SDOF model is the simplest possible model of a structure which can be used for

seismic response analysis. The dynamic behaviour of a SDOF system is limited to

first mode responses and global behaviour only. Very few real structures are

accurately modelled as SDOF systems. Since light-frame structures are

predominantly low-rise, with relatively high natural frequencies of vibration, the

majority of the response contribution is usually from the fundamental racking mode.

A SDOF approximation is therefore sometimes a reasonable assumption when

determining global responses. In the dynamic impact testing in Chapter 2, it was

shown that the test house can be reasonably approximated as a SDOF model, with all

mass lumped at the ceiling level. The SDOF model is useful for obtaining a basic

understanding about how the dynamic and global hysteretic characteristics affect the

global response of a light-frame structure, however no information about local

responses or failure mechanisms can be obtained. The SDOF modelling approach has

been used widely in many different applications by researchers. The results of these

studies have provided many valuable insights into structural behaviour under seismic

loads, and have laid the foundation knowledge for the development of simplified

design and analysis techniques such as the Response Spectrum Analysis method.

4.3.2 Formulation

Consider the hysteretic SDOF model shown schematically in Figure 3.12. In physical

terms, this model may be considered to be a structure with a lumped mass

concentrated at the roof level, and a massless lateral load resisting structure

underneath consisting of a non-linear and a linear-elastic spring in parallel with a

viscous damper. This particular SDOF idealisation is implemented in Chapter 3 using

the modified Bouc-Wen hysteretic force-displacement relation for the non-linear

spring. The equation of motion for this system is outlined in detail in section 3.4, and

this is the formulation for the SDOF hysteretic model for non-linear time-history


190

analysis. The dynamic response history for this system, for a prescribed input

function can be solved by using a numerical solution technique for solving the

equation of motion, such as the Wilson-θ method (Wilson et al., 1973). The equations

can also be formulated in a state vector format, which reduces the problem to a series

of ordinary differential equations (ODEs), which are then solved using a numerical

technique appropriate for solving stiff sets of first order equations such as the

predictor-corrector method. The SDOF hysteretic model formulated in section 3.4,

has been implemented in a computer program. In Chapter 5, it is used for seismic

response analysis of the single-storey L-shaped house described earlier.

4.3.3 Equivalent Linearisation of SDOF model

One of the major advantages that the modified Bouc-Wen hysteresis model has over

other models of hysteresis, is that it is mathematically tractable for approximate RVA

using the EQL technique. The basic concepts of RVA and EQL are outlined in section

4.2.7. In short, EQL is a fast method to calculate response statistics to an input

function described as a stochastic process, without the need to use MCS (see section

4.2.6).

Wen (1976) first formulated an EQL scheme for the differential hysteresis model

proposed by Bouc (1967), which is the ancestor of the hysteresis model used

extensively in this work. Baber and Wen (1981) enhanced the differential model to

include stiffness and strength degradation and extended it to MDOF systems while

maintaining the EQL capabilities. Then Baber and Noori (1986), and Foliente (1993)

added pinching capability whilst maintaining its tractability for EQL, but for SDOF

systems only. Details of the EQL scheme for the SDOF model outlined in section 3.4

and are given in Foliente (1993) and Foliente et al. (1996). This scheme for the

SDOF model, has been implemented in a computer program. In Chapter 5, it is used

for stochastic seismic response analysis, of the single-storey L-shaped test house

described in Chapter 2. The results are compared with those calculated using MCS.


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4.3.4 Extension of SDOF Model to MDOF Systems.

In this work, the modified Bouc-Wen hysteretic SDOF model developed by Foliente

(1993), is extended to a MDOF shear-building and shear-wall type model, which are

presented in sections 4.4 and 4.5, respectively. The EQL scheme is also extended to

MDOF systems and is formulated and presented for the shear-building model in

section 4.4.

4.4 Hysteretic Shear-Building Model

4.4.1 Background

The previous section in this chapter outlined a hysteretic SDOF model for non-linear

dynamic response analysis of light-frame structures. This approach may be suitable

for analysis of the global response of low-rise light-frame construction, but will not be

suitable for taller structures with multiple levels, especially if the mass and/or stiffness

distribution is not uniform over the height of the structure. A simple and practical

MDOF model which may be more appropriate for multi-storey construction is the

shear-beam or shear-building lumped mass model. In this approach, the total mass of

the structure is assumed to be lumped at the floor or roof level, and the floor and

ceiling diaphragms are assumed to be infinitely rigid compared to the shear-walls.

The relative lateral deformation between the levels in the structure is the only

deformation considered, and hence the lateral stiffness of the shear-walls at each

level, are also lumped into a single element. This approach has similar limitations to

the SDOF model discussed earlier. It is not suitable for modelling of structures which

are torsionally sensitive as no torsional response is included, and only the global

response of each level in the structure is obtainable. Later in this chapter, a hybrid

response analysis technique is outlined in which local responses can be obtained by

linking simple and detailed models through system identification. Although quite a

crude modelling approach, the shear-building model is better than the SDOF model

for characterising the behaviour of multi-storey light-frame construction, especially if

the distribution of mass and/or stiffness is not uniform over the height of the structure.

If this is the case, then vibration modes other than the fundamental mode can


192

significantly contribute to the overall response, and this should be taken into account

in the modelling. Since many light-frame structures have car-parking areas in the

lower-floor, with large openings in the lower-storey walls, a non uniform distribution

of mass and stiffness is quite possible. Structures with this configuration were found

to be prone to collapse or severe damage during the Northridge earthquake (Hall,

1996; Andreason and Rose, 1994) and hence models for seismic response analysis

should have some means of accounting for it.

4.4.2 Matrix Formulation

Generally, a shear-building model may have up to r degrees of freedom, each

representing the mass and stiffness characteristics of one floor (or a group of floors) in

a building. Consider the general model of a shear-building shown in Figure 4.5

where: xi is the displacement of the ith mass with respect to the ground; ui = xi - xi-1 is

the relative displacement between each floor; and giAi xxx �� += is the absolute

acceleration of the ith mass under a ground acceleration gx��

The total restoring force of the ith mass is therefore given by

iiiiiiiiiii zkxxkxxcQ )1()()( 11 αα −+−+−= −−�� (4.1)

or in terms of ui

iiiiiiiii zkukucQ )1( αα −++= � (4.2)

Using D'Alembert's principle, the equation of motion is obtained as

01 =−+ +iiAii QQxm �� (4.3)

substituting for Aix�� , Qi and Qi+1 and then rearranging,


193

m2

mr

mr-1

x1

x2

xr-1

xr

m1

ur

f t( )

(a) r DOF shear-building

m ui i��

( )1-a i i ik zc ui i�

( )1 1 1 1-+ + +

a i i ik z c ui i+ +1 1�

a i i ik u+ + +1 1 1

a i i ik u

mi

(b) Forces acting on the i-th mass

Figure 4.5 – Shear-building model.


194

1111 )( +++− −++− iiiiiiiii xcxccxcxm ��

11111111 )( +++++− −++− iiiiiiiii xkxkkxk αααα

giiiiiii xmzkzk ��−=−−−+ +++ 111)1()1( αα

(4.4)

where i = 1, 2, .....r. Assemblage of the equations of motion for all r degrees of

freedom results in the following matrix formulation of the shear-building model

ˆ[ ]{ } [ ]{ } [ ]{ } [ ]{ } [ ]{ } gM X C X K X H Z M I xα α+ + + = −�� (4.5)

where {X} is the absolute displacement vector (relative to ground), {Z} is the relative

hysteretic displacement vector (relative to the floor below) and ˆ{ }I is the influence

vector (all entries are 1 for seismic excitation).

The hysteretic displacement, zi of the ith mass is related to the relative displacement

between floors, ui, through the hysteretic law that was described earlier, but in MDOF

form.

+−

=−

i

niiii

niiiii

iiii zuzzuuA

zhzη

γβν )()(

1��

� (4.6)

The hysteresis model parameters are defined for each element:

]sgn[1

22

2

e0.1)( iiuiiii

ζ/)zq)u((zizh −−−= �ζ (4.7)

]0.1[)( )(1

iipisii e εζεζ −−= (4.8)

))(()( 12 iii iioii ζλεδψεζ ψ ++= (4.9)

iii iεδεν ν+= 0.1)( (4.10)

iii iεδεη η+= 0.1)( (4.11)

0

ft

i i itz u dtε = ∫ � (4.12)


195

1/

( )

ini

uii i i

AZν β γ

= + (4.13)

Note that ui = xi - xi-1 and i = 1, 2, ..., r. All symbols are described in Table 3.1

All the matrices in Eq. 4.5 have dimension (r x r). [M] is the diagonal mass matrix

and [C] is the damping matrix with the following nonzero entries:

1 2 2

2 2 3 3

1 1

( )( )

[ ]( )n n n n

n n

c c cc c c c

Cc c c c

c c− −

+ − − + − = − + − −

� � � (4.14)

where iooi mc ωξ2= .

The [Kα] matrix contains the linear component of the structure stiffness and has the

following nonzero entries:

−

−+−

−+−

−+

=

−−αα

αααα

αααα

ααα

α

nn

nnnnkkkkkk

kkkkkkk

K)(

)()(

][

11

3322

221

�� (4.15)

where iii kk αα = .

The [Hα] matrix contains the hysteretic elements (the non-linear components) and has

the following nonzero entries:


196

−

−

−

=

−α

αα

αα

αα

α

n

nnhhh

hhhh

H

1

32

21

][ �� (4.16)

where iii kh )1( αα −= .

As for the SDOF model presented earlier, the hysteretic shear-building model has

been implemented in a computer program, The dynamic response history for this

system, for a given excitation is solved for by using the Wilson-θ method (Wilson et

al., 1973), which is a generic method for solving the equations of motion for non-

linear systems. The model is used for seismic response analysis of an example three-

storey light-frame structure in Chapter 5.

4.3.4 State Vector Formulation

In order to formulate an EQL scheme for the MDOF hysteretic shear-building model,

it is necessary to represent the system equations in a state vector format, which

reduces the problem to a series of ODEs.

Consider the state vector {Y}

=

=

i

i

i

i

i

i

zuu

yyy

Y �

,3

,2

,1}{ (4.17)

The time derivative, { �}Y results in a set of 3xr ODEs

� , ,y yi i1 2= (4.18)


197

�

( ) ( ) ( )

(

, , , ,

, , ,

, ,

y km

y km

km

y km

y

cm

y cm

cm

y cm

y

km

y km

km

y

ii i

ii

i i

i

i i

ii

i i

ii

i

ii

i

i

i

ii

i

ii

i i

ii

i i

i

i i

ii

i

21 1

11 1

11

1 11 1

1

12 1

12

12 1

1 1

13 1

13

1 1 1

1

= + − +FHG

IKJ +

+ − +FHG

IKJ +

+ − − − + −FHG

IKJ

+ −

− −

−−

−

+ ++

−

−−

−

++

− −

−−

−

α α α α

α α α

α + ++ −1 1

3 1 1) ( ),k

my f ti

ii iδ

(4.19)

� ( )( )

,, , , , , ,y h z

y y y y y yi i

i i i i in

i i i in

i

i i

32 2 3

13 2 3=

− +RS|T|

UV|W|

−ν β γη

(4.20)

where δ i1 is a Kronecker delta (i.e. δ ir =1 when i = r, 0 otherwise)

The 3xr first order equations completely define the shear-building model, and can be

solved simultaneously using a range of techniques for solving systems of ODEs, such

as ‘predictor-corrector’ methods. The formulation above results in a 'stiff' set of

equations in which some of the equations have orders of magnitude more significance

on the solution than others. Stiff sets of equations are not suitably solved by Runge-

Kutta or Taylor's series methods (which are commonly used methods for solving

ODEs) because very high order derivatives are required for an accurate solution. This

formulation of the hysteretic shear-building model has also been implemented in a

computer program, the dynamic response history for this system, for a given

excitation is solved by using an Adams-Bashforth-Moulton predictor-corrector

method. Full details of the method, including the theoretical basis can be found in

Khon (1987). The shear-building model is used for seismic response analysis of an

example three-storey light-frame structure in Chapter 5.

4.4.4 Equivalent Linearisation of Hysteretic Shear-Building Model

In the following section, the EQL scheme, developed by Foliente (1993) and Foliente

et al. (1996), for the SDOF modified Bouc-Wen hysteretic model, is extended to the

MDOF hysteretic shear-building system presented above.


198

Consider the state vector formulation of the shear-building model, from the previous

section given by Equations 4.17 to 4.20.

If the hysteretic relationship in Equation 4.20 is replaced by its linear equivalent

� , , ,y C y K yi e i e i3 3 2 3 3= + (4.21)

where Ce3 and Ke3 are statistical linearisation coefficients for the damping and

stiffness terms, respectively, then the governing ODEs for the linearised MDOF

system can be rewritten in matrix form as

fGYY +=� (4.22)

where

=][][]0[][][][

]0[][]0[

33

***

ee KCHCK

IG (3r x 3r) (4.23)

=}0{)(}0{

1tff (3r x 1) (4.24)

and

=

0

0)( 1

�

�� gx

tf (r x 1) (4.25)

All sub-matrices of G have dimension (r x r), [I] is the identity matrix and [0] is the

null matrix, [K*], [C*], [H*], [Ce3] and [Ke3] are given below


199

+−+

+

+−+

+

+−+

+−

=

−−

−

+

−−

−

11

1

1

11

1

2

3

1

2

2

2

1

1

1

2

1

1

* ][

n

n

n

n

n

n

n

n

n

n

n

n

n

n

mk

mk

mk

mk

mk

mk

mk

mk

mk

mk

mk

mk

mk

K

ααα

αααα

αααα

αα

��

(4.26)

where iii kk α=

+−+

+

+−+

+

+−+

+−

=

−−

−

+

−−

−

11

1

1

11

1

2

3

1

2

2

2

1

1

1

2

1

1

* ][

n

n

n

n

n

n

n

n

n

n

n

n

n

n

mc

mc

mc

mc

mc

mc

mc

mc

mc

mc

mc

mc

mc

C ��

(4.27)

+−+

+

+−+

+

+−+

+−

=

−−

−

+

−−

−

11

1

1

11

1

2

3

1

2

2

2

1

1

1

2

1

1

*][

n

n

n

n

n

n

n

n

n

n

n

n

n

n

mh

mh

mh

mh

mh

mh

mh

mh

mh

mh

mh

mh

mh

H

ααα

αααα

αααα

αα

��

(4.28)

where iii kh )1( α−=


200

=

ne

e

e

e

C

CC

C

,3

2,3

1,3

3�

(4.29)

=

ne

e

e

e

K

KK

K

,3

2,3

1,3

3�

(4.30)

The equations used to calculate the linearisation coefficients Ce3 and Ke3 for each

DOF are given in Appendix D.

If the matrix equation for Y� (Eq. 4.22) is post multiplied by TY , the expectation

values of the terms are taken and the resulting equation is added to its transpose, this

results in the covariance equation for the zero-mean time lag covariance matrix

][ TYYES = .

BSGGSS T ++=� (4.31)

where

=

z

zuu

uzuuu

EEEEEE

S ��

�

(3r x 3r) (4.32)

and

][ TT YffYEB += (3r x 3r) (4.33)

If the forcing function f(t)1 is a zero mean Gaussian white noise with constant power

spectral density S0 then B has only one nonzero term and can be written as

0)1)(1( 2 SB rr πδ ++= (4.34)


201

where irδ is a Kronecker delta (i.e. δ ir =1 when i = r, 0 otherwise).

The sub-matrices of S are all of dimension (r x r) and contain the system response

statistics in their diagonal terms. These are found by numerical integration of the

covariance equation (Eq. 4.31) to solve for the zero mean time lag covariance matrix

(S).

=

][

][][

2,1

22,1

21,1

n

u

yE

yEyE

E�

(4.35)

=

][

][][

2,2

22,2

21,2

n

u

yE

yEyE

E�

� (4.36)

=

][

][][

2,3

22,3

21,3

n

z

yE

yEyE

E�

(4.37)

=

][

][][

,2,1

2,22,1

1,21,1

nn

uu

yyE

yyEyyE

E�

� (4.38)

=

][

][][

,3,1

2,32,1

1,31,1

nn

uz

yyE

yyEyyE

E�

(4.39)


202

=

][

][][

,3,2

2,32,2

1,31,2

nn

zu

yyE

yyEyyE

E�

� (4.40)

The mean square response statistics of the ith degree-of-freedom are then given by

][ 2,1

2,1

2, iiiu yE== σσ (4.41)

][ 2,2

2,2

2, iiiu yE== σσ � (4.42)

][ 2,3

2,3

2, iiiz yE== σσ (4.43)

The EQL scheme for the hysteretic shear-building model has been implemented in a

computer program. In Chapter 5, it is used to calculate the response statistics of an

example three-storey light-frame building. The response statistics are compared with

those calculated using MCS.

4.5 Hysteretic Shear-Wall Model

4.5.1 Background

The previous two models presented in this chapter (SDOF and shear-building) are

formulated for prediction of global response quantities only, and do not consider any

torsional or bi-directional components of the response. Excitations and responses are

considered for one direction only. The main drawback with this approach is that no

insight can be gained into the response of the individual components of the light-

frame structure under an earthquake. The local response of the structure’s major

components, such as its walls and its ceiling and roof diaphragms, are of interest in a

seismic analysis, since it is the most highly stressed or deformed components which

are critical when assessing a given structural configuration for its behaviour during an

earthquake.


203

A simple and practical MDOF model which provides some insight into the response

of individual walls in a light-frame structure, and considers the bi-directional nature of

ground excitations and responses, is the shear-wall model which is shown

diagrammatically in Figure 4.6. In this approach, the in-plane characteristics of each

of the walls in the structure are represented by a single modified Bouc-Wen hysteretic

element for each wall. Out-of-plane stiffness of the walls is assumed to be negligible

and is not considered. The walls are connected together at the ceiling level by a rigid

or semi-rigid elastic frame, which is configured in accordance with the plan layout of

the structure. The properties of this frame can be adjusted to represent the stiffness of

the entire roof and ceiling diaphragm. Alternatively, a braced truss configuration or a

plate element configuration (as in Fischer et al., 2001), or a combination of these

could be employed to represent the roof system, depending on the structural

configuration and the objectives of the analysis. In this work, the ceiling level

diaphragm is assumed to be rigid compared to the walls and is modelled as a rigid

frame. The racking deformation of each of the walls is the only deformation

considered, and is identical at either end of the wall, hence the in-plane stiffness of

each of the shear-walls can be represented by a single element.

The approach used herein, as shown in Figure 4.6, is sufficiently sophisticated for

response prediction at the wall level, and is still simple enough to be appropriate for

small-scale simulation-based stochastic response studies, provided that the number of

simulations required is not too large. Responses of individual nails, screws, bolts or

plate connectors can be approximated from the wall responses, or can be directly

determined from more sophisticated FE models of light-frame structures (as described

in sections 4.2 and 4.6), and from hybrid analyses (as described in section 4.7).

4.5.2 Formulation

The shear-wall model formulation and solution for non-linear dynamic analysis is

similar to that for the shear-building model given in section 4.4. The structural

matrices are assembled for the idealised shear-wall configuration shown in Figure 4.6.


204

All walls modelled as single hysteretic springs with

modified Bouc-Wen hysteresis

Internal X-Walls lumped into single element

Rigid elastic frame for ceiling and roof diaphragm

(a) Box-type representation

W1 W3 W4

W5

W678

W9

N

S

EW

N

S

EWZero –length hysteretic spring elements with modified Bouc-Wen

hysteresis

Rigid Elastic Frame

W2

X

YZ

X

YZ

(b) Planar representation

Figure 4.6 – Shear-wall model.


205

The system stiffness and hysteretic matrices, [K] and [H], are cumbersome to define

analytically in general terms, and are assembled numerically, using a FE approach,

but could also be assembled using any other appropriate method such as the direct

stiffness method. Full details of the various structural analysis techniques and

associated matrix methods can be found in structural analysis textbooks such as

Holzer (1985). Only the hysteretic springs used for each of the walls in the house

require hysteretic parameters assigned to them, all other elements are elastic. The

mass matrix is assumed to be diagonal, with the masses lumped at the nodes which

are at the intersection points of the rigid ceiling frame. The damping matrix is based

on an assumption of Rayleigh damping, with 5% equivalent viscous damping in the

first two modes.

As for the SDOF and shear-building models, the hysteretic shear-wall model has been

implemented in a computer program, and its dynamic response history is calculated

using the Wilson-θ method (Wilson et al., 1973). In Chapter 5, it is used for seismic

response analysis of the single-storey L-shaped house described in Chapter 2.

4.6 Finite Element Model

The strategies for seismic response analysis of light-frame structures presented in this

chapter so far have focussed on predicting the response of the entire structure, or of a

significant sub-system of the structure such as its shear-walls. None of these models

can directly predict failure mechanisms in individual components. Responses of

individual nails, screws, bolts or plate connectors can be approximated from the

global structural or wall responses, but can only be determined directly from more

sophisticated FE models.

Kasal (1992) and Kasal et al. (1994) developed a three-dimensional FE model using

the ANSYS FE software. This model will be verified using the results of the full-

scale testing described in Chapter 2, in a separate study. The FE modelling strategy

used is outlined here to give the reader an appreciation of the wider project goals.


206

The FE model by Kasal (1992) utilises a hybrid modelling strategy, where individual

walls are modelled separately in fine detail. The detailed individual wall models

include elements to represent individual members, sheathing panels, and hysteretic

elements to represent sheathing to frame connections. The hysteretic elements are

load-history dependent and include stress and stiffness degradation (Kasal and Xu,

1997), and have been incorporated into the ANSYS FE software. The detailed wall

models are then reduced into a simplified form, using system identification, where a

single hysteretic spring is used to represent the in-plane wall behaviour. The

simplified and detailed wall models are shown in Figures 4.7 (a) and (b), respectively.

The simplified wall elements are then assembled into three-dimensional models,

reducing the number of degrees-of-freedom significantly. The roof and ceiling

diaphragms are modeled using equivalent elastic plate and bracing elements. The

simplified wall elements are then connected together, and to the diaphragms with non-

linear hysteretic springs. The three dimensional model is shown in Figure 4.7 (c)

This type of model can be used to study load-sharing/interaction and distribution

between the structural sub-systems, and to track the formation of local failure

mechanisms in individual components. This particular model has been used

previously to evaluate load-sharing in a full structure (Kasal, 1992), based on the full-

scale experiments on a simple box-style house by Phillips (1990). In a similar

fashion, the experiments described in Chapter 2 of this thesis are to be used to validate

the FE models, and study the load-sharing, but for a ‘real’ house with irregular plan

layout and different wall configurations. The model will primarily be validated

against the measured displaced shapes and distributions of reaction forces under static

and cyclic loading, to verify its ability to predict the load paths within light-frame

structures under wind and earthquake loading.

Although FE models are powerful in their response analysis capabilities, most have

only been used for static analysis due to their complexity. Because of the reduced

degrees-of-freedom in Kasal's (1992) three-dimensional model, it may be suitable for

limited deterministic dynamic analyses, however the model is still too complex for

significant simulation analyses to be performed directly. In section 4.7, a hybrid

modelling approach, which can be used to integrate deterministic FE models and

simplified stochastic response analysis models is described.


207

Nonlinear springsfor individual sheathing to frame connectors

Sheathing:plate elements

Framing:plate elements

(a) Detailed FE model of stud wall (Kasal, 1992)

(b) Equivalent simplified FE representation of stud wall (Kasal, 1992)

Figure 4.7 – FE model of house.


208

Equivalent simplified wall FE sub-assembies

Roof and ceiling diaphragm:plate elements

(c) Three-dimensional Finite-Element model of the test house

Figure 4.7 (cont’d) – FE model of house.

4.7 Hybrid Response Analysis

4.7.1 Background

So far in this chapter, a range of structural models of light-frame structures, covering

the full spectrum of sophistication, from SDOF to FE models, have been presented.

The choice of an appropriate model for static and/or dynamic analysis may vary from

one analysis to another due to different objectives (Seible and Kingsley 1991; Foliente

1997a).


209

Ideally, all analysis objectives would be catered for using a single (monolithic)

analytical model which was structured so that only data relevant to the analysis

objectives need be supplied. Analysis objectives would include prediction of the

whole structure response (global), and of the individual elements in the structure

(local) in both deterministic and probabilistic terms. Unfortunately, this is not

currently possible, mainly due to computational limitations.

Thus, it is important that 1) a variety of modelling tools, covering a range of

complexity and analysis capability be available for response analysis of light-frame

buildings, and 2) analysis results are able to be transferred between different models

so that different analysis domains can be crossed, i.e. (local to global, and

deterministic to probabilistic).

The idea of linking global and local response analyses has been used extensively in

analysis of light-frame structures (Kasal 1992; CUREE, 2001), but the extension of

this to hybrid deterministic and probabilistic analyses is new. The feasibility of this

approach has been demonstrated in Kasal et al. (1999).

Here, it is discussed how the FE model (section 4.6) and the shear-building model

(section 4.5) can be integrated into a hybrid stochastic/deterministic modelling

approach for seismic response analysis of light-frame buildings. A similar strategy

can be adopted to link the SDOF and shear-wall models as well.

4.7.2 Formulation

In order to accurately model the seismic response of a complex structural system such

as a light-frame structure, whilst also considering the inherent uncertainties in

structural and excitation properties, a hybrid analysis approach is preferred, and is

shown diagrammatically in Figure 4.8. Under this approach, material and system

variability are obtained from laboratory experiments or from known historical values.

These are used to determine the characteristic values (i.e. 5th percentile) of the

properties of the elements that constitute a detailed, three-dimensional FE model of


210

the building. The FE model is then subjected to a very simple loading history that

produces a cyclic response of the entire structure, which is then fitted to a much

simpler model (e.g., a two-dimensional shear-wall model as shown in section 4.5 or

shear-building model as shown in section 4.4), using system identification techniques

(see Chapter 3).

The new identified model has only a few DOF and thus can be used in MCS, where

both system properties (e.g., natural period of the building) and the earthquake

excitation can be treated probabilistically. Response statistics can be calculated to

determine reliability or failure probability under an ensemble of loads, and a critical

load history can be determined out of a possible range of loading scenarios. The

displacement response vector from this critical excitation can then be used to ‘re-load’

the original three-dimensional FE model to determine the detailed response of any of

the elements of the FE model. The feasibility of this approach has been demonstrated

by Kasal et al. (1999).

The key aspect of this hybrid analysis approach is the use of system identification to

move from one structural model into another, to enable the transition from one

analysis domain to another. System identification is also used in determining

hysteresis model parameters directly from experimental data as shown in section 3.5.

A hybrid approach is ideal for light-frame buildings because of the high level of

redundancy and complexity in the system, and the large uncertainty in the key system

parameters and the loading.

The proposed approach provides a framework which exploits the advantages of the

various modelling approaches whilst minimising their inherent disadvantages when

used in isolation. It has tremendous versatility because it facilitates transition from

global and local, and from deterministic to probabilistic analysis domains.


211

Figure 4.8 – Hybrid Response Analysis (Kasal et al., 1999).


This chapter presented an overview of seismic response analysis techniques and a

review of whole-building structural modelling of light-frame structures. A range of

structural models for seismic analysis of light-frame structures were formulated and

presented. The key points and findings are summarised as follows.

• Hysteretic SDOF and shear-building models of light-frame structures were

formulated using the differential Bouc-Wen hysteresis elements presented in

Chapter 3, and implemented in computer programs. These models are suitable

for prediction of global responses under uni-directional earthquake excitations.

• An Equivalent Linearisation scheme, formulated for the hysteretic SDOF

model by Foliente (1993), was extended to a MDOF shear-building model.

This technique can be used as a faster alternative to MCS, to estimate the

response statistics of multi-storey structures, under white-noise-based

excitations.


212

• A hysteretic shear-wall model for light-frame structures, incorporating

differential Bouc-Wen hysteresis elements, was formulated and implemented

in a computer program. The model is suitable for prediction of individual wall

responses under bi-directional excitations, and is simple enough to be suitable

for limited MCS studies.

• A FE modelling strategy, based on work by Kasal (1992), and a hybrid

modelling strategy, for seismic response analysis of light-frame structures has

been outlined. The hybrid modelling strategy facilitates the transition from

global to local response predictions, complex to simple models, and from the

deterministic domain to the stochastic domain.

Chapter 5 – Seismic Response Analysis

213

CHAPTER 5

Seismic Response Analysis

5.1 Introduction

In Chapters 3 and 4 of this thesis, hysteresis and structural models, appropriate for

seismic analysis of light-frame structures were presented. In this chapter, these

models are used to conduct deterministic and stochastic seismic response analyses on

the L-shaped test house described in Chapter 2, and an example three-storey light-

frame building. All of the structural models use modified Bouc-Wen hysteretic

elements, to incorporate the non-linear structural behaviour, which inevitably occurs

in a light-frame structure under extreme seismic loading. The parameters of these

hysteretic elements were derived from the experimental data presented in Chapter 2,

using a non-linear gradient system identification technique. The modified Bouc-Wen

hysteretic element and the system identification of the hysteresis parameters is

described in detail in Chapter 3.

This chapter begins with a deterministic seismic response and sensitivity analysis of

the single-storey L-shaped house described in Chapter 2, using the hysteretic SDOF

model described in section 4.3. The response of an example three-storey light-frame

structure is then examined using the hysteretic shear-building model described in

section 4.4. Next, stochastic response analyses of the single-storey and three-storey

structures are performed using the EQL technique described in sections 4.3 and 4.4.

The response statistics calculated using EQL are compared to those calculated using

MCS methods. Following this, the hysteretic shear-wall model described in section

4.5 is used to calculate the distribution of seismic forces to individual walls of the test

house under seismic loads. The shear-wall model is subjected to cyclic lateral loading

similar to the experiment in Chapter 2, and the response is compared to the

experimental response. Predictions from the SDOF and shear-wall models are


214

compared, and finally, the hysteretic shear-wall model is used to examine the seismic

demands on the individual walls of the test house under bi-directional earthquake

excitation.

5.2 Ground Motions

5.2.1 Introduction

In the seismic response analyses performed in this chapter, two different suites of

ground-motions are used for the excitations. The first suite was developed for the

CUREE-Caltech Woodframe Project by Krawinkler et al. (2000) and is specifically

targeted at timber structures. The second suite was developed for the SAC steel

project (Somerville et al., 1997) and is targeted at steel buildings. More information

on each of these suites is given in the following.

5.2.2 CUREE Ground Motions

A suite of ground motions appropriate for timber-frame structures has been developed

as part of the CUREE-Caltech Woodframe Project (Hall, 2000). The ground motion

suite was developed specifically as part of a research project on experimental testing

protocols for timber-frame structures. The suite of ground motions consists of twenty

‘ordinary’ ground motions and six ‘near-fault’ ground motions. The ordinary ground

motions are representative of a probability of exceedance of 10 percent in 50 years

(10/50) for Los Angeles site conditions. ‘Ordinary’ implies that the ground motions

are recorded far enough away from the fault such that they are free of typical near-

fault pulse characteristics. The 10/50 records were scaled to a target response

spectrum for Los Angeles conditions (firm soil), within a period range of T=0.2 to

T=1.0 seconds, which was assumed appropriate for timber-frame structures. Random

horizontal components were chosen without regard to the faulting mechanism, to

eliminate any bias in the suite.


215

The six near-fault records are representative of a probability of exceedance of 2

percent in 50 years (2/50) for Los Angeles conditions. Only the fault-normal

components were used. A pilot study (Alavi and Krawinkler, 1999) indicated that the

fault-normal component was a reasonable representation of the larger of the two

orthogonal components for a randomly chosen direction. The 2/50 records have not

been scaled, and were chosen on the basis of their relatively short pulse-period, which

is likely to be close to the period range of interest for timber-frame structures. Details

of the ordinary and near-fault ground motions in the CUREE suite are given in Tables

5.1 and 5.2, respectively, and full details of the development of the ground motions is

given in Krawinkler et al. (2000).

The CUREE ground motions were developed specifically for timber-frame structures,

and are suitable for the seismic analysis in this research. However, they contain only

one horizontal component for each of the 26 records, and are therefore only suitable

for analysis under uni-directional excitation. These records are used primarily for the

SDOF and shear-building modelling in section 5.3.

5.2.3 SAC Suite of Ground Motions

The SAC steel project is a joint venture between the Structural Engineers Association

of California, the Applied Technology Council, and CUREE. The specific goal of this

project is to investigate the damage to welded steel moment-frame buildings in the

1994 Northridge earthquake. Various suites of ground motions were developed for

topical investigations which form part of this project. Full details on the development

of the suites of earthquake records can be found in Somerville et al. (1997).

Although the SAC ground motion suites are not targeted at light-frame systems, they

are still useful in this study as they are extensive in number, they are site-specific,

they are grouped probabilistically, and most importantly, the records contain both

fault-normal and fault-parallel components. Since the CUREE records contain only

one horizontal component, the SAC ground motions are used in all analysis under bi-

directional excitation, such as for the hysteretic shear-wall modelling presented in

section 5.5.


216

Table 5.1 – CUREE ground motions: Set of 20 ordinary ground motions with 10% probability of exceedance in 50 years.

Record ID Description Magnitude(Richter)

Distance (km)

Duration (s)

PGA (g)

sup1 Superstition Hills, 1987, Brawley 6.7 18.2 22 0.116

sup2 Superstition Hills, 1987, El Centro Imp. Co. Cent. 6.7 13.9 40 0.258

sup3 Superstition Hills, 1987, Plaster City 6.7 21.0 22.25 0.186

nor2 Northridge, 1994, Beverly Hills 6.7 19.6 30 0.416

nor3 Northridge, 1994, Canoga Park 6.7 15.8 25 0.356

nor4 Northridge, 1994, Glendale - Las Palmas 6.7 25.4 30 0.357

nor5 Northridge, 1994, LA - Hollywood 6.7 25.5 40 0.231

nor6 Northridge, 1994, LA - N Faring Rd 6.7 23.9 30 0.273

nor9 Northridge, 1994, N. Hollywood - Coldwater 6.7 14.6 21.95 0.271

nor10 Northridge, 1994, Sunland - Mt Gleason Ave 6.7 17.7 30 0.157

lp1 Loma Prieta, 1989, Capitola 6.9 14.5 39.9 0.529

lp2 Loma Prieta, 1989, Gilroy Array # 3 6.9 14.4 39.9 0.555



lp5 Loma Prieta, 1989, Hollister Diff. Array 6.9 25.8 39.6 0.279

lp6 Loma Prieta, 1989, Saratoga - W Valley Coll. 6.9 13.7 39.9 0.332

cm1 Cape Mendocino, 1992, Fortuna Fortuna Blvd 7.1 23.6 44 0.116

cm2 Cape Mendocino, 1992, Rio Dell Overpass 7.1 18.5 36 0.385

lan1 Landers, 1992, Desert Hot Springs 7.3 23.3 50 0.154

lan2 Landers, 1992, Yermo Fire Station 7.3 24.9 44 0.152

Average 6.86 19.73 35.21 0.29

Median 6.8 19.05 39.78 0.27


217

Table 5.2 – CUREE ground motions: Set of 6 near-fault ground motions with 2% probability of exceedance in 50 years.

Record ID Description Magnitude (Richter)

Distance (km)

Duration (s)

PGA (g)

lp89lex Loma Prieta, 1989 7 6.3 40 0.67

nr94rrs Northridge, 1994 6.7 7.5 14.95 0.873

nr94newh Northridge, 1994 6.7 7.1 60 0.71

kb95kobj Kobe, 1995 6.9 0.6 60 1.07

kb95tato Kobe, 1995 6.9 1.5 40 0.77

mh84cyld Morgan Hill, 1984 6.2 0.1 60 0.71

Average 6.73 3.85 45.83 0.8 Median 6.80 3.90 50.00 0.74


218

Table 5.3 – SAC ground motions for Los Angeles with 10% probability of exceedance in 50 years.

Record ID Description Earthquake Magnitude

Distance (km)

Duration (sec)

PGA (cm/sec2)

LA01 Imperial Valley, 1940, El Centro 6.9 10 39.38 452.03

LA02 Imperial Valley, 1940, El Centro 6.9 10 39.38 662.88

LA03 Imperial Valley, 1979, Array #05 6.5 4.1 39.38 386.04




LA07 Landers, 1992, Barstow 7.3 36 79.98 412.98

LA08 Landers, 1992, Barstow 7.3 36 79.98 417.49

LA09 Landers, 1992, Yermo 7.3 25 79.98 509.7

LA10 Landers, 1992, Yermo 7.3 25 79.98 353.35

LA11 Loma Prieta, 1989, Gilroy 7 12 39.98 652.49

LA12 Loma Prieta, 1989, Gilroy 7 12 39.98 950.93

LA13 Northridge, 1994, Newhall 6.7 6.7 59.98 664.93

LA14 Northridge, 1994, Newhall 6.7 6.7 59.98 644.49

LA15 Northridge, 1994, Rinaldi RS 6.7 7.5 14.945 523.3

LA16 Northridge, 1994, Rinaldi RS 6.7 7.5 14.945 568.58

LA17 Northridge, 1994, Sylmar 6.7 6.4 59.98 558.43

LA18 Northridge, 1994, Sylmar 6.7 6.4 59.98 801.44

LA19 North Palm Springs, 1986 6 6.7 59.98 999.43


Average 6.76 11.56 51.27 576.53

Median 6.70 7.10 49.98 540.87


219



Distance (km)

Duration (sec)

PGA (cm/sec2)

LA21 1995 Kobe 6.9 3.4 59.98 1258

LA22 1995 Kobe 6.9 3.4 59.98 902.75

LA23 1989 Loma Prieta 7 3.5 24.99 409.95

LA24 1989 Loma Prieta 7 3.5 24.99 463.76

LA25 1994 Northridge 6.7 7.5 14.945 851.62

LA26 1994 Northridge 6.7 7.5 14.945 925.29

LA27 1994 Northridge 6.7 6.4 59.98 908.7

LA28 1994 Northridge 6.7 6.4 59.98 1304.1

LA29 1974 Tabas 7.4 1.2 49.98 793.45

LA30 1974 Tabas 7.4 1.2 49.98 972.58

LA31 Elysian Park (simulated) 7.1 17.5 29.99 1271.2






LA37 Palos Verdes (simulated) 7.1 1.5 59.98 697.84




Average 7.02 6.44 41.98 863.76

Median 7.1 4.95 39.99 877.19


220



Distance (km)

Duration (sec)

PGA (cm/sec2)

LA41 Coyote Lake, 1979 5.7 8.8 39.38 578.34

LA42 Coyote Lake, 1979 5.7 8.8 39.38 326.81

LA43 Imperial Valley, 1979 6.5 1.2 39.08 140.67

LA44 Imperial Valley, 1979 6.5 1.2 39.08 109.45

LA45 Kern, 1952 7.7 107 78.6 141.49

LA46 Kern, 1952 7.7 107 78.6 156.02

LA47 Landers, 1992 7.3 64 79.98 331.22

LA48 Landers, 1992 7.3 64 79.98 301.74

LA49 Morgan Hill, 1984 6.2 15 59.98 312.41

LA50 Morgan Hill, 1984 6.2 15 59.98 535.88

LA51 Parkfield, 1966, Cholame 5W 6.1 3.7 43.92 765.65

LA52 Parkfield, 1966, Cholame 5W 6.1 3.7 43.92 619.36

LA53 Parkfield, 1966, Cholame 8W 6.1 8 26.14 680.01

LA54 Parkfield, 1966, Cholame 8W 6.1 8 26.14 775.05



LA57 San Fernando, 1971 6.5 1 79.46 248.14

LA58 San Fernando, 1971 6.5 1 79.46 226.54

LA59 Whittier, 1987 6 17 39.98 753.7

LA60 Whittier, 1987 6 17 39.98 469.07

Average 6.41 25.53 54.65 417.54

Median 6.15 9.20 51.95 351.44


221

In this work the SAC ground motion suite for Los Angeles site conditions is used, so

as the results are comparable with the CUREE-based results. The SAC suite for Los

Angeles contains 60 records which are summarised in Tables 5.3 to 5.5. Records

la01-la20 are representative of a probability of exceedance of 10 percent in 50 years

(10/50) for the Los Angeles conditions. Records la21-la40 are representative of a

probability of exceedance of 2 percent in 50 years (2/50), and records la41-la60 are

representative of a probability of exceedance of 50 percent in 50 years (50/50). The

suite is constructed of fault-normal – fault-parallel pairs, with the odd-numbered

records representing fault-normal excitations, and the even numbered records

representing fault-parallel excitations. Details of the records for the 10/50, 2/50 and

50/50 probability of exceedance are given in Tables 5.3, 5.4 and 5.5, respectively.

5.3 Deterministic Seismic Response Analyses Using

SDOF and Shear-Building Models

5.3.1 Introduction

In the following section, the responses of SDOF and shear-building models of light-

frame structures are examined using the CUREE and SAC ground motion suites as the

excitation. The SDOF model is representative of the L-shaped timber-frame test

house described in Chapter 2, and the shear-building model is representative of an

example three-storey timber-frame apartment block with a relatively soft lower storey.

5.3.2 Response Analysis of Test House Using SDOF Model

A hysteretic SDOF model of the test house described in Chapter 2, for responses in

the North-South-direction (directions shown in Figure 2.3) is used in the following to

examine the global response of the test house under uni-directional earthquake

loading. The model is derived from the experimental force-displacement results for


222

the whole house destructive test, given in Chapter 2. The parameters of the Bouc-

Wen differential hysteresis rule outlined in section 3.4 were fitted to the ‘lumped’

global experimental data using a reduced-gradient system identification technique. A

comparison between the experimental and model responses is shown in Figure 3.19

The fit to the experimental data is good in terms of both the hysteretic shape, and also

the dissipated energy. The SDOF model of the test house, shown in Figure 5.1, was

used to examine the sensitivity of the assumed structural period, equivalent viscous

damping ratio, and ground motion on the seismic response. The model was subjected

to the CUREE and SAC suites of ground motions described in section 5.2. The

hysteresis parameters used in the model are given in the first column of Table 3.4.

Equivalent viscous damping ratios of 2%, 5%, and 10%, and two different values of

building mass were used (M=5.0 and M=6.5 tonne corresponding to T=0.129 and

T=0.147 sec, respectively).

M

Mass•5t (T=0.129 sec)•6.5t (T=0.147 sec)

Damping •2%•5%•10%

Stiffness•119 kN/cm

Figure 5.1 – Details of SDOF model used for single-storey test house sensitivity study.

The resulting displacement demands from these analyses are summarised in Figures

5.2 and 5.3. The median and 90th percentile values of the displacement demand for

the different cases examined are summarised in Table 5.6

The two different values of building mass used in the analyses, represent the lower

and upper bounds of what the likely mass of the test house would be, in its fully

constructed and finished state. The measured mass of the house in the laboratory was

around 5.0 tonne, as is shown in Figure 2.10, however the house had no roof tiles,


223

cladding, trims or doors and windows. A reasonable range of the mass for the SDOF

model could be anywhere between 5.0 and 6.5 tonnes, depending on the type of

roofing (may be vinyl, asphalt or slate tiles) cladding, trimming and timber which is

used, the moisture content of the timber, and the assumed vertical distribution of the

mass. The corresponding period range for these bounds is between T=0.13 to T=0.15

seconds, which is consistent with the expected period for this type of house, and

measured values in the field (Beck et al., 2001; Foliente and Zacher 1994).

The main features of the analysis results in Figures 5.2 and 5.3, and Table 5.6, are that

the displacement demands at the 90th percentile levels can be:

1. Up to 150% higher for a T=0.15 sec model than for a T=0.13 sec. model.

These are both reasonable values of period for the test house, given the

variability in stiffness and mass appropriate for this type of structure.

2. Up to 190% higher for system with 2% damping compared to a system with

10% damping. Based on the available data, it can be argued that these are both

reasonable equivalent viscous damping values for the test house.

3. Up to three times higher under the SAC ground motions than for the

corresponding CUREE ground motions. The ground motion suites are both

deemed appropriate for Los Angeles site conditions, and are scaled for the

same target response spectrum (firm soil), but for different target period range.

The response difference can be explained by the differing characteristics of the

records selected for the suites.

The median displacement demand under the SAC and CUREE ground motions could

range from 0.2 to 3.03cm, depending on the assumed mass and damping, and the

ground motion suite used in the analysis. This demand range is representative of

damage states from completely undamaged to moderately damaged. The 90th

percentile displacement demand ranges from 0.31cm to 6.84cm, which is

representative of damage states from completely undamaged to severely damaged.

These results indicate the sensitivity of the seismic response of the test house to the

ground motion, and also the assumed mass and damping (this is examined in more

detail later in this section).


224

0.0

0.2

0.4

0.6

0.8

1.0

1.2

sup1

sup2

sup3

nor2

nor3

nor4

nor5

nor6

nor9

nor1

0

lp1

lp2

lp3

lp4

lp5

lp6

cm1

cm2

lan1

lan2

lp89

lex

nr94

rrs

nr94

new

h

kb95

kobj

kb95

tato

mh8

4cyl

d

Earthquake

Max

. Dis

pl (c

m)

10% damping 5% damping 2% damping

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

la01

la02

la03

la04

la05

la06

la07

la08

la09

la10

la11

la12

la13

la14

la15

la16

la17

la18

la19

la20

Earthquake

Max

. Dis

pl (c

m)


0.0

1.0

2.0

3.0

4.0

5.0

6.0

la21

la22

la23

la24

la25

la26

la27

la28

la29

la30

la31

la32

la33

la34

la35

la36

la37

la38

la39

la40

Earthquake

Max

. Dis

pl (c

m)


Figure 5.2 – Displacement demands for SDOF model (T=0.129 sec) for different levels of assumed equivalent viscous damping, under 10/50 and 2/50 CUREE and

SAC earthquakes.

CUREE 10/50 CUREE 2/50

SAC 10/50

SAC 2/50


225

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

sup1

sup2

sup3

nor2

nor3

nor4

nor5

nor6

nor9

nor1

0

lp1

lp2

lp3

lp4

lp5

lp6

cm1

cm2

lan1

lan2

lp89

lex

nr94

rrs

nr94

new

h

kb95

kobj

kb95

tato

mh8

4cyl

d

Earthquake

Max

. Dis

pl (c

m)


0.0

1.0

2.0

3.0

4.0

5.0

6.0

la01

la02

la03

la04

la05

la06

la07

la08

la09

la10

la11

la12

la13

la14

la15

la16

la17

la18

la19

la20

Earthquake

Max

. Dis

pl (c

m)


0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

la21

la22

la23

la24

la25

la26

la27

la28

la29

la30

la31

la32

la33

la34

la35

la36

la37

la38

la39

la40

Earthquake

Max

. Dis

pl (c

m)


Figure 5.3 – Displacement demands from SDOF model (T=0.147 sec) for different levels of assumed equivalent viscous damping, under 10/50 and 2/50 CUREE and

SAC earthquakes.


SAC 10/50

SAC 2/50


226

Table 5.6 – Statistics of displacement demand predictions from SDOF model of test house, under SAC and CUREE ground motions.

Displacement Demand (cm) M = 5 tonne M=6.5 tonne

CUREE SAC CUREE SAC EQ

Level

Equivalent Viscous Damping

Ratio Median 90th Pctl Median 90th Pctl Median 90th Pctl Median 90th Pctl 10% 0.20 0.31 0.36 1.27 0.28 0.52 0.53 1.53

5% 0.26 0.42 0.41 1.82 0.36 0.70 0.69 2.29 10/50

2% 0.32 0.58 0.57 2.27 0.41 1.66 0.89 3.54

10% 0.48 0.67 0.64 1.43 0.83 1.03 1.08 2.88

5% 0.55 0.72 0.85 2.59 0.99 1.12 1.99 4.65 2/50

2% 0.75 1.01 1.26 4.13 1.09 2.54 3.03 6.84

If we consider median response values, it can be said that with an assumed mass of

5.0t, the structure performs very well under all but the most severe earthquakes,

regardless of the assumed damping. For this case, the structure is unlikely to suffer

even moderate damage. However, with an assumed mass of 6.5t, the structure is

likely to incur severe damage under a 2/50 earthquake or moderate damage under the

10/50 earthquakes.

Given the range of responses of the SDOF model of the test house shown in Figures

5.2 and 5.3, and the range of building period and damping examined, this type of

structure is highly unlikely to collapse due to direct shaking, during an earthquake.

The 90th percentile displacement demand of the 6.5 tonne house with 2% damping,

under the SAC 2/50 earthquakes is 6.8cm and the maximum predicted response under

any record is 13cm – this is the extreme worst-case response prediction, but is still

probably less than the displacement capacity of the house (which is at least 12cm).

However the level of damage incurred is uncertain, and could range anywhere from

undamaged to severely damaged. This is important, because the less severe damage

states are increasingly becoming important in performance assessment, due to the

large economic cost of damage repair. It is therefore desirable that structural models

can predict these damage states, and given the range of the predictions above, also

quantify the variability in the predictions.


227

The SDOF model used here gives an indication of the likely global responses,

however different parts of the structure may have different damage status under the

same earthquake, depending on the shear-wall layout and configuration. For example,

in the test house, wall W3 is gypsum clad on both sides and may incur greater damage

than other walls at the same displacement level. Because of this, a more detailed

modelling approach, which considers the individual wall characteristics, as well as bi-

directional excitation and response, may be more appropriate for predicting the

damage status of light-frame construction. The hysteretic shear-wall model presented

in section 4.5 has these capabilities and is used to further examine the seismic

response of the test house, later in this chapter, in section 5.5. For now, the

sensitivity of the SDOF model response to the assumed period, equivalent viscous

damping ratio and ground motion are addressed in more detail in the following sub-

sections.

Sensitivity to Assumed Structural Period

To examine the sensitivity of the response to the assumed structural period further,

inelastic displacement ‘response spectra’ were constructed, over the range of building

periods which are reasonable for single-storey houses, from 0.08 to 0.2 seconds

(12.5Hz to 5Hz) using the CUREE and SAC ground motions. In constructing the

spectra, the strengths of the SDOF systems used in the analyses were assumed to be

proportional to the stiffness, and were scaled according to the experimentally

determined value of 100 kN for the T=0.129 sec system. The strength values which

were used, for selected values of period, are given in Table 5.7.

The individual inelastic displacement response spectra, for the 10/50 and 2/50 SAC

and CUREE ground motions for assumed equivalent viscous damping values of 2%,

5% and 10% are plotted in Figures 5.4, 5.5 and 5.6, respectively. Typical hysteresis

responses of the SDOF model are shown in Figure 5.7. The spectra in Figures 5.4 to

5.6 demonstrate the large range of the responses, due to the variability in the ground

motions and indicate the sensitivity of the response to the assumed period, damping

and ground motion. The median of these inelastic response spectra, for each ground

motion grouping, show a clearer picture, and are given in Figure 5.8. The 90th

percentile of the spectra are given in Figure 5.9. Also marked on Figures 5.8 and 5.9


228

are the periods corresponding to the values, which could be appropriate for the test

house. The ordinates of the median spectra, for selected values of structural period,

are given in Table 5.7, and the ordinates of the 90th percentile spectra are given in

Table 5.8.

It can be seen on the spectra in Figures 5.8 and 5.9, that inelastic behaviour starts to

occur at around 0.8cm displacement (this is equivalent to a drift ratio of around

1/300). All the spectra are basically linear below the 0.8cm displacement level, but

once this threshold is crossed, they begin to increase exponentially. This is consistent

with the results from the destructive experiment described in Chapter 2, where

yielding of the whole house response occurs at around 0.8cm. This is also consistent

with the individual calculated hysteresis responses (such as in Figure 5.7), which do

not exhibit significant yielding until around 0.8cm displacement.

On the median spectra shown in Figure 5.8, a displacement demand of 0.8cm, at

which inelastic behaviour first occurs, falls inside the critical period range for the test

house (i.e. 0.13 to 0.15 sec) on all except the CUREE 10/50 spectra. On the 90th

percentile spectra in Figure 5.9, the critical period range also corresponds with

significant variability in displacement demand – indicating sensitivity to the assumed

period. The predicted median demand can be up to 150% higher for a T=0.15 sec

model than for a T=0.13 sec. model. These results highlight the importance in

determining an accurate value of the fundamental period of the house in order to

accurately assess its seismic response, and the likely damage status, since the

structural period for single-storey houses, may fall in the range where transition from

elastic to inelastic response occurs, or where the response spectrum is rising

significantly.

Sensitivity to Assumed Equivalent Viscous Damping

Equivalent viscous damping is included in the SDOF model, in an idealised lumped

element (see Figure 5.1), to account for all of the non-hysteretic damping which is

present in a real structure. Damping in structures is not well understood, and hence

sensitivity studies are important to understand its influence on response. Field

measurements of damping by Beck et al. (2000), taken from earthquake response


229

measurements on instrumented low-rise light-frame structures, and forced vibration

tests, indicate damping values between 5% and 10%. Another study by Suzuki et al.

(1996) on light-frame timber houses indicates values between 1% and 5%. These

results highlight the large variability in ‘real’ damping in light-frame structures.

The choice of damping value to use in seismic response modelling is complex. This

is because the ‘target’ total damping in the model (which, as explained above, is very

difficult to determine) depends on the value of equivalent viscous damping used, and

also on how the model incorporates the hysteretic component of the total damping. A

model which uses elasto-plastic hysteretic elements has no pre-yield hysteretic

damping contribution. More accurate hysteresis models (e.g. DE or differential

models – see Chapter 3) do account for the pre-yield damping to varying degrees, and

hence the equivalent viscous component of the damping in the model should be

adjusted accordingly to achieve the desired total damping.

The data plotted on Figures 5.8 and 5.9, and presented in Tables 5.7 and 5.8, indicate

that predicted seismic response is sensitive to the assumed equivalent viscous

damping in the SDOF differential hysteretic model. Equivalent viscous damping

values of 2%, 5% and 10% of critical were examined, and the models with smaller

damping consistently predict higher displacement demand, over all periods, and under

all earthquakes considered.

It is difficult to make any recommendations on what equivalent viscous damping

value is best to use for response prediction purposes, because of the uncertainty of the

‘target’ damping, and the variability in the contribution of hysteretic damping arising

from different modelling approaches. The only definitive recommendation which can

be given, is that response predictions are sensitive to the assumed equivalent viscous

damping. For the purposes of the modelling and analysis conducted herein, it will be

assumed that 2% is the lower bound, and 10% is the upper bound of the non-

hysteretic viscous damping, and that 5% is a reasonable value to use, if only a single-

value of equivalent viscous damping is used in a seismic response analysis.


230

0

2

4

6

8

10

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Period (sec)

sup1 sup2 sup3 nor2

nor3 nor4 nor5 nor6

nor9 nor10 lp1 lp2

lp3 lp4 lp5 lp6

cm1 cm2 lan1 lan2

Displacement (cm)

0

6

12

18

24

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Period (sec)

lp89lex

nr94rrs

nr94newh

kb95kobj

kb95tato

mh84cyld

Displacement (cm)

0

2

4

6

8

10

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Period (sec)

la01.dat la02.dat la03.datla04.dat la05.dat la06.datla07.dat la08.dat la09.datla10.dat la11.dat la12.datla13.dat la14.dat la15.datla16.dat la17.dat la18.datla19.dat la20.dat

Displacement (cm)

0

6

12

18

24

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Period (sec)


Displacement (cm)

Figure 5.4 – Displacement demand predictions from SDOF model (with 2% damping) under 10/50 and 2/50 CUREE and SAC ground motions.


SAC 10/50 SAC 2/50


231

0

2

4

6

8

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Period (sec)

sup1 sup2 sup3 nor2

nor3 nor4 nor5 nor6

nor9 nor10 lp1 lp2

lp3 lp4 lp5 lp6

cm1 cm2 lan1 lan2

Displacement (cm)

0

4

8

12

16

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Period (sec)

lp89lex

nr94rrs

nr94newh

kb95kobj

kb95tato

mh84cyld

Displacement (cm)

0

2

4

6

8

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Period (sec)


Displacement (cm)

0

4

8

12

16

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Period (sec)


Displacement (cm)



SAC 10/50 SAC 2/50


232

0

2

4

6

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Period (sec)

sup1 sup2 sup3 nor2

nor3 nor4 nor5 nor6

nor9 nor10 lp1 lp2

lp3 lp4 lp5 lp6

cm1 cm2 lan1 lan2

Displacement (cm)

0

2

4

6

8

10

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Period (sec)

lp89lex

nr94rrs

nr94newh

kb95kobj

kb95tato

mh84cyld

Displacement (cm)

0

2

4

6

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Period (sec)


Displacement (cm)

0

2

4

6

8

10

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Period (sec)


Displacement (cm)


CUREE 10/50

CUREE 2/50

SAC 10/50 SAC 2/50


233

-120

-80

-40

0

40

80

120

-8 -6 -4 -2 0 2 4

Displacement (cm)

Force (kN)

(a) T=0.2 sec system with 5% damping under nr94newh ground motion

-120

-80

-40

0

40

80

120

-4 -2 0 2 4

Displacement (cm)

Force (kN)

(b) T=0.129 sec system with 5% damping under LA30 ground motion

Figure 5.7 – Example hysteretic responses of SDOF model under selected ground motions.


234

0.0

0.4

0.8

1.2

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Period (sec)

Displacement (cm) 2% 5% 10%

CUREE 10/50

M=5t M=6.5t

0

1

2

3

4

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Period (sec)


SAC 10/50

M=5t M=6.5t

Figure 5.8 – Comparison of median displacement demands for different assumed equivalent viscous damping levels.


235

0

2

4

6

8

10

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Period (sec)


CUREE 2/50

M=5t M=6.5t

0

4

8

12

16

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Period (sec)


SAC 2/50

M=5t M=6.5t

Figure 5.8 (cont’d ) – Comparison of median displacement demands for different assumed equivalent viscous damping levels.


236

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Period (sec)


CUREE 10/50

M=5t M=6.5t

0

2

4

6

8

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Period (sec)


SAC 10/50M=5t M=6.5t

Figure 5.9 – Comparison of 90th percentile displacement demands for different assumed equivalent viscous damping levels.


237

0

4

8

12

16

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Period (sec)


CUREE 2/50M=5t M=6.5t

0

4

8

12

16

20

24

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Period (sec)


SAC 2/50

M=5t M=6.5t

Figure 5.9 (cont’d) – Comparison of 90th percentile displacement demands for different assumed equivalent viscous damping levels.


238

Table 5.7 – Median displacement demands for SDOF models with different periods and strengths under SAC and CUREE ground motions.

Hysteretic SDOF System Median Displacement demand (cm)

10/50 CUREE 10/50 SAC 2/50 CUREE 2/50 SAC T sec

F Hz

K kN/cm

Fu kN

2% 5% 10% 2% 5% 10% 2% 5% 10% 2% 5% 10%

0.08 12.50 308.4 264.1 0.08 0.06 0.06 0.14 0.12 0.11 0.18 0.16 0.15 0.21 0.19 0.18

0.10 10.00 197.4 169.0 0.14 0.12 0.10 0.27 0.22 0.20 0.31 0.28 0.25 0.41 0.33 0.30

0.12 8.50 142.6 122.1 0.22 0.20 0.16 0.39 0.34 0.30 0.53 0.47 0.39 0.76 0.60 0.46

0.12 8.00 126.3 108.2 0.27 0.23 0.18 0.49 0.38 0.34 0.67 0.51 0.45 1.02 0.73 0.56

0.13 7.50 111.0 95.1 0.33 0.27 0.22 0.70 0.45 0.40 0.75 0.64 0.56 1.66 1.02 0.74

0.14 7.00 96.7 82.8 0.39 0.33 0.26 0.88 0.62 0.49 0.88 0.82 0.73 2.60 1.50 0.96

0.15 6.50 83.4 71.4 0.51 0.42 0.32 1.59 0.86 0.59 2.30 1.27 1.02 4.50 2.57 1.43

0.17 6.00 71.1 60.8 0.71 0.53 0.39 2.24 1.17 0.75 4.22 2.55 1.60 5.85 3.53 2.45

0.18 5.50 59.7 51.1 0.79 0.66 0.50 3.30 1.95 1.10 4.87 3.86 2.62 7.52 5.18 3.29

0.20 5.00 49.3 42.2 1.22 0.76 0.61 3.81 2.90 1.83 8.98 6.02 3.67 15.79 7.50 4.44

0.22 4.55 40.8 34.9 1.72 1.00 0.72 5.44 3.70 2.16 16.15 8.81 5.49 18.69 12.99 7.08

0.25 4.00 31.6 27.0 2.22 1.57 1.03 7.94 6.23 3.92 38.42 26.64 14.85 24.86 17.46 11.84

0.30 3.33 21.9 18.8 3.66 2.70 1.67 13.16 9.61 6.63 48.66 36.79 23.46 36.78 27.96 18.89

0.35 2.86 16.1 13.8 4.86 3.51 2.34 17.42 10.67 7.93 45.48 34.88 24.93 51.47 30.90 22.85

0.40 2.50 12.3 10.6 5.57 4.57 3.07 25.45 14.50 9.73 42.19 34.94 25.84 50.81 40.01 26.21

0.45 2.22 9.7 8.3 6.69 4.80 3.55 25.86 18.69 12.44 41.24 33.93 27.00 53.47 37.39 29.04

0.60 1.67 5.5 4.7 9.58 6.57 4.92 32.04 24.79 18.34 46.25 38.51 30.85 52.07 38.25 30.32

0.80 1.25 3.1 2.6 12.14 9.01 7.11 38.94 28.52 21.14 56.32 47.23 37.50 66.23 55.95 43.64

1.00 1.00 2.0 1.7 12.36 9.38 7.48 43.97 35.50 26.20 61.60 50.71 41.60 70.69 59.14 51.13


239

Table 5.8 – 90th percentile displacement demands for SDOF models with different periods and strengths under SAC and CUREE ground motions.

Hysteretic SDOF System 90th Percentile Displacement demand (cm)

10/50 CUREE 10/50 SAC 2/50 CUREE 2/50 SAC T sec

F Hz

K kN/cm

Fu kN

2% 5% 10% 2% 5% 10% 2% 5% 10% 2% 5% 10%

0.08 12.50 308.4 264.1 0.14 0.12 0.11 0.28 0.26 0.24 0.20 0.19 0.18 0.40 0.35 0.29

0.10 10.00 197.4 169.0 0.30 0.26 0.20 0.90 0.58 0.49 0.35 0.34 0.33 0.97 0.69 0.54

0.12 8.50 142.6 122.1 0.43 0.36 0.28 2.31 1.34 0.79 0.63 0.55 0.52 3.02 1.56 0.90

0.12 8.00 126.3 108.2 0.51 0.37 0.30 1.93 1.73 1.14 0.72 0.66 0.62 3.63 2.25 1.26

0.13 7.50 111.0 95.1 0.64 0.48 0.36 2.80 1.58 1.41 1.07 0.80 0.74 4.22 3.13 1.78

0.14 7.00 96.7 82.8 1.12 0.62 0.47 3.02 2.11 1.45 1.65 0.98 0.93 4.84 3.35 2.70

0.15 6.50 83.4 71.4 1.94 0.83 0.58 3.57 2.75 1.71 3.97 1.84 1.24 10.28 7.46 4.45

0.17 6.00 71.1 60.8 2.23 1.35 0.74 4.99 3.43 2.33 5.15 3.55 2.39 12.29 8.27 5.71

0.18 5.50 59.7 51.1 2.12 1.87 0.91 6.89 5.16 3.15 8.57 8.15 5.03 16.96 9.69 6.83

0.20 5.00 49.3 42.2 2.60 2.03 1.51 10.04 6.35 4.36 25.1 14.8 8.8 25.8 18.8 10.8

0.22 4.55 40.8 34.9 3.58 2.26 1.54 11.25 7.17 4.77 43.8 25.4 15.5 37.8 22.4 13.4

0.25 4.00 31.6 27.0 5.58 3.63 2.68 14.27 10.40 6.34 64.6 37.2 22.8 50.3 32.4 18.0

0.30 3.33 21.9 18.8 8.05 5.43 3.83 24.55 17.24 10.84 62.2 47.2 34.5 70.2 40.2 25.5

0.35 2.86 16.1 13.8 12.86 8.24 5.42 31.74 20.64 13.26 69.2 51.3 37.0 86.2 57.2 35.1

0.40 2.50 12.3 10.6 17.55 10.32 7.42 35.85 25.21 17.04 62.3 52.3 38.9 95.0 64.1 41.3

0.45 2.22 9.7 8.3 16.03 12.45 8.34 39.83 27.02 19.13 61.9 46.0 37.7 104.3 75.8 52.0

0.60 1.67 5.5 4.7 19.14 14.12 10.51 53.67 31.31 22.75 76.8 64.1 46.3 134.7 92.4 57.9

0.80 1.25 3.1 2.6 26.21 17.66 14.04 79.15 51.11 29.87 76.6 64.3 52.8 155.6 102.9 69.3

1.00 1.00 2.0 1.7 27.91 20.67 14.92 69.19 54.79 40.05 74.6 63.2 54.0 129.1 110.8 84.9


240

Sensitivity to Ground Motion

It can be seen in Figure 5.2, that under the CUREE suite of earthquakes, no significant

inelastic response is predicted by the SDOF models, under any of the 26 earthquakes

for the T=0.129 sec system. The maximum predicted displacement demand, for all

damping values, is around 1.0cm (nr94newh earthquake). Under the SAC ground

motions however, the displacement demand is predicted to be well into the inelastic

range for some of the 10/50 earthquakes, and about half of the 2/50 earthquakes. A

similar discrepancy between the CUREE and SAC based predictions is also apparent

for the T=0.147 sec system (see Figure 5.3). A summary of the statistics of the

predicted displacement demands for the two suites is given in Table 5.6. This

highlights the difference in the demand predictions, with the SAC suite consistently

resulting in larger demands than the CUREE suite, in terms of the median value and

the 90th percentile displacement demand, especially for the 10/50 earthquakes.

In Figure 5.10 (a), the median displacement demand spectra, for the 5% damping case

are plotted together, over the period range from T=0.08 to T=0.2 seconds, which is

the period range of interest for the structures examined in this thesis (i.e. one- to three-

storey). It is shown that the predicted displacement demands are significantly higher

for the SAC ground motion suite than for the corresponding CUREE ground motions

over this entire period range. This observation also applies to the 2% and 10%

damping cases as well. The large difference in predicted median displacement

demand is curious seeing as the ground motion suites were both developed for Los

Angeles site conditions (although for different building types), and for the same target

response spectrum, for firm soil.

Some difference in the predicted median responses for the two suites is to be

expected, since the CUREE records are targeted at timber-frame structures, which

were assumed to have a period range from 0.2 to 1.0 seconds by the developers

(Krawinkler et al., 2000). The SAC records are targeted at steel buildings with an

assumed period range from 0.3 to 4.0 seconds. In Figure 5.10 (b) the difference in

median response predictions over the period range T=0.1 to T=1.0 seconds is shown,

so as to compare the spectra within a target period range that is common to both

suites.


241

0

2

4

6

8

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Period (sec)

Displacement (cm) CUREE 10/50 SAC 10/50 CUREE 2/50 SAC 2/50

5% damping

M=5t M=6.5t

(a) Period range 0.08 to 0.2 seconds

0

10

20

30

40

50

60

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Period (sec)

Displacement (cm) CUREE 10/50 SAC 10/50 CUREE 2/50 SAC 2/50

5% damping

(b) Period range 0.1 to 1.0 seconds

Figure 5.10 – Comparison of median displacement demands for SAC and CUREE ground motions.


242

This also shows that the CUREE spectra predict smaller displacement demands than

the SAC spectra over this period range for the 10/50 earthquakes, but that the spectra

are reasonably close for the 2/50 earthquakes. The ordinates of all the spectra, and

strength values used in the analyses, for selected periods, are given in Table 5.7.

The difference in predicted responses can be explained by closer inspection of the

characteristics of the earthquake records which make up the SAC and CUREE suites.

Although both suites were developed for Los Angeles site conditions, and for the

same target response spectrum (firm soil), the record selection, and characteristics of

the records are quite different. Most importantly, the SAC suites have a higher

average and median PGA, especially for the 10/50 suite, which is double that of the

equivalent CUREE suite (see statistics at bottom of Tables 5.1 to 5.4). The major

differences between the SAC and CUREE suites, which may influence response

prediction, include the following:

- Different structure type and target period range.

- Average PGA for the SAC 10/50 records is double the average for the CUREE

10/50 records

- The SAC 10/50 records are predominantly near-fault records (16 out of 20),

whereas none of the CUREE 10/50 records are from near-fault recordings

- The SAC 2/50 records were all derived from recordings on firm soil whereas

the CUREE 10/50 suite contains a mix of records from soft rock and firm soil

- Average PGA for the SAC 2/50 records is 10% higher than for the equivalent

CUREE records

- The CUREE 2/50 suite contains only six records, compared to twenty records

for the SAC 2/50 suite

For the purposes of this study, both sets of ground motions are assumed to be valid for

the seismic performance prediction of light-frame structures, even though they predict

different damage levels ranging from completely undamaged, to severely damaged,

under the same probability level excitations, with the same target spectrum. The

rationale for this is that this range of damage predictions is consistent with field

observations. During the Northridge earthquake, around 1000 single houses were

‘red-tagged’ (rendered uninhabitable), and around 6000 were ‘yellow tagged’ (limited

entry) in the Los Angeles area (Hall, 1996). Of course many of the single houses


243

were also relatively undamaged, indicating that a wide range of damage states were

indeed present, so having a similarly large spread of damage predictions in the

analytical modelling is reasonable.

The study by Hall (1996) also found that many multi-level timber buildings were

severely damaged or destroyed during the Northridge earthquake. These types of

buildings are the subject of the next section of this thesis.

5.3.3 Response of Three-Storey Building Using Shear-Building Model

A model of an example three-storey timber-frame building was developed to examine

the global response of a multi-level light-frame building under earthquake loading.

The shear-building model described in section 4.4 was used in the study. The mass

and stiffness characteristics are shown in Figure 5.11, and were chosen to match the

natural frequency of a typical three-storey timber building with a relatively soft lower

storey. The lower storey in this type of multi-dwelling apartment building is often

used for car-parking, and therefore requires large openings along one direction, which

can result in a relatively soft storey compared to the upper floors which have many

more shear-walls. These types of timber buildings were found to be prone to collapse

or severe damage during the Northridge earthquake (Hall, 1996).

The fundamental natural frequency of the example building is 4.68 Hz (T=0.21 sec),

which is typical for three-storey light-frame construction and is consistent with the

experimentally determined frequency of the three-storey timber building studied by

Beck et al. (2001) as part of the CUREE-Caltech Woodframe Project. The pinching,

and strength and stiffness degradation characteristics of each of the stories were

assumed to be the same as for the SDOF model (section 5.3.2) and the strengths for

each level were scaled proportionally to the stiffness of the SDOF model (see Table

3.4 for hysteresis parameters). This is reasonable if similar construction is assumed

(i.e. light timber frame with plywood bracing). The equivalent viscous damping ratio

was taken as 5%.


244

The hysteretic shear-building model of the three-storey building was subjected to the

ground motions in the CUREE and SAC suites of records described above. The peak

inter-storey displacement responses from these analyses are summarised in Figure

5.12. Under the 10/50 earthquakes, the median of the inter-storey displacement

demand is 0.58cm and 2.11cm for the CUREE and SAC ground motions,

respectively. For the 2/50 earthquakes the median demand is 3.38cm and 5.83cm,

respectively. As for the single-storey house, the predicted inter-storey demands under

all the earthquakes represent damage states from completely undamaged to severely

damaged. All of the peak displacements were recorded in the lower storey. Examples

of the hysteretic responses predicted by the model, under a ground motion from the

Northridge earthquake, for each of the thee levels, is shown in Figure 5.13 This

shows that the inelastic response occurs primarily in the lower storey, and that the

upper storey responds in the elastic range.

The median and 90th percentile of the responses calculated for this example three-

storey building are given in Table 5.9, and are more severe than for the single-storey

building under the same suite of excitations (compare with Table 5.6). These results

indicate that multi-level light-frame construction can be more prone to damage under

earthquake loading than single level construction, since the displacement demands are

likely to be higher. This will be particularly important if sub-standard construction

details are present in these structures, which may reduce their displacement capacity.

This is consistent with the observations from the damage in timber-frame buildings

observed after the Northridge earthquake, which showed that up to 200 multi-storey

buildings with soft first-storey either collapsed or came close (Office of Emergency

Services, 1995). Many of these failures were due to poor construction details and

practices. The increased demand levels for this type of structure are to be expected,

given the inelastic response spectra of the SDOF systems, depicted in Figures 5.8 and

5.9, which shows a dramatic increase in displacement demand as the structure’s

natural period increases from 0.15 seconds (i.e. representative of single-storey) to 0.2

seconds (i.e. representative of three-storey). It should be noted that the shear-

building model does not consider torsional effects. Torsion may be important for

multi-level light-frame structures with first-storey car parking areas because of the

non-symmetrical wall layout (due to large openings on one side). Inclusion of torsion

is likely to result in larger demand predictions than those presented herein.


245

In Figure 5.14, the predicted demands in terms of drift ratio, are shown for the shear-

building model, and the equivalent SDOF model, with T=0.21 sec and equivalent

viscous damping of 5%. The data indicates the SDOF model can either over-predict

the maximum response compared to the three degree of freedom shear-building model

by up to 150% or under-predict by up to 73%. This highlights that it is important to

characterise the vertical distribution of mass and stiffness when studying the seismic

performance of multi-level light-frame construction, and that analysis methods which

are dependent on assumed SDOF behaviour should be used with caution.

7.2 t

6.5 t

5 t

260 kN/cm

250 kN/cm

220 kN/cm

T= 0.21 sec F = 4.68 HzDamping = 5%

Figure 5.11 – Details of shear-building model for example three-storey building.

Table 5.9 – Statistics of inter-storey displacement demand predictions from shear-building model of example three-storey timber building, under SAC and CUREE

ground motions.

Inter-Storey Displacement Demands (cm)

CUREE SAC Earthquake

Level Median 90th pctl Median 90th pctl

10/50 0.58 1.68 2.11 4.46 2/50 3.38 6.89 5.83 9.01


246

0.0

1.0

2.0

3.0

4.0

sup1

sup2

sup3

nor2

nor3

nor4

nor5

nor6

nor9

nor1

0

lp1

lp2

lp3

lp4

lp5

lp6

cm1

cm2

lan1

lan2

lp89

lex

nr94

rrs

nr94

new

h

kb95

kobj

kb95

tato

mh8

4cyl

d

Earthquake

Max

. Int

er-S

tore

y D

ispl

. (cm

)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

la01

la02

la03

la04

la05

la06

la07

la08

la09

la10

la11

la12

la13

la14

la15

la16

la17

la18

la19

la20

Earthquake

Max

. Int

er-S

tore

y D

ispl

. (cm

)

0.0

4.0

8.0

12.0

16.0

20.0

la21

la22

la23

la24

la25

la26

la27

la28

la29

la30

la31

la32

la33

la34

la35

la36

la37

la38

la39

la40

Earthquake

Max

. Int

er-S

tore

y D

ispl

. (cm

)

Figure 5.12 – Inter-storey displacement demand predictions for example three-storey building under 10/50 and 2/50 CUREE and SAC earthquakes.

CUREE 10/50

CUREE 2/50

SAC 10/50

SAC 2/50


247

-250

-125

0

125

250

-4 -2 0 2 4

Inter-Story Displacement - Level 1 (cm)

Force (kN)

-250

-125

0

125

250

-4 -2 0 2 4


Force (kN)

-250

-125

0

125

250

-4 -2 0 2 4


Force (kN)

Figure 5.13 – Hysteretic responses of example three-storey building under nr94rrs ground motion.


248

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

sup1

sup2

sup3

nor2

nor3

nor4

nor5

nor6

nor9

nor1

0

lp1

lp2

lp3

lp4

lp5

lp6

cm1

cm2

lan1

lan2

lp89

lex

nr94

rrs

nr94

new

h

kb95

kobj

kb95

tato

mh8

4cyl

d

Earthquake

Max

. Drif

t Rat

io

SDOF Shear Building

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

la01

la02

la03

la04

la05

la06

la07

la08

la09

la10

la11

la12

la13

la14

la15

la16

la17

la18

la19

la20

Earthquake

Max

. Drif

t Rat

io

SDOF Shear Building

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

7.0%

la21

la22

la23

la24

la25

la26

la27

la28

la29

la30

la31

la32

la33

la34

la35

la36

la37

la38

la39

la40

Earthquake

Max

. Drif

t Rat

io

SDOF Shear Building

Figure 5.14 – Comparison of maximum drift ratio predictions for SDOF and shear-building models, for example three-storey building under 10/50 and 2/50 SAC and

CUREE ground motions.


SAC 10/50

SAC 2/50


249

5.4 Stochastic Response Analyses Using Equivalent

Linearisation

5.4.1 Introduction

The previous sections in this chapter have examined the response of single and multi-

level light-frame structures under suites of deterministic earthquake loading.

Statistical measures of the responses such as the median and the 90th percentile have

been used as the basis of the observations and conclusions. These statistical response

measures have been derived in a way analogous to simulation using MCS (see section

4.2.6), by performing multiple deterministic analyses, under randomly chosen site-

specific excitations, and measuring the statistics of the response parameters of

interest, such as median displacement demand.

As described in section 4.2.7, a more computationally efficient method for evaluating

the response statistics of a structural system, under random loading, is offered by the

techniques of RVA. Unfortunately, exact solutions for RVA only exist for linear

systems, and the response of light-frame structures to extreme seismic loadings can be

strongly non-linear. In order to apply the RVA technique to hysteretic systems, such

as those examined in the previous sections, one possible approach is to find some

form of equivalent linear system. ‘Statistically’ equivalent linear systems for the

SDOF and shear-building models have been developed for this work, using the EQL

technique, and are formally outlined in section 4.4.4. EQL is much more

computationally efficient for the calculation of response statistics than MCS, since the

analysis is done in a single pass, whereas MCS techniques require multiple (at least

50) time-history response analyses to obtain reasonably accurate results. The benefits

and limitations of MCS and EQL are discussed in detail in section 4.2.

In the following sections, the application of the SDOF and shear-building models in

RVA using EQL is demonstrated and its accuracy is examined. Response statistics

under stationary white noise excitations are calculated using the EQL technique, and

are then compared with the results from MCS, which are assumed to represent the

‘exact’ response statistics.


250

5.4.2 Single-Storey Timber-Frame House

To demonstrate applicability and examine the accuracy of the EQL technique, the

response statistics of the hysteretic SDOF model of the test house described in

Chapter 2, and shown in Figure 5.1, were determined using the EQL technique and

also using MCS with 200 simulations. The natural period of the system was set to

T=0.129 seconds, and the equivalent viscous damping ratio was assumed to be 5%.

The excitation used in both the EQL and MCS analyses was a stationary white noise

acceleration of 20 seconds duration, with a power spectral density of S0=0.5 m2/s,

which has a peak value of around 0.5g. Stationary white noise is not a particularly

good representation of real earthquake excitation, because its envelope is relatively

constant throughout the duration compared to an earthquake. This is much more

severe than a real earthquake which usually only contains a short burst of the

maximum acceleration pulses. Since white noise excitation is likely to be more severe

than a real earthquake, it is therefore appropriate for the purposes of demonstrating

the capabilities of the EQL method in predicting the response statistics of a non-linear

system under extreme dynamic loads. Filtering and modulation of the input excitation

into a more realistic form can be incorporated in the model (e.g. Baber, 1980; Baber

and Wen, 1981) if the technique is shown to work for the extreme case.

Comparisons between the EQL and MCS calculated response statistics of the test

house under stationary white noise excitation (max = 0.5g) are given in Figures

5.15(a) to 5.15(d). Figure 5.15(a) to 5.15(c) show that the EQL method, accurately

predicts the standard deviation of the displacement, velocity and hysteretic restoring

force, over the 20 seconds of the excitation. Figure 5.15(d) shows that the EQL

method over-predicts the mean value of the energy dissipated, compared to the

simulation results. An example from the MCS analysis, of the hysteretic response

under 0.5g white noise excitation is given in Figure 5.16. This highlights the non-

linear nature of the response under this level of excitation.

To interpret the physical meaning of the results from the EQL, it is best to think in

terms of the simulation analyses. The standard deviation of the response parameters,


251

as given on the vertical axis of Figures 5.15(a) to 5.15(c), is the standard deviation,

calculated across the values of all of the 200 responses, at a given point in time. In

the case of displacement response shown in Figure 5.15(a), the standard deviation of

the displacement is increasing as time progresses. This is because the model is

yielding and degrading in strength and stiffness, under the constant amplitude white

noise excitation. The displacements in the individual responses, are therefore

becoming larger and larger as time progresses and hence the variability between the

displacement responses is increasing also. The progression of the standard deviation

of the responses, is similar to the progression of the peak response values versus time.

In fact, the standard deviation is related to the peak. Approximate, but rather complex

methods to calculate the peak value from the standard deviation have been developed

by Davenport (1964), Shinozuka et al. (1968) and Michaelov et al. (2001). The

basic idea can be demonstrated through a simple example.

Under a real suite of random earthquakes, from the same site, and with the same

duration, the standard deviation, and peak value of the response would rise sharply

and then drop off after the strong-motion period of the excitation. To demonstrate this

concept, the standard deviation and envelope of the peak displacement response for

the 20 ordinary ground motions in the CUREE suite are calculated and are shown in

Figure 5.17. The SDOF model used in the analysis is the same as in section 5.3, with

the natural period set to T=0.129 sec and equivalent viscous damping ratio of 5%.

The ground motions were all scaled by a factor of 1.5 to ensure inelastic responses.

These earthquakes are not from the same site or of the same duration, but they all

have strong motion within the first ten seconds of the excitation, and are adequate to

demonstrate the connection between peak value and variability of inelastic responses.

Only the response statistics for the initial 20 seconds are shown in Figure 5.17,

together with polynomial fitted trendlines. It is clear that the peak response is related

to the standard deviation of the response. As a very approximate rule-of-thumb, the

peak value of the response can be approximated as roughly three times the standard

deviation of the response.


252

0.00

0.04

0.08

0.12

0.16

0.20

0 5 10 15 20

Time (secs)

σσσσdisp (cm)

SimulationEQL

(a) Standard deviation of displacement

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0 5 10 15 20

Time (secs)

σσσσvel (cm/s)

SimulationEQL

(b) Standard deviation of velocity

Figure 5.15 – Comparison of response statistics calculated using Monte-Carlo simulation, and Equivalent Linearisation under stationary white noise excitation (max

= 0.5g) for SDOF model of test house.


253

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 5 10 15 20

Time (secs)

σσσσz

SimulationEQL

(c) Standard deviation of hysteretic restoring force

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 5 10 15 20

Time (secs)

µµµµenergy

SimulationEQL

(d) Mean energy dissipation

Figure 5.15 (cont’d) – Comparison of response statistics calculated using Monte-Carlo simulation, and Equivalent Linearisation under stationary white noise

excitation (max = 0.5g) for SDOF model of test house.


254

-50-40-30-20-10

01020304050

-1 -0.5 0 0.5 1

Displacement (cm)

Force (kN)

Figure 5.16 – Example of hysteretic response of SDOF model under stationary white noise excitation (max = 0.5g).

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 5 10 15 20Time (seconds)

Dis

plac

emen

t (cm

)

Peak Response

Standard Deviation Response

Fitted Trendline

(a) Displacement response

Figure 5.17 – Comparison of peak and standard deviation of SDOF model responses under first 20 seconds of CUREE ground motions.


255

The EQL results given above confirm the findings by Foliente (1993) and Foliente et

al. (1996), which verified the accuracy of the EQL technique, for the modified Bouc-

Wen model, for a range of SDOF systems, except Foliente (1993) obtained better

results for mean energy dissipation (see Fig 5.13(d)). However, the SDOF systems

examined by Foliente were not based on experimental data and were unrealistic in the

values of natural period used (Foliente examined periods of T= 1.0, 3.1 and 7.8

seconds). The current work confirms that with the exception of mean energy

dissipation, the EQL technique is quite accurate for a realistic hysteretic system

representative of a light-frame structure, with parameters determined accurately from

experimental data using system identification. The extension of the EQL scheme to

MDOF systems is addressed in the next section of this chapter.

5.4.3 Three-Storey Timber-Frame Building

The EQL scheme for the SDOF Bouc-Wen differential hysteresis model has been

extended to a MDOF shear-building model in this work. The formulation of the shear-

building model, and the associated EQL scheme is given in detail in section 4.4. To

demonstrate the applicability and examine the accuracy of the EQL technique for the

shear-building model, the response statistics of the example three-storey light-frame

structure described in Figure 5.11, were determined using the EQL technique and also

using MCS with 200 simulations. The natural period of the three-storey building is

T=0.21 seconds, and the equivalent viscous damping ratio is 5%.

As for the SDOF system examined earlier, the excitation used in both the EQL and

MCS analyses was a stationary white noise acceleration, of 20 seconds duration, with

a power spectral density of S0=0.5 m2/s, which has a peak value of around 0.5g. The

response of this structure under this level of excitation produces strongly non-linear

responses.


256

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 5 10 15 20

Time (secs)

σσσσdispl (cm)

EQL(1)SIM(1)EQL(2)SIM(2)EQL(3)SIM(3)

(a) Standard deviation of displacement

0

1

2

3

4

5

6

7

8

0 5 10 15 20

Time (secs)

σσσσvel (cm/s)


(b) Standard deviation of velocity

Figure 5.18 – Comparison of response statistics calculated using Monte-Carlo Simulation, and Equivalent Linearisation under stationary white noise excitation

(max = 0.5g) for example three-storey building.


257

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 5 10 15 20

Time (secs)

σσσσz


(c) Standard deviation of hysteretic restoring force

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20

Time (secs)

µµµµenergy


(d) Mean energy dissipation

Figure 5.18 (cont’d) – Comparison of response statistics calculated using Monte-Carlo simulation, and Equivalent Linearisation under stationary white noise

excitation (max = 0.5g) for example three-storey building.


258

-150

-100

-50

0

50

100

150

-1.5 -1 -0.5 0 0.5 1 1.5


Force (kN)

-150

-100

-50

0

50

100

150

-1 -0.5 0 0.5 1


Force (kN)

-80-60-40

-200

2040

6080

100

-0.6 -0.4 -0.2 0 0.2 0.4 0.6


Force (kN)

Figure 5.19 – Example of hysteretic response of shear-building model under stationary white noise excitation (max = 0.5g).


259

Comparisons between the EQL and MCS calculated response statistics of the test

house under stationary white noise excitation (max = 0.5g) are given in Figures

5.18(a) to 5.18(d). Figure 5.18(a) to 5.18(c) show that the EQL method predicts the

standard deviation of the displacement, velocity and hysteretic restoring force, with

reasonable accuracy, for all three storeys in the structure, over the 20 seconds of the

excitation. Figure 5.18(d) shows that the EQL method over-predicts the mean value

of the energy dissipated in the lower floor, compared to the simulation results, but

accurately predicts it in the upper levels in the structure. Examples of the hysteretic

response for each level of the three-storey building, under 0.5g white noise excitation

are given in Figure 5.19, to highlight the non-linear nature of the response of the

building under this level of excitation.

The EQL results for the three-storey example are not quite as good as for the single-

storey case, but approximate the response statistics reasonably well. The results

demonstrate that the technique can be extended to MDOF systems, using the modified

Bouc-Wen hysteresis model, and still produce good results, even for strongly non-

linear systems. If the EQL technique is to be used as a response analysis tool, it

should therefore be extended to include the capability to deal with filtered, modulated

excitation, which is a more realistic representation of earthquake excitation than

stationary white noise.

5.5 Seismic Response Analyses using Hysteretic

Shear-Wall Model

5.5.1 Introduction

Earlier in this chapter, a hysteretic SDOF model was used to predict the response of

the L-shaped test house described in Chapter 2, under earthquake loading appropriate

for Los Angeles site conditions. The SDOF modelling indicated that the response was

sensitive to the assumed natural period and damping of the structure and the ground

motion, and that the level of damage under such earthquakes was highly variable. In a

real house, different parts of the structure may suffer different degrees of damage


260

under the same earthquake, depending on the layout and configuration of the main

shear-walls. In the test house from Chapter 2, wall W3 is gypsum clad on both sides

and may therefore incur greater damage than other walls at the same displacement

level. Because of this, a more detailed modelling approach, which considers the

individual wall characteristics and the geometry of their layout as well as bi-

directional excitation and response, may be more appropriate for damage prediction in

light-frame construction.

The hysteretic shear-wall model presented in section 4.5 has these analysis

capabilities. In this model, the in-plane characteristics of each wall are represented by

a single modified Bouc-Wen hysteretic element. The out-of-plane stiffness of the

walls is assumed to be negligible and is not considered, and the ceiling and roof

diaphragm is assumed to be rigid compared to the walls and is modelled as a rigid

elastic frame. The hysteretic shear-wall model is used to further examine the seismic

response of the test house in the following sections.

5.5.2 Shear-Wall Model Details

Details of the hysteretic shear-wall model, used to analyse the seismic response of the

test house are summarised in Figure 5.20. The hysteresis parameters for each of the

walls, are given in Table 3.4. Where possible, these parameters were determined from

the experimental data obtained during the destructive testing, via system

identification, as outlined in Chapter 3. The cyclic loading in the destructive test was

applied in one direction only, and as a result, the response of the cross walls (walls

W5 to W9), and in particular the central cross-walls (Walls W6 to W8), was much

smaller than the four in-plane walls (walls W1 to W4). Because of this, the ultimate

load was estimated for these walls, based on the data for the other walls, and using the

perforated shear-wall method (NAHBRC, 2000). A comparison between the model

and experimental responses, for each of the walls in the house, in isolation, are given

in Figures 3.20 to 3.25. The model fit for all of the walls is very good, in terms of

both hysteresis shape, and dissipated energy.


261

The stiffness matrix for the model, [K], and the hysteretic matrix, [H], are assembled

using a stiffness-based finite-element approach. Only the hysteretic springs, which

represent the walls, have hysteresis parameters associated with them. The frame

which makes up the ceiling diaphragm is assumed to be elastic, and rigid. This

assumption is not generally valid, as it was shown in section 2.6.5 that the roof and

ceiling diaphragm was not completely rigid compared to the walls. However, it is a

convenient assumption, because the diaphragm flexibility cannot be modelled simply,

due to the complex geometry of the roof system. The aim here is to develop a

relatively simple model, and it is shown later, in section 5.5.3, that this simplistic

approach can still produce reasonable results. More accurate representation of the

roof and ceiling diaphragm can be incorporated in the FE model described in section

4.6.

In accordance with the aim of creating a relatively simple model, the internal walls

were assumed to be rigidly connected to the roof and ceiling diaphragm even although

this was not the case for the test house. This assumption is reasonable for small levels

of load, because in a ‘real’ house the plasterboard lining on the walls is connected to

the ceiling plasterboard using tape and plaster (or glued ‘cornices’ in some cases).

This in effect establishes a rigid connection between the ceiling diaphragm and all the

walls. Thus, internal non-bracing partition walls, which do not normally have a solid

connection to the diaphragm, become engaged in resisting lateral loads applied to the

house. However for very large loadings, the integrity of this non-structural connection

may degrade. These effects are best modelled using a more sophisticated FE model.

The mass matrix used in the model, [M], is diagonal, with the masses lumped at the

nodes, which are at the intersection points of the ceiling-level frame. The mass matrix

is derived from the experimental dead load distribution, which is shown in Figure

2.10. The distribution and the self-weight, as used in the shear-wall model is shown

in Figure 5.20. The damping matrix, [C], is based on an assumption of Rayleigh

damping, with 5% damping in the first two modes, and is defined by Eq. (5.1).

[C] = a1[M] + a2[K] (5.1)


262

The shear-wall model is configured for both uni- and bi-directional excitation and

response. Under uni-directional excitations, only the Y direction (North-South) is

considered. A summary of the natural frequencies and associated Rayleigh damping

coefficients, a1 and a2, for both uni- and bi-directional models is given in Table 5.10.

The only difference between the two models is in the mass matrix. The mass matrix

for the bi-directional model has masses associated with both translational degrees of

freedom at each node. The uni-directional model only has masses associated with the

Y direction (North-South) translational degrees of freedom. The reason for using two

different configurations of the model in this analysis, is to compare the effect on

response prediction of using uni-directional excitation, with that when using bi-

directional excitation. This was also required so that both the SAC and CUREE

ground motions could be used in the shear wall model analysis. The CUREE ground

motion suite contains single direction records only, whereas the SAC suite contains

pairs of orthogonal records.

It should be noted that the fundamental natural frequencies of the house model, in

Table 5.10, are quite different to the measured values derived from the dynamic

impact testing in section 2.5. This is because the natural frequencies derived from the

dynamic impact testing are based on the initial tangent stiffness of the house (see

Table 2.7), and are quite unreliable for dynamic modelling purposes (see section

2.5.3). In the model, the wall stiffness values, as given in Table 3.4, are derived from

the system identification process.

Table 5.10 – Natural frequencies and associated Rayleigh damping coefficients for hysteretic shear-wall model.

1st mode 2nd mode a 1 a 2

Uni-directional model

(North-South only)

T=0.12 sec

F=8.3 Hz

(NS Racking)

T=0.067 sec

F=14.83 Hz

(Torsional)

3.34 0.0007

Bi-directional model

T=0.14 sec

F=7.17 Hz

(NS Racking)

T=0.115 sec

F=8.72 Hz

(EW Racking)

2.47 0.001


263

4 60 35

40

40

45

N

S

EW

N

S

EW

20

2.7 4.7 4.2 4.7

3.3 6.1 4.0 8.2

4.4 3.0 5.5

T (N-S) = 0.12 sec T (E-W) = 0.14 sec Damping = 5%Weight = 50.8 kN

Weight at nodes (kN)

Initial wall stiffness

(kN/cm)

Fault-ParallelExcitation

Fault-NormalExcitation X

YZ

X

YZ

Figure 5.20 – Details of shear-wall model for single-storey test house.


264

They are calculated such that the best-fit hysteresis trace is obtained over the entire

response (not just at the origin), and have little to do with the initial tangent stiffness,

because the hysteresis is non-linear from the origin. The model wall stiffness values

are therefore much smaller than the tangent stiffness, and hence the frequencies of the

model are much lower than those measured from the impact tests. The frequencies

given in Table 5.10 are consistent with field measurements and reasonable for the

type of house considered (Foliente and Zacher, 1994), whereas the frequencies from

the dynamic impact testing may be unreliable (see section 2.5.3).

5.5.3 Comparison Between Shear-Wall Model and Experimental Responses

In order to predict the damage status of the different walls in a light-frame structure, a

structural model must first be able to predict the load path within the structure.

Because of this fact, one of the major objectives of this project was to measure the

load path in the test house described in Chapter 2, and then develop and validate

models which could predict this load path, including a FE model. The aim of the

analysis presented in the following is to verify that the hysteretic shear-wall model

described in section 4.5 is capable of predicting the load path measured during the

destructive experiment from Chapter 2, and its evolution under cyclic loading.

To compare the load-path determined in the experiment, with that predicted by the

model, a static-cyclic load was applied to the model, in the X-direction (North-South)

only, to walls W3 and W4, as was done in the destructive experiment. The load was

applied very slowly in the model, as in the experiment, to eliminate any dynamic

response. However it was applied as a cyclic force, rather than as a cyclic

displacement (as was the case in the experiment). This allowed the diaphragm to

rotate as a rigid body, in a similar manner to the experiment, so as to compare the

model predictions with the experimental results in a more meaningful way. If

displacements were applied to walls W3 and W4, then Walls W1 to W4 would have

experienced identical displacements due to the rigid diaphragm.


265

The percentage of the total applied load, which is resisted by each of the four in-plane

shear-walls (walls W1 to W4) was then calculated at the maximum displacement

point of each loading cycle, and compared with the values measured during the

experiment. Figure 5.21 shows the comparison between the measured and predicted

load distribution to each of the four in-plane walls. The accuracy of the model

prediction for all of the walls is reasonably good, especially given the simplicity of the

model compared to the complexity of the highly redundant system which it is

representing. The results are accurate to within 10% of total load, for displacements

up to 5cm, which is the approximate displacement at maximum load. Importantly, the

model predicts the trend of the progression of the load distribution as the structural

behaviour progresses well into the inelastic range, although after the 5cm

displacement level, when the inelastic behaviour is extreme, the predicted values are

not as accurate.

The intuitive physical interpretation of these model results agrees with the

observations from the experiment. Initially the model predicts that the stiffest wall,

which is the doubly gypsum clad wall W3, resists most of the applied loading. This

wall is the most brittle, so its contribution drops off quickly, after which, the load

resisted by the three more ductile walls (W1, W2, W4) increases accordingly. There

is a slight discrepancy between the experimental and model responses, in the initial

proportion of total load resisted by wall W2. There are two possible sources for this

discrepancy. Firstly, the ceiling diaphragm is not rigid (see section 2.6.5), as was

assumed in the model, and secondly, the relative initial values of the stiffness of each

of the walls could be in error.

Given that the model-predicted results are reasonably good, the assumption of a rigid

roof and ceiling diaphragm in the model seems to be reasonable for this particular

case. This, however, may not be the case for all roof and ceiling diaphragms in

light-frame construction, and cannot be extrapolated to multi-level construction,

where the floor diaphragms may be much more flexible in-plane than the roof and

ceiling system in the test house. Further experimentation and modelling to determine

the effects of the diaphragm rigidity upon building performance and response

prediction are warranted.


266

0%

10%

20%

30%

40%

50%

60%

0 2 4 6 8 10 12Displacement (cm)

Rea

ctio

n Fo

rce

W4 (test) W3 (test) W2 (test) W1 (test)W4 (model) W3 (model) W2 (model) W1 (model)

Figure 5.21 – Comparison of model prediction and experimental results for distribution of in-plane reaction forces in walls W1-W4 under static-cyclic loading.

5.5.4 Comparison Between Shear-Wall Model and SDOF Model Responses

Earlier in this chapter, a hysteretic SDOF model was used to predict the seismic

performance of the L-shaped test house. The SDOF model was based on the global

response data which was obtained during the cyclic testing from Chapter 2. As

described in the previous section, the shear-wall model is also based on the

experimental data, but at the level of individual wall response, rather than global

response. In this section, the SDOF and shear-wall modelling approaches are

compared, by subjecting each model type to the same suites of ground motions, and

then comparing the peak displacement response predictions from each. The shear-

wall model used in this comparison is outlined in section 5.5.2, and shown in Figure

5.20. The SDOF model used is the T=0.129 sec system, with 5% equivalent viscous

damping ratio outlined in section 5.3.


267

Figure 5.22 shows the comparison between maximum predicted displacement demand

from the shear-wall and SDOF models. Only the North-South-direction responses

were considered, and the displacement for the shear-wall model was calculated as the

average displacement of the four main walls in that direction (walls W1-W4). The

average was used in this case, to be consistent with the underlying basis of the SDOF

model, which is derived from averaged data. Comparison of the hysteretic responses

from the SDOF and the shear-wall model (averaged), for selected cases are shown in

Figure 5.23.

It was demonstrated in section 5.3 that the SDOF model is not accurate for global

response prediction of multi-level construction, compared to a MDOF model.

However, Figures 5.22 and 5.23 show that the SDOF model does a reasonably good

job of predicting the maximum global displacement demand of the more detailed

single-storey shear-wall model, under uni-directional excitation, at all response levels

considered. Among the responses which are significantly inelastic, the average error is

8% and the worst case error is 20%. This indicates that a SDOF model, which

accurately incorporates pinching and degrading hysteretic behaviour, may be used as

an approximate tool for studying the global seismic response of single-storey houses.

This level of accuracy, compared to the more complex model, given its simplicity,

means that it can be confidently used in sensitivity and simulation analyses, such as

those in section 5.3, to examine general response trends and determine global

response statistics from a suite of excitations.

It is shown in the following sections, however, that the responses of individual walls

can be quite different to the average global response calculated using a SDOF model,

and that response predictions can also differ depending on whether uni- or bi-

directional excitation is used in the analysis.


268

0.0

0.2

0.4

0.6

0.8

1.0su

p1

sup2

sup3

nor2

nor3

nor4

nor5

nor6

nor9

nor1

0

lp1

lp2

lp3

lp4

lp5

lp6

cm1

cm2

lan1

lan2

lp89

lex

nr94

rrs

nr94

new

h

kb95

kobj

kb95

tato

mh8

4cyl

d

Earthquake

Shear Wall SDOFMax. Displ (cm)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

la01

la02

la03

la04

la05

la06

la07

la08

la09

la10

la11

la12

la13

la14

la15

la16

la17

la18

la19

la20

Earthquake


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

la21

la22

la23

la24

la25

la26

la27

la28

la29

la30

la31

la32

la33

la34

la35

la36

la37

la38

la39

la40

Earthquake


Figure 5.22 – Comparison between displacement demand prediction from shear-wall model and SDOF model under 10/50 and 2/50 CUREE and SAC ground motions.


SAC 10/50

SAC 2/50


269

-100

-75

-50

-25

0

25

50

75

100


Load (kN)

ShearWall Model

-100

-75

-50

-25

0

25

50

75

100

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2Displacement (cm)

Load (kN)

SDOF Model

(a) LA12 Earthquake

-100

-75

-50

-25

0

25

50

75

100


Load (kN)

ShearWall Model

-100

-75

-50

-25

0

25

50

75

100


Load (kN)

SDOF Model

(b) LA32 Earthquake

Figure 5.23 – Comparison of averaged response from walls W1 to W4 in shear-wall model with response from SDOF model, for selected ground motions.


270

5.5.5 Comparison Between Shear-Wall Model Response Under Uni-

Directional and Bi-Directional Excitations

The SAC ground motion suite contains earthquake records with both fault-normal and

fault-parallel components. In this section, the shear-wall model is used to compare

the response predictions under uni- and bi-directional excitations. In the bi-

directional response analyses the building was oriented as shown in Figure 5.20. The

fault-normal records (odd-numbered) were applied in alignment with the North-

South-direction, and the fault-parallel (even-numbered) records were applied in the

East-West-direction. The predicted North-South-direction responses for walls W1-W4

were then averaged, and compared to the results where only the fault-normal

component of the excitation was applied (in the North-South-direction). The results

are shown in Figure 5.24.

The main interest is in comparing the responses which are significantly inelastic

(LA11, 19, 29, 31, 35, 41, 53 and 55). Among these cases, the uni-directional

response predictions can be either smaller (eg. LA11, 19, 29, 31, 35) or larger (eg.

LA41, 53, 55) than the bi-directional predictions. This indicates that a general

correction cannot be applied to the uni-directional results. The average error for

these cases is 18% and the worst case error is 28%. It is therefore concluded that to

minimise this source of error in seismic response prediction for single-storey light-

frame structures, if possible, bi-directional excitation should be used in the analysis.

5.5.6 Analysis of Seismic Demands on Individual Walls Under Bi-Directional

Earthquakes

The analysis in section 5.5.3 showed that the shear-wall model, was able to

reasonably predict the load path in the test house, as measured in the destructive

experiment, well into the inelastic range under cyclic loading. This indicates that the

model should be capable of predicting individual wall responses under severe

earthquake loading. The aim of this section is to examine the response of the

individual walls of the test house under bi-directional seismic excitation.


271

0

0.5

1

1.5

2

2.5

la01

la03

la05

la07

la09

la11

la13

la15

la17

la19

Earthquake

Uni-Directional Excitation Bi-Directional ExcitationMax. Displ (cm)

0

1

2

3

4

5

la21

la23

la25

la27

la29

la31

la33

la35

la37

la39

Earthquake


0

0.5

1

1.5

2

la41

la43

la45

la47

la49

la51

la53

la55

la57

la59

Earthquake


Figure 5.24 – Comparison between displacement demand predictions, averaged for

walls W1 to W4, under bi-directional and uni-directional SAC ground motions.

SAC 10/50

SAC 2/50

SAC 50/50


272

The shear-wall model outlined earlier was subjected to the three different sets of fault-

normal – fault-parallel earthquake pairs in the SAC ground motion suite,

corresponding to probability of exceedance of 10%, 2% and 50% in 50 years.

Individual wall responses, in terms of maximum displacement demand under these

earthquakes are given in Figure 5.25. The predicted hysteresis responses of the

individual walls are shown in Figure 5.26, for the la32 ground motion.

Figure 5.25 shows the displacement demand of the individual walls, under the 10/50,

2/50 and 50/50 SAC earthquakes. This highlights that the different walls within the

structure may have vastly different levels of displacement demand, under the same

earthquake, due to the bi-directional and torsional nature of the response. Animations

of the displacement response time history also highlight this. Under particular

earthquakes (e.g. la27-28, la41-42 & la49-50) some walls in the house respond in the

elastic range, while others respond well into the inelastic range (assuming 0.8cm

displacement indicates transition into inelastic response). Large variation in the

maximum wall response occurs even for walls with the same orientation. This

highlights the fact that there is a torsional component to the response, due to the

structural eccentricity. This variability within the structure indicates that the averaged

response for a given direction, which is the basis of a SDOF model, may not be a

good indicator for individual wall response. This variability can be due to three

factors: 1) the bi-directional nature of the ground motion; 2) the different stiffness and

hysteretic characteristics of the individual walls; and 3) the eccentricity resulting from

differing centre of gravity and stiffness. These are therefore important considerations

when trying to predict damage in a light-frame structure under seismic loading.

The displacement demand results also highlight the large variability of the predicted

displacement demand, even within each of the earthquake ‘probability’ groups. This

again confirms the importance of characterising the variability in the ground motion,

in order to accurately characterise the variability in the response.

Figure 5.27 shows the load distribution predicted by the shear-wall model under a real

earthquake, rather than an arbitrary cyclic load. The transition of the load distribution

under the earthquake, follows a similar trend to that observed in the experiment. (i.e.


273

as the more heavily loaded walls approach their capacity, and begin to respond

inelastically, their load is redistributed to the other parallel walls).

The most commonly used techniques for lateral load distribution in light-frame

structures (NAHBRC, 2000) do not allow for any lateral load-redistribution between

the walls in the structure (see section 1.5). It has been demonstrated here that it may

be appropriate to include this effect into the seismic design procedures of the future.

Full understanding of the lateral load distribution and sharing under seismic loading

will eventually lead to safer and more economical light-frame structures.

Although the plots in Figure 5.25 highlight some interesting characteristics of the bi-

directional seismic response of a light-frame structure, it is difficult to pick any trends

due to the large variability of the response data, which is due to variation in the

ground motion. In order to highlight the trends, and to quantify the demand of

individual walls under the SAC ground motions, the median and 90th percentile

displacement demands of the individual walls were calculated and are shown in

Figures 5.28 and 5.29.


274

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

la01,02 la03,04 la05,06 la07,08 la09,10 la11,12 la13,14 la15,16 la17,18 la19,20Earthquake

Max Displ (cm) W1 W2 W3 W4 W5 W678 W9

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

la21,22 la23,24 la25,26 la27,28 la29,30 la31,32 la33,34 la35,36 la37,38 la39,40Earthquake

Max. Displ (cm) W1 W2 W3 W4 W5 W678 W9

0.00.20.40.60.81.01.21.41.61.8

la41,42 la43,44 la45,46 la47,48 la49,50 la51,52 la53,54 la55,56 la57,58 la59,60

Earthquake

Max. Displ (cm) W1 W2 W3 W4 W5 W678 W9

Figure 5.25 – Displacement demand for individual walls, under bi-directional SAC ground motions.

SAC 10/50

SAC 2/50

SAC 50/50


275

-8-6-4-202468

-6 -4 -2 0 2 4 6

Displacement (cm)

Force (kN)

W1

-30

-20

-10

0

10

20

30

-6 -4 -2 0 2 4

Displacement (cm)

Force (kN)

W2

-40-30-20-10

0102030

-3 -2 -1 0 1 2 3

Displacement (cm)

Force (kN)

W3

-40-30-20-10

010203040

-2 -1 0 1 2 3

Displacement (cm)

Force (kN)

W4

-30

-20

-10

0

10

20

30

-10 -5 0 5 10

Displacement (cm)

Force (kN)

W5

-30

-20

-10

0

10

20

30

-6 -4 -2 0 2 4

Displacement (cm)

Force (kN)

W678

-40-30-20-10

010203040

-6 -4 -2 0 2 4

Displacement (cm)

Force (kN)

W9

Figure 5.26 – Predicted in-plane hysteresis responses of individual walls under bi-directional la32 ground motion.


276

0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20 25 30 35 40

Displacement (mm)

% o

f X R

eact

ion

W1 W2 W3 W4

(a) North-South walls, W1-W4

0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20 25 30 35 40

Displacement (mm)

% o

f Y R

eact

ion

W5 W678 W9

(b) East-West Walls, W5-W9

Figure 5.27 – Distribution of load to individual walls under la32 ground motion.


277

0

0.5

1

1.5

2

2.5

W1 W2 W3 W4 W5 W678 W9

Max

imum

Dis

plac

emen

t (cm

)

50%/50yr 10%/50yr 2%/50yr

Figure 5.28 – Median of displacement demands from shear-wall model, for individual

walls, under bi-directional SAC ground motions.

0

1

2

3

4

5

6

W1 W2 W3 W4 W5 W678 W9

Max

imum

Dis

plac

emen

t (cm

)

50%/50yr 10%/50yr 2%/50yr

Figure 5.29 – 90th percentile of displacement demands from shear-wall model, for

individual walls, under bi-directional SAC ground motions.


278

Figures 5.28 and 5.29 show the median and 90th percentile, respectively, of the

displacement demand on individual walls. The displacements on the East-West

oriented walls (walls W5 to W9) are generally higher at both the median and the 90th

percentile demand level. These walls are aligned with the fault-parallel component of

the earthquakes. As a group, the East-Wast walls have a slightly higher stiffness than

the North-South walls, but have a slightly lower load capacity.

At the median demand level, only the 2/50 earthquakes result in inelastic

displacement demands, except for the gypsum clad wall W3, which has a much lower

elastic limit. At the 90th percentile response level the 10/50 and 2/50 earthquakes

generally do exhibit an inelastic response, and the 50/50 earthquakes generally do not.

These results again highlight the robustness of this type of structure to resist seismic

loads without danger of collapse. It is only under the more severe earthquakes that

significant damage is likely to be incurred.


This chapter presented the results of deterministic and stochastic seismic response

analyses, and sensitivity studies, using models of the single-storey L-shaped house

described in Chapter 2, and a model of an example three-storey light-frame structure.

The results were based on two different suites of ground-motions. The key points and

findings are summarised below, partitioned in accordance with the structure of this

chapter.

5.6.1 Sensitivity Study of Single-Storey House Using SDOF Model

• A hysteretic SDOF model of the test house from Chapter 2, was used to

examine the sensitivity of the assumed structural period, equivalent viscous

damping, and ground motion on the seismic response under CUREE and SAC

ground motion suites. The range of period, damping, and ground motion used

were consistent with the range of ‘reasonable’ values which are appropriate for


279

the L-shaped test house. The range of predicted damage states from the

analyses ranged from completely undamaged to severely damaged.

• The main features of the results were that the predicted displacement demands,

at the 90th percentile levels, using a SDOF model of the single-storey test

house were:

1. Up to 150% higher for a T=0.15 sec model than for a T=0.13 sec.

model. Given the variability in stiffness and mass of the type of structure

examined, these are both reasonable values of period for the test house.

2. Up to 190% higher for a system with 2% equivalent viscous

damping ratio compared to a system with 10%. Based on the available

data, it can be argued that these are both reasonable viscous damping

values for the test house.

3. Up to three times higher under the SAC ground motions than for

the corresponding CUREE ground motions. The ground motion suites are

both deemed appropriate for Los Angeles site conditions, and were scaled

for the same target response spectra (firm soil) but for different period

ranges. The response difference can be explained by the differing

characteristics of the records selected for the suites.

• The sensitivity study highlights the importance of quantifying variability in

model parameters and excitations, and including it in seismic response

modelling.

5.6.2 Response Analysis of Example Three-Storey Building Using Shear-

Building Model

• The global response of an example three-storey light-frame building with

period T=0.21 sec. and equivalent viscous damping ratio of 5%, were

determined under the SAC and CUREE ground motions. The predicted inter-

storey displacement demands were highly variable, and larger than for the

single-storey test house. This finding is consistent with the observations from


280

the damage in timber-frame buildings observed after the Northridge

earthquake, which showed that the type of multi-storey building examined was

potentially vulnerable.

• The displacement demands for the example three-storey building were also

calculated using an equivalent SDOF model (T=0.21 sec damping = 5%). It

was shown that the SDOF model can either over-predict the response by up to

150% or under-predict by up to 73%. This highlights that it is important to

characterise the vertical distribution of mass and stiffness when studying the

seismic performance of multi-level light-frame construction, and demonstrates

that analysis methods which are dependent on assumed SDOF behaviour

should be used with caution.

5.6.3 Stochastic Response Analysis Using Equivalent Linearisation

• The applicability and accuracy of the EQL technique for highly inelastic

SDOF systems was examined. The response statistics of the hysteretic SDOF

model of the test house, under 20-second duration white-noise excitations

(max 0.5g), were determined using the EQL technique and also using MCS

(200 simulations). The natural period of the system was set to T=0.129

seconds, and the equivalent viscous damping ratio was assumed to be 5%.

• It was shown that the EQL method, accurately predicts the standard deviation

of the displacement, velocity and hysteretic restoring force, but over-predicts

the mean value of the energy dissipated, when compared to the simulation

results. This confirmed earlier findings by Foliente (1993) and Foliente et al.

(1996), but for a more realistic system, with appropriate natural frequency for

a light-frame structure, and hysteresis parameters determined accurately from

experimental data using system identification.

• The EQL scheme for the SDOF Bouc-Wen differential hysteresis model was

extended to a MDOF shear-building model in Chapter 4. The applicability and

accuracy of the EQL technique for the MDOF shear-building model, under 20-


281

second duration white noise excitations was examined. The response statistics

of an example three-storey light-frame structure (T=0.21 sec, damping = 5%)

were determined using the EQL technique and also using MCS (200

simulations).

• It was shown that the EQL method, predicts the response statistics of a MDOF

system reasonably accurately, when compared to the simulation results. The

results demonstrate that the EQL technique can be extended to MDOF

systems, using the modified Bouc-Wen hysteresis model, and still produce

good results, even for strongly non-linear systems. It is recommended that the

method be extended to include filtered, modulated excitations, to more

realistically represent earthquakes than stationary white noise.

5.6.4 Response Analysis of Test House Using Shear-Wall Model

• A hysteretic shear-wall model was used to examine the response of individual

walls of the test house under earthquake loading. The model is constructed of

hysteretic elements, which represent each of the walls, and assumes a rigid

roof and ceiling diaphragm. The shear-wall model was configured for both

uni- and bi-directional response analysis. A fundamental period of T=0.12 sec

was used for the uni-directional model, and T=0.14 sec for the bi-directional

model. Rayleigh damping of 5% in the first two modes was assumed.

• The shear-wall model was validated using the results of the static-cyclic

destructive testing presented in Chapter 2. A static-cyclic load was applied to

the model, at walls W3 and W4, as was done in the destructive experiment.

The percentage of the total applied load, which was resisted by each of the

four in-plane shear-walls was compared with the values measured during the

experiment. The accuracy of the model prediction was reasonably good

(within 10% up to ultimate load), especially given the simplicity of the model,

compared to the complexity of the highly redundant system which it is

represents. Given that the model-predicted results are reasonably good, the

assumption of a rigid roof and ceiling diaphragm in the model seems to be


282

reasonable. This, however, cannot be extrapolated to other single-level

configurations or multi-level construction, where the diaphragms may be

much more flexible in-plane than in the test house examined herein.

• Global response estimates for the single-storey test house, using SDOF and

shear-wall modelling approaches were compared, by subjecting each model to

the same suites of ground motions. It was demonstrated that the SDOF model

does a reasonably good job of predicting the global displacement demand

compared to the more detailed model, under uni-directional excitation. The

average error between the results was 8% and the worst case error, 20%. This

indicates that a SDOF model, which accurately incorporates pinching and

degrading hysteretic behaviour, may be used as an approximate tool for

studying the global seismic response of single-storey houses. It was

highlighted that the SDOF model cannot however be used to predict individual

wall response.

• The shear-wall model was used to compare global displacement demand

predictions under uni- and bi-directional excitations. It was shown that the

predicted demands under uni-directional excitations, can be either smaller or

larger than predictions under bi-directional excitations. This indicates that a

general correction cannot be applied to the uni-directional analyses. The

average error was 18% and the worst case error was 28%.

• The shear-wall model was used to examine individual wall responses under

the bi-directional SAC ground motions, in terms of maximum displacement

demand. The analysis highlighted that different walls within the structure may

have very different levels of displacement demand, under the same

earthquake, due to the bi-directional and torsional nature of the response.

Under particular earthquakes, some walls in the house respond in the elastic

range, while others respond well into the inelastic range. Large variation in

wall displacement demand occurs even between walls with the same

orientation. This variability within the structure indicates that the averaged


283

response for a given direction, which is the basis of a SDOF model, may not

be a good indicator of individual wall response.

• Under bi-directional SAC excitations, at the median demand level, only the

2/50 earthquakes result in inelastic displacement levels. At the 90th percentile

response level, the 10/50 and 2/50 earthquakes generally do exhibit a moderate

inelastic response, and the 50/50 earthquakes generally do not. These results

highlight the robustness of this type of structure to resist seismic loads without

danger of collapse. It is only under the more severe earthquakes that

significant damage is likely to be incurred.


284

Chapter 6 – Summary, Conclusions and Recommendations

285

CHAPTER 6

Summary, Conclusions and

Recommendations

6.1 Key Findings

This thesis has presented the results of experimental and analytical investigations into

the performance of light-frame structures under lateral loading (particularly seismic

loading), and a review of the literature relevant to the research. The specific aims of

this research were to: 1) develop simple, experimentally validated numerical models

of light-frame structures, which can be used to predict their performance under

seismic loading; and 2) collect experimental data suitable for validation of detailed

finite-element models of light-frame structures.

To meet these aims, a series of full-scale experiments were conducted on a single-

storey L-shaped timber-frame house. In these experiments, the distribution of the

reaction forces underneath the walls, and the displaced shape of the house were

measured in detail under static and static-cyclic loading. Analytical models of light-

frame structures, were then developed and validated against the experimental results,

and then used to conduct sensitivity and response analysis studies.

The key findings from the experimental and analytical work are as follows:

• The experimental results have provided the most detailed picture of the

reaction forces underneath a non-symmetrical light-frame structure under

lateral loading ever recorded. They have shown that there is potential for

significant load-sharing and redistribution between the external shear-walls of

a light-frame house under both elastic and inelastic response conditions.


286

Under current practice, the most commonly used design techniques for lateral

load distribution in light-frame structures do not accommodate any load-

sharing or redistribution between the walls in the structure.

• The analytical modelling results have shown that relatively simple modelling

strategies can be used to simulate the load-sharing and redistribution

characteristics that were observed in the experiments with reasonable

accuracy. The importance of the interaction between the shear-walls, and the

effect of this interaction on performance prediction was also highlighted by the

modelling work. The results showed that the single-storey test house is highly

unlikely to collapse under earthquake loading due to direct shaking, even for a

large event, but could sustain significant damage. It was demonstrated that the

inherent uncertainties due to the random nature of seismic excitation, and the

assumptions in the modelling process, have a significant effect on the

predicted performance of light-frame structures.

• A modified Bouc-Wen hysteresis model was shown to be a powerful tool for

simulating the load-displacement and energy dissipation characteristics of real

light-frame structures and components, when used in tandem with system

identification. It was also shown that system identification techniques can be

used to determine hysteresis model parameters based on multiple optimisation

criteria, and on multiple experimental data (with different excitations)

simultaneously.

• An Equivalent Linearisation technique, based on a modified Bouc-Wen SDOF

model, was extended to a MDOF shear-building model. This technique can be

used as a faster alternative to MCS, for estimating the response statistics of

multi-storey hysteretic structures, under white-noise excitations, even for

highly non-linear light-frame systems, which exhibit strong pinching and

degradation characteristics.


287

6.2 Detailed Summary and Conclusions

6.2.1 Introduction

A complete summary of the findings presented in this thesis is given in the following

sections. The summary is partitioned in accordance with the structure of the thesis.

6.2.2 Full-Scale Experiments on L-Shaped Test-House

A review of the general behaviour of light-frame systems, and full-scale testing of

light-frame structures was presented. A program of full-scale experiments, conducted

on a single-story L-shaped timber-frame house was then described, and the results

presented. Three types of experiments were conducted:

• elastic tests under a variety of small point loads

• vibration-based dynamic impact tests

• a destructive test under reverse static-cyclic lateral loading

On the basis of the experimental results, the following conclusions can be drawn.

Elastic Testing:

Small point loads, less than 10 kN, were applied to the walls of the house at the

ceiling level, in different configurations, and the displaced shape and the distribution

of the reaction forces under elastic response conditions were measured in detail. It

was found that significant load-sharing and redistribution occurs between the external

shear-resisting walls in the house under elastic response conditions. When a

concentrated load was applied to a single wall, the load was distributed to the non-

loaded external walls, mainly through the roof and ceiling diaphragm. Between 19%

and 78% of the applied load can be shared by the rest of the structure, depending on

the structural configuration and connection detail to the roof and ceiling diaphragm.

Commonly used techniques for lateral load distribution in light-frame structures do

not consider any load-sharing or redistribution.


288

In one of the elastic tests, a uniform displacement field was applied to all four North-

South walls (i.e. walls W1 to W4) until the applied load reached 90% of a hurricane

level design wind load. Under this loading, the response of the structure was

essentially elastic with no damage observed. The results of this test indicate that the

type of house tested should easily withstand this design wind load with minimal or no

structural damage. It is important to highlight that this conclusion is based purely on

the lateral wind load to the walls and does not consider uplift pressure on the roof.

Dynamic Impact Testing:

The natural racking-mode frequencies of the house in the North-South and East-West

directions were obtained by measuring the acceleration response and spectral

characteristics from dynamic impact tests. Tests were conducted before and after the

elastic tests to check for signs of damage. Before the elastic tests, the natural

frequencies determined from the dynamic tests were 13.6 and 14.8 Hz (T=0.074 and

0.075 sec) for the North-South and East-West directions, respectively. After the

elastic tests, the North-South direction frequency had changed to 13.3Hz (T=0.075

sec), with a slightly wider spectral peak than before, indicating slight damage.

However, the change in dynamic characteristics was only small, and may be within

the error bound in this type of experiment.

Calculations of the natural frequency in the North-South direction, derived using the

stiffness from the initial load-cycles of the destructive testing, and the measured mass

of the structure, agree with the value determined from the vibration tests. This

indicated that the global dynamic response exhibits SDOF-type behaviour with all of

the mass lumped at the ceiling level.

Destructive Testing:

Identical static-cyclic displacements up to +/-120mm, were applied at the ceiling level

of the house, on one side of the house, in the North-South direction. The displaced

shape and the distribution of forces throughout the structure under the inelastic

response were measured in detail. The global hysteresis response of the house, and of


289

all the individual walls, is characterised by ductile behaviour, ‘pinching’ at the origin,

and degradation of the strength and stiffness under cyclic loading. The roof and

ceiling diaphragm in the house behaved as a flexible diaphragm under the applied

loading, and absorbed some energy due to hysteretic response, but compared to the

walls it was relatively rigid. The total strength capacity of the house, in the direction

of loading, was around 100 kN with maximum capacity being reached at about 30mm

displacement. Beyond the maximum strength capacity, the house resisted 80 kN at

80mm displacement and 70 kN at 110mm displacement, indicating this type of

construction is highly ductile and not prone to sudden collapse.

As the applied load was increased on walls W3 and W4, the percentage of the load

resisted by walls W1 and W2 also increased, even though no load was applied to these

walls. The applied load was redistributed into these walls mainly via the roof and

ceiling diaphragm. The distribution of the load throughout the structure, changed

significantly during the experiment, with the amount of load resisted by each sub-

system depending on the level of applied displacement, and the structural integrity of

the other subsystems. This was due to the highly ductile nature of the structure.

Under the small loads in the initial stages of the destructive test, 4% of the total in-

plane load was taken by the out-of-plane walls. At peak load, this value increased to

around 9%, and increased further to a maximum of 12% beyond the peak load.

6.2.3 Hysteresis Modelling and System Identification

A general overview of hysteresis modelling and system identification techniques, in

the context of light-frame structures was presented. A modified Bouc-Wen

differential hysteresis element was presented and reviewed in detail. The modified

Bouc-Wen model has pinching and strength and stiffness degradation functions to

accurately simulate the behaviour of light-frame structural components under cyclic

loading.

The GRG method of system identification was used to determine the parameters for

the modified Bouc-Wen hysteresis model, for different experimental data sets, taken


290

from the literature, and for the results of the destructive experiment presented herein.

The model-computed hysteresis closely matched the experimental data in all cases, in

terms of both the hysteresis shape, and energy dissipation.

A ‘parallel’ system identification approach was outlined and illustrated through two

examples. It was demonstrated that a single set of hysteresis model parameters could

be satisfactorily fitted to two different experimental data sets simultaneously, where

the two different data sets were derived from identical test specimens under different

applied loading. The first example involved a timber-frame shear-wall, which was

tested ‘pseudo-dynamically’ under displacement response time histories, obtained

from dynamic analyses using two different scalars of the Northridge earthquake. The

second example involved a Japanese-style timber shear-wall, which was tested under

static-cyclic and then pseudo-dynamic earthquake loading. The feasibility of this

technique, and the implications for analytical modelling and performance prediction

warrant further study, as this finding may have implications for future analytical

modelling and laboratory testing strategies.

6.2.4 Structural Modelling

An overview of seismic response analysis techniques and a review of whole-building

structural modelling of light-frame structures was presented. Hysteretic SDOF and

shear-building models of light-frame structures, which incorporate differential Bouc-

Wen hysteresis elements, were formulated and implemented in computer programs.

These models are suitable for prediction of global responses under uni-directional

earthquake excitations.

An EQL scheme, formulated for the hysteretic SDOF model by Foliente (1993), was

extended to a MDOF shear-building model. This technique can be used as a faster

alternative to MCS, to estimate the response statistics of multi-storey structures, under

white-noise-based excitations.

A hysteretic shear-wall model of a light-frame structure was formulated and

implemented in a computer program. The model incorporates differential Bouc-Wen


291

hysteresis elements to represent individual walls, and is suitable for prediction of

individual wall responses under bi-directional excitations.

A FE modelling strategy, based on work by Kasal (1992), and a hybrid modelling

strategy, for seismic response analysis of light-frame structures was outlined. The

hybrid modelling strategy facilitates the transition from global to local response

predictions, complex to simple models, and from the deterministic domain to the

stochastic domain.

6.2.5 Seismic Response Analysis

The analytical models were used to conduct sensitivity studies, and deterministic and

stochastic response analyses of the single-storey L-shaped test house, and an example

three-storey light-frame structure. Two different suites of ground-motions used in the

analyses were outlined. The first suite was developed for the CUREE project and is

specifically targeted at timber structures. The second suite was developed for the SAC

steel project and is targeted at steel buildings.

SDOF Model Analyses:

The hysteretic SDOF model of the test house was used to examine the sensitivity of

the assumed structural period, equivalent viscous damping, and ground motion on the

predicted seismic response under CUREE and SAC ground motion suites. The range

of period, damping, and ground motion used were consistent with the range of

‘reasonable’ values which are appropriate for the L-shaped test house. The main

features of the results, were that the predicted displacement demands, at the 90th

percentile level using a SDOF model of the single-storey test house were:

1. Up to 150% higher for a T=0.15 sec model than for a T=0.13 sec. model.

Given the variability in stiffness and mass for the type of structure examined,

these are both reasonable values of fundamental period for the test house.

2. Up to 190% higher for a system with 2% equivalent viscous damping ratio

compared to a system with 10%. Based on the available data, it can be argued


292

that these are both reasonable equivalent viscous damping values for the test

house, although this depends on the structural and hysteresis model used.

3. Up to three times higher under the SAC ground motions than for the

corresponding CUREE ground motions. The ground motion suites are both

deemed appropriate for Los Angeles site conditions, and were scaled for the

same target response spectra (firm soil), but for different period ranges. The

response difference can be explained by the differing characteristics of the

records selected for the suites.

The sensitivity study highlighted the importance of quantifying variability in model

parameters and excitations, and including it in seismic response modelling.

Shear-Building Model Analyses:

The global response of an example three-storey light-frame building was determined

under the SAC and CUREE ground motions. The predicted inter-storey displacement

demands were highly variable, and larger than for the single-storey test house. This

finding is consistent with the observations from the damage in timber-frame buildings

observed after the Northridge earthquake, which showed that multi-storey light-frame

buildings with a soft lower storey were vulnerable. The displacement demands for the

example three-storey building were also calculated using an equivalent SDOF model.

It was shown that the SDOF model can either over-predict the response by up to

150% or under-predict by up to 73%. This highlighted the importance of

characterising the vertical distribution of mass and stiffness when studying the seismic

performance of multi-level light-frame construction, and demonstrated that analysis

methods which are dependent on assumed SDOF behaviour should be used with

caution.

Stochastic Response Analyses:

The applicability and accuracy of the EQL technique for highly inelastic SDOF

systems was examined. The response statistics of the hysteretic SDOF model of the

test house, under 20-second duration white-noise excitations (max 0.5g), were

determined using the EQL technique and also using MCS (200 simulations). It was


293

shown that the EQL method, accurately predicts the standard deviation of the

displacement, velocity and hysteretic restoring force, but over-predicts the mean value

of the energy dissipated, when compared to the simulation results. This confirmed

earlier findings by Foliente (1993) and Foliente et al. (1996), but for a more realistic

system, with appropriate natural frequency for a light-frame structure, and hysteresis

parameters determined accurately from experimental data using system identification.

The accuracy of the EQL technique for the MDOF shear-building model, under 20-

second duration white noise excitations was also examined. The response statistics of

an example three-storey light-frame structure were determined using the EQL

technique and also using MCS (200 simulations). It was shown that the EQL method,

predicts the response statistics of a MDOF system reasonably accurately, when

compared to the simulation results. The results demonstrate that the EQL technique

can be extended to MDOF systems, using the modified Bouc-Wen hysteresis model,

and still produces good results, even for strongly non-linear systems. It is

recommended that the method be extended to include filtered, modulated excitations,

to more realistically represent earthquakes than stationary white noise.

Shear-Wall Model Analyses:

A hysteretic shear-wall model was used to examine the response of individual walls of

the test house under earthquake loading. The model was constructed of hysteretic

elements, which represent each of the walls, and assumed a rigid roof and ceiling

diaphragm. The shear-wall model was configured for both uni- and bi-directional

response analysis.

The shear-wall model was validated using the results of the static-cyclic destructive

testing. A static-cyclic load was applied to the model, at walls W3 and W4, as was

done in the destructive experiment. The percentage of the total applied load, which

was resisted by each of the four North-South shear-walls was compared with the

values measured during the experiment. The accuracy of the model prediction was

reasonably good (within 10% up to ultimate load), especially given the simplicity of

the model, compared to the complexity of the highly redundant system which it

represents. Given that the model-predicted results are reasonably good, the


294

assumption of a rigid roof and ceiling diaphragm in the model for this particular house

seems to be reasonable. This however cannot be extrapolated to other single-level

configurations or multi-level construction, where the diaphragms may be much more

flexible in-plane than in the test house examined herein.

Global response estimates for the single-storey test house, using SDOF and shear-wall

modelling approaches were compared, by subjecting each model type to the same

suites of ground motions. It was demonstrated that the SDOF model does a

reasonably good job of predicting the global displacement demand of the single-story

house, compared to the more detailed model, under uni-directional excitation. The

average error between the results was 8% and the worst case error, 20%. This

indicated that a SDOF model, which accurately incorporates pinching and degrading

hysteretic behaviour, may be appropriate for studying the global seismic response of

single-storey houses.

The shear-wall model was used to compare global displacement demand predictions

under uni- and bi-directional excitations. It was shown that the predicted demands

under uni-directional excitations, can be either smaller or larger than predictions

under bi-directional excitations. This indicated that a general correction cannot be

applied to the uni-directional analyses. The average error was 18% and the worst

case error was 28%.

The shear-wall model was then used to examine individual wall responses under the

bi-directional SAC ground motions, in terms of displacement demand. The analysis

highlighted that different walls within the structure may have different levels of

displacement demand, under the same earthquake, due to the bi-directional and

torsional nature of the response. Under particular earthquakes, some walls in the

house respond in the elastic range, while others respond well into the inelastic range.

Large variation in wall displacement demand occurs even between walls with the

same orientation. This variability within the structure indicated that the averaged

response for a given direction, which is the basis of a SDOF model, may not be a

good indicator of individual wall response.


295

Under bi-directional SAC excitations, at the median demand level, only the 2/50

earthquakes result in inelastic displacement levels, and the walls generally reach no

more than half their load capacity. At the 90th percentile response level, the 10/50 and

2/50 earthquakes generally do exhibit an inelastic response, and the 50/50 earthquakes

generally do not. These results highlighted the robustness of this type of structure to

resist seismic loads without danger of collapse. It is only under the more severe

earthquakes that significant damage is likely to be incurred.

6.3 Recommendations for Further Research

The findings presented in this thesis have pointed towards a number of areas that

warrant further study. These are summarised below.

• A limitation of the experimental work is that all the observations are based on

an unfinished house, which consists of the structural elements only and the

plasterboard lining. Inclusion of the non-structural elements in the experiment

(i.e., tapes in the corner joints of interior panel materials, cornices or similar

‘non-structural’ links between roof and wall panels, windows, doors, exterior

wall finish) may alter the initial load paths at small levels of displacement, and

may alter the inelastic behaviour as well. The presence of the non-structural

finishes may also alter the apparent damage status of the structure while

increasing the strength and stiffness. This should be addressed by testing a

finished house to compare and enhance the results presented here. The results

from tests on a finished house can be used to better interpret observed damage

to similar houses in past earthquakes.

• The results of the experimental work also give an insight into how the shear-

walls behave when they are part of a whole structure. Since the current

practice for shear-wall design and analysis is based on isolated wall testing, it

is recommended to test the four main shear-walls W1 to W4 as isolated walls,

in order to examine the link between the isolated wall and the whole system

responses.


296

• The results from the destructive experiment showed that the roof and ceiling

diaphragm was slightly flexible, but relatively rigid compared to the walls. In

the analytical modelling however, the roof diaphragm was assumed to be

completely rigid, yet it still predicted the experimentally determined load-

distribution reasonably well. This cannot necessarily be extrapolated to other

single-level configurations or multi-level construction, where the diaphragms

may have different characteristics than in the test house examined herein. It is

therefore recommended that the effect of the assumed diaphragm rigidity on

response predictions, for a range of different light-frame configurations should

be examined through analytical modelling.

• The literature on hysteresis modelling indicates that there is no general

agreement regarding the effects of hysteretic pinching and degradation on

predicted system response. Some studies have highlighted circumstances

where the hysteresis assumptions can have a significant impact on response

predictions. Comprehensive studies need to be undertaken to determine when

it is appropriate to use simplified hysteresis models, and when it is advisable

to use a more complex model.

• The feasibility of a ‘parallel’ system identification technique was

demonstrated. A single set of hysteresis model parameters was satisfactorily

fitted to two different experimental data sets simultaneously, where the two

different data sets were derived from identical test specimens under different

applied loading. The application of this technique warrants further study as it

may have implications for future analytical modelling and laboratory testing

strategies. It may well be the case that it is preferable to perform repeat

experiments (e.g. on isolated shear-walls) using different loading protocols, so

that a more powerful predictive model can be fitted to the results, using

parallel system identification.

• In this thesis, it was shown that the EQL technique can be extended to MDOF

systems, using the modified Bouc-Wen hysteresis model. The EQL technique


297

still produced good estimates of response statistics for MDOF systems, under

white-noise excitations, even for strongly non-linear systems. It is therefore

recommended that the method be extended to include filtered, modulated

excitations, to more realistically represent earthquakes than stationary white

noise.

• Finally, It is recommend that a three-dimensional FE model be fully calibrated

for this type of house and if possible for another house with a different

configuration than the current test house. Once validated, the FE model, in

combination with the analytical models presented herein, can be used to

conduct sensitivity studies for a wide range of practical house configurations.

This is needed to provide general recommendations for lateral force

distribution in light-frame buildings.


298

Chapter 7 – References

299

CHAPTER 7

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Appendix A – Summary of Full-Scale Elastic Testing Results

315

APPENDIX A

Summary of Full-Scale Elastic

Testing Results

This Appendix contains summaries of the fifteen elastic tests described in Chapter 2.

Notes on Appendix A:

- Each test is portrayed through a series of plots and tables, on a separate page, and

is labelled in accordance with Table 2.4.

- The values of the applied loads, the displacements and the sum of the wall

reaction forces are given in the tables underneath the plots. All loads, reactions

and displacements are in-plane values.

- The loading points and directions are marked on the displaced shape plot. The

wall notation is shown on the first page and the direction conventions are shown

in Figure 2.3


316

Static Test: 1 Dead Load Test



displ (mm) 0 0 0 0 0 - 0


load (N) W1 W2 W3 W4 W5 W678 W9 Tot load (N) W1 W2 W3 W4 W5 W678 W9 Totmax 1962 1328 1472 2519 1983 1458 2505 - sum 0 0 0 0 0 0 0 0min -1040 -673 -430 -3 -1 -98 0 -sum 2758 3829 6476 14055 6485 6370 10802 50775

Gravity Load Only

W1

W2

W3

W4

W5

W9

W8

W6

W7


317

Static Test: 2 load W1 to 0.3T



displ (mm) -1.03 -0.28 -0.03 -0.06 -0.02 - 0.13


load (N) W1 W2 W3 W4 W5 W678 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totmax 955 329 89 183 0 25 273 downforce sum 35 29 -7 -7 624 487 -1206 -45min -861 -174 -73 -159 -170 -266 -10 uplift


318

Static Test: 3 Load W2 to 0.5T



displ (mm) -0.5 -0.59 -0.29 -0.11 -0.01 - 0.06


load (N) W1 W2 W3 W4 W5 W678 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totmax 441 734 111 289 0 175 219 downforce sum 6 29 -28 -14 900 -15 -893 -15min -476 -520 -85 -249 -466 -194 -10 uplift


319




displ (mm) -0.09 -0.08 -0.3 -0.06 -0.01 - 0.01


load (N) W1 W2 W3 W4 W5 W678 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totmax 74 91 551 121 0 21 193 downforce sum 7 3 -21 1 80 -29 39 80min -73 -88 -375 -107 -170 -153 0 uplift


320




displ (mm) -0.11 -0.09 -0.16 -0.42 0.03 - -0.02


load (N) W1 W2 W3 W4 W5 W678 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totmax 106 149 128 751 0 29 198 downforce sum 16 -6 4 -15 -610 -20 676 45min -75 -192 -110 -674 -420 -112 0 uplift


321



W1 W2 W3 W4 W5 W8 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totapplied load (N) 0 0 0 0 0 5183 0 sum 52 78 -132 169 -2 -34 -13 117

displ (mm) -0.03 0.01 -0.02 -0.03 -0.05 - -0.07


load (N) W1 W2 W3 W4 W5 W678 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totmax 303 138 117 71 127 1025 159 downforce sum 52 45 219 278 711 2496 1203 5004min 0 -176 -83 -623 -175 -342 -126 uplift


322

Static Test: 7 Load W8 to 0.5T, W2 to 0.5T


W1 W2 W3 W4 W5 W8 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totapplied load (N) 0 -5070 0 0 0 5134 0 sum -312 -2313 -1287 -838 -34 -93 -81 -4957

displ (mm) 0.44 0.6 0.28 0.07 -0.04 - -0.13


load (N) W1 W2 W3 W4 W5 W678 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totmax 608 667 163 89 626 1113 47 downforce sum 49 10 241 304 -268 2590 2107 5033min -246 -773 -156 -683 0 -372 -348 uplift


323



W1 W2 W3 W4 W5 W8 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totapplied load (N) 0 0 0 0 5160 0 0 sum 90 468 72 -587 -8 -9 3 29

displ (mm) -0.08 -0.07 0 0.13 -0.26 - 0.01


load (N) W1 W2 W3 W4 W5 W678 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totmax 158 369 63 71 353 142 57 downforce sum 21 61 137 50 2784 731 12 3795min -77 -41 -77 -408 -339 -106 -32 uplift


324

Static Test: 9 Load W5 to 0.5T, W4 to 0.8T


W1 W2 W3 W4 W5 W8 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totapplied load (N) 0 0 0 -8040 5170 0 0 sum -39 -318 -1317 -5994 -12 -122 -55 -7857

displ (mm) 0.12 0.11 0.4 0.94 -0.37 - 0.04


load (N) W1 W2 W3 W4 W5 W678 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totmax 187 845 237 1322 796 240 0 downforce sum -7 94 168 75 4801 819 -990 4961min -122 -209 -141 -1417 -149 -107 -417 uplift


325

Static Test: 10 Load W2 & W5 to -0.5T, 0.3T


W1 W2 W3 W4 W5 W8 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totapplied load (N) 0 -5120 0 0 3184 0 0 sum -284 -2061 -1505 -1157 -50 -74 -58 -5189

displ (mm) 0.5 0.59 0.35 0.14 -0.23 - -0.07


load (N) W1 W2 W3 W4 W5 W678 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totmax 538 1003 148 209 793 211 6 downforce sum 20 -7 66 62 1668 411 993 3212min -185 -747 -165 -339 -62 -202 -223 uplift


326

Static Test: 11 Load between W2 & W3 to 0.8T



displ (mm) -0.43 -0.46 -0.07 -0.18 -0.16 - 0.05

btw W2 & W3applied load (N) 6795


load (N) W1 W2 W3 W4 W5 W678 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totmax 435 628 208 444 84 172 266 downforce sum 15 12 -28 -18 369 125 -705 -229min -437 -524 -169 -342 -528 -163 0 uplift


327

Static Test: 12 Push W1, W2, W3, & W4 1.1mm



displ (mm) -1.18 -1.17 -1.1 -1.11 0.03 - 0.07


load (N) W1 W2 W3 W4 W5 W678 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totmax 1155 1382 2016 1808 0 269 955 downforce sum 62 44 -103 -41 726 42 -496 234min -1091 -1135 -1563 -1566 -1286 -618 0 uplift


328

Static Test: 13 Load roof ridge (5 deg. west)


W1 W2 W3 W4 W5 W8 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totapplied load (N) 0 0 0 0 0 0 0 sum -228 -1260 -1137 -2102 -23 -71 -39 -4859

displ (mm) 0.41 0.36 0.3 0.3 0 - -0.01

roof ridge @ 5 deg. Westapplied load (N) -5088


load (N) W1 W2 W3 W4 W5 W678 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totmax 370 393 133 557 398 174 0 downforce sum -6 -4 14 20 -37 137 68 193

min -249 -489 -146 -614 0 -123 -272 uplift


329

Static Test: 14 Load roof ridge (10deg. west)



displ (mm) 0.32 0.26 0.19 0.25 -0.03 - -0.01

roof ridge @ 10 deg. Westapplied load (N) -5208


load (N) W1 W2 W3 W4 W5 W678 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totmax 317 433 191 461 393 164 0 downforce sum -6 3 22 20 386 182 48 656min -246 -415 -202 -522 0 -102 -246 uplift


330

Static Test: 15 Load roof ridge (20 deg. east)



displ (mm) 0.17 0.14 0.1 0.08 0.04 - -0.01

roof ridge @ 20 deg. Eastapplied load (N) -2797


load (N) W1 W2 W3 W4 W5 W678 W9 load (N) W1 W2 W3 W4 W5 W678 W9 Totmax 188 63 91 263 260 61 0 downforce sum -9 -14 -5 -3 -643 -191 74 -792min -185 -220 -103 -210 0 -30 -112 uplift

Appendix B – Summary of Full-Scale Destructive Testing Results

331

APPENDIX B

Summary of Full-Scale Destructive

Testing Results

This Appendix contains snapshot summaries of the applied load, the displacement and

the reaction data from the static-cyclic destructive test described in Chapter 2.

Notes on Appendix B:

- Each snapshot is portrayed through a series of plots and tables, on a separate

page, and is taken at the peaks of selected loading cycles in both directions. Each

snapshot is labelled according to the load-cycle number, and the value of the

applied displacement.

- The values of the applied loads, the displacements and the sum of the wall

reaction forces are given in the tables underneath the graphs. All loads, reactions

and displacements are in-plane values.

- The loading points for the destructive test are indicated on the first two pages in

this appendix. The wall notation and the direction conventions are shown in

Figure 2.3


332

Destructive Test: 3 +2mm


W1 W2 W3 W4 W5 W9 Tot load (kN) W1 W2 W3 W4 W5 W678 W9 Totapplied load (kN) 0 0 11.38 9.17 0 0 20.55 sum 0.26 1.53 10.61 6.92 0.17 0.26 0.49 20.23

displ (mm) -0.33 -0.39 -0.79 -0.935 -0.13 0 -


load (kN) W1 W2 W3 W4 W5 W678 W9 load (kN) W1 W2 W3 W4 W5 W678 W9 Totmax 0.41 0.56 1.35 1.78 0.00 0.12 0.65 downforce sum 0.06 0.00 -0.02 -0.02 -1.03 -0.18 1.11 -0.08min -0.35 -0.60 -1.09 -1.61 -1.01 -0.48 0.00 uplift


333

Destructive Test: 4 -2mm


W1 W2 W3 W4 W5 W9 Tot load (kN) W1 W2 W3 W4 W5 W678 W9 Totapplied load (kN) 0 0 -7.47 -10.16 0 0 -17.63 sum -0.24 -1.57 -7.44 -7.46 -0.16 -0.32 -0.26 -17.44

displ (mm) 0.255 0.315 0.565 0.98 -0.09 -0.09 -


load (kN) W1 W2 W3 W4 W5 W678 W9 load (kN) W1 W2 W3 W4 W5 W678 W9 Totmax 0.32 0.62 0.88 1.71 1.07 0.29 0.00 downforce sum -0.04 0.02 0.02 -0.01 1.24 0.20 -1.25 0.17min -0.35 -0.56 -1.30 -1.92 0.00 -0.12 -0.59 uplift


334




displ (mm) -1.095 -1.23 -2.72 -3.105 -0.3 0.17 -


load (kN) W1 W2 W3 W4 W5 W678 W9 load (kN) W1 W2 W3 W4 W5 W678 W9 Totmax 1.11 1.52 3.10 4.44 0.00 0.37 1.50 downforce sum 0.14 -0.03 -0.09 -0.02 -2.67 -0.37 2.79 -0.26min -0.91 -1.60 -2.14 -4.28 -2.23 -0.87 0.00 uplift


335




displ (mm) 1.055 1.165 2.34 3.09 0.13 -0.3 -


load (kN) W1 W2 W3 W4 W5 W678 W9 load (kN) W1 W2 W3 W4 W5 W678 W9 Totmax 0.82 1.43 1.61 3.62 2.14 0.61 0.00 downforce sum -0.06 0.02 -0.07 0.15 2.22 0.29 -2.30 0.26min -0.62 -1.29 -2.25 -4.02 0.00 -0.33 -1.20 uplift


336




displ (mm) -2.745 -2.855 -6.51 -7.305 -0.81 0.68 -


load (kN) W1 W2 W3 W4 W5 W678 W9 load (kN) W1 W2 W3 W4 W5 W678 W9 Totmax 2.47 3.00 5.18 7.92 0.00 0.68 2.60 downforce sum 0.36 0.01 0.01 -0.11 -4.96 -0.49 4.20 -0.97min -2.03 -3.16 -3.11 -7.16 -3.54 -1.15 -0.13 uplift


337




displ (mm) 2.35 2.77 6.2 7.025 0.73 -0.47 -


load (kN) W1 W2 W3 W4 W5 W678 W9 load (kN) W1 W2 W3 W4 W5 W678 W9 Totmax 1.90 3.07 2.97 7.24 4.16 1.52 0.00 downforce sum -0.01 0.14 -0.11 0.16 4.81 0.59 -5.43 0.16min -1.64 -2.72 -4.94 -8.16 -0.05 -0.84 -2.18 uplift


338




displ (mm) -4.43 -4.63 -11.54 -11.58 -1.41 1.11 -


load (kN) W1 W2 W3 W4 W5 W678 W9 load (kN) W1 W2 W3 W4 W5 W678 W9 Totmax 3.62 4.16 6.32 11.14 0.00 1.02 3.36 downforce sum 0.43 -0.01 0.07 0.00 -6.81 -0.77 5.88 -1.22min -3.07 -4.38 -3.90 -9.36 -4.43 -1.54 -0.18 uplift


339




displ (mm) 3.87 4.45 10.495 11.285 1.37 -0.77 -




340

Destructive Test: 19 +20mm (1)



displ (mm) -6.33 -6.685 -16.25 -16.07 -2.01 1.5 -


load (kN) W1 W2 W3 W4 W5 W678 W9 load (kN) W1 W2 W3 W4 W5 W678 W9 Totmax 4.72 5.06 6.44 13.55 0.00 1.32 4.03 downforce sum 0.49 -0.04 0.07 0.01 -8.23 -1.07 7.27 -1.49min -4.10 -5.44 -4.16 -9.24 -4.94 -1.88 -0.37 uplift


341

Destructive Test: 20 -20mm (1)



displ (mm) 5.51 6.35 15.2 15.82 2.18 -1.11 -


load (kN) W1 W2 W3 W4 W5 W678 W9 load (kN) W1 W2 W3 W4 W5 W678 W9 Totmax 3.74 4.96 4.21 13.56 5.93 2.39 0.00 downforce sum -0.10 0.20 0.00 0.20 7.60 1.05 -8.46 0.50min -3.23 -4.61 -6.01 -12.46 -0.64 -1.64 -4.07 uplift


342




displ (mm) -10.5 -11.27 -26.47 -25.74 -3.29 2.27 -


load (kN) W1 W2 W3 W4 W5 W678 W9 load (kN) W1 W2 W3 W4 W5 W678 W9 Totmax 7.02 6.54 4.93 17.16 0.53 1.90 5.57 downforce sum 0.61 -0.03 -0.11 -0.12 -10.42 -1.49 9.66 -1.90min -5.68 -7.48 -3.59 -11.10 -6.30 -2.52 -0.96 uplift


343




displ (mm) 8.895 10.4 25.245 25.2 3.97 -1.84 -




344




displ (mm) -14.82 -16.1 -36.55 -35.49 -4.53 3.25 -


load (kN) W1 W2 W3 W4 W5 W678 W9 load (kN) W1 W2 W3 W4 W5 W678 W9 Totmax 9.02 7.61 4.16 19.41 1.03 2.38 6.73 downforce sum 0.70 -0.02 -0.09 -0.35 -12.11 -1.68 11.33 -2.21min -7.04 -9.16 -3.52 -12.56 -7.03 -3.57 -1.23 uplift


345




displ (mm) 12.38 14.72 33.93 34.625 5.98 -2.57 -




346




displ (mm) -17.92 -19.67 -42.4 -41.9 -5.38 3.89 -


load (kN) W1 W2 W3 W4 W5 W678 W9 load (kN) W1 W2 W3 W4 W5 W678 W9 Totmax 10.24 7.92 3.57 18.63 1.33 2.64 7.42 downforce sum 0.76 0.11 -0.02 -0.40 -12.80 -2.12 12.01 -2.46min -7.76 -8.84 -3.22 -13.02 -7.57 -4.24 -1.47 uplift


347




displ (mm) 15.81 18.985 42.635 43.315 7.82 -3.16 -




348




displ (mm) -19.91 -21.85 -47.05 -45.69 -5.85 4.28 -




349




displ (mm) 16.7 20.11 45.3 45.24 8.25 -3.33 -




350




displ (mm) -19.89 -21.87 -47.08 -45.73 -5.94 4.28 -




351




displ (mm) 16.8 20.275 45.545 45.485 8.33 -3.33 -




352




displ (mm) -24.7 -27.25 -57 -55.66 -7.01 5.22 -




353




displ (mm) 20.94 25.185 54.405 55.27 10.21 -3.98 -




354




displ (mm) -32.09 -36.2 -71.63 -69.96 -8.97 6.76 -




355




displ (mm) 30.44 35.235 73.82 74.8 13.98 -5.17 -


load (kN) W1 W2 W3 W4 W5 W678 W9 load (kN) W1 W2 W3 W4 W5 W678 W9 Totmax 11.57 10.93 4.36 19.61 13.36 7.08 1.69 downforce sum 0.08 1.01 -0.09 0.04 13.17 4.45 -17.19 1.47min -7.95 -8.47 -3.89 -8.12 -4.25 -4.61 -7.34 uplift


356




displ (mm) -41.18 -45.9 -89.11 -87.21 -11.28 8.64 -




357




displ (mm) 34.81 40.28 82.475 83.47 15.51 -5.77 -


load (kN) W1 W2 W3 W4 W5 W678 W9 load (kN) W1 W2 W3 W4 W5 W678 W9 Totmax 11.70 11.16 2.54 15.40 13.53 6.62 2.31 downforce sum 0.04 1.11 0.01 0.16 13.07 4.89 -17.53 1.75min -7.95 -8.39 -3.13 -6.74 -4.25 -4.35 -7.69 uplift


358




displ (mm) -56.43 -62.15 -115 -113.1 -15 11.46 -




359



W1 W2 W3 W4 W5 W9 Tot load (kN) W1 W2 W3 W4 W5 W678 W9 Totapplied load (kN) 0 0 -42.12 -36.17 0 0 -78.29 sum -6.85 -27.41 -15.26 -18.14 -6.13 -4.97 0.05 -78.70

displ (mm) 48.75 56.555 107.49 109.26 19.74 -7.78 -


load (kN) W1 W2 W3 W4 W5 W678 W9 load (kN) W1 W2 W3 W4 W5 W678 W9 Totmax 12.42 11.96 1.72 4.78 14.81 6.61 3.11 downforce sum 0.00 1.38 -0.24 0.16 13.89 5.20 -18.04 2.34min -7.36 -8.42 -3.31 -6.03 -4.59 -4.76 -8.27 uplift


360

Appendix C – Generalised Reduced Gradient Algorithm

361

APPENDIX C

Generalised Reduced Gradient

Algorithm

This appendix contains a brief summary of the GRG algorithm, reproduced from

Lasdon et al. (1978). Notation in this Appendix is self-contained. The basic GRG

algorithm is to solve nonlinear programs of the form

minimize ( )1mg X+

subject to ( ) 0ig X = , i = 1, neq

( ) ( )0 ig X ub n i≤ ≤ + , i = neq+1,m

( )( ) ilb i X ub i≤ ≤ , i = 1, n

(C1)

where X is a vector of n variables. The number of equality constraints, neq, may be

zero. The functions gi are assumed differentiable. The problem must be formulated in

the form of Eq. (C1). It is then converted to the following equality form by adding m

slack variables Xn+1 … Xn+m

minimize ( )1mg X+

subject to ( ) 0i n ig X X +− = , i = 1, m

( )( ) ilb i X ub i≤ ≤ , i = 1, n+m

( )( ) 0lb i ub i= = , i = n+1, n+neq ( ) 0lb i = , i = n+neq+1, n+m

(C2)

These last two equations are the bounds for the slack variables. The variables X1,…Xn

are called natural variables.

Let X satisfy the constraints of Eq. (C2), and assume that nb of the gi constraints are

binding (i.e. hold as equalities) at X . A constraint gi is taken as binding if

Appendix C – Generalised Reduced Gradient Algorithm

362

( ) ( )i ig ub n i or g lb n iε ε− + < − + < (C3) i.e. if it is within ε of one of its bounds.

GRG uses the nb binding constraint equations to solve for nb of the natural variables,

called the basic variables, in terms of the remaining (n – nb) natural variables and the

nb slacks associated with the binding constraints. These n variables are called

nonbasic. Let y be the vector of nb basic variables and x the vector of n nonbasic

variables, with their values corresponding to X denoted by ( , )y x .

Then the binding constraints can be written

g(y, x) = 0 (C4) where g is the vector of nb binding constraint functions. The basic variables must be

selected so that the nb x nb basis matrix i iB g y= ∂ ∂ is nonsingular at X . Then the

binding constraints (Eq. C4) may be solved for y in terms of x, yielding a function

y(x) valid for all ( , )y x sufficiently near ( , )y x .

This reduces the objective to a function of x only

( )1 ( ), ( )mg y x x F x+ = (C5) and reduces the original problem (at least in the neighborhood of ( , )y x ) to a simpler

reduced problem

minimize ( )F x subject to l x u≤ ≤

(C6)

where l and u are the bound vectors for x. The function ( )F x is called the reduced

objective and its gradient, ( )F x∇ , the reduced gradient. The series of reduced

problems are then solved by a gradient method.

Appendix D – Equivalent Linearisation Coefficients

363

APPENDIX D

Equivalent Linearisation

Coefficients

This appendix contains the equations for the linearisation coefficients used in Chapter

4, as presented in Foliente (1993).

The Linearisation coefficients 3eC and 3eK are given by:

( ) ( )1 13 1 2 3 4 5

1eC C C C C Cξ ξ νν

η η η η

µ µ µµ β γ β γµ µ µ µ

= − + − + + (D1)

( ) ( )

( ) ( )

( )

1

2

1 1

2 2

1

3 1 2 3 42

5 6 7 82 2

9 10

2

2 2

e u

u

K K K K qz K

qz K K K K

n K K

ξν

η η ξ

ξ ν ξ ν

η ξ η ξ

ξ ν

η

µµ β γµ µ µ

µ µ µ µβ γ β γ

µ µ µ µµ µ

β γµ

= − + + −

+ + − +

+ +

(D2)

where:

C n In nsn1

23

1 2 22

=+FHGIKJp

s( ) /G (D3)

C nn n2

23

1 2 12

=+FHGIKJp

s( ) /G (D4)

C e erfc3

32 2

32

2

1

2 2=

+

-FHGIKJ

-m

s m

z

z

D D (D5)

C e I n I nGL GL43

12

1 11= - -

-

ps

D [ ( , ) ( , )] (D6)

C e I n I nGL GL53

12

1 11= + -

-

ps

D [ ( , ) ( , )] (D7)


364

K n nn

In n nsn1

22 3

1232 1 2

232 22

2 1=+FHGIKJ - +LNM

OQP

- +

ps s r r

/ ( )/( )G (D8)

K n nn n2

223 2 3

12 12

=+FHGIKJ

-

pr s s

/G (D9)

K e erfc e3

32 2

2

323 1

212 3

1 222

2

1 32

2 22

=

+

+-FHGIKJ +

LNM

OQP

- -m

s m

s

sr m s

pm

z

z

D DDD( )* * *

/ (D10)

K e erfc e4

32 2

2

323 1

212 3

222

2

1 32

2 22

=

+

+-FHGIKJ +

LNM

OQP

- -m

s m

s

sr m s

p

z

z

D DDD( )* *

/ (D11)

K e I n I n I n I nsum sum GL GL52

3232 23

3

1 1 1 12

1 1 1 11= - - - + + - - +

RSTUVW

-

p

s

sr

p r

s

D [ ( , ) ( , )] [ ( , ) ( , )] (D12)


3232 23

3

1 1 1 12

1 1 1 11= - + - + + - - +

RSTUVW

-

p

s

sr

p r

s

D [ ( , ) ( , )] [ ( , ) ( , )] (D13)


3232 23

3

1 1 1 1 1 12

1 2 1 21= - + + - + + + - - +

RSTUVW

-

p

s

sr

p r

s

D [ ( , ) ( , )] [ ( , ) ( , )] (D14)


3232 23

3

1 1 1 1 1 12

1 2 1 21= - + - - + + + + - +

RSTUVW

-

p

s

sr

p r

s

D [ ( , ) ( , )] [ ( , ) ( , )] (D15)


3232 23

3

1 1 1 1 1 12

1 11= - - + - - + - -

RSTUVW

-

p

s

sr

p r

s

D [ ( , ) ( , )] [ ( , ) ( , )] (D16)


3232 23

3

1 1 1 1 1 12

1 11= - - - - - + + -

RSTUVW

-

p

s

sr

p r

s

D [ ( , ) ( , )] [ ( , ) ( , )] (D17)

I dsnn

= z2 23

2

0

sgn( ) sin/

r q qq

p

(D18)

qr

r0

1 232

23

1=

-FHG

IKJ

-tan (D19)

I f m y f y f erf y y e dyGL ii

im y( , ) exp *

*

= --F

HGIKJ +

LNMM

OQPP +

-

FHG

IKJ

LNMM

OQPP

•

-z 12

12 1

3 1

1

2

30

23 3

3 232 3 3

3m

s

r

s r

(D20)

I f m emk

f k ksum i

k

mm k k k( , ) ( ) ( ) ,*

/* *

( )/sgn

*

*

=FHGIKJ

+FHG

IKJ -

+FHGIKJ

LNM

OQP

-

=

- -

Âs m s gm

s2

2

02 2

1 2 22

22

32

2 12 2

12

DG (D21)

fif fif f

k

m ki

isgn

( )( )

=

-

-

=

= -

RST - -

11

111 (D22)

D1

2 2

32 22

2

=+

q zu

s mz

(D23)

D2 32

232

232

121= - +s r r s( ) * (D24)


365

DD

323 1

2

=r m * (D25)

ms

s mz

132

32 2

22

2

* =+

qzu (D26)

sm s

s m

z

z

13

32 2

2

22

* =

+

(D27)

mm s r

21 3

2232

2

1*

* ( )=

-

D (D28)

ss s r

21 3 23

2

2

1*

*=

-

D (D29)


366

Appendix E – Publications Arising From Research

367

APPENDIX E

Publications Arising From

Research

The following three refereed publications, included in this appendix, were produced

throughout the course of this research:

• Paevere, P. J., G. C. Foliente, and N. H. Haritos. 1998. "On Finding an

Optimum MDOF Inelastic System Model for Dynamic Reliability Analysis."

Pp. 215-222 in Proceedings of the Australasian Conference on Structural

Optimisation. Sydney, Australia, February 11-13. Ed. G. P. Steven, O. M.

Querin, H. Guan, and Xie Y. M. Oxbridge Press, Victoria, Australia.

• Paevere, P. J., N. H. Haritos, and G. C. Foliente. 1998. "A Hysteretic MDOF

Model for Dynamic Analysis of Offshore Towers." Pp. 513-17 in Proceedings

of the Eighth International Offshore and Polar Engineering Conference.

Montreal, Canada, May 24-29.

• Paevere, P. J. and G. C. Foliente. 1999. "Hysteretic Pinching and Degradation

Effects on Dynamic Response and Reliability." Pp. 771-79 in Proceedings of

the Eighth International Conference on the Application of Statistics and

Probability. Sydney, 12-15 December. Ed. R. E. Melchers and M. G. Stewart.

A.A. Dalkema, Rotterdam.


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Documents

Full-Scale Testing, Modelling and Analysis of Light-Frame