a parametric study on the effect of soil...

A PARAMETRIC STUDY ON THE EFFECT OF

SOIL-STRUCTURE INTERACTION ON SEISMIC RESPONSE

OF MDOF AND EQUIVALENT SDOF SYSTEMS

by

BEHNOUD GANJAVI

B.Sc., M.Sc.

THIS THESIS IS PRESENTED FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

OF

THE UNIVERSITY OF WESTERN AUSTRALIA

Structural and Earthquake Engineering

School of Civil and Resource Engineering

June 2012

DECLARATION FOR THESIS CONTAINING PUBLISHED WORK

AND/OR WORK PREPARED FOR PUBLICATION

The thesis contains published work and/or work prepared for publication, which has

been co-authored. The biographical of the work and where it appears in the thesis are

outlined below.

Ganjavi B., and Hao, H. (2011). “A parametric investigation of the influence of soil-structure

interaction on seismic response of MDOF and equivalent SDOF systems” Advances in

Structural Engineering, under review. (Chapter 3)

The estimated percentage contribution of the candidate is 80%.

Ganjavi B., and Hao, H. (2012). “Effect of structural characteristics distribution on strength

demand and ductility reduction factor of MDOF systems considering soil-structure

interaction” Earthquake Engineering and Engineering Vibration; 11(2); 205-220.

(Chapter 4)


Ganjavi B., and Hao, H. (2012). “Strength reduction factor for MDOF soil-structure

systems” The Structural Design of Tall and Special Buildings, DOI: 10.1002/tal.1022;

available online at: http://onlinelibrary.wiley.com/doi/10.1002/tal.1022/abstract. (Chapter 5)


Ganjavi B., and Hao, H. (2012). “A parametric study on evaluation of ductility demand

distribution in Multi-Degree-of-Freedom systems considering soil-structure interaction

effects” Engineering Structures, 43; 88-104. (Chapter 6)


Ganjavi B., and Hao, H. (2011). “Optimum lateral load pattern for elastic seismic design of

buildings incorporating soil-structure interaction effects” Earthquake Engineering and

Structural Dynamics, (In Press), DOI: 10.1002/eqe.2252. (Chapter 7)


Ganjavi B., and Hao, H., and Bolourchi, S. A. (2012). “Optimum seismic-resistant design of

shear buildings considering soil-structure interaction effects and inelastic behavior” Engineering Structures, to be Submitted. (Chapter 8)


Behnoud Ganjavi 06/01/2013

Print Name Signature Date

Hong Hao

Print Name Signature Date

http://onlinelibrary.wiley.com/doi/10.1002/tal.1022/abstract

Abstract The University of Western Australia

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ABSTRACT

Even though extensive researches have been conducted on Soil-Structure Interaction

(SSI) since 1960s, there are several aspects of the complex phenomenon of SSI that have

not been addressed thoroughly. The complex behavior of SSI together with uncertainties

in soil and structure parameters, and in ground motions result in a significant controversy

over the effect of SSI on structural response in both elastic and inelastic states. Recent

studies considered SSI in inelastic response analysis are mainly based on idealized

structural models of single-degree-of-freedom (SDOF) systems. However, due to

neglecting the effects of higher modes, the number of building stories and lateral

strength and stiffness distributions along the height of structures, an SDOF system might

not be able to realistically capture the SSI effects on the inelastic responses of real

buildings.

The primary objective of this research is to advance, through extensive parametric study

and analytical research, knowledge on the effects of SSI on elastic and inelastic

responses of Multi-Degree-Of-Freedom (MDOF) systems, and to develop optimization

techniques for optimum seismic design of elastic and inelastic shear buildings taking into

consideration the SSI effects.

Firstly, the study addresses the effect of SSI on elastic and inelastic response of MDOF

and its equivalent SDOF systems. The adequacy of equivalent SDOF model to estimate

strength and ductility demand of multi-storey soil-structure systems are intensively

investigated. It is concluded that using the common E-SDOF systems for estimating the

strength demands of average and slender MDOF systems when SSI effect is significant

can lead to very un-conservative results.

Secondly, the effect of structural property distribution on strength demand and ductility

(strength) reduction factor of MDOF fixed-base and soil-structure systems has been

investigated. It has been done through intensive parametric analyses of numerous linear

and nonlinear MDOF systems and considering five different shear strength and stiffness

distribution patterns including 3 code-specified patterns as well as uniform and

concentric patterns subjected to a group of earthquakes recorded on alluvium and soft

soils. Results indicate that for both fixed-base and flexible-base models, with exception

of those with very short periods, the averaged total strength demand values of structures

designed based on uniform story strength and stiffness distribution pattern along the

Abstract The University of Western Australia

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height of the structures are significantly greater than those of the other patterns such as

code-compliant patterns. This phenomenon is even more pronounced by increasing the

number of stories. It is concluded that, therefore, using the results of the uniform story

strength and stiffness distribution pattern which has been the assumption of many

previous research works would result in a significant overestimation of the strength

demands, generally from 2 to 4 times, for MDOF systems designed in accordance to the

code-compliant design patterns. Moreover, through a comprehensive parametric study of

numerous MDOF and its equivalent SDOF systems subjected to a large number of

earthquake ground motions effects of SSI on strength reduction factor of MDOF and

equivalent SDOF systems have been intensively investigated. Based on the numerical

results of nonlinear dynamic analyses and statistical regression analyses, a new

simplified equation is proposed to estimate strength reduction factors of MDOF soil-

structure systems.

Subsequently, after extensive parametric studies on the effect of SSI on global (total)

strength and ductility demand of MDOF and the corresponding E-SDOF systems carried

out in Chapters 3 to 5 as the first part of the thesis, the second part of this research

focuses on the effect of SSI on local ductility (damage) demand distribution along the

height of the structures. It is demonstrated that although the structures designed

according to some of the recently proposed optimum load patterns for fixed-base

systems may have generally better seismic performance when compared to those

designed by code-specified load patterns, their seismic performance are far from ideal if

the SSI effects are considered. Therefore, more adequate load patterns incorporating SSI

effects for performance-based seismic design needs to be proposed.

Finally, optimization techniques have been developed for optimum design of elastic and

inelastic shear buildings taking into consideration the SSI effects. An iterative analysis

procedure is introduced to estimate the optimum story shear strength distributions for a

given structure, a given ground motion and soil-structure key parameters, and an

inelastic target level of interest. Based on numerical analyses and statistical regression

analyses new simplified equations are proposed for estimation of lateral load patterns of

elastic and inelastic soil-structure systems. It is shown that the structures designed based

on the proposed pattern, on average, display remarkably better seismic performance (i.e.,

less structural weight and more uniform damage distribution over height) than the code-

compliant and recently proposed patterns by researchers for fixed-base structures.

List of Publications The University of Western Australia

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LIST OF PUBLICATIONS

REFEREED JOURNAL PAPERS:

1) Ganjavi B., and Hao, H. (2011). “A parametric investigation of the influence of

soil-structure interaction on seismic response of MDOF and equivalent SDOF

systems” Advances in Structural Engineering, (Under review).

2) Ganjavi B., and Hao, H. (2012). “A parametric Study on Evaluation of Ductility

Demand Distribution in Multi-Degree-of-Freedom Systems Considering Soil-

Structure Interaction Effects” Engineering Structures, 43; 88-104, October 2012.

3) Ganjavi B., and Hao, H. (2012). “Effect of Structural Characteristics Distribution

on Strength Demand and Ductility Reduction Factor of MDOF Systems

Considering Soil-Structure Interaction” Earthquake Engineering and

Engineering Vibration, 11(2); 205-220.

4) Ganjavi B., and Hao, H. (2012). “Strength Reduction Factor for MDOF Soil-

Structure Systems” The Structural design of Tall and Special Buildings,

(DOI: 10.1002/tal.1022;

http://onlinelibrary.wiley.com/doi/10.1002/tal.1022/abstract).

5) Ganjavi B., and Hao, H. (2012). “Optimum lateral load pattern for Elastic

Seismic Design of Buildings Incorporating Soil-Structure Interaction Effects”

Earthquake Engineering and Structural Dynamics, (in Press). DOI:

10.1002/eqe.2252

6) Ganjavi B., and Hao, H., and Bolourchi, S.A. (2012). “Optimum Seismic-

Resistant Design of Shear Buildings Considering Soil-Structure Interaction

Effects and Inelastic Behavior” Engineering Structures, (To be Submitted).

http://onlinelibrary.wiley.com/doi/10.1002/tal.1022/abstract

List of Publications The University of Western Australia

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REFEREED CONFERENCE PAPERS:

1) Ganjavi B., and Hao, H. (2011). “Elastic and Inelastic Response of Single- and

Multi-Degree-of-Freedom Systems Considering Soil Structure Interaction

Effects” Australian Earthquake Engineering Society Conference. Barossa Valley,

South Australia, 18-20 November.

2) Ganjavi B., and Hao, H. (2011). “Evaluation of the Adequacy of Code Equivalent

Lateral Load Pattern and Ductility Demand Distribution for Soil-Structure

Systems” Australian Earthquake Engineering Society Conference. Barossa

Valley, South Australia, 18-20 November.

3) Ganjavi B., and Hao, H. (2012). “Influence of Structural Property Distribution on

Elastic and Inelastic Strength Demand of Shear Buildings with Soil-Structure

Interaction,” Australasian Structural Engineering Conference. Perth, Western

Australia.11-13 July. Paper N: 006.

4) Ganjavi B., and Hao, H. (2012). “New lateral Force Distribution for Elastic

Seismic Design of Shear Buildings Incorporating SSI Effects,” Australasian

Structural Engineering Conference. Perth, Western Australia.11-13 July. Paper

N: 030.

5) Ganjavi B., and Hao, H. (2012). “An Optimization Technique for Uniform

Damage Distribution in Inelastic Shear Building Incorporation Soil-Structure

Interaction Effects,”15 World Conference on Earthquake Engineering. Lisbon,

Portugal, 24-28 September.

6) Ganjavi B., and Hao, H. (2012). “Ductility Reduction Factor for Multi-Degree-

of-Freedom Systems with Soil-Structure Interaction,” 15 World Conference on

Earthquake Engineering. Lisbon, Portugal, 24-28 September.

Table of contents The University of Western Australia

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TABLE OF CONTENTS

ABSTRACT……………………………………………………………………………....i

LIST OF PUBLICATIONS……………………………………………………………...iv

LIST OF CONTENTS…………………………………………………………………...vi

LIST OF FIGURES……………………………………………………………………..xii

LIST OF TABLES……………………………………………………………………xviii

ACKNOWLEDGEMENTS…………………………………………………………….xix

CHAPTER 1 INTRODUCTION……………………………………………………...1

1.1 Background and Motivation…………………………………………………...1

1.2 Research Goals………………………………………………………………...7

1.3 Outlines………………………………………………………………………..7

1.4 References……………………………………………………………………..9

CHAPTER 2 MODELING AND ANALYSIS PROCEDURES…………………...14

2.1 Introduction…………………………………………………………………...14

2.2 Soil-Foundation- Structure Model……………………………………………15

2.2.1 Soil-Foundation Model………………………………………………….15

2.2.2 Superstructure Models ………………………………………………….18

2. 3 Key Parameters ……………………………………………………………...19

2.4 Methodology and Procedures for Analysis…………………………………...21

2.5 OPTSSI Computer Program…………………………………………………..21

2.5.1 Soil and Structural Modelling and Assumption…………………………21

2.5.2 Structural Damping Modelling………………………………………….24

2.5.3 Earthquake Ground Motion Parameters…………………………………24

2.5.3 Story Shear Strength and Stiffness Distribution Pattern………………..25

2.6 Computational Features of OPTSSI………………………………………….25

2.7 Evaluation of MDOF Soil-Structure Systems Designed Based on Fixed-Base

Assumptions…………………………………………………………………...27

2.7 Database for Parametric Analysis……………………………………………28

2.9 References……………………………………………………………………31


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CHAPTER 3 EFFECT OF SOIL-STRUCTURE INTERACTION ON

ELASTIC AND INELASTIC RESPONSE OF MDOF AND EQUIVALENT SDOF

SYSTEMS ……………………………………………………………………………...33

3.1 Introduction…………………………………………………………………...33

3.2 Methodology and Procedures for Analysis…………………………….……..35

3.2. 1. General Procedure……………………………………………………...36

3.2. 2 Proposed Iterative Procedure…………………………………………...36

3.2.3 Step-by-Step Procedure for Parametric Study…………………………..38

3.3 Effect of SSI on Strength Demands of MDOF and E-SDOF Systems……….39

3.3.1 Strength Demands for E-SDOF Systems Corresponding to Different

Number of Stories …………………………………………………………….39

3.3.2 Strength Demands for MDOF and E-SDOF Soil-Structure Systems…...41

3.3.2 Adequacy of E-SDOF systems in estimating strength demands for MDOF

fixed-base and soil-structure systems…………………………………………46

3.4 Effect of SSI on Ductility Demand of MDOF and E-SDOF Systems………..49

3.5. CONCLUSION………………………………………………………………53

3.6 References…………………………………………………………………….55

CHAPTER 4 Effect of Structural Characteristics Distribution on Strength

Demand and Ductility Reduction Factor of MDOF Systems Considering Soil-

Structure Interaction………………………………………………………………..58

4.1 Introduction…………………………………………………………………...58

4.2 Selected Story Strength and Stiffness Distribution Patterns …………………59

4.3 Analysis Procedure …………………………………………………………..61

4.4 Effect of Structural Characteristics Distribution on Strength Demand of MDOF

Systems…………………………………………………………………………...62

4.5 Comparison between Strength Demands of Fixed-Base and Flexible-base

MDOF Systems …………………………………………………………………..66

4.6 Validation of the Numerical Results …………………………………………68

4.7 Effect of Structural Characteristics Distribution on Ductility Reduction Factor

of MDOF Systems ……………………………………………………………….68

4.7.1 Effect of Structural Characteristics Distribution………………………..69

4.7.2 Effect of Soil Flexibility………………………………………………...71

4. 8. Summary and Conclusions…………………………………………………74

4.9 References……………………………………………………………………76


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CHAPTER 5 Strength Reduction Factor for Multi-Degree-Of-Freedom Systems

Considering Soil-Structure Interaction Effects...........................................................78

5.1 Introduction…………………………………………………………………..78

5.2 Selected Earthquake Ground Motions………………………………………..81

5. 3 Procedure for Analysis………………………………………………………82

5. 4. Effect of SSI on Strength Reduction Factor of E-SDOF Systems………….82

5.4.1 Strength Reduction Factors of E-SDOF Systems for Structures with

Different Number of Stories ………………………………………………….82

5.4. 2 Effect of Ductility Ratio………………………………………………...84

5.4.3 Effect of Dimensionless Frequency……………………………………..84

5.4.4 Effect of Aspect Ratio …………………………………………………..85

5.4.5 Using R of E-SDOF Fixed-base Systems for Soil-Structure

Systems……………………………………………………………………….86

5. 5. Effect of SSI on Strength Reduction Factor of MDOF Systems……………88

5.5.1 Effect of Number of Stories …………………………………………….88

5.5.2 Effect of Dimensionless Frequency……………………………………..92

5.5.3 Effect of Aspect Ratio …………………………………………………..93

5.6 Estimation of the Strength Reduction Factors of MDOF Soil-Structure

Systems…………………………………………………………………………..96

5. 7 Summary and Conclusions…………………………………………………100

5.8 References…………………………………………………………………...102

CHAPTER 6 A Paramteric Study on Evaluation of Ductility Demand

Distribution in MDOF Shear Buildings Considering SSI Effects............................105

6.1 Introduction………………………………………………………………….105

6.2 Lateral Loading Patterns…………………………………………….………107

6.2.1 Code-Specified Seismic Design Lateral Load Patterns……………..…107

6.2.2. Lateral Load Pattern Proposed by Mohammadi et al. (2004)…………107

6.2.3 Lateral Load Pattern Proposed by Park and Medina (2007)…………...107

6.2.4 Lateral Load Pattern Proposed by Hajirasouliha and Moghaddam

(2009)………………………………………………………………………...108

6.3 Analysis Procedure………………………………………………………….109

6.4 Evaluation of Ductility Demand Distribution in Shear-Building Structures

Considering SSI Effect …………………………………………………………110

6.4.1 Effect of Number of Stories ………………...…………………………110


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6.4.2 Effect of Fundamental Period………………………………………….111

6.4.3 Effect of Aspect Ratio and Dimensionless Frequency…………………113

6.4.4 Effect of Damping Model ……………………………………………..115

6.4.5 Effect of Structural Damping Ratio……………………………………115

6.4.6 Effect of Structural Strain Hardening…………….……………………117

6.4.7 Effect of Earthquake Excitation………………………………………..117

6.5 Validation of the Numerical Results ……………………………………..…118

6.6 Adequacy of IBC-2009 Code-Specified Lateral Loading Pattern ……...…..122

6.6.1 Effect of Number of Stories and Target Ductility Demand…………....123

6.6.2 Effect of Dimensionless Frequency and Aspect Ratio ……………..…125

6.7. Adequacy of Conventional Code-Compliant and Recently Proposed Load

Patterns for Soil-Structure Systems…………………………………………….126

6.7.1 Weight-Based Method ………………………………………………...127

6.7.2 COV-Based Method……………………………………………………128

6. 8 Summary and Conclusions…………………………………………………131

6.9 References…………………………………………………………………...134

CHAPTER 7 Optimum Lateral Load Pattern for Elastic Seismic Design of

Buildings Incorporation Soil-Structure Interaction Effects...........................137

7.1 Introduction……………………………………………137

7.2 Selected Earthquake Ground Motions…….…….138

7.3 Optimum Distribution of Elastic Design Lateral Force for Soil-Structure

Systems………140

7.4 Effect of Structural Dynamic Characteristics and SSI Key Parameters on

Optimum Lateral Force Pattern……145

7.4.1 Effect of Convergence Parameter ….145

7.4.2 Effect of Earthquake Excitation …146

7.4.3 Effect of Initial Load Pattern ………………..147

7.4.4 Effect of Fundamental Period…………………149

7.4.5 Effect of Number of Stories ……………………………149

7.4.6 Effect of Dimensionless Frequency …………………..150

7.4.7 Effect of Aspect Ratio …………………..151

7.4.8 Effect of Structural Damping Ratio …………………………..152

7.4.9 Effect of Structural Damping Model……152

7.5 New Lateral Load Pattern for Elastic Soil-Structure Systems……….153

7.6 Monte Carlo Simulation……………………………………………..159


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7. 7 Conclusions…………………………161

7.8 References……………………….164

CHAPTER 8 Optimum Seismic Design of Shear Buildings Considering Soil-

Structure Interaction and Inelastic Behavior………………………………………166

8.1 Introduction………………………………………………………………….166

8.2 Estimation of Optimum Inelastic Lateral Force Distribution for Soil-Structure

Systems………………………………………………………………………….167

8.3 Effect of Structural Dynamic Characteristics and SSI Key Parameters on

Optimum Inelastic Lateral Force Pattern………………………………………..172

8.3.1 Effect of Fundamental Period …………………………………………172

8.3.2 Effect of Target Ductility Demand…………………………………….173

8.3.3 Effect of Number of Stories …………………………………………...174

8.3.4 Effect of Dimensionless Frequency …………………………………...175

8.3.5 Effect of Aspect Ratio ………………………………………………....176

8.3.6 Effect of Structural Damping Ratio and Damping Model……………. 178

8.3.7 Effect of Structural Strain Hardening………………………………….179

8.3.8 Effect of Soil Poisson’s Ratio …………………………………………180

8.3.9 Effect of Earthquake Excitation …………………………….…………180

8.4 New Seismic Load Pattern for Soil-Structure Systems with Inelastic

Behavior…………………………………………………………………………182

8.5 Adequacy of Proposed Optimum Inelastic Lateral Load Pattern ………..…187

8. 6 Conclusions…………………………………………………………………191

8.7 References…………………………………………………………………...193

CHAPTER 9 Concluding Remarks ......................................................................194

9.1 Main Findings………………………………………………………….……194

9.1.1 Effect of Soil-Structure Interaction on Elastic and Inelastic Response of

Equivalent SDOF and MDOF Systems……………………………………...194

9.1.2 Effect of Structural Charactrastics Distribution on Strength Demand and

Ductility Reduction Factor of MDOF Systems Considering Soil-Structure

Interaction ……………………………………………………………..…….195

9.1.3 Strength Reduction Factor For MDOF Systems Considering Soil-

Structure Interaction……………………………………………………….....196

9.1.4 A Paramteric Study on Evaluation of Ductility Demand Distribution

in MDOF Shear Buildings Considering SSI Effects……………………..197


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9.1.5 Optimum Lateral Load Pattern for Elastic Seismic Design of Buildings

Incorporation Soil-Structure Interaction Effects……………….………..…198

9.1.5 Optimum Lateral Load Pattern for Seismic Design of Inelastic Shear-

Buildings Considering Soil-Structure Interaction Effects……………….…199

9.2 Recommendations for Future Works………………………………………200

List of Figures The University of Western Australia

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LIST OF FIGURES

Figure 2-1: Typical 10-storty shear building models (a) fixed-base model and (b) flexible-

base model…………………………………………………………………………………...21

Figure 2-2: Typical MDOF and E-SDOF soil-structure systems (a) MDOF system (b) E-

SDOF system………………………………………………………………………………...21

Figure 2-3: A part of the SSIOPT menu…………………………………………………….27

Figure 2-4: Typical database output for SSIOPT (Strength Demand)………………………29

Figure 2-5: Typical database output for SSIOPT (Strength Reduction Factor)…………….30

Figure 3-1: Comparison of the averaged elastic and inelastic strength demand for different E-

SDOF system with soil-structure interaction (0a = 2)……………………………………….40

Figure 3-2 Comparison between first-mode shape for different number of stories: (a) fixT =1

and (b) fixT =3………………………………………………………………………………..41

Figure 3-3: Comparison of the averaged elastic strength demand for ESDOF and MDOF

soil-structure systems………………………………………………………………………..43

Figure 3-4: Comparison of the averaged inelastic strength demand for ESDOF and MDOF

soil-structure systems for µ =2………………………………………………………………43


soil-structure systems for µ =6………………………………………………………………44

Figure 3-6: Effect of number of stories on the averaged elastic and inelastic strength demand

of fixed-base and soil-structure systems for H r = 3……………………………………….45

Figure 3-7: The ratio of elastic and inelastic strength demands in 10-story building to those

in the corresponding E-SDOF system……………………………………………………….48

Figure 3-8: COV of story ductility demand for different MDOF soil-structure systems……48

Figure 3-9: Height-wise distribution of averaged ductility demand for systems with fixT = 1.5

and μ= 6……………………………………………………………………………………...49

Figure 3-10. Averaged ductility demand for different E-SDOF and MDOF soil-structure

systems for H r = 3…………………………………………………………………………51

Figure 3-11: Averaged ductility demand for different E-SDOF and MDOF soil-structure

systems for µ= 2……………………………………………………………………………..52


systems for µ= 6……………………………………………………………………………..53


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Figure 4-1: Different Lateral force and normalized shear strength patterns for the10-story

building with fixT = 1.5 sec ………………………………………………………………….61

Figure 4-2: Effect of structural characteristics distribution on strength demand for MDOF

systems with N = 5 and H r = 3…………………………………………………………..64


systems with N = 15 and H r = 3………………………………………………………...65

Figure 4-4: Averaged ratio of strength demand in uniform pattern to that of the IBC-2009

pattern for systems with H r = 3…………………………………………………………..65

Figure 4-5: Averaged ratios of strength demands of soil-structures systems with respect to

the fixed-base systems in different story strength and stiffness patterns ( 0a = 3, N= 10)

………………………………………………………………………………………….……67

Figure 4-6: Effect of structural characteristics distribution on averaged ratios of strength

demands of soil-structures systems to the fixed-base systems (N = 10; 0a = 3; H r =5)

……………………………………………………………………………………………….67

Figure 4-7: Comparisons of the averaged strength demands resulted from this study and

OPENSEES for the 15-story building with 0a = 3 (21 earthquakes)………………………..68

Figure 4-8: Effect of structural characteristics distribution on averaged ductility reduction

factor of MDOF fixed-base and soil-structure systems (N = 10 and H r = 3)

……………………………………………………………………………………………….70

Figure 4-9: Comparison of averaged ratios of ductility reduction factor in different load

patterns to the IBC-2009 pattern for systems with N = 10 and H r = 3……………………71

Figure 4-10: Effect of soil flexibility on averaged ductility reduction factor of MDOF

systems ( H r = 3) …………………………………………………………………………..73

Figure 4-11: Averaged ductility demand spectra of MDOF soil-structure systems designed

based on fixed-base ductility reduction factor ( H r = 3 and µ = 6) ……………………….73

Figure 5-1: Comparison of the averaged strength reduction factor for different E-SDOF

systems (µ = 4)………………………………………………………………………………83

Figure 5-2: Averaged strength reduction factor spectra for E-SDOF systems with different

ranges of nonlinearity ( H r = 3)…………………………………………………………....84

Figure 5-3: Effect of dimensionless frequency on Averaged strength reduction factor spectra

of E-SDOF soil-structure systems …………………………………………………………..85

Figure 5-4: Effect of aspect ratio on Averaged strength reduction factor spectra of E-SDOF

soil-structure systems ……………………………………………………………………….86


xiv

Figure 5-5: Averaged ductility demand spectra of E-SDOF soil-structure systems designed

based on fixed-base strength reduction factors ……………………………………………..87

Figure 5-6: Effect of the number of stories on averaged strength reduction factor spectra of

fixed-base and soil-structure systems ( H r = 3) …………………………………………...91

Figure 5-7: Averaged ratios of shear strength demands on MDOF systems to those on E-

SDOF systems for different ranges of nonlinearity (10-story building; H r = 3)…………92

Figure 5-8: Averaged modifying factor for MDOF fixed-base and soil-structure systems (10-

story building; H r = 3) ……………………………………………………………………92

Figure 5-9: Effect of dimensionless frequency on averaged strength reduction factor spectra

of MDOF soil-structure systems ( H r = 1) ……………………………………...…………94

Figure 5-10: Effect of dimensionless frequency on averaged strength reduction factor spectra

of MDOF soil-structure systems ( H r = 5) ……………………………………………...…95

Figure 5-11: Effect of aspect ratio on averaged strength reduction factor spectra of MDOF

soil-structure systems (10-story building) …………………………………………………..95

Figure 5-12: Correlation between Eq. (5-5) and averaged numerical results for strength

reduction factors of MDOF soil-structure systems ( H r = 3) …………...…………………99

Figure 6-1: Different Lateral force and normalized shear strength patterns for 10-story

building with fixT = 1.5 sec and μ= 4……………………………………………………….109

Figure 6-2: Effect of number of stories on height-wise distribution of averaged ductility

demand for systems with fixT = 1 and H r =3…………………………………….………112

Figure 6-3: Effect of fundamental period on height-wise distribution of averaged ductility

demand for systems with N= 10 and H r =3……………………………………………...113

Figure 6-4: Effect of aspect ratio and dimensionless frequency on height-wise distribution of

averaged ductility demand for systems with N= 10 and fixT =1 ……………………...……114

Figure 6-5: Effect of damping model on height-wise distribution of ductility demand for

systems with N= 10, µ = 4 and H r =3subjected to Loma Prieta earthquake (APEEL 2 -

Redwood City) …………………………………………………………………………….116

Figure 6-6: Effect of damping ratio on height-wise distribution of ductility demand for

systems with N= 10, fixT = 1.5, µ = 4 and H r =3 subjected to Loma Prieta earthquake

(APEEL 2 - Redwood City) …………………………………………………………….…116

Figure 6-7: Effect of strain hardening on height-wise distribution of ductility demand for

systems with N= 10, fixT = 1.5, µ = 4 and H r =3 subjected to Loma Prieta earthquake

(APEEL 2 - Redwood City) …………………………………………………………….…118


xv

Figure 6-8: Height-wise distribution of individual and averaged ductility demand for

systems with N= 15, H r =3, fixT =1.5 and µ = 6…………………………….……………118

Figure 6-9: Calibrating the stiffness of the elastic linear springs presented by Grange et al.,

(2011). …………………………………………………………………………………..…121

Figure 6-10: Comparisons of the ductility demand distributions resulted from nonlinear

macro-element and equivalent linear elastic (cone) models for two levels of nonlinearity (µ=

2, 6); 10-story building with fixT =1 sec, H r =5 (Average of 10 earthquake records).

………………………………………………………………………………………...……121

Figure 6-11: Effect of number of stories on averaged COV of story ductility demands for

systems with H r =3 designed according to IBC-2009 load pattern. …..………...………124

Figure 6-12: Effect of maximum ductility on averaged COV of story ductility demands for

systems with H r =3 designed according to IBC-2009 load pattern. …..……...…………125

Figure 6-13: Effect of soil flexibility on averaged COV of story ductility demands for

systems with N=15 and H r =3 designed according to IBC-2009 load pattern …………126

Figure 6-14: Effect of aspect ratio on averaged COV of story ductility demands for systems

with N=15 and 0a =3 designed according to IBC-2009 load pattern. ……………………126

Figure 6-15: Averaged Weight Index of 10-story soil-structure system with H r =3

designed according to different load patterns. ……………………………………………..130

Figure 6-16: Averaged COV of 10-story soil-structure system with H r =3 designed

according to different load patterns. ……………………………………………….………131

Figure 7-1: IBC-2009 (ASCE/SEI 7-05) design spectrum for soil type E and response spectra

of 21 adjusted earthquakes (5% damping) for selected ground motions…………………..140

Figure 7-2: Comparison of IBC-2009 with optimum designed models of fixed-base and soil-

structure system: (a) lateral force distribution; (b) story ductility pattern, 10-story shear

building with fixT = 1.5 sec, H r =3, Kobe (Shin Osaka) simulated earthquake………….145

Figure 7-3: Variation of structural weight index for different values of convergence powers;

10-story soil-structure system with fixT = 1.5 sec, H r = 3, 0a =2, Kobe (Shin Osaka)

simulated earthquake ………………………………………………………………………146

Figure 7-4: Optimum lateral force distribution for different earthquake excitations, 10-story

building with fixT = 1.5 sec: (a) Fixed-base model; (b) Soil-structure model with H r =3

and 0a = 2………………………………………………………………………………..….148


xvi

Figure 7-5: Effect of (a) ground motion intensity and (b) initial load pattern on optimum

lateral force profile for soil-structure systems with fixT = 1.5 sec, H r =3 and

0a = 2; Kobe

(Shin Osaka) simulated earthquake……………………………………………………..….148

Figure 7-6: Effect of initial load pattern on optimization iteration steps; 10-story shear

building; (a) Fixed-base systems (b) soil-structure system with fixT = 1.5 sec, H r = 3,

0 2a , Kobe (Shin Osaka) simulated earthquake. ………………………………………...149

Figure 7-7: Effect of fundamental period (a) and the number of stories (b) on averaged

optimum lateral force profile for soil-structure systems with H r =3 and 0a = 2:

fixT = 1.5

sec. ………………………………………………………………………………………....150

Figure 7-8: Effect of dimensionless frequency on averaged optimum lateral force profile for

10-story soil-structure systems with H r =3: (a) fixT = 1 sec.: (b)

fixT = 2 sec. ………….151

Figure 7-9: Effect of aspect ratio on averaged optimum lateral force profile for a 10-story

soil-structure system with fixT = 1.5 sec…………………………………………………....152

Figure 7-10: Effect of structural damping ratio (a) and damping model (b) on optimum

lateral force profile; 10-story soil-structure system with H r =3, 0a = 2 and fixT = 1.5 sec;

Loma Prieta (APEEL 2 - Redwood City) earthquake…………………………………..….153

Figure 7-11: The spectra of ratio of required to optimum structural weight for the 10-story

soil-structure systems designed according to different load patterns; average of 21

earthquakes………………………………………………………………………………....157

Figure 7-12: The spectra of COV for the 10-story soil-structure systems designed according

to different load patterns; average of 21 earthquakes; 0a = 3…………………………...….157

Figure 7-13: Comparison of different load patterns for 10-story soil-structure systems with

fixT = 1.5 sec, H r =3 and 0a = 3: (a) lateral force distribution; (b) story ductility pattern;

average of 21 earthquakes………………………………………………………………….158

Figure 7-14. Correlation between Eq. (7-4) and numerical results …………………….….158

Figure 7-15: Comparisons of the COV of story ductility demand distribution of the 10-story

building designed based on the proposed optimum pattern and IBC-2009 pattern; ( fixT = 1.5

sec, H r =3, Kobe (Shin Osaka) simulated earthquake)……………………………….….161

Figure 8-1: Comparison of IBC-2009 and fixed-base optimum load patterns with optimum

designed models of soil-structure system: (a) lateral force distribution; (b) story ductility

pattern, 10-story shear building with fixT = 1.5 sec, H r =3, Kobe (Shin Osaka) simulated

earthquake …………………………………………………………………………………172


xvii

Figure 8-2: Effect of fundamental period on averaged optimum lateral force profile for soil-

structure systems with H r =3 and 0a = 2: 10-story building (average of 21

earthquakes)……………………………………………………………………………..….174

Figure 8-3: Effect of target ductility demand on averaged optimum lateral force profile for

soil-structure systems with H r =3 and 0a = 2: 10-story building (average of 21 earthquakes)

Figure 8-4: Effect of the number of stories on averaged optimum lateral force profile for soil-

structure systems with H r =3 and 0a = 2:

fixT = 1.5 sec. (average of 21 earthquakes)…..174

Figure 8-5: Effect of dimensionless frequency on averaged optimum lateral force profile for

10-story soil-structure systems with H r =3, µ= 6: (a) fixT = 0.5 sec.: (b)

fixT = 2 sec…..177

Figure 8-6: Effect of aspect ratio on averaged optimum lateral force profile for a 10-story

soil-structure system with fixT = 1.5 sec, µ= 4…………………………………………..….177

Figure 8-7: Optimum lateral force profile for a 10-story soil-structure system with H r =3,

0a = 2, fixT = 1.5 sec and µ= 6: (a) Effect of structural damping ratio; (b) Effect of structural

damping model, Loma Prieta (APEEL 2 - Redwood City) simulated earthquake………...179

Figure 8-8: Effect of structural post yield behavior on Optimum lateral force profile for a 10-

story soil-structure system with H r =3, 0a = 2,

fixT = 1.5 sec; Loma Prieta (APEEL 2 -

Redwood City) simulated earthquake……………………………………………………...180

Figure 8-9: Effect of soil Poisson ratio on Optimum lateral force profile for a 10-story soil-

structure system with H r =3, 0a = 3, fixT = 1.5 sec; Loma Prieta (APEEL 2 - Redwood City)

simulated earthquake……………………………………………………………………….181

Figure 8-10: Effect of (a) Earthquake excitation and (b) ground motion intensity on optimum

lateral force profile for soil-structure systems with H r =3 and 0a = 2, µ= 4; Kobe (Shin

Osaka) simulated earthquake……………………………………………………………….182

Figure 8-11: Correlation between Eq. (8-5) and numerical results …………………….….187



earthquakes (µ= 2) ………………………………………………………………………....190



earthquakes (µ= 6) ………………………………………………………………………....190

List of Tables The University of Western Australia

xviii

LIST OF TABLES

Table 2-1: Properties of a soil–foundation element based on the cone model concept…..…17

Table 3-1: Selected ground motions recorded on alluvium and soft sites based on USGS site

classification…………………………………………………………………………………36

Table 5-1: Selected ground motions recorded at alluvium and soft sites based on USGS site

classification ………………………………………………………………………………...81

Table 5-2: Constant coefficient ia and ib of Eq. (5-5) ……………………………..………97

Table 5-3: Constant coefficient ia of Eq. (5-5) ………………………………………….…97

Table 5-4: Constant coefficient ib of Eq. (5-5) ……………………………………….……97

Table 5-5: Constant coefficient ia of Eq. (5-5) ……………………………………….……98

Table 5-6: Constant coefficient ib of Eq. (5-5) …………………………………….………98

Table 5-7: Constant coefficient ia of Eq. (5-5) ………………………………….…………98

Table 5-8: Constant coefficient ib of Eq. (5-5) ………………………………………….…99

Table 7-1: Selected ground motions recorded at alluvium and soft sites based on USGS site

classification………………………………………………………………………………..139

Table 7-2: Constant coefficient ia of Eq. (7-4) as function of relative height………….…163

Table 7-3: Constant coefficient ib of Eq. (7-4) as function of relative height………….…163

Table 7-4: Constant coefficient ic of Eq. (7-4) as function of relative height………….…164

Table 8-1: Constant coefficients of Eq. (8-5) as function of relative height (µ= 2) …….…184

Table 8-2: Constant coefficients of Eq. (8-5) as function of relative height (µ= 4) ….……185

Table 8-3: Constant coefficients of Eq. (8-5) as function of relative height (µ= 6) ….……185

Acknowledgments The University of Western Australia

xix

ACKNOWLEDGEMENTS

I would like to express my deep appreciation to my advisor, Winthrop Professor Hong

Hao, for his encouragement and support during the three-year PhD program and for

giving me the opportunity to work under his supervision. Working under his supervision

was always an inspiration and honor.

I am indebted to the staff and postgraduate students from School of Civil and Resource

Engineering and Centre for Offshore Foundation Systems (COFS) for their diverse help

during my PhD study in The University of Western Australia. My special thanks to Dr.

Kaiming Bi for his invaluable suggestions and discussion on various aspects of the

thesis.

Great appreciation is dedicated to The University of Western Australia, School of Civil

and Resource Engineering and Professor Hong Hao, for the financial supports I received

during my candidature, which consisted of an International Postgraduate Research

Award (IPRS) through Australian Government, University Postgraduate Award (U.P.A)

through UWA, a postgraduate top-up scholarship and an AD-Hoc scholarship through

School of Civil and Resource Engineering and Prof. Hong Hao, respectively.

Finally, my boundless and sincere thanks to my wife for all the sacrifices she has

endured and never complained about deficiencies; to my parents for their continual

spiritual and financial supports during my PhD and not only years.

Chapter 1 The University of Western Australia

1

Chapter 1

INTRODUCTION

1.1 BACKGROUND AND MOTIVATION

The inescapable reoccurrence of severe earthquakes around the world has emphasized

the necessity of better understanding of the structural responses subjected to earthquake

ground motions to reduce their vulnerability through better design and retrofitting. As

pointed out by Krawinkler et al., (2006), in performance-based earthquake engineering

framework, a good design is generally based on the philosophy of incorporating

performance target up front in the design process, so that following performance

assessment becomes more of a verification process rather than a design improvement

process. Moreover, a poor initial conceptual design likely will never lead to a good

design even though the initial design to some extent satisfies the performance targets

(Krawinkler et al., 2006).The successful incorporation of performance-based earthquake

engineering in the design process necessitates accurate evaluation of the seismic

demands on structures at different hazard levels to compare with corresponding capacity

criteria.

Seismic demands of building structures are known to be dependent on many factors such

as structural properties, ground motion characteristics, site conditions as well as soil-

structure interaction (SSI). SSI is one of the important factors that can significantly

affect the seismic responses of structures located on soft soils by altering the overall

stiffness and energy dissipation mechanism of the systems. In fact, a soil-structure

system behaves as a new system having longer period and generally higher damping due

to energy dissipation by hysteretic behaviour and wave radiation in the soil. SSI usually

is not an attractive subject for civil engineering community due to its complex

behaviour. The complex behaviour of SSI together with uncertainties in soil and

structure parameters, and in earthquake ground motion result in a significant controversy

over the effect of SSI on structural response in both elastic and inelastic states.


2

The general effects of SSI on elastic response of single-degree-of freedom (SDOF) and

multi-degree-of freedom (MDOF) systems with an emphasis on the former were the

subject of many studies in the 1970s (Perelman et al., 1968; Sarrazin et al., 1972;

Jennings and Bielak, 1973; Chopra and Gutierrez, 1974; Veletsos and Meek, 1974;

Veletsos and Nair, 1975; Veletsos, 1978). In pioneering studies, extensive efforts made

by Jennings and Bielak (1973), Veletsos and Meek (1974) and Veletsos and Nair (1975)

were led to introducing the modification of the seismic demand of elastic SDOF

structures. They found that the effect of inertial interaction on the structural response can

simply be predicted from the response of an equivalent SDOF system through an

increase in the fundamental natural period and a change in the associated damping of a

fixed-base structure. They also concluded that SSI can either increase or decrease the

seismic demand of the structures depending on the system parameters and the

characteristics of the earthquake ground motion. These works led to providing tentative

provisions in ATC3-06 (ATC, 1978), which is actually the foundation of new provisions

on earthquake-resistant design of soil-structure systems (BSSC, 2000; FEMA-440,

2005). Code-compliant seismic designs for SSI systems are, conventionally, based on the

approximation in which the predominant period and associated damping of the

corresponding fixed-base system are modified (Jennings and Bielak, 1973; Veletsos and

Meek, 1974). In fact, the current seismic provisions consider SSI, generally, as a

beneficial effect on seismic response of structures since SSI usually causes a reduction of

total shear strength of building structures (BSSC, 2000; ASCE, 2005). However, the

inelastic behaviour of the superstructure with the influence of SSI, inevitable during

severe earthquakes, has not been well investigated. On the other hand, the current

seismic design philosophy is based on elastic behaviour of structures with SSI effect

when subjected to moderate and severe earthquakes. Hence, there is a necessity to

investigate the effect of SSI on inelastic response of building structures.

One of the pioneering works on inelastic soil-structure systems were made by Veletsos

and Verbic (1974) and Bielak (1978). Utilizing the method of equivalent linearization to

solve the equations of motion Bielak (1978) proposed a simplified approximate formula

for estimation of the fundamental resonant frequency of the system and for an effective

critical damping ratio. They recognized that for non-linear hysteretic structures

compliance of the soil foundation may lead to larger displacements with respect to the

corresponding fixed-base structure. They also pointed out that this behaviour differs

from that generally observed for linear systems, for which the effect of soil-structure

interaction is to reduce the rigid-base response. Muller and Keintzel (1982) subsequently


3

investigated the ductility demands of SDOF soil-structure systems. They showed that the

ductility demand of structures, when considering soil beneath them, could be different

from that of the equivalent SDOF systems without considering SSI. In another study,

Ciampoli and Pinto (1995) have concluded that the inelastic seismic demand of SDOF

systems essentially remains unaffected by SSI in general, and in some cases SSI results

in a decrease in the response. This conclusion, however, contradicts with the results of

Bielak (1978). Rodriguez and Montez (2000) investigated the response and damage of

buildings located on flexible soil and concluded that inelastic displacement demand in

soil-structure system can be approximated by using an equivalent fixed-base system

having an elongated period.

The effects of SSI in yielding systems, including both kinematic and inertial interaction,

were investigated by Aviles and Perez-Rocha (2003). They developed the concepts of

equivalent elastic soil-structure system to include the nonlinear behaviour of the

structure by means of a nonlinear replacement SDOF oscillator defined by an effective

ductility together with the effective period and damping of the system for the elastic

condition. In further works, considering aforementioned nonlinear replacement SDOF

oscillator, they also studied the effect of SSI on strength-reduction and displacement-

modification factors as well as damage index of structures (Aviles and Perez-Rocha,

2005; Aviles and Perez-Rocha, 2007). Ghannad and Ahmadnia (2006) assessed the

adequacy of ATC3-06 (1978) regulation when considering the SSI effect on inelastic

response of structures using simplified SDOF system with elastic-perfectly plastic

behaviour. They concluded that using this provision leads to higher ductility demands in

the structure, especially for the case of short period buildings located on soft soils. In

more recent years, more studies have been reported by researchers to investigate the SSI

effect on inelastic behaviour of SDOF systems (Mahsuli and Ghannad, 2009;

Moghaddasi et al., 2011; Aviles and Perez-Rocha, 2011).

As mentioned in the literature, almost all researches made on nonlinear soil-structure

systems focused on SDOF systems while the SSI effect on inelastic response of MDOF

systems due its more complexity has not been investigated in detail. A few studies of SSI

effects on MDOF systems are those conducted by Dutta et al. (2004), Barcena and Steva

(2007) , Chouw and Hao (2008a, 2008b), Raychowdhury (2011) and Tang and Zhang

(2011). These studies concentrated on investigating the SSI effects on specific structures.

Systematic studies of SSI effects on seismic demands of MDOF systems cannot be

found in the literature yet. Current practice and research often adopt the studies based on


4

SDOF systems to model the performance of MDOF system. However, SDOF systems

having only one DOF may not be able to correctly reflect the realistic behaviour of

common building structures interacting with soil beneath them when subjected to

earthquake ground motions. This can be due to the lack of incorporating the effects of

number of stories and higher modes as well as, more importantly the effect of height-

wise distribution of lateral strength and stiffness on inelastic response of real soil-

structure systems.

In the first part of this dissertation (Chapters 3-5), an intensive parametric study is

performed to investigate the effect of inertial SSI on both elastic and inelastic seismic

strength and ductility demands of MDOF and its equivalent SDOF (E-SDOF) systems

using simplified soil-structure model for surface (shallow) foundation in which the

kinematic interaction is zero. This is carried out for a wide range of non-dimensional

parameters to investigate the adequacy of E-SDOF systems on estimation of seismic

strength and ductility demand of MDOF soil-structure systems (Chapter 3). Moreover, in

Chapter 4 taking into consideration the different shear strength and stiffness distribution

patterns for MDOF systems subjected to a group of earthquake ground motions recorded

on alluvium and soft soils, the effect of structural property distributing on strength

demand and strength (ductility) reduction factor of MDOF fixed-base and soil-structure

systems are parametrically investigated. Chapter 5 parametrically study the effects of

SSI on strength reduction factor of MDOF and its equivalent SDOF systems. A new

simplified equation is proposed to estimate the strength reduction factors of MDOF soil-

structure systems.

In almost all current seismic design codes in the world, lateral-load resisting systems for

regular structures are primarily designed based on the equivalent static lateral force

procedure. This procedure is generally regarding the seismic effects as lateral inertia

forces, which is called force-based design procedure. Therefore, the distribution of story

stiffness and strength along the height of the structures are designed primarily based on

these static forces that are mainly derived according to elastic structural behaviour

analyses of fixed-base structures under seismic loading. The inelastic behaviour is only

accounted approximately in an indirect manner. The height-wise distribution of these

lateral load patterns from various standards such as Euro Code 8 (CEN, 2003), Mexico

City Building Code (Mexico, 2003), Uniform Building Code (UBC, 1997), NEHRP

2003 (BSSC, 2003), ASCE/SEI 7-05 (ASCE, 2005), Australian Seismic code (AS-

1170.4, 2007) and International Building Code, IBC 2009 (ICC, 2009) depends on the


5

fundamental period of the structures and their mass. They are derived primarily based on

elastic dynamic analysis of the corresponding fixed-base structures without considering

soil-structure interaction (SSI) effect. In other words, the seismic lateral load patterns in

all aforementioned provisions are based on the assumption that the soil beneath the

structure is rigid, and hence the influence of SSI effect on load pattern is not considered.

The efficiency of using the code-specified lateral load patterns for fixed-base building

structures have been investigated during the past two decades (Anderson et al., 1991;

Gilmore and Bertero, 1993; Chopra, 1995, Moghaddam and Mohammadi, 2006,;

Ganjavi et al., 2008, Hajirasouliha and Moghaddam, 2009). Leelataviwat et al. (1999)

evaluated the seismic demands of mid-rise moment-resisting frames designed in

accordance to UBC 94. They proposed improved load patterns using the concept of

energy balance applied to moment-resisting frames with a pre-selected yield mechanism.

Lee and Goel (2001) also proposed new seismic lateral load patterns for high-rise

moment-resisting frames up to 20-story with the same concept which Leelataviwat et al.

(1999) used. However, they used SDOF response modification factor as well as

structural ductility factors and dealt with a limited number of ground motions. Their

proposed load pattern fundamentally follows the shape of the lateral load pattern in the

code provisions (i.e., UBC 1994, 1997) and is a function of mass and the fundamental

period of the structure. In a more comprehensive research, Mohammadi et al. (2004) and

Mohammadi and Moghaddam (2006) investigated the effect of lateral load patterns

specified by the United States seismic codes on drift and ductility demands of fixed-base

shear building structures under 21 earthquake ground motions, and found that using the

code-specified design load patterns do not lead to a uniform distribution and minimum

ductility demands. Ganjavi et.al (2008) investigated the effect of equivalent static and

spectral dynamic lateral load patterns specified by the major seismic codes on height-

wise distribution of drift, hysteretic energy and damage subjected to severe earthquakes

in fixed-base reinforced concrete buildings. They concluded that in strong ground

motions, none of the lateral load patterns will lead to uniform distribution of drift,

hysteretic energy and damage, and an intense concentration of the values of these

parameters can be observed in one or two stories especially in equivalent static method.

More recently, several studies have been conducted by researchers to evaluate and

improve the code-specified design lateral load patterns based on the inelastic behaviour

of the structures (Moghaddam and Hajirasouliha, 2006; Park and Medina, 2007;

Hajirasouliha and Moghaddam, 2009; Goel et al., 2010). However, all researches have


6

been concentrated on the different types of structures with rigid foundation, i.e., without

considering SSI effects.

In the second part of this dissertation (Chapters 6-8), through performing intensive

parametric analyses of nonlinear multi-degree-of freedom (MDOF) systems with SSI

subjected to a family of earthquakes recorded on alluvium and soft soils the effect of

SSI on height-wise distribution of ductility demands are investigated (Chapter 6). Effect

of many parameters including fundamental period, level of inelastic behaviour, the

number of stories, damping model, damping ratio, structural strain hardening, earthquake

excitation, level of soil flexibility, and structure slenderness ratio on height-wise

distribution of damage (ductility demand) are intensively investigated. In addition, the

adequacy of three code-complaint lateral loading patterns, namely UBC-97, IBC-2009

and EuroCode-8 as well as three recently proposed optimum loading patterns derived

from analysing fixed-base structures are parametrically investigated for soil-structure

systems. In further work (Chapter 7), using the uniform distribution of damage over the

height of structures, as the design target, an optimization algorithm for seismic design of

elastic soil-structure systems is developed. Consequently, utilizing the proposed

optimization approach a new load pattern for elastic soil-structure systems is proposed

for practical purpose. In Chapter 8, Optimization algorithm developed in Chapter 7 for

elastic soil-structure systems is modified to incorporate the inelastic behaviour. By

performing intensive numerical simulations of responses of inelastic soil-structure shear

buildings with various dynamic characteristics and SSI parameters, the effects of

fundamental period of vibration, ductility demand, earthquake excitation, damping ratio,

damping model, structural post yield behavior, the number of stories, soil flexibility and

structure aspect ratio (slenderness ratio) on the optimal lateral load pattern of soil-

structure systems are investigated. Based on the results of this study, a new lateral load

pattern for soil-structure systems taking into account for inelastic behaviour is proposed.

It is shown that the structures designed based on the proposed pattern, on average, lead

to remarkably better seismic performance (i.e., less structural weight and more uniform

damage distribution over height) than the code-compliant and recently proposed patterns

by researchers for fixed-base structures.


7

1.2 REASERCH GOALS

The present study has been undertaken with the specific aims of:

1. Developing a comprehensive computer program to perform parametric studies

on MDOF and SDOF systems subjected to earthquake ground motions with and

without consideration of the SSI effects;

2. Investigating the effect of inertial SSI on both elastic and inelastic seismic

strength and ductility demands of MDOF and its equivalent SDOF (E-SDOF)

systems using simplified soil-structure model;

3. A comprehensive parametric study to investigate the effect of structural property

distribution on strength demand and ductility reduction factor of MDOF systems

considering soil-structure interaction;

4. Proposing a new simplified equation to estimate strength reduction factors of

MDOF soil-structure systems;

5. Performing parametric study to evaluate the ductility demand distribution in

MDOF shear buildings with SSI effects;

6. Developing optimization techniques for optimum seismic design of elastic and

inelastic shear-building structures incorporating SSI effects; and

7. Proposing new lateral force patterns for seismic design of shear buildings

incorporating SSI effects.

1.3 OUTLINE

This dissertation is composed of nine chapters. The eight chapters subsequent to this

introductory chapter are organized as follows:

Chapter 2 presents a brief classification of soil-structure interaction analysis methods

and then presents the simplified soil-structure model utilized in this study. The

superstructure modelling and assumptions as well as soil-structure key parameters and

analysis procedure are elaborated and discussed. An outline of the comprehensive

computer program written and developed for conducting intensive parametric studies

with consideration of SSI effects is presented.


8

Chapter 3 addresses the effect of SSI on elastic and inelastic response of MDOF and its

equivalent SDOF systems. The adequacy of equivalent SDOF model to estimate strength

and ductility demand of multi-storey soil-structure systems are investigated.

Chapter 4 studies the effect of structural property distributing on strength demand and

ductility (strength) reduction factor for MDOF fixed-base and soil-structure systems. It

has been done through intensive parametric analyses of numerous linear and nonlinear

MDOF systems and considering five different shear strength and stiffness distribution

patterns including 3 code-specified patterns as well as uniform and concentric patterns

subjected to a group of earthquakes recorded on alluvium and soft soils.

Chapter 5 through a comprehensive parametric study of numerous MDOF and its

equivalent SDOF systems subjected to a large number of earthquake ground motions

recorded on alluvium and soft soils, effects of SSI on strength reduction factor of MDOF

and equivalent SDOF systems have been intensively investigated. Based on the

numerical results of nonlinear dynamic analyses and statistical regression analyses, a

new simplified equation is proposed to estimate strength reduction factors of MDOF

soil-structure systems.

Chapter 6 parametrically studies the ductility demand distributions in MDOF shear-

building structures with SSI effects. Effect of many parameters including fundamental

period, level of inelastic behaviour, the number of stories, damping model, damping

ratio, structural strain hardening, earthquake excitation, level of soil flexibility, structure

aspect ratio on height-wise distribution of damage (ductility demand) are intensively

investigated. In addition, the adequacy of three different code-complaint lateral loading

patterns including UBC-97, IBC-2009 and EuroCode-8 as well as three recently

proposed optimum loading patterns for fixed-base structures are parametrically

investigated for soil-structure systems by two methods associated to the economy of the

seismic-resistant system.

Chapter 7 and 8 develop optimization techniques for optimum design of elastic and

inelastic shear buildings taking into consideration the SSI effects. An iterative analysis

procedure is introduced to estimate the optimum story shear strength distributions for a

given structure, a given ground motion and soil-structure key parameters, and an

inelastic target level of interest. Based on numerical analyses and statistical regression


9

analyses new simplified equations are proposed for estimation of lateral load patterns of

elastic and inelastic soil-structure systems.

Finally, the main outcomes of this research are summarized in Chapter 9. The major

research findings are highlighted and discussed. Suggestions are included for issues

requiring further investigation.

1.4 REFERENCES

Anderson JC, Miranda E and Bertero VV (1991). “Evaluation of the seismic

performance of a thirty-story RC building,” UCB/EERC-91/16, Earthquake

Engineering Research Centre, Univ. of California, Berkeley.

Applied Technology Council. (1978). Tentative provisions for the development of

seismic regulations for buildings, ATC-3-06, California.

AS-1170.4. (2007). Structural design actions: Earthquake actions in Australia.

ASCE/SEI 7-05 (2005). Minimum Design Loads for Buildings and Other Structures.

American Society of Civil Engineers: Reston, VA.

Aviles J. and Perez-Rocha L. (2003) “Soil–structure interaction in yielding systems,”

Earthquake Engineering and Structural Dynamics, 32(11): 1749–1771.

Aviles J. and Perez-Rocha JL. (2005) „Influence of foundation flexibility on Rμ and Cμ

factors‟” Journal of Structural Engineering (ASCE) 131(2); 221–230.

Aviles J. and Perez-Rocha J. L. (2007) “Damage analysis of structures on elastic

foundation,” Journal of Structural Engineering (ASCE) 133(10); 1453–1461.

Aviles J. and Perez-Rocha J. L. (2011) “Use of global ductility for design of structure–

foundation systems,” Soil Dynamics and Earthquake Engineering 31(7): 1018–

1026.

Barcena A. and Esteva L. (2007) “Influence of dynamic soil–structure interaction on the

nonlinear response and seismic reliability of multistorey systems,” Earthquake

Engineering and Structural Dynamics 36(3): 327-346.

Bielak J. (1978) “Dynamic response of non-linear building–foundation systems,”


Building Seismic Safety Council (BSSC). (2000) NEHRP Recommended Provisions for

Seismic Regulations for New Buildings and Other Structures, Federal

Emergency Management Agency, Washington, DC.


10

CEN (2003). EuroCode 8: Final draft of EuroCode 8: Design of structure for earthquake

resistance – Part 1: General rules for buildings: European Committee for

Standardization.

Chopra A. K. and Gutierrez J. A. (1974) “Earthquake response analysis of multistory

buildings including foundation interaction,” Earthquake Engineering and

Structural Dynamics 3(1): 65–77.

Chouw N and Hao H. (2008a). “Significance of SSI and non-uniform near-fault ground

motions in bridge response I: effect on response with conventional expansion

joint.” Engineering Structures 30(1): 141–153.

Chouw N and Hao H. (2008a). “Significance of SSI and non-uniform near-fault ground

motions in bridge response II: effect on response with modular expansion joint.”

Engineering Structures 30(1): 154–162.

Ciampoli, M., and Pinto, P. E. (1995). “Effects of soil-structure interaction on inelastic

seismic response of bridge piers.” Journal of Structural Engineering, 121(5):

806-814.

Diaz O., Mendoza E, and Esteva L. Seismic ductility demands predicted by alternate

models of building frames. Earthquake Spectra 1994 10(3):465–487.

Dutta, C. D., Bhattacharya K. and Roy R. (2004) “Response of low-rise buildings under

seismic ground excitation incorporating soil–structure interaction,” Soil

Dynamics and Earthquake Engineering 24(12): 893-914.

Ganjavi, B Vaseghi Amiri, J., Ghodrati Amiri, G and Yahyazadeh Ahmadi, Q., (2008).

“Distribution of Drift, Hysteretic Energy and Damage in Reinforced Concrete

Buildings with Uniform Strength Ratio.” The 14th World Conf. on Earthquake

Engineering, Beijing, China, October 12-17.

Ghannad, M. A. and Ahmadnia A. (2006) “The effect of soil–structure interaction on

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of Solids and Structures 41(22): 6597–6612.

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14

Chapter 2 MODELLING AND ANALYSIS PROCEDURE

2.1 INTRODUCTION

How to treat the behaviour at infinity of the unbounded soil computationally is the main

and important subject in modelling soil-structure interaction problems. It is known that

the emitted wave from the vibration structure-soil interface will finally propagate in the

soil towards infinity, and at a sufficient distance from the structure only outgoing waves

exist in the real radiation problem (Wolf, 1994). This prevents an infinite energy

accumulation. In fact, no energy associated with the waves may radiate from infinity

towards the soil-structure system. This radiation condition will lead to a boundary-value

problem formulated in the frequency domain for an unbounded domain with a unique

solution (Wolf, 1994). Generally two analysis methods are available to solve soil-

structure problem: the direct method and the substructure method.

In the direct method, the region of the soil adjacent to the soil-structure interface is

explicitly modelled with sophisticated finite element method in the same way as the

structure (Wolf, 1994). The artificial boundary needs to be introduced, as there is no

possibility to model the unbounded soil domain with a finite number of elements with

bounded dimensions. The transient equilibrium equations are solved simultaneously for

both structure and continuum elements that model surrounding soil. The boundary

conditions should be employed for boundaries, such that they should be able to dissipate

energy of incident waves. Owing to the discretization of the soil region adjacent to the

structure, a large number of degrees of freedom appear, leading to a significant

computational effort. In fact, although the direct method is analytically straightforward,

it is computationally expensive and not feasible for parametric studies. In order to utilize

such models, one needs excessive data preparation time, and experience, to evaluate the

results. It is believed that this rigorous method should be only used for special or

important structures such as nuclear power plants or dams.


15

In the substructure method, the soil-structure system will be divided in two parts: the

first part is the structure resting on the foundation, and is usually modelled by masses,

dashpots, and possibly nonlinear spring or equivalently by finite elements; the second

part is the soil-structure interface. In this method, first the interaction force-displacement

(dynamic stiffness) in the nodes located on the structure-soil interface is calculated. This

dynamic stiffness representing the rigorous boundary condition with the unbounded soil

can be physically modelled by some springs and dashpots whose coefficients are

dependent on frequency of the excitation. Then the superstructure resting on these

springs and dashpots is analysed subjected to a given earthquake ground motion. In the

substructure method, soil and the structure can be separately analyzed with two different

approaches. Therefore, depending on the specific case and the importance of the problem

each of them can be considered more precisely with respect to another one. This method

is appropriate for parametric studies as well as for standard projects of moderate and

small sizes, and hence will be utilized in this dissertation.

2.2 SOIL-FOUNDATION- STRUCTURE MODEL

2.2.1 Soil-foundation model

In this study, substructure method is used to model soil-structure systems. As stated in

the literature, using the sub-structure method, the soil can be modeled separately and

then combined to establish the soil-structure system. There are various simplified

analysis procedures based on substructure method that can be used to model the soil-

structure systems. Cone model based on the one-dimensional wave propagation theory

has been extensively used by researchers during the past decade to investigate the elastic

and inelastic response of soil-structure systems subjected to earthquake ground motions

(Ghannad and Ahmadnia, 2006; Ghannad and Jahankhah, 2007, Nakhaei and Ghannad,

2008; Mahsuli, and Ghannad, 2009; Khoshnoudian and Behmanesh, 2010; Moghaddasi

et al., 2011a and 2011b).

Cone models have been developed during the past four decades, which can be divided

into three stages (Wolf and Deeks, 2004): The first stage included the pioneer work until

the mid-1970s. In this stage, a surface foundation resting on a homogenous half-space

was analyzed. The second stage was from the 1970s until the mid-1990s, with major

developments in the mid-1990s. In that time, the reflection and refraction of waves on

material discontinuities, as exists in the case of a layer on a half-space, was addressed.

Surface and embedded foundations on a layer resting on a homogenous half-space were


16

modelled. Finally in the third stage that dates back from the mid-1990s to present, the

foundations resting on multi-layered sites were introduced and modelled. A general

approach with sufficient accuracy for a large range of practical cases was developed. At

the incipient stage, cone model was only used to model a foundation on the surface of a

homogenous half-space for vertical and horizontal motions. Later, rocking motion was

addressed by Meek and Veletsos (1974) and, then, torsional motion was examined by

Veletsos and Nair (1974). In another development, Meek and Wolf (1991, 1992)

investigated the behaviour of material discontinuities at the interface of a layer to a half-

space. Reflected and refracted waves at the boundaries of layers to a half-space were

traced by their developed cone model with cross-section properties increasing in the

direction of the wave propagations. Further, an embedded foundation was also treated

using stacks of embedded disks modelled with double cones.

Owing to many uncertainties in earthquake engineering such as determining the dynamic

properties of soil and structures, and earthquake characteristics the accuracy of any

analysis will always be limited. A deviation of maximum 20% of the results of physical

models from those of the rigorous solutions for one set of input parameters is, in general,

sufficient as engineering accuracy criterion (Wolf, 1994). The Cone model used in the

present parametric study provides sufficient accuracy for engineering designs. In

addition to the aforementioned reasons, cone models have some advantages that can be

considered for soil-structure modelling. Cone models satisfy physical features. For

example, for a layer fixed at its base, no radiation damping occurs below the cut-off

frequency. Cone models can be used for sites with general layering (Meek and Wolf,

1993). They can also be used for a surface foundation and an embedded foundation for

all degrees of freedom and for various foundation shapes. Cone models in which wave

patterns are clearly postulated utilize simple physical and exact mathematical solutions.

The wave reflections and refractions at the material discontinuities such as a layer on a

half-space are captured using cone model.

The soil-shallow-foundation element, in which the kinematic interaction is zero, is

modeled by an equivalent linear discrete model based on the cone model with earthquake

frequency-independent coefficients and equivalent linear model (Wolf, 1994; Ghannad

and Jahankhah, 2007, Nakhaei and Ghannad). However, to consider the material

damping viscous soil impedances, i.e., stiffness and damping coefficients, are dependent

on the natural frequency of the system (i.e., soil-structure system) through an iterative

method. Cone model based on the one-dimensional wave propagation theory represents


17

circular rigid foundation with mass fm and mass moment of inertia fI resting on a

homogeneous half-space. As mentioned earlier, in lieu of the rigorous elasto-dynamic

approach, the simplified cone model can be used with sufficient accuracy in engineering

practice (Wolf, 1994). The sway and rocking DOFs are defined as representatives of

translational and rotational motions of the shallow foundation, respectively, disregarding

the slight effect of vertical and torsional motion. The stiffness and energy dissipation of

the supporting soil are represented by springs and dashpot, respectively. In addition,

while being hysteretic inherently, soil material damping is assumed as commonly used

viscous damping so that more intricacies in time-domain analysis are avoided. All

coefficients of springs and dashpots for sway and rocking used to define the soil-shallow

foundation model are summarized in Table 2-1.

Table 2-1: Properties of a soil–foundation element based on the cone model concept

Motion Stiffness Viscous Damping

Added Mass

Horizontal 28 , 2

sh

v rk ρυ

=−

h s fc v Aρ=

------

Rocking

1 / 3υ <

2 38 ,

3(1 )sv rk ϕ

ρυ

=−

p fc v Iϕ ρ= -------

1/3 1 / 2υ≤ ≤

(2 )s fc v Iϕ ρ=

50.3 ( 1 / 3)m rϕ π υ ρ∆ = −

Internal Mass Moment of inertia

1 / 3υ < 9 (1 )( )32

pf

s

vm I r

vϕπ ρ υ= −

1/3 1 / 2υ≤ ≤ 9 (1 )8 fm I rϕπ ρ υ= −

Material Damping

Additional Parallel Connected Element (i= 0 or ϕ )

Viscous Damping to Stiffness

ik Viscous Damping to Mass iC

0

0

2 ( )i iC k ζω

=

0

0

( ) i im c ζω

=

The parameters utilized in Table 2-1 defined as hk , hc , k ϕ and cϕ are sway stiffness,

sway viscous damping, rocking stiffness, and rocking viscous damping, respectively.

Equivalent radius and area of cylindrical foundation are denoted by r and fA . Besides,

ρ , υ , pv and sv are respectively the specific mass density, Poisson’s ratio, dilatational


18

and shear wave velocity of soil. The relationship between pv and sv in the above

equations is defined as follows:

2(1 ) 1 / 3,(1 2 )

2 1/3 1/ 2

p s

p s

v v if

v v if

υ υυ

υ

−= <

−

= ≤ ≤

(2.1)

To consider the soil material damping, 0ζ , in the soil-foundation element, each spring

and dashpot is respectively augmented with an additional parallel connected dashpot and

mass. Also, to modify the effect of soil incompressibility, an additional mass moment of

inertia M ϕ∆ equal to 50.3 ( 1/ 3) rπ υ ρ− can be added to the foundation for υ greater than

1/3 (Wolf, 1994). It is clear that the shear modulus of the soil will change with soil strain

such that it decreases as soil strain increases. Thus, a reduced shear wave velocity which

is compatible with the corresponding strain level in soil should be considered to

incorporate soil nonlinearity. Incorporating soil nonlinearity to the soil-foundation

element, however, may be approximated through conventional equivalent linear

approach in which a degraded shear wave velocity, compatible with the estimated strain

level in soil, is utilized for the soil medium (Moghaddasi et al., 2011a). This is currently

used in the modern seismic provision such as NEHRP 2000 (BSSC, 2000) and FEMA-

440 (2005) where the strain level in soil is implicitly related to the peak ground

acceleration (PGA). In the present study, by considering a range of reasonable values for

dimensionless frequency, this point has been approximately incorporated.

2.2.2 Superstructure models

MDOF superstructure: To incorporate the effects of higher modes, the number of

stories and lateral strength and stiffness distribution on inelastic response of MDOF

buildings interacting with soil beneath them, the well-known shear-beam model is

utilized in this study. Due to its simplicity, shear beam is indeed one of the most

frequently used models that facilitate performing a comprehensive parametric study

(Diaz et al., 1994; Moghaddam and Mohammadi, 2001; Mohammadi et al., 2004;

Moghaddam and Mohammadi, 2006; Hajirasouliha and Moghaddam, 2009). In the

MDOF shear-building models utilized in the present study, each floor is assumed as a

lumped mass to be connected by elasto-plastic springs. Story heights are 3 m and total

structural mass is considered as uniformly distributed along the height of the structure. A



19

bilinear elasto-plastic model with 2% strain hardening in the force-displacement

relationship is used to represent the hysteretic response of story lateral stiffness. This

model is selected to represent the behavior of non-deteriorating steel-framed structures

of different heights. In all MDOF models, lateral story stiffness is assumed as

proportional to story shear strength distributed over the height of the structure in

accordance with the 2009 IBC load pattern (IBC, 2009). Five percent Rayleigh damping

was assigned to the first mode and the mode in which the cumulative mass participation

was at least 95%.

Equivalent SDOF superstructure: For each MDOF building an equivalent SDOF (E-

SDOF) system is introduced in the analysis for this study. The properties of these E-

SDOF systems are set such that the mass of the SDOF system is the same as the total

mass of the MDOF building; similarly, the period of vibration, damping ratio and

effective height of the E-SDOF systems are the same as the fundamental mode properties

of the MDOF building. A typical MDOF of fixed-base and flexible-base models as well

as the corresponding E-SDOF soil-structure system are illustrated in Figures 2-1 and 2-2.

2.3 KEY PARAMETERS

It is well known that the response of the soil-structure system essentially depends on the

size of structure, dynamic characteristics of the soil and structure, the soil profile as well

as the applied excitation. In other words, for a specific earthquake ground motion, the

dynamic response of the structure can be interpreted based on the properties of the

superstructure relative to the soil beneath it. It has been shown that the effect of these

factors can be best described by the following dimensionless parameters (Veletsos, 1977;

Ghannad and Jahankhah, 2007; Mahsuli and Ghannad, 2009):

1. A dimensionless frequency as an index for the structure-to-soil stiffness ratio

defined as:

0 fix

s

Hav

ω= (2.2)

where fixω is the natural frequency of the corresponding fixed-base structure. It

can be shown that the practical range of 0a for conventional building structures is

from zero for the fixed-base structure to about 3 for the case with severe SSI

effect (Ghannad and Ahmadnia, 2006). Besides, H which is the effective height


20

of structure corresponding to the fundamental mode properties of the MDOF

building can be obtained from the following equation:

11 1

11

jn

j j ij i

n

j jj

m hH

m

ϕ

ϕ

= =

=

=∑ ∑

∑ (2.3)

where jm is the mass of the jth story; ih is the height from the base level to level j;

and 1jϕ is the amplitude at jth story of the first mode.

2. Aspect ratio of the building defined as H r , where r is the equivalent foundation

radius.

3. Interstory displacement ductility demand of the structure defined as:

m= y

δµδ

(2.4)

where mδ and yδ are the maximum interstory displacement demand resulted from

a specific earthquake ground motion excitation and the yield interstory

displacement corresponds to the structural stiffness of the same story,

respectively. Note that for the MDOF building µ is referred to as the greatest

value among all the story ductility ratios.

4. Structure-to-soil mass ratio defined as:

2= totmmr Hρ

(2.5)

where H and totm are total height and mass of the structure, respectively.

5. Foundation-to-structure mass ratio f totm m .

6. Poisson’s ratio of the soil denoted by υ .

7. Material damping ratios of the soil 0ζ and the structure Sζ .

The first two factors, affecting the responses more prominently are usually considered as

the key parameters which define the main SSI effect. The third one controls the inelastic

behavior of the structure. The other parameters, having less importance, may be set to

some typical values for conventional buildings (Veletsos and Meek, 1974; Wolf, 1994).

In the present study, the foundation mass ratio is assumed to be 0.1 of the total mass of

the MDOF buildings. However, the effect of this ratio will be investigated in next


21

chapters. The Poisson’s ratio is considered to be 0.4 for the alluvium soil and 0.45 for

the soft soil. Also, a damping ratio of 5% is assigned to the soil material.

Figure 2-1: Typical 10-storty shear building models (a) fixed-base model and (b) flexible-base model

Figure 2-2: Typical MDOF and E-SDOF soil-structure systems (b) MDOF system (a) E-SDOF system

H �

(a) (b)

(b) (a)


22

2.4 METHODOLOGY AND PROCEDURE FOR ANALYSIS

The adopted soil-foundation-structure models introduced in the previous sections are

used directly in the time domain nonlinear dynamic analysis. Step-by-step solution

scheme in which dynamic imposed loads are incrementally applied to the model of the

structure is utilized for all MDOF and E-SDOF models. Variable load increments by

considering events within steps are defined in order to control the equilibrium errors in

each analysis step. An event is considered as any kind of state change that causes a

change in the structural stiffness. To conduct parametric studies for both MDOF and

SDOF systems with consideration of SSI effects subjected to a given earthquake ground

motion, a comprehensive computer program package, “OPTSSI”, has been written

specifically for this thesis. The software has the capabilities of performing parametric

analysis automatically to investigate the influence of many parameters such as elastic

and inelastic strength demand, maximum drift, residual drift, strength reduction factors,

MDOF modifying factor as well as optimization based on uniform damage distribution

over the height of the structure, which will be briefly introduced in the next part.

2.5 OPTSSI COMPUTER PROGRAM

This program has been written by FORTRAN and visual basic programing languages

with more than 70,000 lines for E-SDOF and MDOF shear-building structures of fixed-

base and soil-structure systems. Many verification processes have been conducted, and

the results have been compared with those generated by OPENSEES (2011). The

accuracy of this program will be demonstrated in the next chapters. The main features of

the software can be summarized as follows:

2.5.1 Soil and structural modelling and assumption

1. The structure can be modelled as shear building structures with equal story height up

to 25 stories as described in Section 2.2.2 for elastic and inelastic ranges of response.

The corresponding E-SDOF structure will also be created automatically by the software.

Therefore, the user only needs to select the number of stories from the program menu as

shown below:


23

2. A bilinear elasto-plastic model with 2% strain hardening in the force-displacement

relationship is used to represent the hysteretic response of story lateral stiffness.

However, the program has the capability of specifying different value of strain hardening.

Other types of hysteretic behavior will be added to the program in the next version of the

software.

3. Soil-foundation element is modelled by an equivalent linear discrete model based on

the cone model for an equivalent linear elastic half space (Wolf, 1994) as stated in the

previous section. Any amount of soil density, Poisson’s ratio, dimensionless frequency

and aspect ratio can be selected for a parametric study.

It should be noted that by considering a very small value for dimensionless frequency

(e.g., 0a = 0.001) the system, in a very good approximation, represents a fixed-base

structure.

4. Foundation mas can be considered (i.e., Found. Mass= 1) or ignored (i.e., Found.

Mass= 0):

5. Any amount of target ductility ratio and fundamental period can be specified for both

MDOF and E-SDOF systems with consideration of SSI effects.


24

2.5.2 Structural damping modelling

There are three options to model structural damping. They are the conventional viscous

damping models for MDOF systems including stiffness-proportional damping (ST),

mass-proportional damping (MA) and Rayleigh-type damping (RA) in which damping

matrix is composed of the superposition of a mass-proportional damping term and a

stiffness-proportional damping term. The E-SDOD systems, having only one degree-of-

freedom, are modeled by mass-proportional damping (MA).

For Rayleigh-type damping model, the program automatically checks to find the mode at

which the cumulative mass participation exceeds 95%. It will, then, show the associated

mode number. In addition, any coefficient of structural damping (denoted as Str. Damp

in the program) can be defined by user.

2.5.3 Earthquake ground motion parameters

User can easily define the earthquake record properties such as any type of data format (EQ Format), number of acceleration data (EQ Data), scale factor (EQ S.Fact), acceleration time step, time step for time history analysis.


25

2.5.3 Story shear strength and stiffness distribution pattern

One of the excellent capabilities of the software for research work and parametric study

is that the program has more than 12 options for story shear strength and stiffness

distribution pattern. In other word, any kind of the strength and stiffness distribution

along the height of structure will affect the response of the structure such as ductility

reduction factor, displacement amplification factor, height-wise distribution of seismic

demands and etc. Therefore, the program has been written such that user can choose

more than 12 different load patterns including uniform pattern, concentric, triangular,

trapezoid, rectangular, 2 types of parabolic patterns, UBC-97, IBC- 2009, Euro-Code 8

and those optimum patterns recently proposed by researches for fixed-base buildings. All

patterns have been predefined in the software and the user just need to choose the

specific load pattern:

2.6 COMPUTATIONAL FEATURES OF OPTSSI

The program has various computational features for both fixed-base and soil-structure

systems with elastic and inelastic behaviour. The main features can be described as

follows:

1. OPTSSI can optimize different shear buildings ranging from 2 to 25 stories for

both fixed-base and soil-structure systems for any specified values of ductility

ratio, fundamental period, aspect ratio, and dimensionless frequency.

Optimization is based on the uniform distribution of damage (ductility or drift)

along the height of the structure subjected to a given earthquake ground motion.

It will be shown that the structure will have the least structural weight at this

state. In this approach, the structural properties are automatically modified

through an iterative process so that inefficient material is gradually shifted from

strong to weak parts of the structure. This process is continued until a state of

uniform deformation is achieved. This optimization technique will be discussed

in Chapters 7 and 8 of this thesis in detail.


26

2. Coefficient of variation (COV) for different parameters such as story ductility

demand, maximum drift, residual drift are calculated in each step as well as in

the final step of analysis.

3. The absolute values and the distribution patterns of elastic drift, maximum drift

in inelastic state, and residual drift and ductility demands along the height of the

structures are computed.

4. Effective mass, effective height and structural weight index are computed. In

addition, periods and damping ratios of the soil-structure system in the first 5

modes are calculated by the program. The mode number and the total cumulative

effective mass for Rayleigh-type damping are computed as well.

5. Base shear coefficient for both E-SDOF and MDOF systems are computed for

fixed-base and soil-structure systems in elastic and inelastic ranges of response.

The effects of SSI and number of degrees of freedom can be easily investigated

by comparing the results.

6. In most of the seismic design provisions, the concept of strength reduction factor

has been developed to account for inelastic behaviour of structures under seismic

excitations. Most recent studies considered soil-structure interaction (SSI) in

inelastic response analysis are mainly based on idealized structural models of

SDOF systems. However, an SDOF system might not be able to well capture the

structural response characteristics of real MDOF systems. Another feature of

“OPTSSI’ is that the program can compute strength reduction factors of MDOF

and E-SDOF systems for both fixed-base and soil-structure systems. MDOF

modifying factor for strength reduction factor are also computed for both fixed-

base and soil-structure systems. A part of SSIOPT menu is shown in Figure 2-3.


27

Figure 2-3: A part of the SSIOPT menu

2.7 EVALUATION OF MDOF SOIL-STRUCTURE SYSTEMS DESIGNED

BASED ON FIXED-BASE ASSUPMTION

There is another purpose for the investigation of the effect of SSI on the seismic

demands of structures. Usually structures are designed without considering SSI effects.

Therefore, it is necessary to evaluate the influence of SSI on these structures that were

already designed based on fixed-base assumption with considering the effect of soil

flexibility. A separate computer programs has been written such that the structure that

first designed based on the fixed-based assumption are again analyzed with consideration

of the underlying soil flexibility for the different target periods, ductility ratios and SSI

key parameters for both MDOF and E-SDOF systems. At this state, the maximum

ductility demand, height wise distribution of the ductility demand and its COV will be


28

calculated for the SSI system. Four different analyses are considered for this

investigation as follows:

1. Use of base shear of fixed-base system (strength demand) for SSI system.

2. Use of strength reduction factor of MDOF fixed-base systems for MDOF soil-

structure systems

3. Use of strength reduction factor of SDOF fixed-base System for MDOF fixed-

base System

4. Use of strength reduction factor of SDOF soil-structure systems for MDOF soil-

structure system

2.8 DATABASE FOR PARAMETRIC ANALYSIS

Although all the aforementioned parameters can be calculated by the software,

processing the large numbers of the output data for different earthquakes and structural

parameters are really difficult and maybe impossible for an intensive parametric study.

Therefore, a database program has been written to transfer all the completed data to the

predesigned spread sheet files. A database is a collection of information that is organized

so that it can be easily accessed, managed, and updated. This will act as a database such

that after each analysis the results will be transferred to the specific place of the spread

sheet. Two examples of this database are shown in Figures 2-4 and 2-5 for strength

demands and strength reduction factors of MDOF and E-SDOF systems, respectively.

http://whatis.techtarget.com/definition/0,289893,sid9_gci212343,00.html


29

Figure 2-4: Typical database output for SSIOPT (Strength Demand)


30

Figure 2-5: Typical database output for SSIOPT (Strength Reduction Factor)


31

2.9 REFERENCE:


Seismic Regulations for New Buildings and Other Structures, Federal Emergency

Management Agency, Washington, DC.

Diaz O, Mendoza E, Esteva L. (1994) Seismic ductility demands predicted by alternate

models of building frames. Earthquake Spectra 10(3):465–487.

FEMA 440. (2005). Improvement of nonlinear static seismic analysis procedures. Report


Technology Council.



Ghannad, M. A., And Jahankhah, H. (2007). “Site dependent strength reduction factors

for soil–structure systems.” Soil Dynamics & Earthquake Engineering. 27(2), 99–

110.


design of structures.” J. Struct. Eng., 135(8), 906–915.

IBC-2009. (2009). International Building Code, International Code Council, Country

Club Hills, USA.

Khoshnoudian F. and Behmanesh I. (2010), “Evaluation of FEMA-440 for including

soil-structure interaction”, Journal of Earthquake engineering and engineering

vibration, 9(3): 1-12.

Meek, J.W. and Veletsos, A.S. (1974). “Simple Models for Foundations in Lateral and

RockingMotion”, Proceedings of the 5th World Conf. on Earthquake Engineering,

IAEE, Rome, 2, 2610-2631.

Meek, J.W. and Wolf, J.P. (1992). “Cone Models for Homogeneous Soil, I”, Journal of

the Geotechnical Engineering Division, ASCE, 118(5), 667-685.

Meek, J.W. and Wolf, J.P. (1991). “Insights on Cut off Frequency for Foundation on

Soil Layer”, Earthquake Engineering and Structural Dynamics, 20, 651-665, Also

in Proceedings of the 9th European Conference on Earthquake Engineering,

EAEE, Moscow 1990, 4-A, 34-43.

Meek, J.W. and Wolf, J.P. (1993). “Why Cone Models can Represent the Elastic Half-

Space”, Earthquake Engineering & Structural Dynamics, 22: 759–771.

Mohamadi. K. R., El-Naggar, M. H., and Moghaddam, H. (2004). “Optimum strength

distribution for seismic resistant shear buildings.” International Journal of Solids

Structures., 41(21-23), 6597–6612.



32

Moghaddam, H., and Mohammadi, R. K. (2006). “More efficient seismic loading for

multidegrees of freedom structures.” Journal. Structural Engineering (ASCE).,

132(10), 1673–1677.

Nakhaei, M., Ghannad, M.A. (2008) The effect of soil–structure interaction on damage

index of buildings. Engineering Structures 30(6); 1491–1499.

Mahsuli, M. and Ghannad, M. A. (2009) The effect of foundation embedment on

inelastic response of structures. Earthquake Engineering & Structural Dynamics

38(4); 423–437.

Moghaddasi, M., Cubrinovski, M., Chase, J. G., Pampanin, S. and Carr, A., (2011)

Probabilistic evaluation of soil–foundation–structure interaction effects on seismic

structural response. Earthquake Earthquake Engineering & Structural Dynamics

40(2); 135–154.


Effects of soil–foundation–structure interaction on seismic structural response via

robust Monte Carlo simulation Engineering Structures 33(4); 1338-1347.

OPENSEES, (2011), OpenSees Command Language Manual. Open System for

Earthquake Engineering Simulation. Mazzoni, S., McKenna, F., Scott. M. H.,

Fenves, G. L. Available at http://opensees.berkeley.edu/

Veletsos, A. S. and Meek, J. W. (1974) “Dynamic behavior of building–foundation

system,” Earthquake Engineering and Structural Dynamics 3(2), 121–138.

Veletsos, A.S. and Nair, V.D. (1974). “Response of Torsionally Excited Foundations”,

Journal of the Geotechnical Engineering Division, ASCE, 100(3), 476-482.




Wolf JP (1994), “Foundation Vibration Analysis using Simple Physical Models.”

Prentice-Hall: Englewood Cliffs, NJ.

Wolf, J.P. and Deeks, A.J. (2004). “Foundation Vibration Analysis: A Strength-of-

Materials Approach”, Elsevier Oxford.

http://www.sciencedirect.com/science/article/pii/S0141029607001721




http://opensees.berkeley.edu/


33

Chapter 3 EFFECT OF SOIL-STRUCTURE INTERACTION ON ELASTIC AND

INELASTIC RESPONSE OF EQUIVALENT SDOF AND MDOF SYSTEMS

3.1 INTRODUCTION

The general effects of SSI on elastic response of SDOF and MDOF systems with an

emphasis on the former were the subject of many studies in the 1970s (Perelman et al.,

1968; Sarrazin et al., 1972; Jennings and Bielak, 1973; Chopra and Gutierrez, 1974;

Veletsos and Meek, 1974; Veletsos and Nair, 1975; Veletsos, 1977). These works led to

providing tentative provisions in ATC3-06 (ATC, 1978), which is actually the

foundation of new provisions on earthquake-resistant design of soil-structure systems

(BSSC, 2000; FEMA-440, 2005). As stated in the first chapter of this thesis, code-

compliant seismic designs for soil-structure systems are, conventionally, based on the

approximation in which the predominant period and associated damping of the

corresponding fixed-base system are modified (Jennings and Bielak, 1973; Veletsos and

Meek, 1974). In fact, the current seismic provisions consider SSI, generally, as a

beneficial effect on seismic response of structures since SSI usually causes a reduction of

total shear strength demand of building structures (BSSC, 2000; ASCE, 2005). However,

the coupled effect of SSI and inelastic behavior of the superstructure, inevitable during

severe earthquakes, has not been well investigated. On the other hand, nearly in all

seismic codes, the current seismic design philosophy is based on inelastic behavior of

structures when subjected to moderate and severe earthquakes. Hence, there is a

necessity to investigate the effect of SSI on inelastic response of building structures.

Two of the pioneering works on inelastic soil-structure systems were made by Veletsos

and Verbic (1974) and Bielak (1978). Muller and Keintzel (1982) subsequently

investigated the ductility demands of SDOF soil-structure systems. They showed that the

ductility demand of structures, when considering soil beneath them, could be different

from that of the equivalent SDOF systems without considering SSI. During the last

decade, more studies have been conducted by researchers to investigate the SSI effect on

inelastic behavior of SDOF systems (Aviles and Perez-Rocha, 2003 and 2005; Ghannad


34

and Jahankhah. 2007; Mahsuli and Ghannad, 2009; Moghaddasi et al., 2011a and 2011b;

Aviles and Perez-Rocha, 2011). However, owing to the inherent complexity, the

common approach in analyzing nonlinear structural response with SSI adopts equivalent

SDOF model, which is the foundation of current seismic provisions.

On the other hand, the relationship between MDOF and SDOF system response of fixed-

base systems was first studied by Veletsos and Vann (1971) by considering some shear-

beam models with equal story masses connected by weightless springs in series from one

degree of freedom (DOF) to five DOFs. They concluded that for systems having more

than three DOFs the proposed design regulations for SDOF systems were not sufficiently

accurate and could lead to unconservative estimates of the required inelastic lateral

strength, and that errors tended to increase as the number of degrees of freedom

increased. Another study was conducted by Nassar and Krawinkler (1991) on three types

of simplified fixed-base MDOF models to estimate the modifications required to the

inelastic strength demands obtained from bilinear SDOF systems in order to limit the

story ductility demand in the first story of the MDOF systems to a predefined value.

They found that the deviation of MDOF story ductility demands from the SDOF target

ductility ratios increased with period and target ductility ratio. More examples of the

works conducted on the subject can be found in the reference (Seneviratna and

Krawinkler, 1997; Santa-Ana and Miranda, 2000; Moghaddam and Mohammadi, 2001).

However, all of the works were performed on fixed-base systems, i.e. based on an

assumption that soil beneath the structure is rigid.

As mentioned in the literature, almost all researches made on nonlinear soil-structure

systems focused on SDOF systems while the SSI effect on inelastic response of MDOF

systems due its more complexity has not been investigated in detail. A few studies of SSI

effects on MDOF systems are those conducted by Dutta et al. (2004), Barcena and Steva

(2007), Tang and Zhang (2011) , and Ganjavi and Hao (2011a and 2011b). However, the

lack of clarity in SSI effects on seismic demands of MDOF systems deserved special

attention. In fact, SDOF systems having only one DOF may not be able to correctly

reflect the realistic behavior of common building structures interacting with soil beneath

them when subjected to strong ground motions. This can be due to the lack of

incorporating the effects of number of stories and higher modes as well as, more

importantly, the effect of height-wise distribution of lateral strength and stiffness on

inelastic response of real soil-structure systems. Here, in this chapter an intensive

parametric study has been performed to investigate the effect of inertial SSI on seismic


35

strength and ductility demands of MDOF as well as the corresponding equivalent SDOF

(E-SDOF) systems using simplified soil-structure model for surface (shallow)

foundation in which the kinematic interaction is zero. This is carried out by analyzing

6400 linear and nonlinear MDOF and E-SDOF models with SSI subjected to 21

earthquake records to investigate the SSI effects on elastic and inelastic response of

MDOF and E-SDOF systems. In addition, the adequacy of E-SDOF systems in

estimation of seismic strength and ductility demand of MDOF soil-structure systems are

also parametrically investigated.

3. 2. METHODOLOGY AND PROCEDURE FOR ANALYSIS

3.2. 1. General procedure

The adopted soil-foundation structure models introduced in Chapter 2 are used directly

in the time domain nonlinear dynamic analysis. A series of 5-, 10-, and 15-story shear

buildings and also their equivalent SDOF models, introduced in Chapter 2, are

considered to investigate the effect of SSI on strength and ductility demands of both

MDOF and E-SDOF systems. In this investigation, an ensemble of 21 earthquake ground

motions with different characteristics recorded on alluvium and soft soil deposits (soil

type C, with shear wave velocity between 180 and 360 m/s, and D, with shear wave

velocity lower than 180 m/s, based on the USGS site classification) are compiled and

utilized in the nonlinear dynamic time history analyses. All selected ground motions are

obtained from earthquakes with magnitude greater than 6 having closest distance to fault

rupture more than 15 km without pulse type characteristics. The main parameters of the

selected ground motions are given in Table 3-1. In this regard, for a given earthquake

ground motion, a family of 6400 different soil-structure models including MDOF as

well as E-SDOF models and various predefined key parameters are considered. This

includes MDOF and E-SDOF models with 30 fundamental periods of fixed-base

structures, ranging from 0.1 to 3 sec with an interval of 0.1, three values of aspect ratio (

H r =1, 3, 5), four values of dimensionless frequency ( 0a = 0, 1, 2, 3), and three values

of target interstory displacement ductility ratio ( t = 1, 2, 6), where t =1 corresponds to

the elastic state. It should be noted that the range of the fundamental period and aspect

ratio, considered in the present study, are wider than those of the most practical

structures. They are considered here, however, to cover all possible conditions and to

compare the results obtained from MDOF systems of different number of stories with

those obtained from their equivalent SDOF systems. For each earthquake ground


36

motion, the total normalized elastic and inelastic shear strength of the MDOF and E-

SDOF system are computed by a proposed iterative procedure, which will be explained

in the next subsection, in order to reach the target t in the structure within a 0.5%

error. Total normalized shear strength is defined as the total shear strength demands

divided by the total structural mass and then normalized to the peak ground acceleration

(PGA).

Table 3-1 Selected ground motions recorded on alluvium and soft sites based on USGS site

classification

3.2. 2 Proposed iterative procedure

Having more number of DOFs in comparison with the corresponding equivalent SDOF

system, MDOF systems needs generally more computational efforts if the strength and

ductility demand ratio subjected to a specified earthquake ground motion are to be

calculated. In contrary to SDOF systems, strength and ductility demand of an MDOF

system are also dependent on the presumed design lateral load distribution. In other

words, considering the same total base shear strength demand any predefined lateral load

distribution may change the amount of maximum ductility ratio ( max ). Therefore, for an

Event Year Station Distance

(km)

Soil type

(USGS)

Component PGA (g)

Imperial Valley 1979 Compuertas 32.6 C 15, 285 0.186, 0.147

Imperial Valley 1979 El Centro Array #12 18.2 C 140 0.143

Loma Prieta 1989 Agnews State Hospital 28.2 C 0 0.172

Loma Prieta 1989 Gilroy Array #4 16.1 C 0 0.417

Loma Prieta 1989 Sunnyvale - Colton Ave 28.8 C 270 0.207

Northridge 1994 LA - Centinela St 30.9 C 155, 245 0.465, 0.322

Northridge 1994 Canoga Park - Topanga

Can

15.8 C 196 0.42

Kobe 1995 Kakogawa 26.4 D 0, 90 0.251, 0.345

Kobe 1995 Shin-Osaka 15.5 D 0, 90 0.243, 0.212

Loma Prieta 1989 APEEL 2 - Redwood City 47.9 D 43 0.274

Loma Prieta 1989 Foster City - 355

Menhaden

51.2 D 360 0.116

Superstitn

Hills(B)

1987 5062 Salton Sea Wildlife

Refuge

27.1 D 315 0.167

Morgan Hill 1984 Gilroy Array #2 15.1 C 90 0.212

Northridge 1994 LA - N Faring Rd 23.9 C 0, 90 0.273, 0.242

Northridge 1994 LA - Fletcher Dr 29.5 C 144, 234 0.162, 0.24


37

MDOF system, ductility demand for each story needs to be computed and the greatest

value among all stories is then considered as the ductility demand of the MDOF system.

In the present study, the following method is proposed to calculate the ductility demand

of MDOF systems in order to minimize the iteration steps:

1. IBC 2009 lateral load distribution (IBC, 2009) is adopted for the distribution of shear

strength as well as stiffness along the height of the structure with specified fixed-

base target period.

2. The MDOF structure is excited by the given earthquake ground motion, and the

maximum interstory displacement ductility ratio is computed and compared with the

specified target value. If the computed ductility ratio is equal to the target value

within the 0.5% of the accuracy, no iteration is necessary. Otherwise, total base shear

strength must be scaled (by either increasing or decreasing) until the target ductility

ratio is resulted. To do this the following relation is proposed:

1 i( ) ( ) Res i sV V l (3-1)

where ( )s iV is the total base shear strength of MDOF system at ith iteration and Re l

can be defined as:

maxRet

l

(3-2)

in which β is an iteration power larger than zero. Results of this study indicate that β

power for 1t (elastic state) can be taken as a constant value for all MDOF and SDOF

shear-building structures when subjected to any earthquake excitation. For 1t

(inelastic state), however, the β power value is generally more dependent on the amount

of fundamental period and less on the level of inelasticity as well as earthquake

excitation characteristics, and thus usually lower values of β are used for fast

convergence. It is found that for elastic MDOF and SDOF shear-building structures a

very fast convergence, i.e. less than 5 iterations, can be obtained for β equal to 0.8. For

Inelastic state ( 1t ) β value, depending on the structure fundamental period, can be

approximately defined as:


38

fix

fix

fix

0.05 0.1 T 0.5

0.1 0.25 0.5 <T 1.5

0.25 0.4 T 1.5

(3-3)

It is worth mentioning that for inelastic state taking a constant value of β for all

fundamental periods may not be possible. For example, using a constant value of 0.4 for

1.5fixT cannot guarantee that iterative procedure converges when the structure is

subjected to different ground motions having different frequency contents. This can be

the nature of nonlinearity. Therefore, an average value in each range, in most cases, will

lead to a good convergence.

3.2.3 Step-by-step procedure for parametric study

The procedures described above are summarized as follows:

1. Define the MDOF model depending on the prototype structure height and number of

stores.

2. Assign an arbitrary value for total stiffness and strength and then distribute them

along the height of the structure based on the IBC lateral load pattern (IBC, 2009).

3. Select an earthquake ground motion listed in Table 3-1.

4. Consider a presumed set of aspect ratio, H r , and dimensionless frequency, 0a , as

the predefined key parameters for SSI effects.

5. Select the fundamental period of fixed-base structure and scale the total stiffness

without altering the stiffness distribution pattern such that the structure has a

specified target fundamental period. Note that this approach is just for MDOF

structures while for an SDOF structure; stiffness can simply be scaled to reach the

target period.

6. Refine H r based on the fundamental modal properties of the fixed-base MDOF

structure.

7. Select a target interstory-displacement ductility demand ratio for the MDOF fixed-

base structure.

8. Perform nonlinear dynamic analysis for the fixed-base MDOF structure subjected to

the selected ground motion and compute the total shear strength demand, ( )s iV .

9. Follow the proposed iterative procedure based on equations 3-2 and 3-3 to reach the

target ductility demand within a 0.5% tolerance error.


39

10. Perform nonlinear dynamic analysis for the MDOF soil-structure system with the

same total shear strength as obtained from step 9 subjected to the same ground

motion and compute the ductility demand ratio, without any iteration.

11. Repeat steps 7–10 for different target ductility demand ratios.

12. Repeat steps 5–11 for different presumed target periods.

13. Repeat steps 4–12 for different sets of H r , and 0a .

14. Repeat steps 4–13 for different equivalent SDOF systems.

15. Repeat steps 3–14 for different earthquake ground motions.

16. Repeat steps 1–15 for different number of stories.

3.3 EFFECT OF SSI ON STRENGTH DEMANDS OF MDOF AND E-SDOF

SYSTEMS

3.3.1 Strength demands for E-SDOF systems corresponding to different number of

stories

A series of different E-SDOF soil-structure systems corresponding to the first-mode

shape of 5-, 10- and 15-story buildings are analyzed to compare the elastic and inelastic

strength demands for different SDOF soil-structure systems subjected to an ensemble of

21earthquake ground motions listed in Table 3-1. Instead of the first-mode effective

modal mass, total mass of each MDOF systems is considered to model the corresponded

E-SDOF soil-structure system. As an example, the average values of strength demand

for three different E-SDOF soil-structure systems, i.e. E-SDOF of 5-, 10- and 15-story

buildings, are depicted in Figure 3-1. The results are shown for E-SDOF soil-structure

systems with two different aspect ratios, H r = 1, 5 as the representative of squat and

slender buildings for dimensionless frequency 0a = 2 as well as for three values of target

interstory displacement ductility ratios ( t = 1, 2, 6) where t = 1 corresponds to the

elastic case. The abscissa in all figures is the first-mode period of the fixed-base

structure, fixT , and the vertical axis is the strength demand normalized by the total

structural mass times PGA for each earthquake ground motion. It can be clearly seen that

the strength demands of E-SDOF systems are independent of the number of stories such

that all the strength demand curves for E-SDOF soil-structure systems corresponding to

the 5-, 10- and 15-story buildings are completely coincident. The reason of this

similarity goes back to the first-mode shape of the shear buildings which is independent

of the number of stories as illustrated in Figure 3-2. The figure is plotted to compare the


40

normalized first-mode shape of the fixed-base shear buildings for three presumed

number of stories with two different target periods, fixT =1 and 3 sec. The horizontal and

vertical axes are normalized mode shape and relative height, respectively. As seen, for

each presumed target fundamental period, the normalized mode shapes of all three

MDOF shear buildings completely coincide and hence are not dependent on the number

of stories. It is important to note that this result is true when (1) total structural mass is

uniformly distributed along the height of the structures and (2) lateral stiffness in all

MDOF buildings with different number of stories is distributed based on the same

specified pattern which here is IBC load pattern.

Figure 3-1: Comparison of the averaged elastic and inelastic strength demand for different E-

SDOF system with soil-structure interaction ( 0a = 2)

N= 5 N= 10 N= 15

(Fe

or Fy

) /

M.P

GA

0

0.4

0.8

1.2

1.6

0 1 2 3

H̅/r =1

0

0.4

0.8

1.2

1.6

0 1 2 3

H̅/r =1

0

0.4

0.8

1.2

1.6

0 1 2 3

H̅/r =1

(Fe

or Fy

) /

M.P

GA

0

0.4

0.8

1.2

1.6

0 1 2 3

H̅/r =5

0

0.4

0.8

1.2

1.6

0 1 2 3

H̅/r =5

0

0.4

0.8

1.2

1.6

0 1 2 3

H̅/r =5

µ = 1 µ = 2 µ = 6

Tfix Tfix Tfix


41

Figure 3-2: Comparison between first-mode shape for different number of stories: (a) fixT =1

and (b) fixT =3

3.3.2 Strength demands for MDOF and E-SDOF soil-structure systems

Here the effects of SSI on total strength demand for both E-SDOF and MDOF soil-

structure systems are studied and compared with each other. Figure 3-3 illustrates the

elastic strength demand spectra for E-SDOF systems as well as for 5- and 15-story

buildings which are respectively representatives of common SDOF and MDOF

buildings. The results are provided for systems with two aspect ratios, H r = 1, 5

respectively representing the squat and slender buildings and for two dimensionless

frequencies ( 0a = 1 and 3) in comparison with the corresponding fixed-base structures.

As stated before, 0a is an index for the structure-to-soil stiffness ratio controlling the

severity of SSI effects, and a value of 3 for this parameter represents strong SSI effect

for common building structures. The results exhibit a same trend for both E-SDOF and

MDOF buildings in elastic state such that strength demands for soil-structure systems are

lower than those for fixed-base structures. This is consistent with the results of the study

carried out for SDOF systems by Ghannad and Jahankha (2007). However, a significant

difference is observed between the strength demands for SDOF systems and the

corresponding MDOF systems for the case of 0a = 3, i.e., when SSI effect is significant.

This reveals that for fixed-base systems, the difference between the results of SDOF and

MDOF systems are significantly lower than those of the soil-structure systems. By

increasing the SSI effect, i.e., larger 0a , elastic strength demands of MDOF systems can

be remarkably larger than those of SDOF systems especially when the structure has

longer periods. This phenomenon is intensified for slender structures with H r = 5, as

well as by increasing the number of stories, i.e., increasing the number of DOFs. As an

Normalized Mode Shape Normalized Mode Shape

Rel

ativ

e H

eig

ht

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

N=5

N=10

N=15

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

(a) (b)


42

instance, for the system with 0a = 3 and fixT =2, the values of strength demands of 5- and

15-story buildings are respectively 1.78 and 2.57 times the associated E-SDOF systems

for squat structure with H r = 1, and 5.5 and 8.16 times for very slender systems with

H r = 5 while these values are 1.2 and 1.55 for the fixed-base systems. Hence, it can be

concluded that in elastic case, using E-SDOF base shear strength for the associated

MDOF system particularly when SSI effect is significant may result in a significant

underestimation of strength demand in the MDOF soil-structure system when compared

to fixed base systems. This point will be more discussed in the next part.

The same calculations are carried out for two aforementioned target ductility ratios ( t =

2, 6) as representatives of low and high inelastic behaviours, and for all earthquake

ground motions listed in Table 3-1. The mean values are computed for each fundamental

period of fixed-base structure and the results are then plotted in Figures 3-4 and 3-5 in

the same format as Figure 3-3. Generally, somewhat the same trend as elastic case can be

found for the set of curves associated to the target ductility ratio equal to 2 in Figure 3-4.

The exception is for very short periods with high aspect ratio known as slender buildings

such that soil-structure systems have greater strength demands in comparison to the

fixed-base systems, especially for E-SDOF and low-rise MDOF building (N= 5). The

trend is intensified for all E-SDOF and MDOF buildings and is thus more obvious by

increasing the target ductility ratio as shown in Figure 3-5 for t = 6. As seen, the

amounts of strength demands of both E-SDOF and MDOF soil-structure systems,

irrespective of the number of stories, are greater than those of the corresponding fixed-

base systems for the case of slender structures having very short period. However,

considering the fact that real slender MDOF buildings usually do not have such very

short periods, it may be concluded that generally SSI reduces the lateral structural

strength demands. The same results concluded by Ghannad and Jahankhah (2007) for the

case of SDOF soil-structure systems. Moreover, it is seen that for the practical range of

periods the SSI effect decreases as target ductility demands increases, which is more

prominent for E-SDOF and low-rise MDOF systems. Looking at the set of curves for t

= 6, however, by increasing the number of stories, the reduction of lateral strength

demands with respect to the corresponding fixed-base structure is still significant for the

case with significant SSI effect, 0a = 3, which is more prominent for the case of 15-story

building in Figure 3-5. It can be concluded that although for E-SDOF systems the SSI

effect may become less prominent as the structure experiences more inelastic


43

deformations and therefore negligible, this phenomenon, irrespective of the aspect ratio,

can be significant for MDOF soil-structure systems with predominant SSI effect with 0a

= 3 as the number of stories increases.

Figure 3-3: Comparison of the averaged elastic strength demand for ESDOF and MDOF

soil-structure systems


soil-structure systems for µ =2

N= 1 N= 5 N= 15

Tfix Tfix Tfix

Fixed base a0 =1 a0 =3

(Fe

or Fy

) /

M.P

GA

0

0.5

1

1.5

2

2.5

0 1 2 3

H̅/r=1

0

0.5

1

1.5

2

2.5

0 1 2 3

H̅/r=1

0

0.5

1

1.5

2

2.5

0 1 2 3

H̅/r=1

(Fe

or Fy

) /

M.P

GA

0

0.5

1

1.5

2

2.5

0 1 2 3

H̅/r=5

0

0.5

1

1.5

2

2.5

0 1 2 3

H̅/r=5

0

0.5

1

1.5

2

2.5

0 1 2 3

H̅/r=5

Tfix Tfix Tfix

N= 1 N = 5 N = 15


(Fe

or Fy

) /

M.P

GA

(F

e o

r Fy

) /

M.P

GA

0

0.5

1

1.5

2

0 1 2 3

H̅/r=1

0

0.5

1

1.5

2

0 1 2 3

H̅/r=1

0

0.5

1

1.5

2

0 1 2 3

H̅/r=1

0

0.5

1

1.5

2

0 1 2 3

H̅/r=5

0

0.5

1

1.5

2

0 1 2 3

H̅/r=5

0

0.5

1

1.5

2

0 1 2 3

H̅/r=5


44


soil-structure systems for µ =6

The above observations indicate that by increasing the SSI effect, i.e., larger 0a , strength

demands of MDOF systems can be remarkably larger than those of E-SDOF systems

especially in the long periods range for both elastic and inelastic structures. In order to

get better understanding of this observation, Figure 3-6 is provided to illustrate the

difference between the strength demands of MDOF and the associated E-SDOF models

for both fixed-base and soil-structure systems. The plot can better show the effect of

number of stories, i.e., number of DOFs, on the strength demands spectra for both fixed-

base and soil-structure systems when undergoing different levels of deformations. The

results are provided for E-SDOF and three MDOF systems (N= 5, 10 and15) with H r =

3, and for two different dimensionless frequencies, 0a = 1, 3 as well as for fixed-base

structures. As seen, in elastic state, i.e. t = 1, except for short periods, the values of

strength demands increase as the number of stories increases. This trend is intensified by

increasing the value of 0a such that for the severe SSI effect ( 0a =3), the difference

between the strength demands of E-SDOF system and those of the corresponding 15-

story building increases remarkably. In the inelastic state, however, the trend is

somewhat different in a way that; (1) nearly in all periods, especially for higher level of

inelasticity, strength demand increases as the number of stories increases, but the rate of

increment becomes smaller with the increase of the number of stories; and (2) as the

level of inelasticity increases, the difference between the strength demand values of

(Fe

or Fy

) /

M.P

GA

(F

e o

r Fy

) /

M.P

GA

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3

H̅/r=5

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3

H̅/r=5

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3

H̅/r=5

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3

H̅/r=1

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3

H̅/r=1

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3

H̅/r=1


N= 1 N = 5 N = 15

Tfix Tfix Tfix


45

MDOF systems and those of the corresponding E-SDOF systems for the case with strong

SSI effect with 0a =3 reduces. It should be noted that although for the case of 0a =3, the

growth rate of strength demands with increasing the number of stories reduces as

structure undergoes more level of inelasticity, it is still significant particularly when the

strength demands of MDOF systems are to be compared with those of the associated E-

SDOF systems.

Figure 3-6: Effect of number of stories on the averaged elastic and inelastic strength demand

of fixed-base and soil-structure systems for H r = 3

0

0.5

1

1.5

2

2.5

0 1 2 3

µ=1

0

0.5

1

1.5

2

0 1 2 3

µ=1

0

0.3

0.6

0.9

1.2

1.5

0 1 2 3

µ=1

0

0.3

0.6

0.9

1.2

1.5

1.8

0 1 2 3

µ=2

0

0.3

0.6

0.9

1.2

1.5

0 1 2 3

µ=2

0

0.3

0.6

0.9

1.2

1.5

1.8

0 1 2 3

µ=2

0

0.2

0.4

0.6

0.8

1

0 1 2 3

µ=6

0

0.2

0.4

0.6

0.8

1

0 1 2 3

µ=6

0

0.2

0.4

0.6

0.8

1

0 1 2 3

µ=6

(Fe

or Fy

) /

M.P

GA

(F

e o

r Fy

) /

M.P

GA

(F

e o

r Fy

) /

M.P

GA

Fixed base a0 = 1 a0= 3

Tfix Tfix Tfix

N= 1 N= 5 N = 10 N = 15


46

3.3.3. Adequacy of E-SDOF systems in estimating strength demands for MDOF

fixed-base and soil-structure systems

To better investigate the adequacy of common E-SDOF systems in estimating strength

demands of corresponding MDOF systems for both fixed-base and soil-structure systems

Figure 3-7 is plotted. This figure presents the mean ratio of strength demands for the 10-

story building to those of the E-SDOF systems. The results are provided for systems

with three ductility demands (µ =1, 2, 6), three aspect ratios ( H r = 1, 3, 5) and for three

different dimensionless frequencies, 0a = 1, 2, 3, as well as for fixed-base structures.

Based on the results presented in this figure the following conclusion can be drawn:

1. For fixed-base systems, there is no significant difference between the strength

demands of MDOF and those of the corresponding E-SDOF systems in elastic

range of response. However, for soil-structure systems the ratios of strength

demands in MDOF systems to those of the corresponding E-SDOF systems are

significant. This phenomenon is intensified as aspect ratio increases i.e., for

medium and slender buildings. As an instance, for the structure with fundamental

period of 2.5 sec, the ratio is1.63 for the fixed-base system while 2.53, 5.63 and

9.26 for squat, medium and slender soil-structure buildings, respectively. This

implies that opposed to fixed-base systems, using strength demands of SDOF

soil-structure systems for MDOF soil-structure systems in elastic range of

response can lead to very un-conservative results.

2. Except for structures with very short periods, by increasing the level of

inelasticity the ratios of strength demands in MDOF systems to those of the

corresponding E-SDOF systems increase for fixed-base systems but decrease for

soil-structure systems. Nevertheless, for the cases of medium and especially

slender soil-structures systems, these ratios are still greater than those of the

fixed-base systems. As an instance, for the case with fixT = 2 sec, the required

strength demands of the MDOF system are 2.05 and 2.4 times the strength

demands of corresponding E-SDOF system for fixed-base systems with µ= 1 and

2, respectively while they are 5.5 and 3.6 times of E-SDOF systems for slender

soil-structure systems with severe SSI effect (i.e. H r = 5, and 0a = 3 ).

3. There is no significant difference between the results of fixed-base and squat

soil-structure systems for the cases of low and high inelastic response (i.e. H r =

1, and µ = 2 and 6).


47

It can be concluded that, opposed to fixed-base systems, using the common E-SDOF

soil-structure systems for estimating the strength demands of medium and slender

MDOF soil-structure systems when SSI effect is significant can lead to very un-

conservative results. Moreover, from Figure 3-6 it is also seen that for MDOF systems it

seems that the strength demands usually increase with the number of stories. This

phenomenon may be justified by studying the coefficient of variation (COV) and the

distribution pattern of story ductility demand ratios along the height of MDOF

structures. The COV is a statistical measure of the dispersion of data points, here

ductility demand ratio along the building height. It is defined as the ratio of the ductility

demand standard deviation to the mean ductility among all stories. As mentioned before,

the definition of ductility demand for MDOF systems is somewhat different from that for

SDOF systems. The ductility demand for MDOF systems is conventionally referred to as

the greatest value among all the story ductility ratios, hence the values of ductility ratios

in all other stories are lower than the presumed target ductility value. This may result in

a greater strength demand when compared to the same MDOF building in which all

stories have identical ductility ratio equal to the presumed target value.

To better interpret this justification, the averaged COV of ductility ratios for MDOF

systems having three different numbers of stories, (N= 5, 10, 15) for all earthquake

ground motions used in this study are computed and the results are plotted in Figure 3-8.

The results are provided for systems with aspect ratio of 3, target ductility ratio of 6 and

for 2 values of dimensionless frequency, 0a = 1, 3 as well as the fixed-base structures.

As seen, except in the short period range, COV of ductility ratios, in all cases, increases

as the number of story increases. Figure 3-9 also shows height-wise distribution of the

averaged ductility demands for the same MDOF systems with fixT = 1.5. The abscissa in

all figures is the averaged ductility demands and the vertical axis is relative height of the

structure. It can be seen that by increasing the number of stories more stories have the

ductility demands lower than the target value. In addition, it is observed that the

averaged maximum ductility ratios are not exactly close to the target one. This is because

the maximum ductility ratio depends on a given earthquake ground motion, and it may

happen in different stories.


48

Figure 3-7: The ratio of elastic and inelastic strength demands in 10-story building to those

in the corresponding E-SDOF system;

Figure 3-8: COV of story ductility demand for different MDOF soil-structure systems

0

2

4

6

8

10

0 1 2 3

µ = 1

0

2

4

6

8

10

0 1 2 3

µ = 1

0

2

4

6

8

10

12

0 1 2 3

µ = 1

0

1

2

3

4

5

6

7

8

0 1 2 3

µ = 2

0

1

2

3

4

5

6

7

8

0 1 2 3

µ = 2

0

1

2

3

4

5

6

7

8

0 1 2 3

µ = 2

0

1

2

3

4

5

0 1 2 3

µ = 6

0

1

2

3

4

5

0 1 2 3

µ = 6

0

1

2

3

4

5

0 1 2 3

µ = 6

Fixed base a0 =1 a0 =2 a0 =3 Fy

(MD

OF

/ E-

SDO

F)

Fy (M

DO

F /

E-SD

OF)

Fy

(MD

OF

/ E-

SDO

F)

= 1 = 3 = 5

Tfix Tfix Tfix

N= 5 N= 10 N= 15


Tfix Tfix Tfix

CO

V

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3

µ= 6

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3

µ= 6

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3

µ= 6


49

Figure 3-9: Height-wise distribution of averaged ductility demand for systems with fixT = 1.5

and μ= 6

3.4. EFFECT OF SSI ON DUCTILITY DEMAND OF MDOF AND E-SDOF

SYSTEMS

To investigate the effect of SSI on the ductility demand of the MDOF and E-SDOF

structures the common procedure is utilized. First the elastic and inelastic total shear

strength, as representatives of elastic and inelastic strength demands, for each of MDOF

and E-SDOF systems without considering the effect of soil beneath them are computed

to reach a presumed target ductility ratio, t , when subjected to a designated earthquake

ground motion. Subsequently, using the same total shear strength of the fixed-base

structure, the ductility demand of the soil-structure system with different values of 0a and

H r is computed to investigate the SSI effects. The effect of SSI on ductility demands

of MDOF and E-SDOF systems can then be examined by comparing the difference

between the ductility demand of the fixed-base model and that of the soil–structure

system. This comparison reflects the controversial point existing in conventional design

methodology such that if a predesigned fixed-base model is to be located on flexible soil,

based on reality, what structural behaviour maybe expected?

The step-by-step procedure outlined in Section 3.2.3 is used here to investigate the effect

of SSI on ductility demand of E-SDOF and MDOF building structures. As an example,

Figure 3-10 shows the averaged ductility demand spectra of all earthquake ground

motions used in this study for the structural model with an aspect ratio of 3. Results

Rel

ativ

e H

eigh

t


μ μ μ

N= 5 N= 10 N= 15


50

include both elastic and inelastic ductility demand spectra for three values of

dimensionless frequency, 0a = 1, 2, 3, all for both MDOF and E-SDOF soil-structure

systems. The vertical axis in all plots is the ratio of ductility demand in flexible-base

structures to that of the fixed-based structure. As seen, the results of MDOF structures

are different from those of the corresponding E-SDOF systems such that the results can

be classified into two parts; first, the set of curves associated with low value of

dimensionless frequency ( 0a = 1); second, the curves corresponded to the large amount

of dimensionless frequency ( 0a = 2, 3) which are the representatives of the cases with

substantial SSI effects. For the E-SDOF systems, irrespective of the dimensionless

frequency or the level of inelasticity, there is a threshold period before that the ductility

demand of the structure with SSI is larger than that of the fixed-base one; subsequently,

this tendency is reversed. The larger is the dimensionless frequency, the greater is the

difference between the ductility demands of the fixed- and flexible-base systems. The

variation of ductility demands for MDOF systems, however, can be completely different

from that of the E-SDOF systems depending on the amount of dimensionless frequency

as well as the level of inelasticity. It can be observed that for MDOF systems with 0a = 1,

i.e., the curves related to the first column in Figure 3-10, the ratios of ductility demands

in almost all periods are greater than unity. Also, this trend is intensified as the number

of stories increases, which is more obvious for the case of 15-story building. Looking at

the second and third columns of the same figure which are associated to the cases with

significant SSI effects ( 0a = 2, 3), it can be seen that although like E-SDOF systems there

is still a threshold period before and after that the ratios of ductility demands are

respectively greater and lower than unity, the trend do not continue for the longer period

like E-SDOF systems in a way that for MDOF systems after reaching to a minimum

level, the ratio again rises as period increases.


51


systems for H r = 3

To better understand the difference between ductility demands of E-SDOF systems and

those of the MDOF structures Figures 3-11 and 3-12 are provided in another format in

which the results for different aspect ratios are also included. The results are presented

for two levels of ductility, t = 2 and 6. As seen, the trends are the same as those

discussed above while the effect of aspect ratio can also be observed here. For the E-

SDOF systems it can be observed that the larger the aspect ratio, the grater is the

difference between ductility demands of fixed- and flexible-base systems. The results for

E-SDOF systems are the same with recent studies carried out in SDOF systems by

Ghannad and Ahmadnia (2006) and Mahsuli and Ghannad (2009). For MDOF systems,

however, the difference between ductility demands of the structure with or without

considering SSI is less significant. By increasing the aspect ratio the difference between

the ratios of ductility demands for different dimensionless frequencies decrease with

E-SDOF N= 5 N= 15

a0 = 1 a0 = 2 a0= 3

Tfix Tfix Tfix

µss

i / µ

fix

µss

i / µ

fix

0

0.5

1

1.5

2

2.5

0 1 2 3

µ= 1

0

0.5

1

1.5

2

2.5

0 1 2 3

µ= 1

0

0.5

1

1.5

2

2.5

0 1 2 3

µ= 1

0

0.5

1

1.5

2

2.5

0 1 2 3

µ= 2

0

0.5

1

1.5

2

2.5

0 1 2 3

µ= 2

0

0.5

1

1.5

2

2.5

0 1 2 3

µ= 2

µss

i / µ

fix

0

0.5

1

1.5

2

0 1 2 3

µ= 6

0

0.5

1

1.5

2

0 1 2 3

µ= 6

0

0.5

1

1.5

2

0 1 2 3

µ= 6


52

periods. This is more obvious for the case of t = 6. Moreover, as discussed above in

contrary to the E-SDOF systems, the ratios of ductility demands of MDOF systems for

the case of low dimensionless frequency, 0a = 1, are greater than unity for almost all

periods and are intensified by increasing the number of stories and aspect ratio. It can

also be observed that by increasing the level of target ductility ratio, the ratios of

ductility demands even for the case with high dimensionless frequency values ( 0a = 2, 3)

increase in a way that for some practical periods, the amounts of ratios of ductility

demands might be more than unity as shown in Figure 3-12.


systems for µ= 2

E-SDOF N = 5 N = 15

Tfix Tfix Tfix

µss

i / µ

fix

0

0.5

1

1.5

2

0 1 2 3

H̅/r= 1

0

0.5

1

1.5

2

0 1 2 3

H̅/r= 1

0

0.5

1

1.5

2

0 1 2 3

H̅/r= 1

a0 =1 a0 =2 a0 =3

µss

i / µ

fix

0

0.5

1

1.5

2

2.5

0 1 2 3

H̅/r= 3

0

0.5

1

1.5

2

2.5

0 1 2 3

H̅/r= 3

0

0.5

1

1.5

2

2.5

0 1 2 3

H̅/r= 3

µss

i / µ

fix

0

0.5

1

1.5

2

2.5

0 1 2 3

H̅/r= 5

0

0.5

1

1.5

2

2.5

0 1 2 3

H̅/r= 5

0

0.5

1

1.5

2

2.5

0 1 2 3

H̅/r= 5


53


systems for µ= 6

3.5. CONCLUSION

An intensive parametric study was carried out to investigate the effect of SSI on the

strength and ductility demands for MDOF as well as its equivalent SDOF systems

considering both elastic and inelastic behaviours. It was demonstrated that strength and

ductility demands of MDOF soil-structure systems depending on the number of stories,

dimensionless frequency, aspect ratio and the level of inelasticity can be very different

from those of the corresponding equivalent SDOF ones. Based on the comprehensive

nonlinear dynamic analyses the results are summarized as follows:

1. Elastic strength demands of E-SDOF and MDOF soil-structure systems are lower

than those of the fixed-base structures for both squat and slender structures.

Tfix Tfix Tfix

E-SDOF N = 5 N = 15

a0 =1 a0 =2 a0 =3 µ

ssi /

µfi

x

0

0.5

1

1.5

2

0 1 2 3

H̅/r= 1

0

0.5

1

1.5

2

0 1 2 3

H̅/r= 1

0

0.5

1

1.5

2

0 1 2 3

H̅/r= 1 µ

ssi /

µfi

x

0

0.5

1

1.5

2

0 1 2 3

H̅/r= 5

0

0.5

1

1.5

2

0 1 2 3

H̅/r= 5

0

0.5

1

1.5

2

0 1 2 3

H̅/r= 5

µss

i / µ

fix

0

0.5

1

1.5

2

0 1 2 3

H̅/r= 3

0

0.5

1

1.5

2

0 1 2 3

H̅/r= 3

0

0.5

1

1.5

2

0 1 2 3

H̅/r= 3


54

2. For the low level of inelasticity, slender E-SDOF and MDOF soil-structure systems

with short periods have greater strength demands in comparison to the fixed-base

systems. However, by increasing the number of stories the difference considerably

decreases and thus is negligible. For the high level of inelasticity, the amounts of

strength demands of both E-SDOF and MDOF soil-structure systems, irrespective of

the number of stories, are greater than those of the corresponding fixed-base systems

for the case of slender structures having very short period.

3. Opposed to fixed-base systems, using the common E-SDOF soil-structure systems for

estimating the strength demands of medium and slender MDOF soil-structure

systems when SSI effect is significant can lead to very un-conservative results. This

phenomenon is more pronounced for the cases of elastic and low level of inelasticity.

However, current seismic regulations for considering SSI effect are mainly based on

the SDOF systems. Care should be taken when using E-SDOF strength demand in

estimating the strength and ductility demands of multi-story soil-structure systems.

4. For the E-SDOF systems, irrespective of the dimensionless frequency or the level of

inelasticity, there is a threshold period before that the ductility demand of the

structure with SSI is larger than that of the fixed-base one; subsequently, this

tendency is reversed. These are consistent with recent studies (Ghannad and

Ahmadnia, 2006). However, for MDOF systems with less SSI effect, i.e., lower

dimensionless frequency, the ratios of ductility demands in almost all periods are

greater than unity, and will increase for slender structures as the number of stories

increases. For the cases of the predominate SSI effects, although like E-SDOF

systems there is still a threshold period before and after that the ratios of ductility

demands are respectively greater and lower than unity, the trend do not continue for

the longer period like E-SDOF systems in a way that for MDOF systems after

reaching to a minimum level, the ratio again rises as period increases.

Results of this study show that the commonly adapted equivalent SDOF systems cannot

accurately estimate the strength and ductility demands of MDOF soil-structure systems,

especially for the cases of mid- and high-rise building, due to their higher mode and

number of DOFs effects. Consequently, more detailed investigations need to be

conducted for the cases of ductility reduction and strength modification factors for

MDOF soil-structure systems, which will be done in the next chapters.


55

3.6 REFERENCES

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seismic regulations for buildings, ATC-3-06, California.



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factors‟” Journal of Structural Engineering (ASCE) 131(2); 221–230.

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foundation,” Journal of Structural Engineering (ASCE) 133(10); 1453–1461.

Aviles J. and Perez-Rocha J. L. (2011) “Use of global ductility for design of structure–

foundation systems,” Soil Dynamics and Earthquake Engineering 31(7): 1018–

1026.

Barcena A. and Esteva L. (2007) “Influence of dynamic soil–structure interaction on the

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Bielak J. (1978) “Dynamic response of non-linear building–foundation systems,”



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buildings including foundation interaction,” Earthquake Engineering and

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Ganjavi, B., and Hao, H. (2011a). “Elastic and Inelastic Response of Single- and Multi-

Degree-of-Freedom Systems Considering Soil Structure Interaction Effects.”

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Australia.

Ganjavi, B., and Hao, H. (2011b). “Evaluation of the Adequacy of Code Equivalent

Lateral Load Pattern and Ductility Demand Distribution for Soil-Structure

Systems.” Australian Earthquake Engineering Society Conf., Barossa Valley,


56

South Australia.



IBC-2009. (2009), International Building Code, ICC, Birmingham, AL.

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Jennings, P. C. and Bielak, J. (1973) “Dynamics of buildings–soil interaction,” Bulletin

of Seismological Society of America 63(1): 9–48.

Mahsuli, M., and Ghannad. M. A., (2009). “The effect of foundation embedment on

inelastic response of structures” Earthquake Engineering & Structural Dynamics.,

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Moghaddam, H. and Mohammadi, R. K. [2001] “Ductility reduction factor of MDOF

shear-building structures,” Journal of Earthquake Engineering 5(3), 425-440.

Moghaddasi, M., Cubrinovski, M., Chase, J. G., Pampanin, S., and Carr, A. (2011).


seismic structural response” Earthquake Engineering & Structural Dynamics.,

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“Effects of soil–foundation–structure interaction on seismic structural response via

robust Monte Carlo simulation,” Engineering Structure 33(4); 1338-1347.

Muller, F. P. and Keintzel, E. (1982) “Ductility requirements for flexibly supported anti-

seismic structures,” Proceedings of the Seventh European Conference on

Earthquake Engineering, Athens, Greece, vol. 3, 20–25 September, 27–34.

Nassar, A. and Krawinkler, K. (1991) Seismic Demands for SDOF and MDOF Systems,.

Report No.95, Department of Civil Engineering, Stanford University, Stanford,

California.

Perelman, D. S., Parmelee, R. A. and Lee, S. L. (1968) “Seismic response of single-story

interaction system,” Journal of the Structural Division (ASCE) 94(ST11): 2597–

2608.

Santa-Ana, P. R. and Miranda, E. (2000) “Strength reduction factors for multi-degree of

freedom systems,” Proceedings of the 12th world conference on Earthquake

Engineering: Auckland, Paper No.1446.

Sarrazin, M. A., Roesset, J. M. and Whittman, R. V. (1972) “Dynamic soil–structure

interaction,” Journal of the Structural Division (ASCE) 98(ST7): 1525–1544.




57

Seneviratna, G. D. and Krawinkler, H. (1997) “Evaluation of inelastic MDOF effects for

seismic design,” Report No.120, Department of Civil Engineering, Stanford

University, Stanford, California.

Tang, Y. and Zhang, J. (2011) “Probabilistic seismic demand analysis of a slender RC

shear wall considering soil–structure interaction effects,” Engineering Structure

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Veletsos, A. S. and Vann, P. (1971) “Response of ground-excited elastoplastic systems,”

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58

Chapter 4

EFFECT OF STRUCTURAL CHARACTERISTICS DISTRIBUTION ON

STRENGTH DEMAND AND DUCTILITY REDUCTION FACTOR OF MDOF

SYSTEMS WITH SOIL-STRUCTURE INTERACTION

4.1 INTRODUCTION

In the previous chapter, an intensive parametric study has been performed to investigate the

effect of SSI on both elastic and inelastic seismic strength and ductility demands of

equivalent SDOF (E-SDOF) and MDOF systems designed according to IBC-2009 code-

compliant lateral load pattern (IBC-2009). It was concluded that depending on the number

of stories, soil flexibility and structure aspect ratio, and also the level of inelasticity, E-

SDOF models may not lead to accurate estimation of strength and ductility demands for

multi-story soil-structure systems. For the cases of medium and slender buildings with

predominant SSI effect it can result in a significant underestimation of strength demand for

MDOF soil-structure systems. It is also known that structural stiffness and strength

distributions have an important role in seismic response of structures. In many previous

parametric studies such as those conducted by Veletsos and Vann (1971), Sirvastav and Nau

(1987) and Mobasseri et al. (1992), it was assumed that stories stiffness or strength were

distributed uniformly along the height of the MDOF systems. Thus, in this idealization, the

shear resistance is constant throughout the height while the required seismic shear resistance

according to the current building codes decreases from bottom to top. Although in the

practical seismic design of low-rise building frames, i.e., buildings with less than 5 stories,

story stiffness or strength may often be uniform, the assumption of uniformity may be

questionable for mid- and high-rise buildings. Consequently, since the results of many

previous studies are based on this assumption, the adequacy of this idealization should be

investigated for elastic and inelastic behavior of fixed-base and flexible-base building

structures.


59

The effect of different code-specified lateral load patterns on seismic performance of fixed-

base buildings have been investigated by researchers during the past two decades (Anderson

et al., 1991; Gilmore and Bertero, 1993; Chopra, 1995; Leelataviwat et al., 1999;

Mohammadi et al. 2004; Park and Medina, 2007). However, no investigation has been

carried out for the case of soil-structure systems yet. In this chapter, considering 5 different

shear strength and stiffness distributions, which will be explained in the next section, the

effect of SSI on strength demand and ductility (strength) reduction factor ( R ) of MDOF

shear-building structures are parametrically investigated. This is carried out for a wide range

of structural and non-dimensional parameters of MDOF soil-structure systems subjected to a

group of earthquake ground motions recorded on alluvium and soft soils as listed in Table 3-

1 of chapter 3.

4.2. SELECTED STORY STRENGTH AND STIFFNESS DISTRIBUTION

PATTERNS

The general formula of the lateral load pattern specified by the most current seismic codes

such as Euro-Code 8 (CEN, 2003), Mexico City Building Code (Mexico, 2003), Uniform

Building Code (UBC, 1994 and 1997), NEHRP 2003 (BSSC, 2003), ASCE/SEI 7-05

(ASCE, 2005), Australian Seismic code (AS-1170.4, 2007) and International Building Code

(IBC-2009) is defined as:

1

. k

x xx n

k

i i

i

w hF V

w h

(4-1)

where xF and V are respectively the lateral load at level x and the design base shear; iw and

xw are the portion of the total gravity load of the structure located at the level i or x; ih and

xh are the height from the base to the level i or x; n is the number of stories; and k is an

exponent that differs from one seismic code to another. In IBC-2009, k is related to

fundamental period of the structure, which is equal to 1 and 2 for structures having a period

of 0.5 sec or less, and for structures having a period of 2.5 sec or more, respectively. For

structures having a period between 0.5 and 2.5 sec, k is computed by linear interpolation

between 1 and 2. It should be mentioned that, the distribution of lateral force based on IBC


60

2009 is identical to that of NEHRP 2003 and ASCE/SEI 7-05 provisions. Note that when k

is equal to 1, the pattern corresponds to an inverted triangular lateral load distribution and

the response of building, thus, is assumed to be controlled primarily by the first mode. While

k = 2 corresponds to a parabolic lateral load pattern with its vertex at the base in which the

response is assumed to be influenced by higher mode effects. In UBC-97, k is a constant

and equal to 1. However, for structures having fundamental period greater than 0.7 sec, the

force at the top floor calculated from Eq. (4-1) is increased by adding a concentrated force

0.07tF TV . In this case, the base shear V in Eq. (4-1) is replaced by ( )tV F . It should

be noted that tF should not exceed 0.25 V and may be considered as zero when the

fundamental period of vibration is 0.7 sec or less. Finally for EuroCode-8, k is also a

constant and equal to 1 for all period ranges. In fact, the seismic lateral load in height of the

structure according to EuroCode-8 is an inverted triangular pattern, which is identical to

UBC-97 and IBC 2009 load patterns when fundamental period is less than or equal to 0.7

and 0.5, respectively. In the present study, besides the above three mentioned code-

specified lateral strength and stiffness patterns, two more patterns including uniform and

concentric patterns are also considered to investigate the effect of structural characteristics

distributions on strength demand and R of MDOF soil-structure systems. Uniform and

concentric patterns (i.e., total shear strength is concentrated in top story) can be defined by

considering exponent k equals and close to zero and infinity, respectively. Note that in

concentric pattern the total shear force is concentrated on the roof story. Figure 4-1

illustrates a comparison of all the above-mentioned lateral force and normalized shear

strength patterns for the 10-story building with T = 1.5 sec. As mentioned earlier, lateral

story stiffness is assumed as proportional to story shear strength distributed over the height

of the structure.


61

Figure 4-1: Different Lateral force and normalized shear strength patterns for the10-story

building with fixT = 1.5 sec

4.3 ANALYSIS PROCEDURE

Step-by-step algorithm proposed in chapter 3 is utilized for all MDOF models designed

according to the aforementioned different lateral load patterns. A series of 5-, 10-, 15-story

shear buildings are considered to investigate the effect of structural characteristics

distribution on strength demand and ductility (strength) reduction factor of MDOF soil-

structure systems subjected to a group of earthquake ground motions recorded on alluvium

and soft soils. In this regard, for a given earthquake ground motion, a large family of 21600

different MDOF soil-structure models including various predefined key parameters are

considered. This includes MDOF models of three different number of stories (N= 5, 10, and

15) with 30 fundamental periods of fixed-base structures, ranging from 0.1 to 3 sec with an

interval of 0.1, three values of aspect ratio ( H r =1, 3, 5), three values of dimensionless

frequency ( 0a =1, 2, 3) as well as the corresponding fixed-base model (i.e., 0a = 0), four

values of target interstory displacement ductility ratio ( t = 1, 2, 4, 6) where t =1

corresponds to the elastic state, and 5 different lateral strength and stiffness distribution

patterns. For each earthquake ground motion, strength demand and therefore ductility

Normalized Shear Strength Lateral Force / Base Shear

1

2

3

4

5

6

7

8

9

10

11

0 0.2 0.4 0.6 0.8 1

1

2

3

4

5

6

7

8

9

10

11

0 0.2 0.4 0.6 0.8 1 1.2

Sto

ry

IBC-2009 UBC-97 EuroCode-8 Concentric Uniform


62

reduction factor for different patterns are computed by a proposed iterative procedure

adopted in chapter 3 in order to reach the target ductility ( t ) in the structure, as a part of

the soil–structure system, within a 0.5% error.

4.4 EFFECT OF STRUCTURAL CHARACTERISTICS DISTRIBUTION ON

STRENGTH DEMAND OF MDOF SYSTEMS

To study the effect of structural characteristics distribution on strength demand of MDOF

fixed-base and flexible-base buildings, systems of 5 and 15 stories are considered. These

buildings are representatives of the common building structures of relatively low- and high-

rise buildings. Results illustrated in Figures 4-2 and 4-3 are the mean response values from

21 earthquake ground motions for systems with H r = 3, corresponding to three ductility

ratios ( t = 1, 2, 6) representing respectively elastic, low and high inelastic behaviors, and

soil-structure system with dimensionless frequency of 3, as well as the fixed-base structures.

As stated before, 0a is an index for the structure-to-soil stiffness ratio controlling the

severity of SSI effects, and also the value of 3 for this parameter is representative of systems

in which SSI effect is significant for common building structures. The vertical axis in all

figures is the mean strength demands normalized by the total structural mass times PGA for

each earthquake ground motion and the horizontal axis is the fixed-base fundamental period

of the structure. Based on the results presented in Figures 4-2 and 4-3, it can be observed

that:

1. In elastic and low level of inelastic behavior of both fixed-base and flexible-base

low-rise buildings (i.e., 5-story building in Figure 4-2), with exception of short

periods, there is a significant difference among the strength demand values of the

structures designed in accordance to the different lateral strength and stiffness

distribution patterns, especially for the case of uniform pattern which yields

completely different strength demand. However, the results corresponding to IBC-

2009 and UBC-97 are to some extent coincident.

2. In high level of inelastic behavior for both fixed-base and flexible-base low-rise

buildings, except for uniform pattern, the strength demand values corresponding to

all patterns considered in this study are somewhat coincident and thus independent

of the lateral story strength and stiffness pattern.


63

3. In the 15-story building (Figure 4-3) which represents high-rise buildings in this

study, except for short periods, the difference among the results corresponding to the

different patterns are more pronounced than those of the 5-story building for both

fixed-base and flexible-base buildings. It can also be seen that even in the high level

of inelasticity region, the differences among the results of UBC-97, EuroCode-8,

IBC-2009 and concentric patterns are very prominent for structures having long

periods.

4. Different from the low-rise buildings, except in the regions with short periods, there

is a significant difference between the strength demand spectra of IBC-2009 and

UBC-97 for both fixed-base and flexible-base 15-story buildings, especially in the

longer periods region. As an instance, for the case of severe SSI effect (i.e., 0a =3)

with fundamental period of 1.5 sec, the strength demand values of IBC-2009 pattern

are respectively 33%, 24% and 46% greater than those of UBC-97 pattern for target

ductility demands of 1, 2 and 6, respectively. This is because of the difference

between the two code-specified load patterns which in turn reflects the effect of

higher modes on high-rise buildings.

5. Generally, with exception of short period structures, EuroCode-8 pattern regardless

of the level of inelasticity has the greatest strength demand values among the three

code-specified strength and stiffness patterns for both fixed-base and flexible-base

models. The concentric pattern, except in the short period region, has generally the

least strength demand values among all the patterns considered in this study.

Figure 4-4 shows the effect of number of stories on the ratios of strength demand spectra of

structures designed in accordance to the uniform strength and stiffness distribution pattern

with respect to IBC-2009 code-specified pattern. Results are provided as mean values of 21

earthquake ground motions for systems of 5-, 10- and 15-strory buildings with H r =3, two

ductility ratios ( t = 2, 6) as well as two values of dimensionless frequencies ( 0a =1, 3) and

the fixed-base models. The vertical axis in all figures is the mean ratio of strength demand in

uniform pattern to that of the IBC-2009 pattern and the horizontal axis is the fundamental

period of the corresponding fixed-base structure. As seen, in both the fixed-base and

flexible-base models, with exception of very short periods, the ratios generally increase with

the number of stories. The ratios are generally greater than 2 and even in some cases will

reach to the value of 4. It is also obvious that the ratio in 10- and 15-story buildings which


64

represent respectively the mid- and high-rise buildings are significantly larger than that of

the 5-story building. This means that using the results of the uniform story strength and

stiffness distribution pattern as it was commonly assumed in many previous research works

would result in a significant overestimation of the strength demands, generally from 2 to 4

times, for MDOF systems designed in accordance to the code-compliant design patterns.


systems with N = 5 and H r = 3

IBC-2009 UBC-

EuroCode-8 Concentric Uniform

µ = 1 µ = 2 µ = 6 Tfix Tfix Tfix

(Fe

or Fy

) /

M.P

GA

0

1

2

3

4

0 1 2 3

Fixed Base

0

0.5

1

1.5

2

2.5

0 1 2 3

Fixed Base

0

0.5

1

1.5

0 1 2 3

Fixed Base

0

0.5

1

1.5

2

2.5

0 1 2 3

a0 = 3

0

0.5

1

1.5

2

0 1 2 3

a0 = 3

0

0.5

1

1.5

0 1 2 3

a0 = 3

(Fe

or Fy

) /

M.P

GA


65


systems with N = 15 and H r = 3

Figure 4-4: Averaged ratio of strength demand in uniform pattern to that of the IBC-2009 pattern

for systems with H r = 3

N= 5 N= 10 N= 15

0

1

2

3

4

5

0 1 2 3

μ = 6

0

1

2

3

4

0 1 2 3

μ = 6

0

1

2

3

4

0 1 2 3

μ = 6

FU

nif

orm

/ F

IBC

-20

09

0

1

2

3

4

0 1 2 3

μ = 2

0

1

2

3

4

0 1 2 3

μ = 2

0

1

2

3

4

0 1 2 3

μ = 2

FU

nif

orm

/ F

IBC

-200

9


Tfix Tfix Tfix

0

1

2

3

4

5

0 1 2 3

Fixed Base

0

1

2

3

4

0 1 2 3

Fixed Base

0

0.5

1

1.5

2

0 1 2 3

Fixed Base

0

0.5

1

1.5

2

2.5

0 1 2 3

a0 = 3

0

0.5

1

1.5

2

0 1 2 3

a0 = 3

0

0.5

1

1.5

0 1 2 3

a0 = 3

µ = 1 µ = 2 µ = 6 Tfix Tfix Tfix

(Fe

or Fy

) /

M.P

GA

(F

e o

r Fy

) /

M.P

GA

IBC-2009 UBC-

EuroCode-8 Concentric Uniform


66

4. 5. COMPARISON BETWEEN STRENGTH DEMANDS OF FIXED-BASE AND

FLEXIBLE-BASE MDOF SYSTEMS

In this section, to study the effect of SSI on strength demands of MDOF systems designed in

accordance to different strength and stiffness patterns, the 10-story building is considered.

The averaged ratios of the strength demands of soil-structure systems to those of the fixed-

base systems for 3 different story strength and stiffness patterns, i.e., IBC-2009, EuroCode-8

and the uniform pattern, subjected to 21 ground motions are computed and the results are

illustrated in Figure 4-5. Results are provided for systems with three values of aspect ratios

( H r =1, 3, 5) which respectively represent squat, medium and slender buildings, and with

three values of ductility ratios ( t = 1, 2, 6) for the case of severe SSI effect (i.e., 0a = 3). It

can be observed that in elastic range of vibration, except for slender structures with very

short periods in which strength demand values of soil-structure systems are nearly equal to

those of the fixed-base ones, the strength demands of soil-structure systems are remarkably

lower than those of the fixed-base models. However, for inelastic response by increasing the

level of inelastic behavior the strength demands of medium and slender soil-structure

systems (i.e. H r = 3, 5) with short periods of vibration are generally greater than those of

the fixed-base systems. This trend becomes more pronounced for the case of slender

buildings with high level of inelastic behavior, which is more apparent in structures

designed in accordance to the uniform pattern. This finding is consistent with the results of

SDOF systems investigated by Ghannad and Jahankhah (2007). It is also seen that the effect of

aspect ratio on the strength demands of soil-structure systems with respect to the fixed-based

models is reversed in long periods range; however, it is still less than unity. Figure 4-6 is

also plotted to better show the effect of the three aforementioned strength patterns on the

averaged ratios of strength demands of soil-structures systems to those of the fixed-base

systems for slender buildings. The results are provided in the same format as Figure 4-5. It

can be found that for the cases of elastic and low inelastic response ( t = 1, 2) there is no

significant difference between the results of three patterns while the difference is significant

for the case with high level of inelastic behavior (i.e., t = 6).


67

Figure 4-5: Averaged ratios of strength demands of soil-structures systems with respect to the

fixed-base systems with different story strength and stiffness patterns ( 0a = 3, N= 10)

Figure 4-6: Effect of structural characteristics distribution on averaged ratios of strength

demands of soil-structures systems to the fixed-base systems (N = 10; 0a = 3; H r =5)

= 1 = 3 = 5

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3

μ =1

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3

μ =1

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3

μ =1

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3

μ =2

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3

μ =2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3

μ =2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3

μ =6

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3

μ=6

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3

μ =6

FSS

I /

Ffix

F

SSI

/ Ff

ix

FSS

I /

Ffix

IBC-2009 EuroCode-8 Uniform

Tfix Tfix Tfix

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3

μ =1

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3

μ =2

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3

μ =6

IBC-2009 EuroCode-

8

Uniform

FSS

I /

Ffix

Tfix Tfix Tfix


68

4.6 VALIDATION OF THE NUMERICAL RESULTS

In this section, to validate the accuracy of the numerical results of this study the 15-story

building with H r = 3, three ductility ratios ( t = 1, 2, 6) representing respectively elastic,

low and high inelastic response corresponding to severe SSI effect ( 0a = 3) have been

considered and analyzed using OPENSEES (2011). All the soil-structure systems considered

here were designed in accordance with the IBC-2009 lateral load pattern. Figure 4-7 shows a

comparison of the averaged strength demands for all earthquake ground motions. As seen,

there is an excellent agreement between the results obtained with the computer program

developed for this study and OPENSEES for both elastic and inelastic ranges of response,

demonstrating the accuracy of the developed computer program.

Figure 4-7: Comparisons of the averaged strength demands resulted from this study and

OPENSEES for the 15-story building with 0a = 3 (21 earthquakes)

4.7 EFFECT OF STRUCTURAL CHARACTERISTICS DISTRIBUTION ON

DUCTILITY REDUCTION FACTOR OF MDOF SYSTEMS

In this part, effect of lateral strength and stiffness distributions on ductility (strength)

reduction factor ( R ) of MDOF systems are investigated. For an MDOF system R is

defined as:

( )

( )

eMDOF i

yMDOF i

FR

F

(4-2)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3

µ = 1

This Study

OPENSEES

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3

µ = 2

This Study

OPENSEES

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3

µ = 6

This Study

OPENSEES

(Fe

or Fy

) /

M.P

GA

Tfix Tfix Tfix


69

where eMDOFF and yMDOFF are respectively elastic and inelastic strength demands of the

MDOF system subjected to a given ground motion for presumed target ductility demand.

This parameter will be fully investigated in Chapter 5 for both MDOF and its equivalent

SDOF systems of soil-structure systems.

4.7.1 Effect of structural characteristics distribution

To parametrically investigate the effect of presumed structural characteristics distributions

on ductility reduction factors ( R ) of fixed-base and flexible-base buildings the 10-story

buildings with H r = 3 and 0a = 1, 3, three ductility ratios ( t = 2, 4, 6) representing

respectively low, medium and high inelastic behaviours, as well as fixed-base structures are

considered. The results illustrated in Figure 4-8 are the average values of responses to all the

selected ground motions. The vertical axis in all figures is the averaged ductility (strength)

reduction factor and the horizontal axis is the fundamental vibration period of the associated

fixed-base structure. Based on the results presented in Figure 4-8, it is seen that for both

fixed-base and flexible-base structures, by increasing the level of inelasticity the difference

between the results of different patterns increases. However, for the case of severe SSI effect

(i.e., 0a = 1, 3), except for the concentric pattern, there is no significant difference between

the results obtained with other patterns considered in this study for structures with short and

medium periods. As an instance, for the case of severe SSI effect and with high level of

inelasticity ( t = 6), the averaged values of R for the structures designed in accordance to

different story strength and stiffness patterns including concentric, UBC-97, EuroCode-8,

IBC-2009 and uniform patterns are respectively 2.51, 3.94, 4.3, 4.6 and 4.9. As seen, for this

case the most dispersion is associated to the concentric pattern. This trend also has been

observed for models of 5- and 15- story buildings. Overall, it can be concluded that in low

level of inelastic behavior effect of story strength and stiffness distribution patterns on the

values of R is not significant and hence practically negligible for both fixed-base and

flexible-base models. Moreover, in all patterns considered in this study, generally,

increasing the fundamental period of vibration is always accompanied by an increase in

averaged value of R , which is intensified by increasing the inelastic range of vibration.

Figure 4-9 shows the variation of the ratio of R for different patterns with respect to that of

the IBC-2009 pattern for the same 10-story building with two levels of ductility ratio ( t = 2,


70

6). Besides confirmation of the above observations, it may be concluded that generally for

both fixed-base and flexible-base models with low level of inelastic behavior there is no

significant difference between the values of ductility reduction factor of the structures

designed in accordance to the aforementioned code-compliant patterns. For cases of fixed-

base and less SSI-effect models ( 0a = 1), by increasing the level of inelastic behaviour this

difference could become significant for some periods. This phenomenon, however, is

negligible as SSI effect becomes pronounced.

Figure 4-8: Effect of structural characteristics distribution on averaged ductility reduction factor

of MDOF fixed-base and soil-structure systems (N = 10 and H r = 3)

0

0.5

1

1.5

2

2.5

0 1 2 3

a0 = 1

0

1

2

3

4

0 1 2 3

a0 = 1

0

1

2

3

4

5

6

7

8

0 1 2 3

a0 = 1

Rµ

0

0.5

1

1.5

2

2.5

0 1 2 3

a0 = 3

0

1

2

3

4

0 1 2 3

a0 = 3

0

1

2

3

4

5

6

7

0 1 2 3

a0 = 3

Rµ

µ = 2 µ = 4 µ = 6

Tfix Tfix Tfix

0

0.5

1

1.5

2

2.5

0 1 2 3

Fixed base

0

1

2

3

4

5

0 1 2 3

Fixed base

Rµ

0

1

2

3

4

5

6

7

0 1 2 3

Fixed base

IBC-2009 UBC-97 EuroCode-8 Concentric Uniform


71

Figure 4-9: Comparison of averaged ratios of ductility reduction factor corresponding to

different load patterns to that of the IBC-2009 pattern for systems with N = 10 and H r = 3

4.7.2 Effect of soil flexibility

The effect of soil flexibility on ductility reduction factor of MDOF systems of 5 and 15

stories designed in accordance to IBC-2009 load pattern is examined in this section. Figure

4-10 shows the mean values of responses subjected to 21 earthquake ground motions for

systems with H r = 3, three ductility ratios ( t = 2, 4, 6), soil-structure systems with two

dimensionless frequencies ( 0a = 1, 3), and the corresponding fixed-base structures. For the

case of 5-strory building, it is seen that by increasing the inelastic behavior SSI effect on

ductility reduction factor becomes more important such that increasing SSI effect is always

accompanied by decreasing in the value of R . This finding is consistent with the results of

the study carried out for SDOF systems by Ghannad and Jahankhah (2007). However, the

results of 15-story building show that SSI effect decreases such that in low level of inelastic

behavior there is no prominent difference between the results of fixed-base and soil-

structure systems. By increasing the level of inelastic behavior, although the difference

again increases, it is still to a large extent less than that of the 5-story building. Hence, it


Tfix Tfix Tfix

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3

μ= 2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3

μ= 2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3

μ= 2

Rµ

i / R

µ (

IBC

-20

09

)

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3

μ= 6

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3

μ= 6

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3

μ= 6

Rµ

i / R

µ (I

BC

-20

09

)

EuroCode-8 UBC-97 Concentric Uniform


72

may be concluded that the results of SDOF soil-structure systems for ductility reduction

factor may not be directly applicable to MDOF soil-structure systems, and some

modifications such as those carried out for fixed-base systems should be taken into account

for soil-structure systems. This point will be further investigated in Chapter 5. It should be

mentioned that in some periods the mean R of fixed-base and less SSI effect cases are

equal or even less than those of the models with the severe SSI effect for the 15-story

building.

To have a better understanding of SSI effect on R of MDOF systems another procedure is

utilized here. First, the elastic shear strength for each MDOF soil-structure system is

computed when subjected to a designated earthquake ground motion. Subsequently, using

the same ductility reduction factor of MDOF fixed-base structure, the inelastic strength

demand of the soil-structure system with presumed target ductility ratio is reduced and

computed. Finally, each MDOF soil-structure system is again analyzed subjected to the

same earthquake ground motion and the new ductility demand is calculated. The effect of

SSI on ductility reduction factor of MDOF systems can then be examined by comparing the

difference between the new resulted ductility demand and that of the target one. To

investigate this phenomenon Figure 4-11 is illustrated. Results are plotted for 5-, 10- and 15-

story buildings with severe SSI effect, H r =3 and with high level of inelastic behaviour.

As seen, using R of MDOF Fixed-base systems for soil-structure systems will result in

large values of ductility demand which in some cases are 3 times that of presumed target one.

This phenomenon is less prominent as the number of stories increases but still significant at

some periods.


73

Figure 4-10: Effect of soil flexibility on averaged ductility reduction factor of MDOF systems

( H r = 3)

Figure 4-11: Averaged ductility demand spectra of MDOF soil-structure systems designed based

on fixed-base ductility reduction factor ( H r = 3 and µ = 6)

0

0.5

1

1.5

2

2.5

0 1 2 3

N = 5

0

1

2

3

4

5

0 1 2 3

N = 5

0

2

4

6

8

0 1 2 3

N = 5

Rµ

0

0.5

1

1.5

2

2.5

0 1 2 3

N = 15

0

1

2

3

4

0 1 2 3

N = 15

0

2

4

6

0 1 2 3

N = 15

Rµ

µ = 2 µ = 4 µ = 6

Tfix Tfix Tfix

Fixed

base

a0 =1 a0 =3

a0 =1 a0 =3 Target

0

3

6

9

12

15

18

21

0 1 2 3

N = 5

0

3

6

9

12

15

18

21

0 1 2 3

N =1 0

0

3

6

9

12

15

18

21

0 1 2 3

N = 15

µ

Tfix Tfix Tfix


74

4. 8 SUMMARY AND CONCLUSIONS

A parametric study has been performed to investigate the effect of different strength and

stiffness distribution patterns, including three code-specified and two arbitrary patterns, on

strength demand and ductility reduction factor of MDOF fixed-base and soil-structure

systems. The results of this study can be summarized with the following broad conclusions:

1. In elastic and low level inelastic response, both fixed-base and flexible-base low-rise

buildings, with exception of those having short periods, show significant differences

among the strength demand values of the structures designed in accordance to the

different lateral strength and stiffness distribution patterns, especially those designed

based on the uniform distribution pattern. However, the results of IBC-2009 and

UBC-97 are to some extent coincident. In high level inelastic response, except for

uniform pattern, the results of all patterns are somewhat coincident and thus

independent of the lateral story strength and stiffness pattern. However, by

increasing the number of stories, differences among strength demand values of all

patterns increase.

2. For both fixed-base and flexible-base models, with exception of those with very

short periods, the averaged strength demand values of uniform pattern are

significantly greater than those of the other patterns considered in this study. This

phenomenon is even more pronounced by increasing the number of stories. The

ratios of strength demand in uniform pattern to those of the code-specified patterns

are generally greater than 2 and in some cases will reach to the value of 4. It can be

concluded that, therefore, using the results of the uniform story strength and stiffness

distribution pattern which has been the assumption of many previous research

works would result in a significant overestimation of the strength demands,

generally from 2 to 4 times, for MDOF systems designed in accordance to the code-

compliant design patterns.

3. In elastic range of vibration, except for slender structures with very short periods in

which strength demand values of soil-structure systems are nearly equal to those of

the fixed-base ones, the strength demands of soil-structure systems are remarkably

lower than those of the fixed-base models. This is compatible with the current

seismic-code regulation (BSSC, 2003; ASCE, 2005) on SSI effects based primarily


75

on the elastic analysis. However, for inelastic state by increasing the level of

inelastic response the strength demands of average and slender soil-structure systems

with short periods of vibration are usually greater than those of the fixed-base

systems. This trend is more significant for the case of slender buildings with high

level of inelasticity, and the most serious happens for the case of uniform pattern

where the strength demand value reaches about 60% greater than those of the fixed-

base models.

4. Overall, in low level of inelastic behavior the effect of story strength and stiffness

distribution patterns on the values of R is not significant and hence practically

negligible for both fixed-base and flexible-base models. By increasing the level of

inelastic behavior the difference between the results of different patterns increases.

Nevertheless, for the case with severe SSI, except for the concentric pattern which is

the most different pattern from other patterns, the difference is insignificant for

structures with short and intermediate periods.

5. A comparison between the mean results of ductility reduction factor of MDOF

fixed-base and soil-structure systems shows that for the case of 5-strory building,

SSI effect on R becomes more significant with increasing inelastic response, and

thus increasing SSI effect is always accompanied by decreasing in value of R . This

finding is compatible with the results of the study carried out for SDOF systems by

Ghannad and Jahankhah (2007). However, by increasing the number of stories SSI

effect decreases such that in low level of inelastic response there is no significant

difference between the results of fixed-base and soil-structure systems. By

increasing the level of inelastic response, although the difference again increases, it

is still to a large extent less than that of the 5-story building.. Hence, it may be

concluded that the results of SDOF soil-structure systems for ductility reduction

factor may not be directly applicable to MDOF soil-structure systems, and some

modifications such as those carried out for fixed-base systems should be taken into

account for soil-structure systems. It is also shown that using R of MDOF fixed-

base systems for soil-structure systems when SSI effect is predominant will result in

large values of ductility demand which in some cases are three times that of the

presumed target one. This phenomenon is less prominent as the number of stories

increases but still significant in some periods.


76

4.9 REFERENCES

Anderson JC, Miranda E and Bertero VV (1991), Kajima Research Team. ―Evaluation of

the seismic performance of a thirty-story RC building,‖ UCB/EERC-91/16,

Earthquake Engineering Research Center, University of California, Berkeley.

AS-1170.4. (2007), Structural design actions: Earthquake actions in Australia.

ASCE/SEI 7-05 (2005), Minimum Design Loads for Buildings and Other Structures.


Building Seismic Safety Council (BSSC) (2003), National Earthquake Hazard Reduction

Program (NEHRP) Recommended Provisions for Seismic Regulations for 348 New

Buildings and Other Structures—Part 2: Commentary (FEMA 450-2). Federal

Emergency Management Agency, Washington, D. C.

CEN (2003), EuroCode 8: Final draft of EuroCode 8: Design of structure for earthquake

resistance – Part 1: General rules for buildings. Bruxelles: European Committee for

Standardization.

Chopra AK (1995), Dynamics of Structures- Theory and Applications to Earthquake

Engineering, 1st edition, Prentice Hall, Englewood Cliffs, New Jersey.

Ghannad M.A And Jahankhah H (2007), ―Site dependent strength reduction factors for

soil–structure systems,‖ Soil Dynamics and Earthquake Engineering 27(2): 99–110.

Gilmore TA and Bertero VV (1993), ―Seismic performance of a 30-story building located

on soft soil and designed according to UBC 1991‖. UCB/EERC-93/04. Earthquake

Engineering Research Center, University of California, Berkeley.

IBC-2009. (2009). International Building Code, ICC, Birmingham, AL.

Leelataviwat S, Goel SC and Stojadinovic´ B (1999), ―Toward performance-based seismic

design of structures,‖ Earthquake Spectra 15: 435–461.


Mobasseri M, Roesset JM and Klingner RE (1992) ―The relation between local and overall

ductility demands in multi-degree-of-freedom framed type structures,‖ The 10th

World Conference on Earthquake Engineering, Madrid, Spain, Vol. 1 pp. 849-858.

Mohammadi R. K, El-Naggar MH and Moghaddam H (2004) ―Optimum strength

distribution for seismic resistant shear buildings,‖ International Journal of Solids and

Structures 41: 6597–6612.


77

OPENSEES, (2011), OpenSees Command Language Manual. Open System for Earthquake

Engineering Simulation. Mazzoni, S., McKenna, F., Scott. M. H., Fenves, G. L.

Available at http://opensees.berkeley.edu/

Park, K. and Medina, R. A., 2007. ―Conceptual seismic design of regular frames based on

the concept of uniform damage,‖ ASCE Journal of Structural Engineering, 133 (7),

945-955.

Srivastav S and Nau JM (1988), ―Seismic analysis of elastoplastic MDOF structures,‖ ASCE

Journal of Structural Engineering, 1114(6): 1339-1353.

UBC (1994), Uniform Building Code, International Conference of Building Officials,

Whittier, California.

UBC (1997), Uniform Building Code, International Conference of Building Officials,

Whittier, California.

Veletsos AS and Vann P (1971) ―Response of ground-excited elastoplastic systems,‖

Journal of the Structural Division, (ASCE) 97(4): 1257-1281.

http://opensees.berkeley.edu/


78

Chapter 5

STRENGTH REDUCTION FACTOR FOR MULTIPLE-DEGREE-OF-FREEDOM

SYSTEMS CONSIDERING SOIL-STRUCTURE INTERACTION

5.1 INTRODUCTION

The primary seismic design of buildings in most of the conventional seismic codes is based

on force-based procedure. These codes permit structures to behave inelastically during

moderate and severe earthquake ground motions. In strong earthquake ground motions, the

design base shear strength recommended in seismic provisions are typically much lower

than the base shear strength that are required to sustain the structure in the elastic range.

Strength reductions from the elastic strength demand are prevalently accounted for through

the use of strength reduction factor, R, which is one of the most controversial issues in the

seismic-resistant design provisions. This factor, strongly dependent on the energy

dissipation capacity of the structural systems, is used to reduce the elastic design force

spectra in earthquake-resistant design. The code-specified values of strength reduction

factors in different seismic provisions even for the same type of structure are usually

different, reflecting the fact that the recommended values could be to a large extent based on

judgments, experiences and observed behaviors of structures during past earthquake events

besides of analytical results. For an idealized elasto-plastic SDOF system, R corresponds to

the seismic force at the predefined design level and can be considered as a product of the

conventional reduction factor R , reflecting the nonlinear hysteric behavior in a structure,

and R that account for other reduction factors such as reductions due to element

overstrength, redundancy, strain hardening and etc.

.R R R (5-1)

During the past four decades, extensive studies have been conducted on strength reduction

factor. The pioneering investigations performed by Veletsos and Newmark (1960) and


79

Newmark and Hall (1973) may be regarded as the first renowned studies on R . Based on

elastic and inelastic response spectra of NS component of El Centro earthquake as well as

previous studies on SDOF systems to pulse-type excitations, Newmark and Hall (1973)

proposed simplified expressions for R as a function of target period and ductility ratio of

the structure. In another study, based on mean inelastic spectra of 20 artificial ground

motions compatible with the Newmark-Hall (1973) elastic design spectra, Lai and Biggs

(1980) proposed alternative expressions as a function of also the target ductility, period as

well as period ranges. Many more studies were made by researchers to propose simplified

equations for strength reduction factor of fixed-base SDOF systems (Elghadamsi and

Mohraz, 1978; Fischinger et al., 1994; Miranda and Bertero, 1994; Lam et al., 1998; Ordaz

and Perez-Rocha, 1998; Karmakar and Gupta, 2007). Elghadamsi and Mohraz (1987) may

be one of the first researchers who studied the influence of soil condition on R . With

further investigations Krawinkler and Rahnama (1992) and Miranda (1993) demonstrated

the significant effect of soil conditions, especially for the case of soft soils, on strength

reduction factors. However, effect of SSI on R has not been explicitly considered in their

works. Recent studies on elastic and inelastic responses of SDOF soil-structure systems

indicated that SSI could have significant effects on ductility demand of structures (Aviles

and Perez-Rocha, 2003 and 2005; Ghannad and Jahankha, 2007; Mahsuli and Ghannad,

2009, Ganjavi and Hao, 2011).

Ghannad and Jahankha (2007) investigated the effect of site condition and SSI on R of

SDOF systems. They concluded that SSI reduces the R values, especially for the case of

buildings located on soft soils; therefore, using the fixed-base strength reduction factors for

soil-structure systems leads to underestimation of seismic design forces. These studies are

mainly based on the dynamic response of SDOF systems while real structures have MDOF

and, thus, more realistic representation of real structures needs MDOF models. Moreover,

complex behaviors such as contributions to structural responses from higher modes cannot

be captured with an SDOF system especially in the inelastic response range. The

relationship between MDOF and SDOF system responses of fixed-base systems was first

studied by Veletsos and Vann (1971) by considering some shear-beam models with equal

story masses connected by weightless springs in series from one degree of freedom (DOF)

to five DOFs. They concluded that for systems having more than three DOFs the proposed


80

design regulations for SDOF systems were not sufficiently accurate and could lead to non-

conservative estimates of the required inelastic lateral strength, and that errors tended to

increase as the number of degrees of freedom increased. Another study was conducted by

Nassar and Krawinkler (1991) on three types of simplified fixed-base MDOF models to

estimate the modifications required to the inelastic strength demands obtained from bilinear

SDOF systems in order to limit the story ductility demand in the first story of the MDOF

systems to a predefined value. They found that the deviation of MDOF story ductility

demands from the SDOF target ductility ratios increased with structural vibration period and

target ductility ratio. More examples of the works conducted on the subject can be found in

the reference (Seneviratna and Krawinkler, 1997; Santa-Ana and Miranda, 2000;

Moghaddam and Mohammadi, 2001). However, all of the works were performed on fixed-

base systems, i.e. based on a presumed assumption that soil beneath the structure is rigid.

Moreover, as demonstrated in Chapter 3, the common SDOF systems may not lead to

accurate estimation of the strength and ductility demands of MDOF soil-structure systems,

especially for the cases of mid- and high-rise buildings, due to the significant contributions

from high vibration modes.

Halabian and Erfani (2010), by considering some limited generic RC frame models resting

on flexible foundations, evaluated the effects of stiffness and strength of the structure on

strength reduction factors. They concluded that the foundation flexibility could significantly

change the strength reduction factors of the RC frames and neglecting this phenomenon may

lead to erroneous conclusions in the prediction of seismic performance of flexibly supported

RC frame structures. However, despite the observations of the SSI effects, no practical

equation to estimate the strength reduction factors of MDOF soil-structure systems has been

presented yet. In the present study, an intensive parametric study has been performed to

investigate the effects of SSI on R values of MDOF and its equivalent SDOF (E-SDOF)

systems. This is carried out for a wide range of structural dynamic characteristics and non-

dimensional key parameters to investigate the relationship between R values of MDOF

and SDOF soil-structure systems. Finally, based on numerical results a new simplified

equation which is functions of fixed-base fundamental period, ductility ratio, the number of

stories, structure slenderness ratio and dimensionless frequency is proposed to estimate

strength reduction factors for MDOF soil-structure systems.


81

5.2 SELECTED EARTHQUAKE GROUND MOTIONS

In this investigation, because of the variability in ground motion characteristics which

affects structural responses an ensemble of 30 earthquake ground motions with different

characteristics recorded on alluvium and soft soil deposits (soil type C, with shear wave

velocity between 180 and 360 m/s, and D, with shear wave velocity lower than 180 m/s,

based on the USGS site classification) are compiled and utilized in the nonlinear dynamic

time history analyses. Like in the previous chapters, all selected ground motions are

obtained from earthquakes with magnitude greater than 6 having the closest distance to fault

rupture more than 15 km without pulse type characteristics. The main parameters of the

selected ground motions are given in Table 5-1.

Table 5-1: Selected ground motions recorded at alluvium and soft soil sites

Event Year Station Distanc

e (km)

Soil type

(USGS)

Component PGA (g)

Imperial Valley 1979 Compuertas 32.6 C 15, 285 0.186, 0.147

Imperial Valley 1979 El Centro Array #12 18.2 C 140, 230 0.143, 0.116

Loma Prieta 1989 Agnews State Hospital 28.2 C 0, 90 0.172, 0.159

Loma Prieta 1989 Gilroy Array #4 16.1 C 0, 90 0.417, 0.212

Loma Prieta 1989 Sunnyvale - Colton Ave 28.8 C 270, 360 0.207, 0.209


Northridge 1994 Canoga Park - Topanga

Can

15.8 C 196, 106 0.42, 0.356

Kobe 1995 Kakogawa 26.4 D 0, 90 0.251, 0.345

Kobe 1995 Shin-Osaka 15.5 D 0, 90 0.243, 0.212

Loma Prieta 1989 APEEL 2 - Redwood

City

47.9 D 43, 133 0.274, 0.22


Menhaden

51.2 D 360, 270 0.116, 0.107

Superstitn

Hills(B)


Refuge

27.1 D 315, 225 0.167. 0.119

Morgan Hill 1984 Gilroy Array #2 15.1 C 0, 90 0.162, 0.212




82

5. 3. PROCEDURE FOR ANALYSIS

A series of 3-, 5-, 7-, 10-, 15- and 20-story shear buildings and also their equivalent SDOF

(E-SDOF) models are considered to investigate the effect of SSI on strength reduction

factors of both MDOF and E-SDOF systems. In this regard, for a given earthquake ground

motion, a family of 21600 different soil-structure models including MDOF as well as E-

SDOF models with various predefined key parameters are considered. This includes MDOF

and E-SDOF models with 30 fundamental periods of the corresponding fixed-base structures,

ranging from 0.1 to 3 sec with intervals of 0.1, three values of aspect ratio ( H r =1, 3, 5),

four values of dimensionless frequency ( 0a = 0, 1, 2, 3), and five values of target interstory

displacement ductility ratio ( t =1, 2, 4, 6, 8) where t =1 corresponds to the elastic state.

For each earthquake ground motion, the total normalized elastic and inelastic shear strength

of the MDOF and E-SDOF system are computed in order to reach the t in the structure, as

a part of the soil–structure system, within a 0.5% error. Total normalized shear strength is

defined as the total shear strength demands divided by the total structural mass and then

normalized to the peak ground acceleration (PGA). Therefore, strength reduction factors of

both MDOF and E-SDOF soil-structure models can be computed by dividing the elastic

shear strength to the inelastic shear strength corresponding to the presumed target ductility

ratio. In this regard, SSIOPT software, developed by the author for this thesis and introduced

in Chapter 2, is used to calculate the R spectra for MDOF and E-SDOF systems. Generally,

the iterative procedure proposed in Chapter 3 can also be used to calculate elastic and

inelastic strength demands required for computing strength reduction factors of MDOF and

E-SDOF systems.

5.4 EFFECT OF SSI ON STRENGTH REDUCTION FACTOR OF E-SDOF

SYSTEMS

5.4.1 Strength reduction factors of E-SDOF systems for structures with different

number of stories

A series of different E-SDOF soil-structure systems corresponding to the first-mode shape

of 3-, 5-, 10- 15- and 20-story buildings are analyzed to compare the R of different E-


83

SDOF soil-structure systems subjected to an ensemble of 30 earthquake ground motions

listed in Table 5-1. As mentioned earlier, instead of the first-mode effective modal mass,

total mass of each MDOF systems is considered to model the corresponding E-SDOF soil-

structure system. As an example, the average values of strength demand for five different E-

SDOF soil-structure systems corresponding to 3-, 5-, 10- 15- and 20-story buildings, are

depicted in Figure 5-1. The results are shown for fixed-base and soil-structure systems with

aspect ratios of 3 and dimensionless frequency 0a = 2 as well as target interstory

displacement ductility ratios of 4 ( t = 4). The abscissa in all figures is the first-mode period

of the fixed-base structure, fixT , and the vertical axis is the averaged strength reduction

factors R resulted from 30 earthquake ground motions. It can be clearly seen that the R

spectra of E-SDOF systems for shear-buildings are independent of the number of stories

such that all spectra for E-SDOF soil-structure systems corresponding to MDOF systems

with different number of stories are completely coincident. As stated in Chapter 3, the

reason of this similarity goes back to the first-mode shape of the shear buildings which is

independent of the number of stories in shear-building structures. For each target

fundamental period, the normalized mode shapes of MDOF shear buildings with different

number of stories are completely coincident and hence are not dependent on the number of

stories. It is important to note that this result is true when (1) total structural mass is

uniformly distributed along the height of the structures and (2) lateral stiffness in all MDOF

buildings with different number of stories is distributed based on the same specified pattern

which here is IBC load pattern.

Figure 5-1: Comparison of the averaged strength reduction factor for different E-SDOF systems

(µ = 4)

Rµ

Tfix Tfix

0

2

4

6

0 1 2 3

Fixed Base

N= 5

N= 10

N= 15

N= 20

0

1

2

3

0 1 2 3

Flexible Base, a0 = 2

N= 5

N= 10

N= 15

N= 20


84

5.4.2. Effect of ductility ratio

Figure 5-2 shows the effect of ductility ratio on averaged strength reduction factor spectra of

E-SDOF soil-structure systems with H r = 3, four ductility ratios ( t = 2, 4, 6, 8), and three

dimensionless frequencies ( 0a = 1, 2, 3), and subjected to 30 earthquake ground motions. As

seen, the R spectra for the cases with less SSI effect ( 0a = 1) are more sensitive to the

variation of ductility ratio in comparison to those of the severe SSI effect. Significant

reduction of R values for soil-structure systems with strong SSI effects is observed, which

results in less variations of R values as the SSI effect becomes more pronounced. As an

instance the variations of values for ductility ratios from 8 to 2 are from 7.4 to 1.8 and

2.7 to 1.2 for the cases of less ( 0a = 1) and severe SSI effect ( 0a = 3), respectively.

Figure 5-2: Averaged strength reduction factor spectra for E-SDOF systems with different

ranges of nonlinearity ( H r = 3)

5.4.3. Effect of dimensionless frequency

Figure 5-3 shows the mean values of R from 30 earthquake ground motions for systems

with two aspect ratios ( H r = 1, 5) representing respectively squat and slender buildings,

three ductility ratios ( = 2, 4, 8), soil-structure systems with three dimensionless

frequencies ( = 1, 2, 3), and the corresponding fixed-base structures. It is seen that by

increasing the inelastic response level, SSI effect on ductility reduction factor becomes more

pronounced such that increasing SSI effect is always accompanied by a decrease in R . The

phenomenon is more pronounced for slender buildings ( H r = 5). This indicates that SSI

significantly affects the strength reduction factor spectra of E-SDOF systems. Hence, it may

be concluded that using R of fixed-base systems leads to significant underestimation of

R

t

0a

0

2

4

6

8

10

0 1 2 3

a0= 1

0

1

2

3

4

5

6

0 1 2 3

a0= 2

0

1

2

3

4

5

0 1 2 3

a0= 3 µ=2µ=4µ=6µ=8

Rµ

Tfix Tfix Tfix


85

inelastic strength demands of soil-structure systems especially for the case of severe SSI

effects i.e., 0a = 2, 3. This finding is consistent with the results reported by Ghannad and

Jahankhah (2007) on SDOF systems.

5.4.4. Effect of aspect ratio

Figure 5-4 investigates the effect of aspect ratio on strength reduction factor of E-SDOF

systems. It shows the mean values of R from 30 earthquake ground motions for soil-

structure systems with two dimensionless frequencies ( 0a = 1, 3), three aspect ratios ( H r =

1,3, 5) and three ductility ratios ( t = 2, 4, 8). It can be seen that, except for E-SDOF

systems with very short periods, increasing the aspect ratio is always accompanied by a

decrease in the values of R . The trend is more pronounced for the cases with pronounced

SSI effect ( 0a =3) and long periods. As an instance, for the case of 0a =3, the values of R

for slender E-SDOF systems ( H r = 5) are 36%, 59% and 95% less than those of the squat

E-SDOF systems ( H r = 1) for target ductility ratios of 2, 4 and 8, respectively.

Figure 5-3: Effect of dimensionless frequency on Averaged strength reduction factor spectra of

E-SDOF soil-structure systems

Fixed base a0 =1 a0 =2 a0 =3

Tfix Tfix Tfix

Rµ

0

1

2

3

0 1 2 3

µ = 2, H̅/r =1

0

2

4

6

0 1 2 3

µ = 4, H̅/r =1

0

2

4

6

8

10

12

14

0 1 2 3

µ = 8, H̅/r=1

Rµ

0

1

2

3

0 1 2 3

µ = 2, H̅/r =5

0

2

4

6

0 1 2 3

µ = 4, H̅/r =5

0

2

4

6

8

10

12

14

0 1 2 3

µ =8, H̅/r =5


86

Figure 5-4: Effect of aspect ratio on Averaged strength reduction factor spectra for E-SDOF soil-

structure systems

5.4.5 Using R of E-SDOF fixed-base systems for soil-structure systems

To have a better understanding of SSI effect on R of E-SDOF systems another procedure

is utilized here, which is the procedure used in the current regulations on seismic design of

soil-structure systems (BSSC, 2000). First, the elastic shear strength for each of E-SDOF

soil-structure system is computed when subjected to a designated earthquake ground motion.

Subsequently, using the same ductility reduction factor of the corresponding fixed-base

system, the inelastic strength demand of the soil-structure system with presumed target

ductility ratio is reduced and computed. Finally, each E-SDOF soil-structure system is again

analyzed subjected to the same earthquake ground motion and SSI key parameters (i.e., 0a ,

and H r ), and then the new ductility demand is calculated. The effect of SSI on ductility

reduction factor of E-SDOF system can then be examined by comparing the difference

between the new resulted ductility demand and that of the target one. Figure 5-5 shows the

results corresponding to three dimensionless frequencies ( 0a = 1, 2, 3), three aspect ratios

( H r =1, 3, 5) and two target ductility ratios ( t = 2, 6). As seen, using the R values of

fixed-base systems for soil-structure systems will overestimate the ductility demand

especially for the cases of slender buildings with severe SSI effects. As an instance, for the

Rµ

0

1

2

3

0 1 2 3

µ = 2, a0 =1

0

2

4

6

0 1 2 3

µ = 4, a0 =1

0

2

4

6

8

10

0 1 2 3

µ = 8, a0 =1

H̅/r = 1

H̅/r = 3

H̅/r = 5

Rµ

0

1

2

3

0 1 2 3

µ = 2 a0 =3

0

1

2

3

0 1 2 3

µ = 4, a0 =3

0

1

2

3

4

5

0 1 2 3

µ = 8, a0 =3

Tfix Tfix Tfix


87

case with a presumed target ductility ratio of 6, dimensionless frequency of 3 and

fundamental period of 2 sec, the ductility demands of soil-structure systems are 16.3, 28.3

and 41.8 for respectively squat, medium and slender structures. These results indicate that

using the strength reduction factors of E-SDOF fixed-base systems for soil-structure systems

may result in a significant underestimation of inelastic shear strength demands of soil-

structure systems, and, as a result, the structure would experience much higher deformation

demands than expected.

Figure 5-5: Average ductility demand spectra for E-SDOF soil-structure systems designed based on

fixed-base strength reduction factors

0

6

12

18

24

30

36

42

48

54

0 1 2 3

μ=6, H̅/r=1

0

6

12

18

24

30

36

42

48

54

0 1 2 3

μ=6, H̅/r=3

0

6

12

18

24

30

36

42

48

54

0 1 2 3

μ=6, H̅/r=5

µ

0

2

4

6

8

10

12

14

0 1 2 3

μ=2, H̅/r=1

0

2

4

6

8

10

12

14

0 1 2 3

μ=2, H̅/r=3

0

2

4

6

8

10

12

14

0 1 2 3

μ=2, H̅/r=5

µ

Tfix Tfix Tfix

Target a0 =1 a0 =2 a0 =3


88

5.5 EFFECT OF SSI ON STRENGTH REDUCTION FACTOR OF MDOF SYSTEMS

5.5.1. Effect of number of stories

To study the effect of number of stories on strength reduction factors for fixed-base and

flexible-base structures, buildings of 3, 5, 10, 15 and 20 stories as well as the corresponding

E-SDOF systems are considered which represent the common building structures from low-

to high-rise models. Results illustrated in Figure 5-6 are the mean values of 30 earthquake

ground motions for systems with H r = 3, corresponding to two ductility ratios ( t =2, 6)

representing respectively low and high inelastic behaviors, and soil-structure system with

two dimensionless frequencies ( 0a = 1 and 3), as well as the fixed-base structures. As stated

before, 0a is an index for the structure-to-soil stiffness ratio controlling the severity of SSI

effects, and also the value of 3 for this parameter corresponds to significant SSI effect. It is

observed that for fixed-base systems, regardless of the level of nonlinearity, increasing the

number of DOFs (stories) always results in a reduction in the averaged values of R . For

soil-structure systems, the effect of the number of stories are, however, very different from

the fixed-base models. For the cases with significant SSI effect, R spectra become less

sensitive to the variation of the number of stories. This is more apparent in cases with low

level of inelasticity. In addition, an interesting point can be observed for the case of E-SDOF

soil-structure systems with severe SSI effect ( 0a = 3) in which R values are significantly

lower than those of the MDOF systems in almost all ranges of period. Therefore, it can be

concluded that the modifying factors for strength reduction factors of MDOF soil-structure

systems could be completely different from those of the fixed-base systems. For fixed-base

structures, it has been proposed to multiply R of SDOF systems by a modifying factor that

takes into account the possible concentration of displacement ductility demands in specific

floors (Miranda, 1997; Santa-Ana and Miranda, 2000) for use of the reduction factor in

seismic analysis of MDOF systems. This factor was defined by Santa-Ana and Miranda

(2000) for fixed-base systems as:

( )

( )

SDOF iM

MDOF i

VR

V

(5-2)


89

where SDOFV and MDOFV are the strength demands of SDOF and MDOF systems subjected to

a given ground motion and presumed target ductility demand, respectively. MR represents a

modification factor to the strength reduction factor of SDOF systems so it can be applied to

MDOF structures. Therefore, the strength reduction factor of MDOF systems ( ( )MDOFR )

can be computed from following equation:

( ) ( ).MDOF SDOF MR R R (5-3)

As seen, this modification factor just considers the difference between the inelastic demands

of MDOF and the corresponding SDOF systems. Santa-Ana and Miranda (2000) and

Moghaddam and Mohammadi (2001) showed that for fixed-base systems the values of this

factor are approximately equal to one regardless of the number of stories. This means that

for μ =1 the lateral strength of the MDOF systems is on average nearly equal to that of the

SDOF system. However, results of this study indicate that this finding is not correct for soil-

structure systems. To show the importance of this problem, the averaged ratios of strength

demands on MDOF to those on E-SDOF systems for different ranges of nonlinearity are

computed and the results are depicted in Figure 5-7 for both the fixed-base and soil-structure

systems of the10-story building. It is seen that different from the fixed-base systems, the

ratios of strength demands in elastic range of response (i.e., μ =1) is significant for soil-

structure systems. In fact, in elastic range of response the ratios remarkably increase with

SSI effect such that the more SSI effect (larger values of 0a ), the more significant difference

between the strength demands of MDOF and SDOF systems. As an instance, for the

structure with long period of vibration, the value of this ratio can be greater than 5 when SSI

effect is predominant while it is about 1.3 for the fixed-base system. Results of this study

show that this phenomenon is more pronounced as the value of aspect ratio ( H r ) increases.

For inelastic range of response, however, the effect of SSI becomes less important in a way

that in high level of inelasticity the averaged ratios of strength demands are approximately

insensitive and thus independent of the soil flexibility. It can be concluded that for soil-

structure systems the values of both elastic and inelastic strength demands must be taken

into account for calculation of the modification factor to the strength reduction factor.

Therefore, the modification factor for soil-structure systems or for more precise analyses of

fixed-base systems should be defined as:


90

( )

( )

MDOF

M

SDOF

RR

R

(5-4)

where MR represents a modification factor to the strength reduction factor of SDOF systems

so it can be applied to both MDOF fixe-base and soil-structure systems.

To parametrically examine this modification factor for both fixed-base and soil-structure

systems, results for the 10-story building with three levels of nonlinearity ( t = 2, 4, 8)

corresponding to three values of dimensionless frequency ( 0a = 1, 2, 3) as well as the fixed-

base structures with aspect ratio of 3 are shown in Figure 5-8. It is seen that, regardless of

the level of nonlinearity, the values of MR are generally less than one for the case of fixed-

base systems. However, these factors increase as SSI effect increases (i.e. increasing the

amount of 0a ). It is also obvious that different from the fixed –base systems, MR values are

sensitive to the level of nonlinearity for soil-structure systems such that they increase with

ductility ratio and are generally larger than one especially for structures with longer periods

and severe SSI effects. As an example, for the case with high level of inelasticity ( t =8)

and fundamental period of 2 sec, the values of MR are 0.78, 0.94, 1.71 and 2.4 for fixed-

base, and soil-structure system with 0a = 1, 0a = 2, and 0a = 3, respectively. As discussed

above (Figure 5-7), the large differences among the MR values are because of the large

difference between the values of elastic strength demands of soil-structure systems and

fixed-base models.


91

Figure 5-6: Effect of the number of stories on averaged strength reduction factor spectra for

fixed-base and soil-structure systems ( H r = 3)

Tfix Tfix

Rµ

0

0.5

1

1.5

2

2.5

3

0 1 2 3

Fixed base, µ = 2

E-SDOF N = 3N = 5 N = 10N = 15 N = 20

0

2

4

6

8

10

0 1 2 3

Fixed base, µ = 6

Rµ

0

1

2

3

0 1 2 3

a0 = 1, µ = 2

0

2

4

6

8

0 1 2 3

a0 = 1, µ = 6

Rµ

0

1

2

3

0 1 2 3

a0 = 3, µ = 2

0

2

4

6

0 1 2 3

a0 = 3, µ = 6


92

Figure 5-7: Averaged ratios of shear strength demands on MDOF systems to those on E-SDOF

systems for different ranges of nonlinearity (10-story building; H r = 3)

Figure 5-8: Averaged modifying factor for MDOF fixed-base and soil-structure systems (10-

story building; H r = 3)

5.6.2 Effect of dimensionless frequency

Figures 5-9 and 5-10 examine the effect of dimensionless frequency on averaged strength

reduction factor of MDOF soil-structure systems in different ranges of inelastic response

( t = 2, 4, 8). From these two figures, the following observations can be made:

In low level of nonlinearity (i.e., t =2), regardless of the aspect ratio, the SSI

effects decrease as the number of stories increases. In other words, only in low-rise

buildings (i.e., 5-story building) the values of R reduces as the dimensionless

frequency increases, but in mid- and high-rise building they can approximately be

considered as insensitive to the variation of 0a .

Vy

(MD

OF)

/ V

y (S

DO

F)


Tfix Tfix Tfix

0

1

2

3

4

5

6

0 1 2 3

µ= 1

0

1

2

3

4

5

6

0 1 2 3

µ= 2

0

1

2

3

4

5

6

0 1 2 3

µ= 8 R͂

M =

Rµ

(M

DO

F) /

Rµ

(SD

OF)

0

0.5

1

1.5

2

2.5

3

0 1 2 3

µ= 2

0

0.5

1

1.5

2

2.5

3

0 1 2 3

µ= 4

0

0.5

1

1.5

2

2.5

3

0 1 2 3

µ= 8

Tfix Tfix Tfix



93

By increasing the level of nonlinearity, the SSI effects become more pronounced for

low- and mid-rise buildings such that the greater the frequency, the more reduction

in the values of R . The phenomenon is more pronounced in squat structures ( H r

=1). The same trend but with less intensity can be observed for high-rise building

(i.e., 20-story building) with H r =1.

For slender high-rise building (i.e., 20-story building and H r =5) the SSI effect

significantly reduces even for high level of nonlinearity. It can be seen that the

values of R , except for structure with short periods, are not very sensitive to the

variations of 0a . In some ranges of period the R values for the fixed-base system

and the system with low SSI effect ( 0a = 1) are lower than those with severe SSI

effect. Therefore, it may be concluded that the SSI effects on strength reduction

factors of MDOF systems become less important as the number of stories increases.

For the case of slender high-rise buildings the R values can be approximately

considered as independent of the variation of dimensionless frequency especially in

low and medium elastic response ranges.

5.6.3 Effect of aspect ratio

In order to examine the effect of aspect ratio on strength reduction factor of MDOF-soil

structure systems the 10-story building with three values of aspect ratio ( H r = 1, 3, 5) and

with three ductility ratios ( t = 2, 4, 8) as well as two dimensionless frequencies ( 0a = 1, 3)

are considered and analyzed subjected to the selected ground motions listed in Table 5-1.

The results are plotted in Figure 5-11. It is clear that for the case of less SSI effect, the

values of averaged R are insensitive to the variation of aspect ratio but significant for E-

SDOF systems as shown in Figure 5-4. For the case with severe SSI effect and high inelastic

response, except in short period ranges, the values of mean R increase with the aspect ratio,

which is completely different from the results obtained for the E-SDOF system, where

increasing the aspect ratio is always accompanied by decreasing the R values. This finding

indicates that SSI affects the strength reduction factors of MDOF and E-SDOF systems in a

different manner. The same results have been observed in this study for MDOF soil-

structure systems with different number of stories.


94

Figure 5-9: Effect of dimensionless frequency on averaged strength reduction factor spectra for

MDOF soil-structure systems ( H r = 1)

0

1

2

3

0 1 2 3

µ = 2, N =10

0

1

2

3

4

0 1 2 3

µ = 4, N =10

0

2

4

6

8

10

0 1 2 3

µ = 8, N =10

Rµ

0

1

2

3

0 1 2 3

µ = 2, N =5

0

1

2

3

4

5

0 1 2 3

µ = 4, N =5

0

2

4

6

8

10

0 1 2 3

µ = 8, N =5

Rµ

0

1

2

3

0 1 2 3

µ = 2, N =20

0

1

2

3

4

0 1 2 3

µ = 4, N =20

0

3

6

9

0 1 2 3

µ = 8, N =20

Rµ

Tfix Tfix Tfix



95

Figure 5-10: Effect of dimensionless frequency on averaged strength reduction factor spectra for

MDOF soil-structure systems ( H r = 5)

Figure 5-11: Effect of aspect ratio on averaged strength reduction factor spectra for MDOF soil-

structure systems (10-story building)

0

1

2

3

0 1 2 3

µ = 2, a0 = 3

0

1

2

3

4

0 1 2 3

µ = 4 a0 = 3

0

2

4

6

8

0 1 2 3

µ = 8 a0 = 3

Rµ

Tfix Tfix Tfix

0

1

2

3

0 1 2 3

µ = 2, a0 = 1

0

2

4

6

8

10

0 1 2 3

µ = 8, a0 = 1

Rµ

0

1

2

3

4

0 1 2 3

µ = 4 a0 = 1

H̅/r= 1

H̅/r= 3

H̅/r= 5

Rµ

0

1

2

3

0 1 2 3

µ = 2, N =10

0

1

2

3

4

0 1 2 3

µ = 4, N =10

0

2

4

6

8

10

0 1 2 3

µ = 8, N =10

Rµ

0

1

2

3

0 1 2 3

µ = 2, N =20

0

1

2

3

4

0 1 2 3

µ = 4, N =20

0

3

6

9

0 1 2 3

µ = 8, N =20

Rµ

0

1

2

3

0 1 2 3

µ = 2, N =5

0

1

2

3

4

5

0 1 2 3

µ = 4, N =5

0

2

4

6

8

10

0 1 2 3

µ = 8, N =5

Tfix Tfix Tfix



96

5.6 ESTIMATION OF THE STRENGTH REDUCTION FACTORS FOR MDOF

SOIL-STRUCTURE SYSTEMS

As stated in the literature, the code-specified values of strength reduction factors in different

seismic provisions are usually based on judgments, experiences and observed behaviors of

structures during past earthquake events rather than on analytical results. On the other hand,

it is believed that the strength reduction factors obtained by neglecting the SSI effect can be

utilized to estimate the inelastic strength demands of soil-structure systems. This is, in fact,

the foundation for current regulations on seismic design of soil-structure systems (BSSC.

2000). However, based on the results of this study, it has been demonstrated that this

assumption can lead to significant underestimation of inelastic strength demands of MDOF

soil-structure systems especially for the cases of low- and mid-rise buildings with high level

of nonlinearity. In addition, it is also concluded that using the values of strength reduction

factors of SDOF systems could result in significant underestimation or overestimation of

strength reduction factors for fixed-base and soil-structure systems, respectively.

In earthquake-resistant design and, in general, for practical purpose it is desirable to have a

simplified expression to estimate strength reduction factors of MDOF systems. Here, based

on nonlinear dynamic analyses of 10800 MDOF soil-structure systems the following simple

equation is proposed:

( )ib

MDOF i fixR a T (5-5)

where fixT is the fundamental period of the corresponding fixed-based structure; ia and ib

are constants depending on the interstory displacement ductility ratio, the number of stories,

aspect ratio, and dimensionless frequency, and can be obtained from Tables 5-2 to 5-8. In

addition, the values of strength reduction factor corresponding to the different values of

ductility ratio, the number of stories, aspect ratio and dimensionless frequency specified in

Tables 5-2 to 5-8 can be easily obtained by linear interpolation. To show the capability of

the proposed equation in estimating the strength reduction factors of MDOF soil-structure

systems Figure 5-12 is provided. This figure shows the comparison of the proposed equation

in predicting the strength reduction factors of 5- and 20-story buildings with different ranges

of nonlinearity obtained from Eq. (5-5) with the averaged numerical results. As seen, there is


97

a good agreement between Eq. (5-5) and the averaged numerical results for strength

reduction factors of MDOF soil-structure systems.

Table 5-2: Constant coefficient ia and

ib of Eq. (5-5)

µ = 2 ia

µ = 2 bi

H r = 1, 3, 5

H r = 1, 3, 5

N 0a = 0

0a = 1 0a = 2

0a = 3 0a = 0

0a = 1 0a = 2

3 1.794 1.66 1.57 1.53 0.106 0.107 0.116

5 1.645 1.57 1.51 1.51 0.0976 0.105 0.136

7 1.573 1.573 1.48 1.48 0.092 0.092 0.132

10 1.48 1.48 1.48 1.48 0.1 0.1 0.1

15 1.44 1.44 1.44 1.44 0.095 0.095 0.095

20 1.42 1.42 1.42 1.42 0.092 0.092 0.092

Table 5-3: Constant coefficient ia of Eq. (5-5)

ia µ = 4

H r = 1, 3

H r = 5

N 0a = 0 0a = 1 0a = 2 0a = 3 0a = 1 0a = 2 0a = 3

3 3.39 2.91 2.51 2.32 2.91 2.51 2.32

5 2.98 2.65 2.43 2.29 2.65 2.43 2.37

7 2.72 2.47 2.26 2.21 2.47 2.36 2.31

10 2.49 2.29 2.173 2.11 2.29 2.28 2.22

15 2.28 2.16 2.03 2.03 2.16 2.136 2.136

20 2.16 2.07 1.97 1.97 2.07 2.06 2.06

Table 5-4: Constant coefficient ib of Eq. (5-5)

ib µ = 4

H r = 1, 3

H r = 5

N 0a = 0

0a = 1 0a = 2

0a = 3 0a = 1

0a = 2 0a = 3

3 0.251 0.237 0.246 0.243 0.237 0.246 0.243

5 0.245 0.248 0.28 0.292 0.248 0.28 0.326

7 0.225 0.231 0.266 0.293 0.231 0.27 0.31

10 0.21 0.213 0.238 0.272 0.213 0.252 0.28

15 0.165 0.184 0.251 0.251 0.184 0.248 0.248

20 0.151 0.164 0.235 0.235 0.164 0.23 0.23


98


ia µ = 6

H r = 1

H r = 3, 5

N 0a = 0

0a = 1 0a = 2

0a = 3 0a = 1

0a = 2 0a = 3

3 4.86 4.25 3.33 3.08 4.25 3.48 3.3

5 4.56 3.98 3.33 3.08 3.98 3.48 3.3

7 4.29 3.72 3.24 3.08 3.72 3.48 3.3

10 3.8 3.36 2.9 2.9 3.36 3.2 3.2

15 3.33 3.065 2.79 2.79 3.065 3 3

20 3.09 2.83 2.61 2.61 2.83 2.89 2.89


ib µ = 6

H r = 1

H r = 3, 5

N 0a = 0 0a = 1 0a = 2 0a = 3 0a = 1 0a = 2 0a = 3

3 0.336 0.33 0.36 0.377 0.33 0.4 0.434

5 0.365 0.366 0.36 0.377 0.366 0.4 0.434

7 0.373 0.365 0.378 0.377 0.365 0.4 0.434

10 0.332 0.34 0.394 0.394 0.34 0.427 0.427

15 0.271 0.284 0.369 0.369 0.284 0.394 0.394

20 0.243 0.263 0.345 0.345 0.263 0.369 0.369


ia

µ = 8

H r = 1

H r = 3

H r = 5

N 0a = 0

0a = 1 0a = 2

0a = 3 0a = 1

0a = 2 0a = 3

0a = 1 0a = 2

0a = 3

3 6.27 5.065 4.22 3.83 5.065 4.35 4.05 5.3 4.53 4.234

5 5.92 5.065 4.22 3.83 5.065 4.35 4.05 5.3 4.53 4.234

7 5.75 4.98 4.24 3.87 4.98 4.31 4.09 5.11 4.58 4.31

10 5.45 4.78 4.09 3.81 4.78 4.225 4.08 4.78 4.51 4.28

15 4.94 4.32 3.61 3.61 4.32 3.81 3.81 4.32 4.09 4.09

20 4.49 3.98 3.56 3.56 3.98 3.73 3.73 3.98 3.98 3.98


99


ia

µ = 8

H r = 1

H r = 3

H r = 5

N 0a = 0

0a = 1 0a = 2

0a = 3 0a = 1

0a = 2 0a = 3

0a = 1 0a = 2

0a = 3

3 0.393 0.41 0.408 0.42 0.41 0.446 0.463 0.436 0.476 0.501

5 0.427 0.41 0.408 0.42 0.41 0.446 0.463 0.436 0.476 0.501

7 0.438 0.425 0.438 0.45 0.425 0.465 0.473 0.452 0.487 0.514

10 0.451 0.44 0.456 0.474 0.44 0.482 0.494 0.44 0.5 0.515

15 0.425 0.4 0.461 0.461 0.4 0.461 0.461 0.4 0.503 0.503

20 0.39 0.377 0.408 0.408 0.377 0.437 0.437 0.377 0.454 0.454

Figure 5-12: Correlation between Eq. (5-5) and averaged numerical results for strength reduction

factors of MDOF soil-structure systems ( H r = 3)

Rµ

0

0.5

1

1.5

2

2.5

0 1 2 3

a0 = 1, µ = 2

0

1

2

3

4

5

0 1 2 3

a0 = 1, µ = 4

0

2

4

6

8

0 1 2 3

a0 = 1, µ = 6

Rµ

0

0.5

1

1.5

2

2.5

0 1 2 3

a0 = 3, µ = 2

0

1

2

3

4

0 1 2 3

a0 = 3, µ = 4

0

2

4

6

0 1 2 3

a0 = 3, µ = 6

Tfix Tfix Tfix

N= 5 N= 5, Eq. (5-5) N = 20 N = 20, Eq. 5-5)


100

5.7 CONCUSIONS

An intensive parametric study has been performed to investigate the effect of SSI on

strength reduction factor of E-SDOF and MDOF fixed-base and soil-structure systems. The

results of this study are summarized in the following:

Strength reduction factor spectra of E-SDOF systems for shear-buildings with

uniformly distributed structural mass along the height are independent of the number

of stories.

In E-SDOF systems SSI effect is always accompanied by decreasing in values of R .

Using R of fixed-base systems leads to significant underestimation of inelastic

strength demands of soil-structure systems.

Except for E-SDOF systems with very short periods, increasing the aspect ratio is

always accompanied by decreasing in the values of R , which is more pronounced

for the cases with significant SSI effect and long vibration periods.

For MDOF fixed-base systems, regardless of the level of nonlinearity, increasing the

number of DOFs (stories) always reduces the averaged values of R . This

phenomenon is more pronounced for low- to mid-rise buildings. However, for soil-

structure systems, as the SSI effect becomes more significant, R spectra become

less sensitive to the number of stories, especially in the low inelastic response range.

With severe SSI effect the R values of E-SDOF systems are significantly lower

than those of the MDOF systems in almost all ranges of periods. The MDOF

modifying factors for strength reduction factors of soil-structure systems could be

completely different from those of fixed-base systems. The more significant is the

SSI effect, the more difference between the elastic strength demands of MDOF and

SDOF systems. The phenomenon is more pronounced as aspect ratio ( H r )

increases. A new modification factor ( MR ) for soil-structure and fixed-base systems

that account for both elastic and inelastic strength demands has been introduced.

MDOF modification factor values are sensitive to the level of nonlinearity for soil-

structure systems such that they increase with ductility ratio and are generally larger

than “1” especially for structures with long periods and severe SSI effects.


101

SSI effects on strength reduction factors of MDOF systems become more important

in squat low- and mid-rise building and less important as the number of stories

increases such that for the case of slender high-rise buildings they can be

approximately considered as independent of the dimensionless frequency.

For the case of less SSI effect, the values of averaged R are insensitive to the

variation of aspect ratio of MDOF soil-structure systems but very sensitive to the

aspect ratio of E-SDOF systems. For the case with severe SSI effect and high

inelastic response, except for short period ranges, the values of mean R increase

with the aspect ratio, which is completely different from the E-SDOF results in

which increasing the aspect ratio is always accompanied by decreasing the R values,

indicating the SSI can affect strength reduction factors of MDOF and E-SDOF

systems in a different manner.

A new simplified equation which is functions of fixed-base fundamental period,

ductility ratio, the number of stories, aspect ratio and dimensionless frequency is

proposed to estimate the strength reduction factors of MDOF soil-structure systems.


102

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105

Chapter 6

A PARAMTERIC STUDY ON EVALUATION OF DUCTILITY DEMAND

DISTRIBUTION IN MDOF SHEAR BUILDINGS CONSIDERING SSI EFFECTS

6.1 INTRODUCTION

In the first part of this thesis (i.e., Chapters 3-5), effect of SSI on global (total) strength and

ductility demand of MDOF and the corresponding E-SDOF systems have been extensively

investigated. The results of these investigations can be beneficial for performance-based

earthquake resistant design through controlling the amount of total (global) structural

damage based on an appropriate selection of total stiffness, strength, and ductility

requirements in the conceptual seismic design. However, the distribution of damage, which

is mainly caused by lateral force redistribution characteristics of inelastic structural

responses, is not controlled. In fact, structures with inappropriate distributions of story

strength and stiffness have performed poorly in recent earthquake events that most of the

observed collapses have been related, somewhat, to inappropriate distribution of story

strength and stiffness along the height of the structures. It is also well recognized that

inappropriate strength and stiffness distributions could be responsible for a deficient

structural behaviour such as concentrated drift or ductility demand (damage) in some stories.

Therefore, damage distributions along the building height will be extensively investigated

and addressed for soil-structure systems in the second part of the thesis (Chapters 6-8).

Nearly most of the seismic design procedures in current major seismic codes for regular

structures in the world are mainly based on elastic structural response analyses under

seismic lateral forces and account for inelastic behavior in an indirect manner. As stated in

Chapter 1, the shape of these lateral load patterns along the height of structures from various


106

standards such as Euro Code-8 (CEN, 2003), Mexico City Building Code (Mexico, 2003),

Uniform Building Code (UBC, 1997), NEHRP 2003 (BSSC, 2003), ASCE/SEI 7-05 (ASCE,

2005), Australian Seismic code (AS-1170.4, 2007) and International Building Code, IBC

2009 (ICC, 2009) depends on the fundamental period of the structures and their mass. They

are derived primarily based on elastic dynamic analysis of fixed-base structures without

considering soil-structure interaction (SSI) effect. In the United States, the current code-

specified seismic design procedures are mainly based on the NEHRP Recommended

Provisions published in 2003 (BSSC, 2003). It should be mentioned that the seismic design

criteria in ASCE/SEI 7-05 (ASCE, 2005), exclusively based on the NEHRP 2003 (BSSC,

2003), is also adopted in IBC-2009 (ICC, 2009) for minimum design load criteria. The

seismic lateral load patterns in all aforementioned provisions are based on the assumption

that the soil beneath the structure is rigid, and hence the influence of SSI effect on load

pattern is not considered. The efficiency of using the code-specified lateral load patterns for

fixed-base building structures have been investigated during the past two decades (Anderson

et al., 1991; Gilmore and Bertero, 1993; Chopra, 1995, Moghaddam and Mohammadi, 2006,;

Ganjavi et al., 2008, Hajirasouliha and Moghaddam, 2009). However, all researches have

been concentrated on the different types of structures with rigid foundation, i.e., without

considering SSI effects. In fact, it is necessary to clarify the influence of structural

characteristics distributions on the local (story) ductility demands (damage) when SSI is to

be considered. This is because the pattern of local plastic deformation is definitely

influenced by soil characteristics as well as the distribution of stiffness and strength along

the building height. Here, in this chapter a comprehensive parametric study has been

performed to investigate the effect of SSI on height-wise distribution of ductility demands in

shear-building structures with different structural properties. Effect of many parameters

including fundamental period, level of inelastic behavior, number of stories, damping model,

damping ratio, structural strain hardening, earthquake excitation, level of soil flexibility,


investigated. In addition, the adequacy of three different code-compliant lateral loading

patterns including UBC-97, IBC-2009 and EuroCode-8 as well as three recently proposed

optimum loading patterns for fixed-base structures are parametrically investigated for soil-

structure systems by two methods associated to the economy of the seismic-resistant system.


107

6.2 LATERAL LOADING PATTERNS

To investigate the adequacy of different loading patterns for soil-structure systems, various

patterns including code-specified seismic design load patterns of IBC-2009, UBC-97 and

EuroCode-8, as well as those recently proposed by researchers are considered.

6.2.1 Code-specified seismic design lateral load patterns

In this case, three current code-compliant lateral load patterns including IBC-2009 (ICC,

2099), UBC-97 and Euro Code-8 (CEN, 2003) are considered. The general formula and

distribution patterns of theses selected codes have been already discussed and addressed in

Chapters 4 of the thesis. As stated in the literature, generally, the seismic design load

patterns from various standards are mainly a function of the fundamental period of the

structures and their mass, which are only based on elastic dynamic analysis concepts, and,

hence, the level of inelastic behavior and soil flexibility (SSI effect) are not accounted for in

the distribution of lateral loads.

6.2.2 Lateral load pattern proposed by Mohammadi et al. (2004)

Mohammadi et al.(2004), based on the nonlinear dynamic analyses on fixed-base shear

building models subjected to 21 earthquake ground motions, introduced a new lateral load

pattern as a function of the fundamental period of the structure and target ductility. Their

proposed pattern is a rectangular pattern accompanied by a concentrated force λTV at the top

floor, where λ is a coefficient depending on the fundamental period T and the target ductility

μ, and is defined as:

(0.6 0.03 )(0.9 0.04 ).

Te

(6-1)

6.2.3 Lateral load pattern proposed by Park and Medina (2007)

Park and Medina (2007), based on the nonlinear dynamic analyses on fixed-base regular

moment-resisting frames subjected to 40 far-field ground motions from California

earthquakes recorded on stiff-soil sites, proposed a new lateral load pattern which is


108

consistent, in format, with the lateral load distributions of IBC 2009 and UBC 97. They

concluded that the proposed approach provides, on average, a more uniform distribution of

story ductility ratios and story drift ratios, when compared to the distributions obtained using

current seismic code provisions, i.e., the 2006 IBC which is the same as IBC 2009. Their

proposed lateral load pattern is given by the following expression:

1

.( ) k

x xx topn

k

i i

i

w hF V F

w h

(6-2)

where topF and k are respectively consistent, in definition, with tF and exponent k in UBC

97and IBC 2009 which are defined as:

0.32 0.001 0.13 22 66topF

H k HV

(6-3)

0.56 0.17 1 5 tk

(6-4)

In fact, topF is an additional force applied to the top (roof) story to incorporate higher mode

effects, and t is the target global ductility demand of the structure.

6.2.4 Lateral load pattern proposed by Hajirasouliha and Moghaddam (2009)

Following the research carried out by Mohammadi et al. (2004), Hajirasouliha and

Moghaddam (2009), based on the nonlinear dynamic analyses on fixed-base shear building

models subjected to 15 earthquake ground motions recorded on alluvium soil, proposed a

new lateral load pattern as a function of the fundamental period of the structure and target

ductility which is defined as:

( ) ( ) i i

i i i t

c T dF a T b

(6-5)


109

where iF is the optimum load component at the ith story; T is the fundamental period of the

structure; t

is the target ductility demand; and ia , ib , ic and id are constant coefficients at

the ith story. These coefficients can be obtained at each level of the structure by

interpolating the values given in their paper.

As reviewed above, none of the above load patterns explicitly considered the influences of

SSI. In the present study, the adequacy of all the above-mentioned lateral load patterns on

height wise distribution of ductility demand in soil-structure systems is investigated and

discussed. Figure 6-1 illustrates a comparison of the above-mentioned lateral force and

normalized shear strength patterns for a 10-story building with fixT = 1.5 sec and μ= 4.

Figure 6-1: Different Lateral force and normalized shear strength patterns for 10-story building

with fixT = 1.5 sec and μ= 4

6.3 ANALYSIS PROCEDURE

A series of 5-, 10-, 15- and 20-story shear buildings are considered to investigate the effect

of SSI on height-wise distribution of ductility demand in shear-building structures with

different structural properties. In this regard, for a given earthquake ground motion, a family

1

2

3

4

5

6

7

8

9

10

11

0 0.2 0.4 0.6

Sto

ry

1

2

3

4

5

6

7

8

9

10

11

0 0.2 0.4 0.6 0.8 1

Normalized Shear Strength Lateral Force / Base Shear

Park and Medina Mohammadi et al. Hajirasouliha and Moghaddam

EuroCode-8 UBC-97 IBC-2009


110

of 7200 different MDOF soil-structure models including various predefined key parameters

are considered. This includes MDOF models of four different number of stories (N= 5, 10,

15 and 20) with 30 fundamental periods of fixed-base structures, ranging from 0.1 to 3 sec

with intervals of 0.1, three values of aspect ratio ( H r =1, 3, 5), three values of

dimensionless frequency ( 0a =1, 2, 3) as well as the fixed-base model, four values of target

interstory displacement ductility ratio ( t = 1, 2, 4, 6) where t =1 corresponds to the elastic

state, and six different lateral loading patterns. For each earthquake ground motion, ductility

demand distribution pattern along the height of the structure is computed by the iterative

procedure proposed in previous chapters in order to reach the t in the structure, as a part of

the soil–structure system, within a 0.5% error. OPTSSI computer program written

specifically for this dissertation is utilized here to compute the ductility demand distribution

pattern along the height of the structure. A large family of 30 earthquake ground motions

with different characteristics recorded on alluvium and soft soil deposits, as utilized in

Chapter 5, are considered. The main parameters of the selected ground motions are given in

Table 5-1.

6.4 EVALUATION OF DUCTILITY DEMAND DISTRIBUTION IN SHEAR-

BUILDING STRUCTURES CONSIDERING SSI EFFECT

A family group of MDOF shear-building structures with different number of stories and

characteristics mentioned earlier are analyzed to comprehensively investigate the influence

of SSI on height-wise distribution of ductility demand in MDOF systems subjected to an

ensemble of 30 earthquake ground motions listed in Table 5-1. Results of this section are

based on IBC-2009 code-specified lateral loading pattern and the adequacy of other load

patterns will be investigated in the following parts.

6.4.1 Effect of number of stories

To study the effect of number of stories on height-wise distribution of ductility demand for

fixed-base and flexible-base structures, shear buildings of 5, 10, 15 and 20 stories are


111

considered. They are selected as representatives of the common building structures from

low- to high-rise models. Results illustrated in Figure 6-2 are mean values of 30 earthquake

ground motions for systems with H r = 3, fixT = 1 sec, and with three ductility ratios ( t =

1,2, 6) as representatives of respectively elastic, low and high inelastic behaviours as well as

for two dimensionless frequencies 0a = 1 and 3 in comparison with fixed-base structures.

The abscissa in all figures is the averaged ductility demands and the vertical axis is relative

height of the structure. The results generally exhibit the same trend for all MDOF buildings

with different number of stories such that the maximum ductility demand usually happens in

the top story (roof). It can be seen that, however, as the number of stories increases, height-

wise distribution of the ductility demand becomes more non-uniform, indicating the

significance of higher-mode effects on height-wise distribution of seismic demands. This

trend is intensified for the case of higher inelastic behavior for both fixed-base and flexible-

base systems. Therefore, it may be concluded that IBC-2009 code-specified load pattern

which is fundamentally based on elastic fixed-base structures may not effectively

incorporate higher-mode effect. It is also worth mentioning that the ductility demands

distributions become more non-uniform when SSI effect becomes prominent ( 0a = 3), which

will be discussed in detail in the next sections. It should be noted that, the averaged

maximum ductility ratios, in some cases, may not be exactly close to the target one. This is

because the maximum ductility ratio depends on a given earthquake ground motion, and it

may happen in different stories.

6.4.2 Effect of fundamental period

To investigate the effect of fundamental period on the height-wise distribution of averaged

ductility demand of all the 30 ground motions, a 10-story shear building model of aspect

ratio H r = 3 without or with SSI effect are considered. The results corresponding to three

target ductility demands ( t = 1, 2, 6), four fundamental periods of 0.2, 0.5, 1 and 3 sec, as

well as two dimensionless frequencies 0a = 1 and 3 in comparison with the corresponding

fixed-base structures are plotted in Figure 6-3. As seen, the ductility demand distribution is

strongly dependent on the fundamental period of the structure especially for the case of

severe SSI effect. In elastic state, ductility demand distributions for cases of fixed-base and


112

less SSI-effect models i.e., 0a = 1, with short period (i.e., rigid structure) are nearly uniform

and thus have optimal performance. However, by increasing the soil flexibility and thus

increasing the fundamental period of the soil-structure system, the height-wise distribution

of the ductility demand becomes non-uniform. In inelastic response, height-wise distribution

of ductility demands for all ranges of period are no-uniform, which are intensified by

increasing the level of inelasticity as well as the SSI effect.

Figure 6-2: Effect of number of stories on height-wise distribution of averaged ductility demand

for systems with fixT = 1 and H r =3

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1

μ =1

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1

μ =1

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1

μ =1

Rel

ativ

e H

eigh

t

0.0

0.2

0.4

0.6

0.8

1.0

0 0.5 1 1.5 2

μ =2

0.0

0.2

0.4

0.6

0.8

1.0

0 0.5 1 1.5 2

μ =2

0.0

0.2

0.4

0.6

0.8

1.0

0 0.5 1 1.5 2

μ =2

Rel

ativ

e H

eigh

t

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 6

μ =6

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 6

μ =6

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 6

μ =6

Rel

ativ

e H

eigh

t


μ μ μ

N= 5 N= 10 N = 15 N = 20


113

Figure 6-3: Effect of fundamental period on height-wise distribution of averaged ductility

demand for systems with N= 10 and H r =3

6.4. 3 Effect of aspect ratio and dimensionless frequency

Figure 6-4 shows the effect of aspect ratio and dimensionless frequency, 0a , on averaged

ductility demand distribution along the height of the soil-structure systems. As stated before,

aspect ratio and dimensionless frequency are two key parameters that affect the response of

the soil-structure systems subjected to earthquake excitation. The results are plotted for the

10-story shear building with fundamental period of 1sec, corresponding to three target

ductility demands ( t = 1, 2, 6), three values of aspect ratio ( H r =1, 3, 5), representing

respectively squat, average and slender building, three values of dimensionless frequency


μ μ μ

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1

μ =1

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1

μ =1

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1

μ =1 R

elat

ive

Hei

ght

0.0

0.2

0.4

0.6

0.8

1.0

0 0.5 1 1.5 2

μ =2

Rel

ativ

e H

eigh

t

0.0

0.2

0.4

0.6

0.8

1.0

0 0.5 1 1.5 2

μ =2

0.0

0.2

0.4

0.6

0.8

1.0

0 0.5 1 1.5 2

μ =2

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

μ =6

Rel

ativ

e H

eigh

t

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

μ =6

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

μ =6

Tfix = 0.2 Tfix = 0.5 Tfix = 1 Tfix = 3


114

( 0a =1, 2, 3), as well as the corresponding fixed-base models. It can be observed that SSI

effect becomes more significant as the aspect ratio increases, i.e., for the case of slender

buildings. The trend is less pronounced as the level of inelasticity increases, but is still

significant in all the cases considered in the current study, leading to more non-uniform

distributions of ductility demand along the height of the structure as compared to the

corresponding fixed-base model.

Figure 6-4: Effect of aspect ratio and dimensionless frequency on height-wise distribution of

averaged ductility demand for systems with N= 10 and fixT =1

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1

μ =1

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1

μ =1

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1

μ =1

Rel

ativ

e H

eigh

t

0.0

0.2

0.4

0.6

0.8

1.0

0 0.5 1 1.5 2

μ =2

0.0

0.2

0.4

0.6

0.8

1.0

0 0.5 1 1.5 2

μ =2

0.0

0.2

0.4

0.6

0.8

1.0

0 0.5 1 1.5 2

μ =2

Rel

ativ

e H

eigh

t

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

μ =6

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

μ =6

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

μ =6

Rel

ativ

e H

eigh

t

Fixed base a0 a0 =2 a0 =3

= 1 = 3 = 5

μ μ μ


115

6.4.4 Effect of damping model

In order to examine the effect of damping models on height-wise distribution of ductility

demand in fixed- and flexible-base structures, three conventional viscous damping models

including stiffness-proportional damping, mass-proportional damping and Rayleigh-type

damping in which damping matrix is composed of the superposition of a mass-proportional

damping term and a stiffness-proportional damping term are considered. In this case, story

ductility demands of the 10-story shear building with fundamental periods of 0.5 and 2 sec,

target ductility ratio of 4, aspect ratio of 3, and two dimensionless frequencies ( 0a = 1 and 3)

as well as the corresponding fixed-base structure subjected to Loma Prieta earthquake

(APEEL 2 - Redwood City) are computed and plotted in Figure 6-5. It can be seen that

while there is no significant difference between the results of mass-proportional and

Rayleigh-type damping models, the difference is pronounced when compared to those of the

stiffness-proportional damping model. In fact, since the story stiffness of shear-building

structures are modeled by only elasto-plastic spring, in the higher modes in which the

structure has shorter period and hence greater structural stiffness, consequently large

amounts of viscous energy may be absorbed after yielding which is unrealistic. This

observation indicates that stiffness-proportional damping model may not lead to reliable

predictions of structural responses. It is always advisable to use Rayleigh-type damping

model to better incorporate the effect of higher modes after yielding. The best option could

be tangent stiffness-based Rayleigh damping, with the stiffness coefficients being updated

regularly. However, the implementation of this method may cause numerical solution

unstable once significant changes in stiffness values take place. Besides, this method is also

computationally more expensive than that in which the initial stiffness matrix is used.

6.5. Effect of structural damping ratio

The effect of structural damping ratio ( st ) on height-wise distribution of ductility demand

is illustrated in Figure 6-6 for a 10-story shear-building structure with a target ductility

demand of 4, fundamental period of 1 sec, aspect ratio of 3, and for two dimensionless

frequencies 0a = 1 and 3 in addition to the fixed-base structure subjected to Loma Prieta

earthquake (APEEL 2 - Redwood City). As seen, except for the case of 10% damping ratio

which is more obvious for the soil-structure systems with respect to the fixed-base model,


116

height-wise distribution of the ductility demand may be considered as somewhat insensitive

to the variations of damping ratio.

Figure 6-5: Effect of damping model on height-wise distribution of ductility demand for systems

with N= 10, µ = 4 and H r =3subjected to Loma Prieta earthquake (APEEL 2 - Redwood City)

Figure 6-6: Effect of damping ratio on height-wise distribution of ductility demand for systems

with N= 10, fixT = 1.5, µ = 4 and H r =3 subjected to Loma Prieta earthquake (APEEL 2 -

Redwood City)

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

Tfix = 0.5

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

Tfix = 0.5

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

Tfix= 0.5

Rel

ativ

e H

eigh

t

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

Tfix= 2

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

Tfix= 2

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

Tfix= 2

Rel

ativ

e H

eigh

t

Mass Stiffness Rayleigh

Fixed base a0 = 1 a0 = 3

μ μ μ

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

μ= 4

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

μ= 4

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

μ= 4

ξst= 0% ξst= 2%

10

ξst= 5% ξst= 10%


μ μ μ

Rel

ativ

e H

eigh

t


117

6.4.6 Effect of structural strain hardening

Effect of different structural strain hardening (SH) values on the ductility demand

distribution along the height of the structure for both fixed-base and flexible-base structure

is presented in Figure 6-7. The results are plotted in the same format as Figure 6-6 and for

the same earthquake ground motion record. It can be seen that, with exception of the case

with a large value of strain hardening (SH= 10%) in which the shape of ductility demands

distribution along the height of the structure is to some extent different from that of the less

amounts of SH, the story ductility demand distribution pattern is not significantly dependent

on the secondary slope of post-yield response for the most practical cases.

6.4.7 Effect of earthquake excitation

To investigate the effect of the ground motion variability on height-wise distribution of

ductility demand for both fixed-base and flexible-base shear-building models, individual

results of all 30 earthquake ground motions listed in Table 1 along with their average values

for a 15- story shear-building structure are presented in Figure 6-8. The results are for

systems with target ductility demand of 4, fundamental period of 1.5 sec, aspect ratio of 3,

and for two dimensionless frequencies ( 0a = 1, 3) as well as the fixed-base structure. As

seen, it is obvious that the height-wise distribution of ductility demand in some cases can be

remarkably sensitive to the earthquake ground motion excitations in both fixed-base and

flexible-base models, and it may vary from one earthquake to another. However, in most

ground motions used in this study, there is not a big discrepancy in the general pattern of the

ductility demand distribution when compared to the corresponding averaged pattern.


118

Figure 6-7: Effect of strain hardening on height-wise distribution of ductility demand for

systems with N= 10, fixT = 1.5, µ = 4 and H r =3 subjected to Loma Prieta earthquake (APEEL

2 - Redwood City)

Figure 6-8: Height-wise distribution of individual and averaged ductility demand for systems

with N= 15, H r =3, fixT =1.5 and µ = 6

6.5 VALIDATION OF THE NUMERICAL RESULTS

As stated in Chapter 2, sub-structure method has been used in this study to model soil-

structure systems in which the soil-foundation element has been represented by an

equivalent linear discrete model based on the cone model (Wolf, 1994). During the past

decade, this method has been extensively used by researchers to investigate the elastic and


μ μ μ

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

μ= 4

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

μ= 4

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

μ= 4

Rel

ativ

e H

eigh

t

SH= 0% SH= 2%

10

SH= 5% SH= 10%

μ μ μ

Rel

ativ

e H

eigh

t

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 6

μ = 6

0.0

0.2

0.4

0.6

0.8

1.0

0 0.5 1 1.5 2

μ = 2

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1

μ =1

Individual Average


119

inelastic response of soil-structure systems subjected to earthquake ground motions

(Ghannad and Jahankhah, 2007; Nakhaei and Ghannad, 2008; Mahsuli and Ghannad, 2009;

Ganjavi and Hao, 2011; Moghaddasi et al., 2011a and 2011b). In a more recent study,

however, Grange et al. (2011) using a recently developed macro-element with consideration

of both material and geometrical nonlinearities and by considering two synthetic ground

motion records having PGA of 0.35g and 0.7g representing respectively weak and strong

earthquakes, as well as two types of soils with shear wave velocities of 360 and 150 m/s

investigated the effects of nonlinear soil-structure interaction on reinforced concrete viaduct.

They concluded that utilizing the equivalent linear model to reflect nonlinear SSI effects

may lead to erroneous results (i.e., conservative) for strong ground motions with high PGA,

which is more pronounced for softer soil (i.e., sv = 150 m/s). Although, all far-field

earthquake ground motions used in the present study have PGA less than 0.47g, and 27 out

of 30 ground motion components have PGA less than 0.35g that would be considered as

nearly weak or average ground motions based on classification made by Grange et al.,

(2011), to validate the numerical results of this study the recently developed macro-element

to simulate dynamic soil-structure interaction for shallow foundation by Grange et al.,

(2009) is applied to the 10-story shear building with fixT =1 sec. For this purpose, a soil with

shear wave velocity of 100 m/s and shear modulus of 19 MPa has been considered. The

detailed characteristics of the selected soil such as soil stiffness and damping, friction angle

and ultimate bearing stress can be found in reference (Grange et al., 2009).

The selected macro-element model has the capability of taking into account the plasticity of

the soil, the uplift of the foundation, P effects and the radiation damping. In this model

the foundation is assumed infinitely rigid while its movement can be described with

generalized forces and displacements at the centre of the foundation. The general

formulation of the selected macro-element can be found in (Grange et al., 2009; Chatzigogos

et al., 2009). The foundation dimensions are selected such that it represents a slender

building ( H r = 5) for the 10-story building which is more critical for severe SSI effects. To

compare the results of this study i.e., equivalent linear discrete model (EL) with those of

macro-element (ME), the stiffness of the linear springs in equivalent linear elastic model

needs to be calibrated using energy criterion such that they accumulate the same energy as


120

the nonlinear SSI macro-element (see Figure 6-9). However, the dissipated energy in the

hysteretic loop of the macro-element has not been considered (Grange et al., 2011). Based

on this assumption, the dimensionless frequency ( 0a ) of the cone model for this example

and the selected soil is approximately equal to 2. Figure 6-10 shows comparisons of the

height-wise distribution of ductility demands resulted from the two SSI models for three

different PGAs of 0.3g, 0.5g and 0.7g, representing respectively weak, average and strong

earthquakes corresponding to two values of target ductility demand (µ= 2, 6). Results are

based on the average of the first 10 earthquake ground motions listed in Table 5-1. As seen,

the mean ductility demand distributions over the height of the structure are nearly coincident

for the cases of weak and average ground motion PGA (i.e., PGA= 0.3g, 0.5g).

Nevertheless, regardless of the value of target ductility demand, by increasing the

earthquake intensity the difference between the ductility demand profiles of macro-element

and equivalent linear elastic model increases and becomes pronounced for the case of strong

ground motion with PGA= 0.7g. Generally, as PGA increases, the mean ductility demands

of top stories for equivalent elastic linear model increase with respect to the nonlinear

macro-element model especially for the case with high PGA. However, the general shapes

of the ductility demand distributions over the height of the structure for two aforementioned

models are nearly similar. It is concluded that using the linear elastic model to take into

account the SSI effects during severe earthquake ground motions with high PGA may result

in a conservative ductility demand for top stories. However, when PGA is less than 0.5g,

both models lead to almost identical results. This observation is generally the same as that

made in (Grange et al., 2011). Considering the fact that all ground motion components

considered in this study have PGAs less than 0.5g, therefore, for practical purpose, the

equivalent linear elastic model can be used to estimate the ductility demand distribution

along the height of the structure with SSI effects.


121

Figure 6-9: Calibrating the stiffness of the elastic linear springs presented by Grange et al.,

(2011).

Figure 6-10: Comparisons of the ductility demand distributions resulted from nonlinear macro-

element and equivalent linear elastic (cone) models for two levels of nonlinearity (µ= 2, 6); 10-

story building with fixT =1 sec, H r =5 (Average of 10 earthquake records).

M

θ

Macro-element Linear spring

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6

µ= 6, PGA= 0.3

EL

ME

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6

µ= 6, PGA= 0.5

EL

ME

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6

µ= 6, PGA= 0.7

EL

ME

Sto

ry

1

2

3

4

5

6

7

8

9

10

0 0.5 1 1.5 2

µ= 2, PGA= 0.3

EL

ME

1

2

3

4

5

6

7

8

9

10

0 0.5 1 1.5 2

µ= 2, PGA= 0.5

EL

ME

1

2

3

4

5

6

7

8

9

10

0 0.5 1 1.5 2

µ= 2, PGA= 0.7

EL

ME

Sto

ry

µ µ µ


122

6.6 ADEQUACY OF IBC-2009 CODE-SPECIFIED LATERAL LOADING

PATTERN

It is believed that the coefficient of variation (COV) of ductility demand distribution along

the building height could be used for assessing the adequacy of design load patterns to

optimize the use of material (Mohammadi et al., 2004). This is because the more uniform

the ductility demand distribution, the better is the seismic performance of the structure. As

stated before, the COV is a statistical measure of the dispersion of data points, here the

ductility demand ratio along the building height. It is defined as the ratio of the ductility

demand standard deviation to the mean ductility demand of all stories. Mohammadi et al.

(2004) and Moghaddam and Hajirasouliha (2006) showed that the seismic loading patterns

suggested by seismic codes do not lead to a uniform distribution of ductility demand in

fixed-base shear-building structure. This founding was confirmed by Park and Medina (2007)

for fixed-base non-deteriorating moment-resisting frames and by Ganjavi et al. (2008) for

fixed-base reinforced concrete frames. However, all these studies considered soil beneath

the structure as rigid without considering the SSI effect. In this section, the influence of SSI

effect on COV of ductility demands along the building height is investigated for shear-

building structures designed according to IBC-2009 code-specified load pattern.

To investigate the efficiency of IBC-2009 load patterns in seismic performance of the soil-

structure systems in comparison to the fixed-base one, 5- , 10-, 15 and 20-story shear-

building models with 30 different fundamental periods, 4 ductility demands ( t = 1, 2, 4, 6),

3 dimensionless frequencies ( 0a = 1, 2, 3), and 3 aspect ratios ( H r =1, 3, 5) are subjected

to the 30 selected ground motions listed in Table 5-1. In each case, shear strength and

stiffness are distributed along the stories according to IBC-2009 code-specified load pattern.

The total structural stiffness then is scaled to adjust the fixed-base fundamental period. With

the iterative procedure and without altering the stiffness and strength distribution pattern, the

total shear strength of the structure is scaled until the target ductility ratio is resulted with

less than 0.5% error. Consequently, COV of the story ductility demands can be calculated

for each case.


123

6.6.1 Effect of number of stories and target ductility demand

Effect of number of stories on COV of story ductility demands in both fixed-base and

flexible-base structures designed according to the IBC-2009 load pattern is presented in Fig.

6-11, where results for structural models with H r = 3, three ductility ratios ( t = 1, 2, 6),

dimensionless frequency of 2, and the fixed-base structures are shown. The vertical axis in

all figures is the averaged COV of story ductility demands and the horizontal axis is the

fixed-base fundamental period of the structure. It can be seen that for both fixed-base and

flexible-base structures, regardless of the number of stories, using IBC-2009 load pattern

leads to nearly uniform ductility demands distribution for the structure with short period

within the elastic range, i.e., a rather small COV value. For the longer periods i.e.,

0.4fixT sec, however, the efficiency of the IBC-2009 load pattern is reduced as the number

of stories and fundamental period increase, which could be interpreted as the effect of

higher modes that has not been considered sufficiently in the IBC-2009 load pattern. In

inelastic response, the performance of the structures designed according to IBC-2009 load

pattern becomes worse with a significant increase in COV of ductility demand even in short

period. The situation becomes more pronounced by increasing the inelastic level of vibration,

especially for the fixed-base structures. It is interesting that in low level of inelastic response

(i.e., t = 2), seismic performance of the structure is better (i.e. smaller COV) by increasing

the number of stories for the structures with short period. This is probably because the

structural response is governed primarily by the fundamental vibration mode in this short

period range; hence the ductility demands along the building height also follow the

fundamental mode pattern. With an increased number of stories, the standard deviation of

COV reduces because of more number of stories to normalize the variations while the

variations are somewhat similar, hence the COV is also reduced accordingly.

To better investigate the effect of target ductility demands on the averaged COV of story

ductility demands, Figure 6-12 is provided. The results are for a 15-story shear building with

H r = 3, and with four ductility ratios ( t = 1, 2, 4, 6) as well as for two dimensionless

frequencies ( 0a = 1, 2) in comparison with the fixed-base structures. It is observed that for

both fixed-base and flexible-base models increasing the target ductility ratio is always

accompanied by an increase in COV of story ductility demands, which is compatible with


124

the results of the study carried out for fixed-base shear-building systems by Hajirasouliha

and Moghaddam (2009). For the fixed-base structures within a large inelastic response

range of vibration the COV of story ductility demands for the structures having short period

is significantly greater than that of the long periods while by increasing the soil flexibility

(i.e. increasing the SSI effect) this phenomenon will be reversed. The reason is that the

fundamental period of soil-structure systems always increases with soil flexibility, which

makes the fundamental period of soil-structure system greater than that of the fixed-base

model. It is also seen that for both fixed-base and flexible-base models with long period of

vibration, COV of story ductility demands is more dependent on the level of inelasticity (i.e.

target ductility demand value) than the fundamental period of the building. As stated before,

however, nearly all code-specified seismic load patterns do not consider the target ductility

demand of the structure.

Figure 6-11: Effect of number of stories on averaged COV of story ductility demands for

systems with H r =3 designed according to IBC-2009 load pattern.

N= 5 N= 10 N = 15 N = 20

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3

Fixed Base

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3

Fixed base

0

0.2

0.4

0.6

0.8

1

0 1 2 3

Fixed base

CO

V

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3

a0 = 2

0

0.2

0.4

0.6

0.8

0 1 2 3

a0 = 2

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3

a0 =2

CO

V

µ = 1 µ = 2 µ = 6

Tfix Tfix Tfix


125

Figure 6-12: Effect of maximum ductility on averaged COV of story ductility demands for

systems with H r =3 designed according to IBC-2009 load pattern.

6.6.2 Effect of dimensionless frequency and aspect ratio

Here, the effects of two key parameters: dimensionless frequency and aspect ratio on the

COV of story ductility demands of soil-structure systems are investigated. Figure 6-13

shows the effect of dimensionless frequency (i.e. soil flexibility) by illustrating the average

of COV obtained with 30 earthquake ground motions versus fixed-base fundamental period

for a 15-story shear-building with H r = 3 and various target ductility demands. It is

observed that, with exception in the very short period range, the COV of story ductility

demands increases with soil flexibility, leading to reduction of the seismic performance of

the soil-structure systems with respect to the fixed-base ones. This trend is less prominent

as the level of inelastic behavior increases but still significant. The effect of aspect ratio is

also presented in Figure 6-14 with the same format as Figure 6-13 for the case of severe SSI

effect (i.e. 0a = 3). As seen, the results can be classified into two parts; first, the set of

curves associated with elastic and low level of inelastic responses (i.e., t = 1 and 2,

respectively), in which, except for very short periods, the COV of ductility demands

increases with the aspect ratio; second, the curves corresponding to the high level of

inelastic behaviour ( t = 6) in which the COV decreases with aspect ratio in short range of

periods; afterwards, this trend is reversed, i.e., the COV increases with the aspect ratio.

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3

N = 15

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3

N = 15

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3

N = 15

CO

V


Tfix Tfix Tfix

μ = 1 μ = 2 μ = 4 μ = 6


126

Figure 6-13: Effect of soil flexibility on averaged COV of story ductility demands for systems

with N=15 and H r =3 designed according to IBC-2009 load pattern,

Figure 6-14: Effect of aspect ratio on averaged COV of story ductility demands for systems

with N=15 and 0a =3 designed according to IBC-2009 load pattern.

6.7 ADEQUACY OF CONVENTIONAL CODE-COMPLIANT AND RECENTLY

PROPOSED LOAD PATTERNS FOR SOIL-STRUCTURE SYSTEMS

In this section the adequacy of six different lateral loading patterns, described in Section 6.2,

including 3 different code-specified and 3 recently proposed lateral loading patterns for

fixed-base structures are parametrically investigated for soil-structure systems. For this

purpose, two methods including the weight-based method (Mohammadi, 2004) and COV-

Tfix Tfix Tfix


0

0.2

0.4

0.6

0.8

0 1 2 3

μ= 1

0

0.2

0.4

0.6

0.8

1

0 1 2 3

μ= 2

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3

μ= 6

CO

V

0

0.2

0.4

0.6

0.8

0 1 2 3

μ= 1

0

0.2

0.4

0.6

0.8

1

0 1 2 3

μ= 2

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3

μ= 6

CO

V

Tfix Tfix Tfix

= 1 = 3 = 5


127

based method, both of them are related to the economy of the seismic resistant system, are

considered.

6.7.1 Weight-based method

In weight-based method, it is assumed that the weight of the lateral load-resisting system at

each story, E iW , is proportional to the story shear strength, iV (Mohammadi et al., 2004).

Therefore, the total weight of the seismic resistant system, EW , can be calculated as:

1 1 1

. . ,n n n

E Ei i i

i i i

W W V V

(6-6)

where is the proportioning coefficient. According to Eq. (6-6), Weight Index of the MDOF

structure related to the presumed lateral load pattern can be defined as:

Weight Index . EW W PGA

(6-7)

where W and PGA are total weight of the structure and peak ground acceleration,

respectively.

To assess the relative adequacy of the different loading patterns for soil-structure systems of

identical period and ductility ratio, the 10-story shear-building structure with aspect ratio

and dimensionless frequency of 3, and with 30 fundamental period as well as two ductility

ratios ( t =2, 6) are considered. The weight index of all structures designed according to the

6 aforementioned loading patterns subjected to the first 20 earthquake ground motions listed

in Table 5-1 are calculated and then the average values are compared to each other as shown

in Figure 6-15. The loading pattern that corresponds to the spectrum with minimum weight

index is considered as the most adequate loading pattern. Based on the results presented in

Figure 6-15, it can be observed that

1. Generally, in short periods range, the values of weight index associated to all three

cod-specified load patterns and the one proposed by Park and Medina (2004) are

greater than those by Mohammadi et al. (2004) and Hajirasouliha and Moghaddam

(2009). For long periods, however, structures designed according to UBC-97,


128

Mohammadi et al. (2004) and Park and Medina (2007) have less weight index

compared to those of other load patterns. As an example, for the case of severe SSI

effect ( 0a = 3) with t = 2 and fixT = 1.5 sec, the values of structural weight

associated to Parke and Medina (2004), UBC-97, IBC-2009, Hajirasouliha and

Moghaddam (2009) and EuroCode-8 load patterns are on average 17.6%, 25.4%,

58%, 60.1% and 81.3% greater than that associated to Mohammadi et al. (2004),

respectively. This shows that the lateral load pattern can significantly affect the

seismic performance of the structures with SSI effect.

2. It should be noted that the load patterns of UBC-97 and EuroCode-8 are similar for

structures having period less than or equal to 0.7, hence their results are coincident

for this range of periods; afterwards the force at the top floor for the case of UBC-97

is increased by adding a concentrated force 0.07tF TV , which makes the

structures designed based on the UBC-97 load pattern have better seismic

performance with respect to those designed based on EuroCode-8. This could be

explained by increasing the influence of higher modes as the period of vibration

increases, which is incorporated by UBC-97 code through concentrated force at top

floor.

3. For soil-structure systems the structures designed according to the optimum load

pattern proposed by Hajirasouliha and Moghaddam (2009) for fixed-base shear

buildings has the best performance for the short periods while along with EuroCode-

8, the worst seismic performance for long periods range. This implies that the

optimal load pattern proposed by Hajirasouliha and Moghaddam (2009) for fixed-

base shear-building structures significantly misses their efficiency in soil-structure

systems regardless of inelastic response ranges. This point will be better

demonstrated in the averaged COV of story ductility demands in the following part.

6.7.2 COV-Based Method

To parametrically investigate the efficiency of all aforementioned lateral loading patterns for

soil-structure systems averaged COV of story ductility demands of all structures subjected to

the same earthquake ground motions of the weight-based method are calculated and the

spectra plotted in Figure 6-16. As seen, although somewhat the same results as weight-


129

based method can be drawn from COV-based method, this figure illustrates the efficiency of

all load patterns with respect to the optimal seismic performance for the soil-structure

systems. Overall, it can be concluded that none of the load patterns lead to the optimal

performance, i.e., uniform ductility demands along the building height, for the soil-structure

systems. Increasing the soil flexibility and the ductility demands are generally accompanied

by an increase in the averaged COV of story ductility demands. As an instance, COV of all

load patterns for the case of severe SSI effect and high level of inelastic behavior is greater

than 0.5, which indicates significant non-uniform distribution of structural damage, i.e.,

ductility demand, for the soil structure systems. Finally, it is demonstrated that although the

structures designed according to some load patterns such as Mohammadi et al. (2004) and

Park and Medina (2007) may have generally better seismic performance when compared to

those designed by code-specified load patterns, their seismic performance is far from ideal if

the SSI effects are considered.


130

Figure 6-15: Averaged Weight Index of 10-story soil-structure system with H r =3 designed

according to different load patterns.

μ = 2 μ= 6

Tfix Tfix

0

2

4

6

8

10

12

0.0 0.5 1.0 1.5 2.0 2.5 3.0

a0 = 1

Wei

gh

t In

dex

0

2

4

6

8

0.0 0.5 1.0 1.5 2.0 2.5 3.0

a0 = 1

Wei

gh

t In

dex

0

2

4

6

8

10

0.0 0.5 1.0 1.5 2.0 2.5 3.0

a0 = 3

0

2

4

6

8

0.0 0.5 1.0 1.5 2.0 2.5 3.0

a0 = 3




131

Figure 6-16: Averaged COV of 10-story soil-structure system with H r =3 designed according

to different load patterns.

6.8 SUMMARY AND CONCLUSIONS

A comprehensive parametric study has been carried out to investigate the effect of SSI on

ductility demands distribution and seismic performance of shear-building structures

designed in accordance to different load patterns including code-specified design lateral load

patterns and those recently proposed by researchers for fixed-base structures. Effect of many

parameters including fundamental period, level of inelastic behavior, the number of stories,

μ = 2 μ= 6

Tfix Tfix

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.0 0.5 1.0 1.5 2.0 2.5 3.0

a0 = 1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

a0 = 1

CO

V

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0

a0 = 3

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

a0 = 3

CO

V




132

damping model, damping ratio, structural strain hardening, earthquake excitation, level of

soil flexibility, and aspect ratio on height-wise distribution of damage (ductility demand) are

extensively investigated. In addition, the adequacy of different lateral loading patterns

proposed for fixed-base structures are parametrically investigated for soil-structure systems

by two methods i.e., weight-based and COV-based methods. Results of this study can be

summarized as follows:

1. For both fixed-base and flexible-base structures, regardless of the number of stories,

using IBC-2009 load pattern leads to nearly uniform ductility demands distribution

for structures with short periods within the elastic response range. For structures

with long periods, i.e., 0.4fixT sec, however, the efficiency of the IBC-2009 load

pattern is reduced as the number of stories and fundamental period increase, because

of contributions to the responses from higher modes that has not been considered

sufficiently in the IBC-2009 load pattern. In inelastic response range, the

performance of the structures is significantly reduced even for structures with short

vibration period. The performance is even worse with increasing the inelastic level

of vibration.

2. For the fixed-base structures in the large inelastic response range, the performance

of the structures having short period is remarkably less ideal than those having long

periods. By increasing the soil flexibility (i.e. increasing the SSI effect) and

consequently increasing the fundamental period of soil-structure systems, this

phenomenon is reversed. For both fixed-base and flexible-base models with long

period of vibration, the performance of the structure is more dependent on the level

of inelasticity (i.e. target ductility demand value) than the fundamental period of the

building although nearly in all current code-specified seismic load patterns the

ductility demands are not considered.

3. Generally, SSI effect is more significant as the aspect ratio increases, i.e., for the

case of slender building, leading to more non-uniform distribution of ductility

demand along the height of the structure as compared to the corresponding fixed-

base structure model. The influence of aspect ratio on SSI effect is less prominent as

the level of inelastic response increases.


133

4. While there is no significant difference between the results of mass-proportional and

Rayleigh-type damping models for fixed-base and soil-structure systems, the

difference is pronounced when compared to the stiffness-proportional damping

model. Stiffness-proportional damping model may not be appropriate when higher

modal contribution to overall response is significant. The story ductility demand

distribution pattern is only weakly dependent on damping ratio and the secondary

slope of post-yield response for the most practical cases.

5. The numerical results of this study have been validated by the recently developed

macro-element to simulate nonlinear dynamic soil-structure interaction for shallow

foundation. It is concluded that using the linear elastic model to take into account the

SSI effects during severe earthquake ground motions with high PGA may result in a

conservative ductility demand for top stories when compared to the macro-element

model. However, when PGA is less than 0.5g, both models yield to almost identical

results.

6. Among the three code-specified design lateral load patterns, i.e., EuroCode-8, UBC-

97 and IBC-2009, UBC-97 leads to the best performance of structures with

consideration of SSI effects.

7. When considering the SSI effect the structures designed according to the optimal

load pattern proposed by Hajirasouliha and Moghaddam (2009) for fixed-base shear

buildings has the best performance for structures with short periods, while along

with EuroCode-8, the worst seismic performance for structures with long vibration

periods. This implies that the optimal load pattern proposed by Hajirasouliha and

Moghaddam (2009) for fixed-base shear-building structures significantly misses

their efficiency in soil-structure systems.

8. It is demonstrated that although the structures designed according to some load

patterns such as those proposed by Mohammadi et al. (2004) and Park and Medina

(2007) may have generally better seismic performance when compared to those

designed by code-specified load patterns, their seismic performance are far from

ideal if the SSI effects are considered. Therefore, more adequate load patterns

incorporating SSI effects for performance-based seismic design needs to be

proposed, which is a course of study that will be carried out in the following

chapters of this dissertation.


134

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136

Park, K., and Medina, R. A. (2007). ―Conceptual seismic design of regular frames based on

the concept of uniform damage‖ Journal of Structural Engineering, ASCE, 133(7),

945-955.


Wolf JP (1994), ―Foundation Vibration Analysis using Simple Physical Models.‖ Prentice-

Hall: Englewood Cliffs, NJ.


137

Chapter 7

OPTIMUM LATERAL LOAD PATTERN FOR ELASTIC SEISMIC DESIGN OF

BUILDINGS INCORPORATING SOIL-STRUCTURE INTERACTION EFFECTS

7.1. INTRODUCTION

As shown in Chapters 4 and 6, the structural configuration in terms of stiffness and strength

distributions has a key role in seismic response and behavior of structures. Through an

intensive parametric study presented in Chapter 6, for the first time, the efficiency of current

code-compliant lateral load patterns as well as those recently developed and proposed by

researchers for fixed-base structures were parametrically investigated for soil-structure

systems. Results indicated that using the code-specified load pattern leads to nearly uniform

(optimum) ductility demands distribution for structures having short periods and within the

elastic range of response. For structures with long periods, however, it loses its efficiency as

the number of stories and soil flexibility as well as the level of inelasticity increase

especially for the cases of severe SSI effects. Moreover, although the structures designed

according to some of the recently proposed lateral load patterns for fixed-base structures

may have generally better seismic performance when compared to those designed by code-

specified load patterns, they lose their efficiency and, thus, their seismic performance are far

from optimum if the SSI effects are considered. Therefore, more adequate load patterns

incorporating SSI effects for performance-based seismic design needs to be developed.

In this chapter, using the uniform distribution of damage over the height of structures, as the

criterion, an optimization technique for seismic design of elastic soil-structure systems is

developed. In this regard, the optimization algorithm proposed by Hajirasouliha and

Moghaddam (2009) for fixed-base buildings based on uniform distribution of deformation

over the height of the structure is developed for elastic soil-structure systems. In the next


138

chapter (Chapter 8), this algorithm will be extended and developed to take into account for

inelastic behaviour. By performing intensive numerical simulations of responses of elastic

soil-structure shear buildings with various dynamic characteristics and SSI parameters, the

effects of fundamental period, number of stories, earthquake excitation, soil flexibility,

aspect ratio, damping ratio and damping model on optimum distribution pattern are

investigated. Based on 30240 optimum load patterns derived from numerical simulations

and nonlinear statistical regression analyses, a new lateral load pattern for elastic soil-

structure systems is proposed. It is a function of the fixed-base period of the structure, soil

flexibility and structural slenderness ratio (aspect ratio). It is shown that the seismic

performance of such a structure is superior to those designed by code-compliant or recently

proposed patterns by researchers for fixed-base structures. Using the proposed load pattern

in this study, the designed structures experience up to 40% less structural weight as

compared with the code-compliant or optimum patterns developed based on fixed-base

structures.

7.2 SELECTED EARTHQUAKE GROUND MOTIONS

The determination of an optimum lateral force distribution needs to take variability in

ground motion data into account to provide a uniform distribution of story ductility ratios

(damage) along the height for similar structures and ground motions. Generally, it is

believed that for design purpose, the design earthquake ground motion should be classified

for each structural performance and soil type category. In this regard, an ensemble of 21

earthquake ground motions with different characteristics are compiled. The selected ground

motions listed in Table 7-1 are components of six earthquakes including Imperial Valley

1979, Morgan Hill 1984, Superstition Hills 1987, Loma Prieta 1989, Northridge 1994 and

Kobe 1995. All the selected ground motions are obtained from earthquakes with magnitude

greater than 6 having closest distance to fault rupture more than 15 km without pulse type

characteristics. To be consistent, using SeismoMatch software (SeismoMatch, 2011) these

seismic ground motions are adjusted to the elastic design response spectrum of IBC-2009

with soil type E. SeismoMatch is an application capable of adjusting earthquake

accelerograms to match a specific target response spectrum using wavelets algorithm. The

ground motions utilized in the present study have the predominant period ranging from 0.5

to 1.35 sec, recorded on sites with shear wave velocity from 90 to 350 m/s, which are


139

approximately consistent to the IBC-2009 elastic response spectrum of soil type E.

Therefore, they were grouped and adjusted to match the design response spectrum of soil

type E corresponding to IBC-2009. Figure 7-1 shows a comparison of the 21 matched

ground motion spectra with the target elastic design response spectrum of IBC-2009 (ICC,

2009).

Table 7-1: Selected ground motions recorded at alluvium and soft soil sites based on USGS site

classification

Event Year Station Distance

(km)

Soil type

(USGS)

Component PGA (g)

Imperial Valley 1979 Cucapah 23.6 C 85 0.309

Imperial Valley 1979 El Centro Array #12 18.2 C 140 0.143

Loma Prieta 1989 Agnews State Hospital 28.2 C 0 0.172

Loma Prieta 1989 Gilroy Array #4 16.1 C 0 0.417

Loma Prieta 1989 Sunnyvale - Colton Ave 28.8 C 270 0.207


Northridge 1994 Canoga Park -Topanga

Can

15.8 C 196 0.42

Kobe 1995 Kakogawa 26.4 D 0, 90 0.251, 0.345

Kobe 1995 Shin-Osaka 15.5 D 0, 90 0.243, 0.212

Loma Prieta 1989 APEEL 2 - Redwood

City

47.9 D 43 0.274


Menhaden

51.2 D 360 0.116

Superstitn

Hills(B)


Refuge

27.1 D 315 0.167

Morgan Hill 1984 Gilroy Array #2 15.1 C 90 0.212




140

Figure 7-1: IBC-2009 (ASCE/SEI 7-05) design spectrum for soil type E and response spectra of

21 adjusted earthquakes (5% damping) for selected ground motions

7.3 OPTIMUM DISTRIBUTION OF ELASTIC DESIGN LATERAL FORCE FOR

SOIL-STRUCTURE SYSTEMS

As stated before, using code-compliant lateral load patterns does not lead to the optimum

seismic performance of structures. Based on the results of Chapter 6, using code-specified

load pattern for soil-structures systems with severe SSI effect and high inelastic response

does not lead to uniform (optimal) ductility demand distribution over the height of

structures. This means that the deformation (ductility) demand in some stories of the

building does not reach the presumed target level of seismic capacity, which indicates that

the structural material has not been entirely exploited over the height of the building. This

chapter deals with the development of the optimization technique to distribute predefined

structural damage in elastic range of response along the height of the structure. In this

regard, the optimization technique adopted by Mohammadi et al., (2004) and Hajirasouliha

and Moghaddam (2009) for fixed-base shear- building structures is utilized to develop the

optimal load pattern for elastic soil-structure systems. In this approach, the structural

properties are modified so that inefficient material is gradually shifted from strong to weak

parts of the structure. This process is continued until a state of uniform deformation is

achieved (Hajirasouliha and Moghaddam, 2009). In the present study, the seismic demand

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.5 1 1.5 2 2.5 3

Individual synthetic earthquakes

IBC-2009 (Soil type E)

Period (sec)

Pse

udo A

ccel

erat

ion (

g)


141

parameter used to quantify the structural damage is the inter-story displacement ductility

ratio ( ). It should be mentioned that although for the elastic response the ductility ratio is

limited to the value equal to one, this parameter can still be used as an index representing the

level of deformation with respect to the predefined target value. Considering the theory of

uniform deformation proposed by Mohammadi et al., (2004) and Hajirasouliha and

Moghaddam (2009) for fixed-base shear- building structures, the following step-by-step

optimization algorithm is developed and proposed for elastic shear-building soil-structure

systems:

1. Define the MDOF shear-building model depending on the prototype structure height and

number of stories.

2. Assign an arbitrary value for total stiffness and strength and then distribute them along

the height of the structure based on the arbitrary lateral load pattern, e.g., uniform

pattern. As mentioned earlier, the lateral story stiffness is assumed as proportional to the

story shear strength distributed over the height of the structure.

3. Select an earthquake ground motion.

4. Consider a presumed set of aspect ratio, H r , and dimensionless frequency, 0a , as the

predefined key parameters for SSI effects.

5. Select the fundamental period of fixed-base structure and scale the total stiffness without

altering the stiffness distribution pattern such that the structure has a specified target

fundamental period. The following equation is used for scaling the stiffness to reach the

target period by just one step:

2

1 1

1arg

( ) ( ) .( )n n

ij i j i

i it et

TK K

T

(7-1)

where jK , iT and argt etT are story stiffness in the jth story, fixed-base period in the ith

step and the target fixed base period, respectively. Refine H r based on the

fundamental modal properties of fixed-base MDOF structure (Eq. 2-3).

6. Perform dynamic analysis for the soil-structure system subjected to the selected ground

motion and compute the total shear strength demand, ( )s iV . If the computed ductility

ratio is equal to the target value within the 0.5% of the accuracy, no iteration is

necessary. Otherwise, total base shear strength is scaled (by either increasing or


142

decreasing) until the target ductility ratio is achieved. To do this the following equation

is proposed:

1 i max( ) ( ) ( )s i sV V (7-2)

where ( )s iV is the total base shear strength of MDOF system at ith iteration and max is

the maximum story ductility ratio among all stories. Parameter β is an iteration power

which is more than zero. As shown in Chapter 3, β power for 1t (elastic state) can

be taken as a constant value of 0.8 for all fixed-base and flexible base shear-building

structures when subjected to any earthquake excitation.

7. Calculate the coefficient of variation (COV) of story ductility distribution along the

height of the structure and compare it with the target value of interest which is

considered here 0.02. If the value of COV is less than the presumed target value, the

current pattern is regarded as optimum pattern. Otherwise, the story shear strength and

stiffness patterns are scaled until the COV decreases below or equal to the target value.

8. Stories in which the ductility demand is less than the presumed target values (i.e., 1t )

are identified and their shear strength and stiffness are reduced. To achieve the fast

convergence in numerical computations, the equation proposed by Hajirasouliha and

Moghaddam (2009) is revised for elastic soil-structure systems as follows:

1[ ] [ ] .[ ]i q i q iS S

(7-3)

where [ ]i qS = shear strength of the ith floor at qth iteration, i =story ductility ratio of

the ith floor and = convergence parameter that has been considered equal to 0.1- 0.2

as the acceptable range by Moghaddam and Hajirasouliha (2006, 2008) and

Hajirasouliha and Moghaddam (2009) for elastic and inelastic fixed-base structures.

The results of this study show that for elastic fixed-base and soil structure systems

can be taken from 0.5 to 1. However, in most cases the value of 0.8 leads to the fastest

convergence (i.e., less than 5 iterations). The effect of the convergence parameter on

optimum elastic design of the shear-building structures will be investigated in the next

section. Now, a new pattern for lateral strength and stiffness distributions is obtained.


143

9. Control the current maximum story ductility ratio ( max ) and refine the total base shear

strength of soil-structure systems if max is not equal to the target value within the 0.5%

of the accuracy based on Eq. 7-2 of step 6. Otherwise, go to the next step.

10. Control the current fixed-base period and modify it if it is not equal to the target value

within the 1% of the accuracy based on Eq. 7-1 of step 5. Otherwise, go to the next step.

11. Control the current effective height ( H ) and refine it if the value is not equal to the

previous value within the 1% tolerance based on Eq. 2-3. Otherwise, go to the next step.

12. Control the current Rayleigh-type damping coefficients and modify them if they are not

equal to the previous values within the 1% tolerance. Otherwise, go to the next step.

13. Convert the optimum shear strength pattern to the optimum lateral force pattern.


15. Repeat steps 4–15 for different sets of H r and 0a .



To show the efficiency of the proposed method for optimum seismic design of soil-structure

systems in elastic range of response, the above algorithm is applied to a 10-story shear

building with fixT = 1.5 sec, H r = 3, and 0 2a subjected to Kobe (Shin Osaka) simulated

earthquake. Figure 7-2a illustrates a comparison of IBC-2009 (ICC, 2009) load pattern with

the optimum patterns of fixed-base and soil-structure systems. As seen, there is a significant

difference between the optimum pattern of soil-structure systems and the other two patterns.

These three patterns are applied to the same 10-story building with consideration of soil

flexibility (SSI effect) and then the height-wise distribution of story ductility demand

resulted from utilizing these lateral load patterns are computed and depicted in Figure 7-2b.

It can be seen that while using the SSI optimum pattern results in a completely uniform

distribution of the deformation, using both the code-specified and fixed-base optimum

patterns lead to a rather non-uniform distribution of ductility demand along the height of the

soil-structure systems in elastic range of vibration. The COV of story ductility demand

distributions resulted from applying IBC-2009 pattern, the fixed-base optimum pattern and

SSI optimum pattern are 0.226, 0.196 and 0.003, respectively. This indicates that SSI

phenomenon through changing the dynamic characteristics of structures can significantly

affect drift distribution along the height of structures. Therefore, utilizing fixed-base


144

optimum load pattern may not result in an optimal seismic performance of soil-structure

systems and, thus, a more adequate load pattern accounting for SSI effects should be defined

and proposed for soil structure systems. This will be discussed more in the next sections.

In another point of view, it may be of interest to compare the required structural weight for

buildings that have been designed for different seismic load patterns. The efficiency of the

selected load patterns, then, can further be evaluated accordingly. In this regard, the weight

index defined by Eqs. 6-6 and 6-7 in Chapter 6 is utulized here to demonstate the efficiency

of the proposed optimization algorithm. The loading pattern that corresponds to the

minimum weight index is considered as the most adequate loading pattern. As an instance,

the weight indices corresponding to the three load patterns considered in the previous

example are calculated here to demonstrate the efficiency of the proposed optimization

algorithm. The weight indices computed from applying IBC-2009 pattern, the fixed-base

optimum pattern and SSI optimum pattern are respectively 8.68, 7.74 and 5.25. As seen, the

required structural weight value corresponding to the proposed SSI optimum load pattern is

respectively 39.5% and 32.2% less than that of the IBC-2009 and the fixed-base optimum

patterns, which means that utilizing the proposed algorithm can remarkably reduce the

required structural weight in elastic range of response. Therefore, to improve the seismic

performance of the structure under this specific earthquake the frame should be designed in

accordance to an equivalent lateral load pattern that is different from the conventional code-

specified and the suggested fixed-base optimum patterns by Hajirasouliha and Moghaddam

(2009).


145

Figure 7-2: Comparison of IBC-2009 with optimum designed models of fixed-base and soil-

structure system: (a) lateral force distribution; (b) story ductility pattern, 10-story shear building

with fixT = 1.5 sec, H r =3, Kobe (Shin Osaka) simulated earthquake

7.4 EFFECT OF STRUCTURAL DYNAMIC CHARACTERISTICS AND SSI KEY

PARAMETERS ON OPTIMUM LATERAL FORCE PATTERN

Before proposing a general load pattern for optimal design of soil-structure systems in

elastic range of response, it is necessary to, firstly, investigate the effects of various

parameters including structural dynamic characteristics, SSI key parameters as well as those

specifically corresponding to the optimization algorithm.

7.4.1 Effect of convergence parameter

In order to examine the effect of power, α, defined in Eq. (7-3) for convergence; the

previous example again is solved for different values of α. Figure 7-3 shows the variation of

structural weight index corresponding to different values of convergence powers of 0.1, 0.2,

0.5, 0.8, 1 and 1.5. It shows how the structural weight index varies with the iteration step

from a presumed initial load pattern, here the uniform load pattern, toward the optimum

pattern. Different from the findings by Hajirasouliha and Moghaddam (2009) who have

proposed the best values of 0.1-0.2 for all ranges of ductility demands, i.e., both elastic and

inelastic response, results of this study indicate that for elastic response, regardless of the

1

2

3

4

5

6

7

8

9

10

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Sto

ry

Lateral Force / Base Shear

1

2

3

4

5

6

7

8

9

10

0.2 0.4 0.6 0.8 1 1.2

Ductility

(a

)

(b)

IBC-2009 Fixed-base

SSI, = 2


146

structural model, i.e., either fixed-base or flexible-base model, the values of 0.5-1 could be

the best ranges for α. In addition, in elastic response, there is no fluctuation in convergence

problem for the power of α ranging from 0.01 to 1 while the fluctuation happens for α

greater than one as seen for the case of α= 1.5 in Figure 7-3. Based on intensive analyses

performed for both fixed-base and soil-structure systems in elastic response, it is concluded

that, on average, the value of 0.8 could be a good value for α in order to achieve the fastest

convergence. As seen, using α= 0.8 the required numbers of iterations to reach the optimum

design are only 2 and 3 steps for respectively the fixed-base and soil-structure system.

However, it will be 32 and 16 steps in fixed-base systems and 32 and 19 steps in soil-

structure systems for α= 0.1 and 0.2, respectively. It is also interesting to note that after only

one iteration, the value of weight index reduces to less than 50% of its initial value i.e., from

14 to 6.66.

Figure 7-3: Variation of structural weight index for different values of convergence powers; 10-

story soil-structure system with fixT = 1.5 sec, H r = 3, 0a =2, Kobe (Shin Osaka) simulated

earthquake

7.4.2 Effect of earthquake excitation

To investigate the effect of varying earthquake ground motions on optimum lateral force

pattern for both fixed-base and flexible-base shear-building models in elastic range of

response, individual results of all the 21 matched earthquake ground motions listed in Table

7-1 along with their average values for the 10- story shear building are presented in Figure

7-4. The results are for systems with fixT = 1.5 sec for fixed-base system (Figure 7-4a) as

well as for soil-structure system with aspect ratio of 3, and dimensionless frequency of 2

WI

Step Step

6

7

8

9

10

11

12

13

14

15

16

17

18

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

(a) Fixed base α= 0.1 α= 0.2

α= 0.5 α= 0.8

α= 1 α= 1.5

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

(b) SSI, a0 = 2 α= 0.1 α= 0.2

α= 0.5 α= 0.8

α= 1 α= 1.5


147

(Figure 7-4b). As seen, it is obvious that the optimum lateral load pattern is sensitive to the

earthquake ground motion characteristics for both the fixed-base and especially the flexible-

base models. However, in most ground motions used in this study, there is not a big

discrepancy in the general pattern of optimum lateral load profile when compared to the

corresponding averaged pattern. Therefore, it is expected that utilizing the mean pattern will

lead to acceptable designs although some inevitable variation is not avoidable depending on

the earthquake ground motion. This will be demonstrated in the next part. It is also worth

mentioning that, in general, the soil-structure systems are more sensitive to the seismic

excitation than fixed-base systems. As seen in Figure 7-4b, in some ground motions the sign

of the lateral force corresponding to the SSI pattern is negative for one or two stories. This

phenomenon could be due to the effect of higher mode contributions in soil-structure

systems with severe SSI effects that are much more flexible than the corresponding fixed-

base building.

Effect of ground motion intensity on the optimum load profile of the elastic 10-story soil-

structure model with fixT = 1.5, H r = 3, and 0 2a subjected to Kobe (Shin Osaka)

simulated earthquake with the PGA multiplied by 0.5, 1, 2, and 3 factors are illustrated in

Figure 7-5a. The results indicate that for a specific fundamental period, aspect ratio and

dimensionless frequency, the optimum lateral load pattern for elastic response is

independent of the ground motion intensity factor (SF), which is consistent with the finding

of Mohammadi et al., (2004) and Hajirasouliha and Moghaddam (2009) for fixed-base

shear-building structures.

7.4.3. Effect of initial load pattern

Considering the proposed optimization algorithm for soil-structure systems, an initial

strength and stiffness distribution is required to reach the optimum answer. Hajirsouliha and

Moghdam (2009) concluded that for inelastic fixed-base shear buildings, the optimum

lateral force pattern is not dependent on the initial strength pattern, and to some extent it

would affect the speed of convergence. This point is investigated here for the case of elastic

soil-structure system of the previous example by considering the same initial patterns

utilized by Moghddam and Hajirsouliha (2008), and the results are depicted in Figure 7-5b.

As seen, the same results can be concluded for the elastic soil-structure systems.

Nevertheless, the results of this study indicate that using the proposed optimization


148

algorithm for both elastic fixed-base and soil-structure systems, initial strength pattern has

no effect on the convergence speed. For example, using all assumed patterns including

rectangular, triangular, IBC-2009 and concentric patterns requires only 3 steps to converge

to the same optimal load pattern as shown in Figure 7-6 for both fixed-base and soil-

structure systems.

Figure 7-4: Optimum lateral force distribution for different earthquake excitations, 10-story

building with fixT = 1.5 sec: (a) Fixed-base model; (b) Soil-structure model; H r =3 and 0a = 2

Figure 7-5: Effect of (a) ground motion intensity and (b) initial load pattern on optimum lateral

force profile for soil-structure systems with fixT = 1.5 sec, H r =3 and 0a = 2; Kobe (Shin

Osaka) simulated earthquake

1

2

3

4

5

6

7

8

9

10

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

(a)

1

2

3

4

5

6

7

8

9

10

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

(b

)

Lateral Force / Base Shear Lateral Force / Base Shear

Sto

ry

Individual

Results Average

1

2

3

4

5

6

7

8

9

10

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

SF= 0.5

SF=1

SF=2

SF=3

1

2

3

4

5

6

7

8

9

10

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Concentric

Triangular

Inverted Triangular

Rectangular

Sto

ry


(a) (b)


149

Figure 7-6: Effect of initial load pattern on optimization iteration steps; 10-story shear

building; (a) Fixed-base systems (b) soil-structure system with fixT = 1.5 sec, H r = 3,

0 2a , Kobe (Shin Osaka) simulated earthquake.


To study the effect of fundamental period on optimum load pattern of elastic soil-structure

systems, the 15-story building models with H r = 3 and 0a = 2, having fixed-base

fundamental periods of 0.3, 0.6, 1, 2 and 3 sec are considered. For each case, the optimum

load patterns are derived for the 21 matched earthquake ground motions listed in Table 7-1

and the average results are plotted in Figure 7-7a. As seen, the averaged optimum load

pattern is strongly dependent on the fundamental period of vibration such that increasing the

fundamental period is generally accompanied by increasing the lateral shear force at top

stories which can be interpreted as the effect of higher modes.


To examine the effect of number of stories on the optimum distribution profile, the proposed

optimization algorithm is applied to analyze 5-, 7-, 10-, 15- and 20-story soil-structure

models with fixT = 1.5, H r = 3 and 0a = 2 subjected to the 21 matched earthquake ground

motions. The average results are depicted in Figure 7-7b. In order to compare the averaged

optimum patterns corresponding to different number of stories, the normalized lateral loads

are plotted. In Figure 7-7b, the vertical and horizontal axes are relative height and

normalized lateral load divided by base shear strength, respectively. From this figure, it can

be concluded that the optimum load patterns are almost independent of the number of

6

7

8

9

10

11

12

13

14

15

16

17

18

0 1 2 3 4

(a) Fixed base Rectangular

Triangular

IBC-2009

Concentric

3

4

5

6

7

8

9

10

11

12

13

14

15

0 1 2 3 4

(b) SSI, a0 = 2 Rectangular

Triangular

IBC-2009

ConcentricW

I

Step Step


150

stories. This finding is also consistent with that by Hajirasouliha and Moghadam (2009) for

fixed-base shear-building structures.


Figure 7-8 shows the effect of dimensionless frequency, 0a on averaged optimum load

pattern of elastic soil-structure systems subjected to 21 matched ground motions. As stated

before, aspect ratio and dimensionless frequency are two key parameters that can affect the

response of the soil-structure systems subjected to earthquake excitation. The results are

plotted for the 10-story shear building with two fundamental periods of 1 and 2 sec, and

H r =3 corresponding to three values of dimensionless frequency ( 0a =1, 2, 3). It can be

observed that dimensionless frequency can significantly affect the averaged optimum load

pattern such that increasing the value of dimensionless frequency is accompanied by

increasing the lateral load at bottom and top stories, and decreasing the load in middle

stories. This phenomenon could be again due to the effect of higher mode effect as a result

of increasing the fundamental period of the soil-structure systems.

Figure 7-7: Effect of fundamental period (a) and the number of stories (b) on averaged optimum

lateral force profile for soil-structure systems with H r =3 and 0a = 2: fixT = 1.5 sec.

.


Sto

ry

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 0.05 0.1 0.15 0.2 0.25 0.3

(a)

T=0.3T=0.6T=1

T= 2

Normalized lateral Force / Base Shear

Rel

ativ

e H

eig

ht

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.05 0.1 0.15 0.2

(b) H̅/r = 3

N=5

N= 7

N= 10

N= 15

N= 20


151

Figure 7-8: Effect of dimensionless frequency on averaged optimum lateral force profile for 10-

story soil-structure systems with H r =3: (a) fixT = 1 sec. (b) fixT = 2 sec.


Figure 7-9 shows the effect of aspect ratio on averaged optimum load pattern of elastic soil-

structure systems. The results are for the 10-story shear building with fixT =1.5 sec, two

dimensionless frequencies ( 0a =1, 3), representing the insignificant and severe SSI effect,

respectively and three values of aspect ratio ( H r = 1, 3, 5) representing respectively squat,

average and slender buildings subjected to 21 matched ground motions. As seen, for the case

of insignificant SSI effect (i.e., Figure 7-9a), increasing aspect ratio will not change the

optimum load profile remarkably. However, by increasing the dimensionless frequency and,

therefore more significant SSI effect, the aspect ratio will greatly affect the averaged

optimum load pattern. The trend is to some extent similar to that of the dimensionless

frequency discussed in the previous section such that increasing the value of aspect ratio is

accompanied by increasing the lateral load at bottom and especially top stories, and

decreasing the load in the middle stories, and this trend is more pronounced for slender

buildings (i.e. H r = 5).

Fixed-base 𝑎0 = 1

𝑎0 = 2

𝑎0 = 3


Rel

ativ

e H

eig

ht

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.05 0.1 0.15 0.2 0.25

(a)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

(b)


152

Figure 7-9: Effect of aspect ratio on averaged optimum lateral force profile for a 10-story soil-

structure system with fixT = 1.5 sec

7.4.8 Effect of structural damping Ratio

The effect of structural damping ratio on optimum load pattern of elastic soil-structure

systems is illustrated in Figure 7-10a for the 10-story shear-building structure with fixT = 1.5,

H r = 3 and 0a = 2 corresponding to four values of 0%, 2%, 5% and 10% of damping

ratios subjected to matched Loma Prieta earthquake (APEEL 2 - Redwood City). As seen,

earthquake loads associated to the top stories reduces by increasing the damping ratio which,

in its turn, reduces higher mode effects. The phenomenon is more pronounced for the case of

damping ratio of 10%. Therefore, one may conclude that for the practical purpose, the

optimum load pattern of elastic soil-structure systems can be considered insensitive to the

variation of damping ratio. The results are consistent with those concluded for fixed-base

systems by Hajirasouliha and Moghaddam (2009).

7.4.9 Effect of structural damping model

In order to examine the effect of damping models on optimum load pattern of elastic soil-

structure systems, three conventional viscous damping models including stiffness-

proportional damping, mass-proportional damping and Rayleigh-type damping in which

damping matrix is composed of the superposition of a mass-proportional damping term and

a stiffness-proportional damping term are considered. In this case, optimum lateral load

pattern of the 10-story soil-structure systems with fixT = 1.5, H r = 3 and 0a = 2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.05 0.1 0.15 0.2 0.25 0.3

(a) a0 = 1

H/r= 1

H/r= 3

H/r =5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

(b) a0 = 3

H/r= 1

H/r= 3

H/r =5Rel

ativ

e H

eig

ht



153

corresponding to aforementioned damping models subjected to Loma Prieta earthquake

(APEEL 2 - Redwood City) are computed and plotted in Figure 7-10b. It can be seen that

while there is no significant difference between the results of mass-proportional and

Rayleigh-type damping models, the difference is pronounced when compared to that of the

stiffness-proportional damping model. The same result has been found for other fixed-base

and soil-structure models subjected to different seismic ground motions. This observation

indicates that stiffness-proportional damping model may lead to quite different predictions

of structural responses as compared to the Rayleigh damping model. It is, therefore,

advisable to use Rayleigh-type damping model to better incorporate the effect of higher

modes.

Figure 7-10: Effect of structural damping ratio (a) and damping model (b) on optimum lateral

force profile; 10-story soil-structure system with H r =3, 0a = 2 and fixT = 1.5 sec; Loma Prieta

(APEEL 2 - Redwood City) earthquake

7.5 NEW LATERAL LOAD PATTERN FOR ELASTIC SOIL-STRUCTURE

SYSTEMS

To generalize the use of the proposed optimization algorithm for conceptual seismic design

of elastic soil-structure systems, it is necessary to develop statistical models for estimating

the optimal design lateral load pattern as a function of relevant structural and soil

characteristics. Because of the variability in ground motion characteristics, it is not

straightforward to determine an equivalent lateral load pattern to provide, on average, a

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

ξ = 0%

ξ = 2% ξ = 5%

ξ = 10%

(a)


Rel

ativ

e H

eig

ht

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Mass

Rayleigh Stiffness

(b)



154

uniform distribution of deformation along the height of the soil-structure system when the

system is subjected to earthquake excitations with different frequency contents. Generally, it

is believe that for design purpose, the design earthquake ground motion should be classified

for each structural performance and soil type category. More reliable load pattern, then, can

be obtained by commuting the mean values of optimum patterns associated to the design

earthquakes compatible with each seismic design spectrum. To be consistent, 21 seismic

ground motions compatible to the elastic design response spectrum of IBC-2009 (ICC,

2009) with soil type E as listed in Table 7-1 are selected. Numerous soil-structure systems

including four different number of stories (i.e., 5-, 10, 15 and 20-story buildings), 30

fundamental periods ranging from 0.1 to 3 sec with intervals of 0.1, three values of aspect

ratio ( H r =1, 3, 5) representing respectively squat, average and slender buildings, and four

values of dimensionless frequency ( 0a =0, 1, 2, 3) are considered. It should be noted that 0a

= 0 and 3 correspond to the systems with fixed base and severe SSI effect, respectively.

Consequently, utilizing the proposed optimization algorithm, 30240 optimum lateral load

patterns are derived for elastic soil-structure systems. For each fundamental period,

dimensionless frequency and aspect ratio, the mean optimum load pattern corresponding to

21 matched earthquake ground motions are obtained. It is expected that designs based on the

mean patterns would exhibit a more uniform damage along the height of soil-structure

systems. Based on the results of this study and nonlinear statistical regression analysis, the

following expression is proposed for optimal design of elastic shear-building soil-structure

systems:

( 0.5 )0.5. ( ) e fixT

i i i fix fix iF a b T Ln T c (7-4)

where iF = optimum load component at the ith story; fixT = fixed-base fundamental period;

and ia , ib and ic = constant coefficients of the ith story which are functions of aspect ratio

and dimensionless frequency and are given in Tables 7-2 to 7-4 for each level of structure. It

should be noted that iF is the optimum lateral load component that must be scaled to the

total load components at the end of calculation. In addition, the optimum load patterns

corresponding to values of the relative height, aspect ratio and dimensionless frequency

corresponding to the specified values in Tables 7-2 to 7-4 can be easily obtained by linear

interpolation of the two associated load patterns.


155

The efficiency of the proposed load pattern can be investigated by comparing the structural

weight indices resulted from using Eq. 7-4 with code-compliant and recently proposed load

patterns for fixed-base structures. Accordingly, two aforementioned lateral load patterns by

Park and Medina (2007) and Hajirasouliha and Moghaddam (2009) as well as IBC-2009

(ICC, 2009) lateral load pattern are considered here and the average results of these patterns

are parametrically compared with results of this study (i.e., Eq. 7-4). For this purpose, the

values of normalized weight index (WI) of 10-story shear buildings designed based on the

four above-load patterns for 30 fundamental periods ranging from 0.1 to 3 sec, three values

of aspect ratio ( H r =1, 3, 5) and two values of dimensionless frequency ( 0a =1, 3) are

calculated subjected to 21 matched earthquake ground motions. Then, the averaged values

of weight index for all the above-load patterns are computed and illustrated in Figure 7-11.

Based on the results presented in Figure 7-11, it can be observed that:

1. For short periods of vibration, the structures designed in accordance to the optimum load

pattern proposed by Park and Medina (2007) has the worst performance among all load

patterns. This implies that this pattern loses their efficiency for this ranges of period

even when SSI effect is not significant (i.e., 0a =1).

2. For all ranges of period and SSI effects, the load pattern proposed in this study gives the

best results in comparison to the results of other load patterns. The superiority is more

pronounced for the cases of long periods with severe SSI effects. As seen, The ratios of

required to the optimum structural weight index for models designed with Eq. 7-4 are,

on average, from 1.02 to 1.15 which can be considered as optimum for practical

purposes.

3. The loading patterns proposed by Hajirasouliha and Moghaddam (2009) and IBC-2009,

on average, give good results for structures with short periods and insignificant SSI

effect, however; they remarkably lose their efficiency with increasing the dimensionless

frequency and aspect ratio.

4. Generally, by increasing the aspect ratio and dimensionless frequency (i.e., increasing

SSI effects) the two previous proposed load patterns for fixed-base structures as well as

code-specified load pattern significantly lose their efficiency while the proposed load

pattern in this study still display superior seismic performance, especially for slender

building ( H r =5) with predominant SSI effect ( 0a =3). As an example, for the case of


156

slender building with severe SSI effect with fixT = 1.5 sec, the values of structural weight

for structures designed with Eq. 7-4, Park and Medina (2007), Hairasouliha and

Moghaddam (2009) and IBC-2009 seismic code are respectively 15%, 44%, 53% and

65% above the optimum weight. This implies that significant improvement is achieved

by utilizing the proposed load pattern of this study for severe soil-structure systems.

The average COV of story ductility ratio for the 10-story soil-structure systems designed

according to different load patterns corresponding to the severe SSI effect with 0a = 3

and three values of aspect ratios are plotted in Figure 7-12. As seen, the average COV

corresponding to the Eq. 7-4 are less than those of other patterns especially for cases of

the slender buildings with long periods of vibration. As an instance, a comparison of the

different lateral load patterns corresponding to all forgoing patterns along with the

resulted ductility demand distributions for the 10-story building with fixT = 1.5 sec, H r

= 3 and 0a = 3 is provided in Figure 7-13. It is clear that there is a significant difference

among the results obtained with different load patterns. Structures designed based on the

proposed load pattern in this study exhibit a much more uniform distribution of damage

along the height as compared to those designed according to other load patterns.

Figure 7-14 shows the comparison of the proposed equation in predicting the optimum

load pattern with the average numerical results. As shown, there is an excellent

agreement between Eq. (7-4) and the average numerical results to estimate the optimum

load pattern in elastic soil-structure systems corresponding to different sets of structural

and soil parameters.


157



earthquakes

Figure 7-12: The spectra of COV for the 10-story soil-structure systems designed according to

different load patterns; average of 21 earthquakes; 0a = 3

1.0

1.1

1.2

1.3

1.4

1.5

0 1 2 3

a0 = 1

1.0

1.1

1.2

1.3

1.4

1.5

0 1 2 3

a0 = 1

1.0

1.1

1.2

1.3

1.4

1.5

0 1 2 3

a0 = 1

WIi

/ W

IOP

T

1.0

1.2

1.4

1.6

1.8

2.0

0 1 2 3

a0 = 3

1.0

1.2

1.4

1.6

1.8

2.0

0 1 2 3

a0 = 3

1.0

1.2

1.4

1.6

1.8

2.0

0 1 2 3

a0 = 3

WIi

/ W

IOP

T

=1

=3

Tfix

Tfix

Tfix

Park and Medina (Eq. 6-2)

10) IBC-2009

Hajirasouliha and Moghaddam (Eq. 6-5)

Proposed (Eq. 7-4)

=5


10) IBC-2009


Proposed (Eq. 7-4)

0.0

0.1

0.2

0.3

0.4

0.5

0 1 2 3

H̅/r =1

0.0

0.1

0.2

0.3

0.4

0.5

0 1 2 3

H̅/r =3

0.0

0.1

0.2

0.3

0.4

0.5

0 1 2 3

H̅/r =5

CO

V

Tfix Tfix Tfix


158

Figure 7-13: Comparison of different load patterns for 10-story soil-structure systems with fixT =

1.5 sec, H r =3 and 0a = 3: (a) lateral force distribution; (b) story ductility pattern; average of

21 earthquakes

Figure 7-14. Correlation between Eq. (7-4) and numerical results

Lateral Force / Base Shear Ductility

Sto

ry

1

2

3

4

5

6

7

8

9

10

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

1

2

3

4

5

6

7

8

9

10

0.2 0.4 0.6 0.8 1 1.2

(a) (b)

Park and Medina (Eq. 6-2) IBC-2009


Proposed (Eq. 7-4)

Tfix= 0.5, = 1, = 5 (Eq. 7-4)

Tfix= 0.5, = 1, = 5

Tfix= 2, = 2, = 3 (Eq. 7-4)

Tfix= 2, = 2, = 3

Tfix= 1, = 2, = 5 (Eq. 7-4)

Tfix= 1, = 2, = 5

Tfix= 3, = 3, = 1 (Eq. 7-4)

Tfix= 3, = 3, = 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45


Rel

ativ

e H

eig

ht


159

7.6 MONTE CARLO SIMULATION

There are numerous uncertainties in seismic design of structures in terms of dynamic

properties of the structures, earthquake ground motion characteristics, soil material, soil

impedance i.e., soil stiffness and damping and etc. Moreover, it is for the first time that an

optimization technique with consideration of SSI effect is developed for seismic-resistant

design of the Shear-building structures; therefore it could be completely natural and

reasonable that the proposed load pattern of this study would not be a perfect pattern to

cover all possibilities. Although the subject of the uncertainty will need a separate detailed

study, it is investigated, here, through one of the most important parameters that have been

utilized in optimization algorithm. Since the adopted optimization technique was based on

the optimum distribution of story shear strength along the height of the structure, the story

shear strength has been selected for sensitivity analysis.

In this section, using Monte Carlo simulation (Fishman, 1995) a sensitivity analysis is

carried out on the optimum lateral load pattern. In this regard, the effect of variation of the

optimum lateral strength pattern on seismic response of the soil-structure systems is

investigated. To do this, it is necessary to define a criterion as reprehensive of optimum

behaviour of the structure subjected to a given earthquake ground motion. It has been shown

that the COV of story ductility demand distribution and structural weight index could be two

of the criteria that are directly related to the optimum behaviour of the structure. For this

sensitivity analysis, the COV of story ductility demand distribution is selected as an

optimization criterion. Based on Monte Carlo simulation method the shear strength values of

all stories are randomly determined by considering the average and specified amount of

dispersion. For each story, then, the mean values of story shear strength corresponding to the

proposed optimum pattern and IBC-2009 load pattern are computed. The dispersions of 1%,

3.33%, 5% and 10% are considered. For each of the aforementioned load patterns, 1500

models were randomly generated under constraints to conform to the adopted fixed-base

fundamental period of 1.5 sec and to produce realistic soil-structure models. The number of

1500 models was chosen with the intention to achieve the best fit distribution for the

randomly selected parameters and increase the accuracy of the Monte-Carlo simulation

(Fishman, 1995). Thus, a large number of sample structures with random story shear

strength values corresponding to the two aforementioned load patterns are produced. Then,


160

each of the soil-structure systems is analyzed subjected to the given ground motion, and the

COV of story ductility demand distribution along the height of the structure is computed. As

stated in the paper, the load pattern corresponding to the least value of the COV is regarded

as the more efficient pattern. Figure 7-15 shows a comparison of results of proposed

optimum pattern and IBC-2009 load pattern for a 10-story shear building with dimensionless

frequency of 2 and H r = 3. The following conclusions can be drawn from Figure 7-15:

1. The COV values of the structures designed based on optimum lateral load patterns

are always less than those of the structures designed in accordance to the IBC-2009

cod-compliant pattern. This implies that even inaccurate estimation of the optimum

story shear strength with reasonable dispersion will lead to a better seismic

performance (i.e., more uniform damage distribution) for structures designed based

on the proposed optimum load pattern with respect to the code-specified pattern.

2. For both considered load patterns (i.e., optimum load pattern and IBS-2009 pattern),

increasing the percentage of the dispersion value is accompanied by increasing the

COV value. Moreover, the results show that as the dispersion value increases the

efficiency of the optimum load pattern decreases and moves toward the code-

specified pattern. However, as mentioned the COV values of optimum pattern are

always less than those of code-specified pattern. This phenomenon is natural and

expectable because the COV values of IBC-2009 pattern are initially large enough,

and, thus, the depression of story shear strength will affect the ductility distribution

less when compared to the optimum pattern.


161

Figure 7-15: Comparisons of the COV of story ductility demand distribution for the 10-story

building designed based on the proposed optimum pattern and IBC-2009 pattern; Monte Carlo

similation (fixT = 1.5 sec, H r =3, Kobe (Shin Osaka) simulated earthquake).

7.7 CONCLUSIONS

This chapter developed an optimization algorithm for optimum seismic design of elastic

soil-structure systems. The adopted method is based on the concept of uniform

deformation proposed by Mohammadi et al. (2004) and Hajirasouliha and Moghadam

(2009) for fixed-base shear building structures. Based on intensive numerical analyses of

structural models with different structural and foundation conditions subjected to 21

selected earthquake ground motions iterated to be compatible to the IBC design

spectrum, an optimum load pattern is proposed for structure design with consideration of

SSI effect. From the numerical results obtained in this study, the following conclusions

can be drawn:

1. A value of 0.8 could be an appropriate value for α in order to achieve the fastest

convergence in optimum analysis. The required number of iterations to reach the

optimum design is, on average, less than 4 steps, while it needs between 15 to 30

steps by using the values of 0.1-0.2 as proposed by other researchers. It is also

0

0.1

0.2

0.3

0.4

0.5

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12%

Ave. (Optimum)

Ave. + St. Dev. (Optimum)

Ave. (IBC-2009)

Ave. + St. Dev. (IBC-2009)

Shear Strength dispersion

CO

V


162

demonstrated that, generally, after only one iteration step, the requied structural

weight reduces to less than 50% of its initial value.

2. The optimum load pattern for elastic shear-building structures with SSI effect is

highly dependent on the fundamental period, dimensionless frequency, aspect ratio,

seismic excitation and structural damping model, but almost independent of the

number of stories, structural damping ratio and earthquake intensity.

3. While there is no significant difference between the results of mass-proportional and

Rayleigh-type damping models, the difference is pronounced when compared to that

of the stiffness-proportional damping model. The stiffness-proportional damping

model may not lead to reliable predictions of structural responses.

4. The proposed load pattern which is a function of fixed-base fundamental period,

dimensionless frequency and structure aspect ratio gives better structural design than

the code-compliant and recently proposed patterns by researchers for fixed-base

structures.

5. Generally, by increasing the aspect ratio and dimensionless frequency (i.e., increasing

SSI effects) the two load patterns by Park and Medina (2004) and Hajirasouliha and

Moghadam (2009) for fixed-base structures as well as the code-specified load

patterns significantly lose their efficiency while the proposed load pattern of this

study leads to better seismic performance (i.e, less structure weight and more uniform

damage distribution over height). Using the proposed load pattern in this study, the

designed structures experience up to 40% less structural weight as compared with the

code-compliant or optimum patterns developed based on fixed-base structures.

6. This study provides a fundamental step towards the development of the more rational

seismic design methodology that explicitly account for the complex phenomenon of

soil-structure interaction and presumed level of drift in elastic response. More

research works for more complex structural configurations and behaviour especially

for inelastic response are deemed necessary for developing a practical methodology

applicable to design and analysis of structures to earthquake ground motions. In the

next chapter, the proposed optimization will be developed to take into account for

inelastic behaviour.


163

Table 7-2: Constant coefficient ia of Eq. (7-4) as function of relative height

Table 7-3: Constant coefficient ib of Eq. (7-4) as function of relative height

H r =1

H r =3

H r =5

Relative Height 0a =0

0a =1 0a =2

0a =3 0a =1

0a =2 0a =3

0a =1 0a =2

0a =3

0.05 14.42 11.66 11.52 19.81 18.84 33.19 37.12 26.00 36.68 18.20

0.10 14.02 12.37 17.01 17.92 22.67 28.86 39.29 32.41 36.39 26.58

0.20 14.80 8.88 13.76 -0.26 25.62 38.22 28.60 33.74 41.52 32.80

0.30 8.16 1.51 -11.24 -14.55 12.23 13.12 21.69 14.90 27.10 28.50

0.40 -0.33 -7.76 -19.14 -5.54 -18.35 -8.30 5.19 -17.38 -5.74 7.84

0.50 -8.42 -9.90 -29.40 -17.97 -27.54 -45.31 -23.89 -42.70 -37.57 -27.24

0.60 -13.88 -12.15 -21.60 -25.45 -23.29 -42.44 -32.83 -36.88 -39.70 -19.17

0.70 -9.14 -2.93 -8.54 -13.58 -13.69 -23.23 -17.79 -17.60 -16.45 -0.80

0.80 -9.94 -18.04 -3.49 -7.03 -14.90 -7.19 -25.51 -8.53 -11.22 -15.19

0.90 -6.16 1.51 16.10 15.01 7.65 4.94 -10.15 9.33 -4.96 -19.70

1.00 13.77 35.03 62.89 63.45 35.05 42.51 14.38 41.38 8.72 -22.17

H r =1

H r =3

H r =5


0a =1 0a =2

0a =3 0a =1

0a =2 0a =3

0a =1 0a =2

0a =3

0.05 -29.25 -17.74 -11.83 -19.92 -31.52 -38.37 -15.05 -43.98 -34.60 50.53

0.10 -21.00 -13.75 -20.06 -15.80 -32.31 -23.73 -18.58 -48.27 -22.64 34.09

0.20 -9.50 0.67 -10.22 22.26 -29.33 -39.63 -3.73 -38.72 -20.95 19.01

0.30 11.21 19.42 39.53 51.49 -0.60 2.62 -13.99 -0.34 -1.97 2.87

0.40 29.20 41.46 60.68 25.50 60.89 30.51 -15.25 57.49 19.99 -34.07

0.50 45.45 50.00 81.38 34.53 81.51 84.61 11.72 100.88 42.95 -20.39

0.60 64.20 61.09 61.49 56.48 76.02 80.59 36.88 90.28 52.20 -12.30

0.70 66.77 51.69 54.20 57.88 66.84 69.10 45.41 69.24 36.57 -4.41

0.80 87.47 104.28 78.30 86.03 96.71 80.04 119.67 81.91 82.42 87.42

0.90 118.23 105.59 88.62 103.89 97.41 122.26 164.23 100.49 145.46 182.15

1.00 156.27 117.54 95.89 123.76 130.43 154.95 238.68 130.73 239.11 328.05


164

Table 7-4: Constant coefficient ic of Eq. (7-4) as function of relative height

H r =1

H r =3

H r =5


0a =1 0a =2

0a =3 0a =1

0a =2 0a =3

0a

=1 0a =2 0a =3

0.05 67.1 54.4 54.3 74.7 73 94.5 70.7 92.9 90.4 -20.7

0.10 61.0 53.7 71.1 70.9 80.7 77.1 79.3 106.9 79.1 6.8

0.20 55.3 41.1 63.2 16.6 86.8 109.5 59.9 104.7 87.1 33.8

0.30 33.7 20.5 -8.9 -24.8 50.8 47.9 73.2 53.5 61. 56.1

0.40 13.9 -5.5 -35.3 16.5 -36.6 6 69.5 -31.4 18.5 92.4

0.50 -3.5 -9.8 -61.8 2.5 -60.7 -77.3 23.8 -94.1 -24.1 57.6

0.60 -22.3 -17.6 -26.8 -24.1 -44. -63.4 -5 -70.9 -27.2 59.9

0.70 -14.6 7.5 -3.0 -11.7 -18.1 -28.0 2.5 -24.7 16. 72.1

0.80 -31.2 -58.8 -21.1 -35.1 -45.6 -21 -82.2 -23.8 -25.9 -36.1

0.90 -54.7 -36.8 -11.0 -34 -19.7 -52.8 -112.9 -23. -83.4 -135.2

1.00 -70.1 -12.1 28.5 -5.8 -23.5 -48.2 -163.2 -20 -164.7 -286

7.8 REFERENCES



Fishman GS. (1995)., Monte Carlo: concepts, algorithms, and applications. New York:

Springer-Verlag.


design of structures.” Journal of Structural Engineering, ASCE, 135(8), 906–915.


AL.

Moghaddam H, and Hajirasouliha I. (2008). “Optimum strength distribution for seismic

design of tall buildings”. The Structural Design of Tall and Special Buildings, 2008;

17: 331–349.

Mohammadi. K. R., El-Naggar, M. H., and Moghaddam, H. (2004). “Optimum strength

distribution for seismic resistant shear buildings.” International Journal of Solids and

Structures. 41(21-23), 6597–6612.


165

Park, K., and Medina, R. A. (2007). “Conceptual seismic design of regular frames based on

the concept of uniform damage” Journal of Structural Engineering, ASCE, 133(7),

945-955.

SeismoMatch . (2011). A computer program for adjusting earthquake records to match a

specific target response spectrum. Available from: http://www.seismosoft.com.

http://www.seismosoft.com/


166

Chapter 8

OPTIMUM SEISMIC DESIGN OF SHEAR BUILDINGS CONSIDERING SOIL-

STRUCTURE INTERACTION AND INELASTIC BEHAVIOR

8.1 INTRODUCTION

In the previous chapter, a new optimization algorithm for optimum seismic design of elastic

shear-building structures with SSI effects has been adopted and developed. The adopted

method has been based on the concept of uniform damage distribution proposed by

Moahammadi et al. (2004) and Hajirasouliha and Moghadam (2009) for fixed-base shear-

building structures as discussed in Chapter 6. Based on numerous optimum load patterns

derived from numerical simulations and nonlinear statistical regression analyses, a new load

pattern for elastic soil-structure systems has been proposed. It has been showed that using

the proposed load pattern could lead to a more uniform distribution of deformations over the

height of structures such that the designed structures experience up to 60% less structural

weight as compared with the code-compliant or aforementioned optimum patterns proposed

for fixed-base structures. On the other hand, almost all current seismic codes allow

structures to behave inelastically during moderate and severe earthquake ground motions

(UBC, 1994; UBC, 1997; BJC, 1997; NEHRP, 2003, Mexico, 2003; CEN, 2003; ASCE,

2005; AS-1170.4, 2007; ICC, 2009). Therefore, the optimization algorithm adopted for

elastic soil-structure systems in Chapter 7 needs to be modified to take into account inelastic

behavior of the structure, which will be addressed in this chapter in detail. In the present

chapter, by performing numerous numerical simulations of responses of inelastic soil-

structure shear buildings with various dynamic characteristics and SSI parameters, the

effects of fundamental period of vibration, ductility demand, earthquake excitation, damping

ratio, damping model, structural post yield behavior, the number of stories, soil flexibility,

structure aspect ratio (slenderness ratio), and soil Poisson’s ratio on the optimum lateral load


167

pattern of soil-structure systems subjected to the same 21 matched earthquake ground

motions as utilized in Chapter 7 are investigated. Based on the results of this study, a new

lateral load pattern for soil-structure systems with inelastic response is proposed which is a

function of the period of the structure, target ductility demand, dimensionless frequency and

structure aspect ratio. It is shown that the structures designed based on the proposed pattern,

on average, display remarkably better seismic performance (i.e., less structural weight and

more uniform damage distribution over height) than the code-compliant and recently

proposed patterns by researchers for fixed-base structures.

8.2 ESTIMATION OF OPTIMUM INELASTIC LATERAL FORCE DISTRIBUTION

FOR SOIL-STRUCTURE SYSTEMS

As shown in Chapter 6, using code-specified load pattern for soil-structures systems with

severe SSI effect and high inelastic response does not lead to uniform (optimum) ductility

demand distribution over the height of structures. This means that the deformation (ductility)

demand in some stories of the building does not reach the presumed target level of seismic

capacity, which indicates that the structural material has not been entirely exploited over the

height of the building. In this section, the optimization algorithm adopted by the writer for

optimum elastic shear-strength distribution of soil-structure systems is modified to take into

account the inelastic behaviour of structures. In this approach, the structural properties are

modified so that inefficient material is gradually shifted from strong to weak parts of the

structure. This process is continued until a state of uniform deformation is achieved

(Hajirasouliha and Moghaddam, 2009). In the present study, the seismic demand parameter

used to quantify the structural damage is the inter-story displacement ductility ratio ( ).

The step-by-step optimization algorithm presented in Chapter 7 is modified for shear-

building soil-structure systems to estimate the optimum inelastic lateral force distribution

under a given earthquake ground motion excitation, the detail steps are given below:

1. Define the MDOF shear-building model depending on the prototype structure height and

number of stories.

2. Assign an arbitrary value for total stiffness and strength and then distribute them along

the height of the structure based on the arbitrary lateral load pattern, e.g., uniform


168

pattern. As mentioned earlier, the lateral story stiffness is assumed as proportional to the

story shear strength distributed over the height of the structure.

3. Select an earthquake ground motion.

4. Consider a presumed set of aspect ratio, H r , and dimensionless frequency, 0a , as the

predefined key parameters for SSI effects.

5. Select the fundamental period of fixed-base structure and scale the total stiffness without

altering the stiffness distribution pattern such that the structure has a specified target

fundamental period. The following equation is used for scaling the stiffness to reach the

target period by just one step:

2

1 1

1

( ) ( ) .( )n n

ij i j i

i it

TK K

T

(8-1)

where jK , iT and argt etT are story stiffness in the jth story, fixed-base period in the ith

step and the target fixed base period, respectively. Refine effective height of the

structure, H based on the fundamental modal properties of fixed-base MDOF structure

(Eq. 2-3).

6. Select an arbitrary target ductility ratio and perform dynamic analysis for the soil-

structure system subjected to the selected ground motion and compute the total shear

strength demand, ( )s iV . If the computed ductility ratio is equal to the target value within

the 0.5% of the accuracy, no iteration is necessary. Otherwise, total base shear strength

is scaled (by either increasing or decreasing) until the target ductility ratio is achieved.

To do this the following equation is proposed:

max1 i( ) ( ) ( )s i s

t

V V

(8-2)

where ( )s iV is the total base shear strength of MDOF system at the ith iteration; t and

max are respectively the target ductility ratio and maximum story ductility ratio among

all stories. Parameter β is an iteration power which is more than zero. As shown in

Chapter 3, β value for 1t (elastic state) can be taken as a constant value for all

MDOF shear-building structures when subjected to any earthquake excitation. For


169

1t (Inelastic state), however, the β power value is generally more dependent on the

fundamental period of the structure, but less dependent on the level of inelasticity and

the earthquake excitation characteristics, and thus usually lower values of β are used

for convergence. It is found that for elastic MDOF shear-building structures a very fast

convergence, i.e. less than 5 iterations, can be obtained for β equal to 0.8. For Inelastic

state ( 1t ) β value, depending on the fundamental period, can be approximately

defined as:

fix

fix

fix

0.05 0.1 T 0.5

0.1 0.25 0.5 <T 1.5

0.25 0.4 T 1.5

(8-3)

7. Calculate the coefficient of variation (COV) of story ductility distribution along the

height of the structure and compare it with the target value of interest which is

considered here 0.02. If the value of COV is less than the presumed target value, the

current pattern is regarded as optimum pattern. Otherwise, the story shear strength and

stiffness patterns are scaled until the COV decreases below or equal to the target value.

8. Stories in which the ductility demand is less than the presumed target value are

identified and their shear strength and stiffness are reduced. To obtain the fast

convergence in numerical computations, the equation proposed by Hajirasouliha and

Moghaddam (2009) for fixed-base systems is revised for soil-structure systems as

follows:

1[ ] [ ] .[ ]ii q i q

t

S S

(8-4)

where [ ]i qS = shear strength of the ith floor at qth iteration, i =story ductility ratio of

the ith floor and = convergence parameter that has been considered equal to 0.1- 0.2

as the acceptable range by Hajirasouliha and Moghaddam (2009) for elastic and

inelastic fixed-base structures. Nevertheless, in Chapter 7, the author showed that for

elastic fixed-base and soil structure systems, the value of 0.8 generally leads to the

fastest convergence (i.e., less than 5 iterations). In addition, it was concluded that in

elastic range of response, there is no fluctuation in convergence problem for the power


170

of α ranging from 0.01 to 1 while the fluctuation happens for α greater than one. Based

on intensive analyses performed in the present study for soil-structure systems with

inelastic response, it is concluded that different from the elastic response, α can be

dependent on earthquake excitation characteristics, soil flexibility and dynamic

properties of structures. It is found that α value is generally more dependent on, in the

order of importance, damping model, earthquake excitation, fundamental period of the

structure and the level of inelasticity, and less dependent on damping ratio, strain

hardening, the number of stories and soil flexibility. Smaller values of α should be used

for fast convergence in inelastic response than that used for elastic response. Results of

this study show that owing to the nature of the nonlinearity, a constant value may not

guarantee achieving the fast convergence for all cases of soil-structure systems. Based

on intensive nonlinear dynamic analyses on shear-building structures in which the

Rayleigh-type damping is used, α= 0.07 for 3t and α= 0.1 for 3t are proposed

for soil-structure systems in inelastic response. After iteration analyses a new pattern

for lateral strength and stiffness distributions is obtained.

9. Control the current maximum story ductility ratio ( max ) and refine the total base shear

strength of soil-structure systems if max is not equal to the target value within the 0.5%

of the accuracy based on Eq. 8-2 of step 6. Otherwise, go to the next step.

10. Control the current fixed-base period and modify it if it is not equal to the target value

within the 1% of the accuracy based on Eq. 8-1 of step 5. Otherwise, go to the next step

11. Control the current effective height ( H ) and refine it if the value is not equal to the

previous value within the 1% tolerance based on Eq. 2-3 (Chapter 3). Otherwise, go to

the next step

12. Control the current Rayleigh-type damping coefficients and modify them if they are not

equal to the previous values within the 1% tolerance. Otherwise, go to the next step

13. Convert the optimum shear strength pattern to the optimum lateral force pattern.

14. Repeat steps 6–13 for different target ductility ratio.


16. Repeat steps 4–15 for different sets of H r and 0a .




171

To show the efficiency of the proposed method for optimum seismic design of soil-structure

systems in inelastic range of response the above algorithm is applied to the 10-story shear

building with fixT = 1.5 sec, H r = 3, and 0 2a subjected to Kobe (Shin Osaka) simulated

earthquake. Figure 8-1a illustrates a comparison of IBC-2009 load pattern with the optimum

patterns of fixed-base and soil-structure systems. As seen, there is a significant difference

between the optimum pattern of soil-structure systems and the other two patterns. These

three patterns are applied to the same 10-story building with consideration of SSI effect and

then the height-wise distribution of story ductility demand resulted from utilizing these

lateral load patterns are computed and depicted in Figure 8-1b. It can be seen that while

using the SSI optimum pattern results in a completely uniform distribution of the

deformation, utilizing both the code-specified and fixed-base optimum patterns lead to a

very non-uniform distribution of ductility demand along the height of the soil-structure

systems in inelastic range of vibration. The COV of story ductility demand distributions

resulted from applying IBC-2009 pattern, the fixed-base optimum pattern and SSI optimum

pattern are 0.94, 0.64 and 0.003, respectively. Similarly, the values of 0.226, 0.196 and

0.003 were obtained for the same earthquake and structure model in Chapter 7 for elastic

response. This indicates that SSI phenomenon through changing the dynamic characteristics

of structures can more significantly affect the damage distribution along the height of

structures in inelastic range of response when compared to that of the elastic state.

Therefore, utilizing fixed-base optimum load pattern may not result in an optimal seismic

performance of soil-structure systems and, thus, a more adequate load pattern accounting for

both SSI effects and inelastic behaviour should be defined and proposed for soil-structure

system. This will be discussed more in the next sections.


172

Figure 8-1: Comparison of IBC-2009 and fixed-base optimum load patterns with optimum

designed models of soil-structure system: (a) lateral force distribution; (b) story ductility pattern,

10-story shear building with fixT = 1.5 sec, H r =3, Kobe (Shin Osaka) simulated earthquake

8.3 EFFECT OF STRUCTURAL DYNAMIC CHARACTERISTICS AND SSI KEY

PARAMETERS ON OPTIMUM INELASTIC LATERAL FORCE PATTERN


To study the effect of fundamental period on optimum load pattern for inelastic soil-

structure systems, the 10-story building models with H r = 3 and 0a = 2 having fixed-base

fundamental periods of 0.5, 1, 2 and 3 sec are considered. The results are shown for two

ranges of nonlinearity ( = 2, 6) representing the low and high level of inelasticity,

respectively. For each case, the optimum load patterns are derived for the 21 matched

earthquake ground motions listed in Table 7-1 and the average results are plotted in Figure

8-2. As seen, the averaged optimum load pattern is significantly dependent on the

fundamental period of vibration for both low and high levels of nonlinearity. Nevertheless,

this effect is somewhat different for two ranges of inelastic response. In low level of

nonlinearity (i.e., µ= 2), increasing the fundamental period is mostly accompanied by

increasing the lateral shear force at top stories which can be interpreted as the effect of

higher modes, and only for the case of very long period ( fixT =3 sec) the lateral force at both

the top and bottom stories increase. In high level of nonlinearity (i.e., µ= 6), however,

IBC-2009 Fixed-base

SSI, = 3

Sto

ry

Lateral Force / Base Shear Ductility

1

2

3

4

5

6

7

8

9

10

0 0.05 0.1 0.15 0.2 0.25 0.3

(a) µ= 6

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7

(b) µ= 6


173

increasing the fundamental period is generally accompanied by increasing the lateral shear

force at both the top and bottom stories for all cases, and the increase in the lateral forces at

the bottom stories is more significant than that at the top stories. Previous studies carried out

by Hajirasouliha and Moghaddam (2009) on fixed-base shear-building structures showed

that increasing the fundamental period is only accompanied by increasing the shear strength

in top stories, which is consistent with the results of elastic soil-structure systems presented

in Chapter 7. However, the results obtained here indicate that SSI can affect the optimum

distribution of the load pattern in inelastic response in a different way when compared with

fixed-base patterns.

8.3.2 Effect of target ductility demand

Figure 8-3 shows the effect of target ductility demand on averaged optimum load pattern of

soil-structure systems in inelastic response subjected to 21 matched ground motions. For this

purpose, the 10-story shear-building models with H r = 3 and 0a = 2, fixed-base

fundamental periods of 0.5 and 2 sec respectively representing the rigid and flexible

structures and target ductility demands of 1, 2, 4 and 6 are considered. As seen, for both

rigid and flexible models the averaged optimum lateral load patterns are significantly

dependent on the target ductility demand while nearly in all code-specified seismic load

patterns for both fixed-base and soil-structure systems this parameter is not considered. It

can also be seen that for soil-structure systems increasing the target ductility demand results

in a decrease or increase in the story shear strength in top or bottom stories, respectively.

This conclusion is consistent with the finding of Hajirasouliha and Moghaddam (2009) for

fixed-base shear-building structures.


174

Figure 8-2: Effect of fundamental period on averaged optimum lateral force profile for soil-

structure systems with H r =3 and 0a = 2: 10-story building (average of 21 earthquakes)

Figure 8-3: Effect of target ductility demand on averaged optimum lateral force profile for soil-

structure systems with H r =3 and 0a = 2: 10-story building (average of 21 earthquakes)


To examine the effect of number of stories on the optimum distribution profile, the proposed

optimization algorithm is applied to 5-, 10-, 15- and 20-story soil-structure models with fixT

= 1.5, H r = 3 and 0a = 2 subjected to the 21 matched earthquake ground motions. The

average results are depicted in Figure 8-4. In order to compare the averaged optimum

patterns corresponding to different number of stories, the normalized lateral loads are

plotted. In Fig. 8-4 which is the same format as Figure 7-7b of Chapter 7, the vertical and

horizontal axes are relative height and normalized lateral load divided by base shear

1

2

3

4

5

6

7

8

9

10

-0.05 0.05 0.15 0.25 0.35

µ = 2

T= 0.5

T= 1

T= 2

T= 3

1

2

3

4

5

6

7

8

9

10

0 0.05 0.1 0.15 0.2 0.25

µ = 6

T= 0.5

T= 1

T= 2

T= 3

Sto

ry


1

2

3

4

5

6

7

8

9

10

0 0.05 0.1 0.15 0.2

(a) Tfix = 0.5

µ= 1

µ= 2

µ= 4

µ= 6

1

2

3

4

5

6

7

8

9

10

0 0.1 0.2 0.3 0.4

(b) Tfix = 2

µ= 1

µ= 2

µ= 4

µ= 6


Sto

ry


175

strength, respectively. From this figure, it can be concluded that the optimum load patterns

are almost independent of the number of stories. This finding is also consistent with those

by Hajirasouliha and Moghadam (2009) for fixed-base shear-building structures and the

results of previous chapter for elastic soil-structural systems.

Figure 8-4: Effect of the number of stories on averaged optimum lateral force profile for soil-

structure systems with H r =3 and 0a = 2: fixT = 1.5 sec. (average of 21 earthquakes)


Figure 8-5 shows the effect of dimensionless frequency, 0a on averaged optimum load

pattern for soil-structure systems subjected to 21 matched ground motions. As stated before,

aspect ratio and dimensionless frequency are two key parameters that can affect the response

of the soil-structure systems subjected to earthquake excitation. To demonstrate the

influence of dimensionless frequency, results of the 10-story shear building with two

fundamental periods of 0.5 and 2 sec and H r =3 corresponding to three values of

dimensionless frequency ( 0a =1, 2, 3) as well as the fixed-base structure are plotted. It is

observed that in both rigid and flexible structures, changing the dimensionless frequency can

Rel

ativ

e H

eigh

t

Normalized Lateral Force / Base Shear

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.05 0.1 0.15 0.2

µ = 4

N= 5

N= 10

N= 15

N= 20


176

significantly affect the averaged optimum load pattern in inelastic range of response. For

rigid model (i.e., fixT = 0.5) increasing the value of dimensionless frequency results in an

increase in the lateral load at top stories and a decrease in the load at bottom stories while

for flexible structures increasing the dimensionless frequency is accompanied by increasing

the lateral load at both the bottom and top stories, and decreasing the load in middle stories.

This phenomenon could be again due to the effect of higher mode effect as a result of

increasing the fundamental period of the soil-structure systems.


Figure 8-6 shows the effect of aspect ratio on averaged optimum load pattern of soil-

structure systems. The results are for the 10-story shear building with fixT =1.5 sec, ductility

demand of 4, two dimensionless frequencies ( 0a =1, 3), and three values of aspect ratio (

H r = 1, 3, 5) subjected to 21 matched ground motions. As seen, for the case of

insignificant SSI effect (i.e., Figure 8-6a), increasing the aspect ratio will not change the

optimum load profile noticeably. However, by increasing the dimensionless frequency and,

therefore more significant SSI effect, the aspect ratio will significantly affect the averaged

optimum load pattern. With severe SSI effect, the trend is to some extent similar to that of

the dimensionless frequency discussed in the previous section such that increasing the value

of aspect ratio is accompanied by increasing the lateral load at bottom and top stories, and

decreasing the load in the middle stories, which is more pronounced for slender buildings

(i.e. H r = 5). It can be concluded that SSI effect on optimum lateral load pattern will

become more significant for the case of slender building with larger dimensionless

frequency. Nearly the same conclusion has been drawn for the elastic soil-structure systems

in Chapter 7.


177

Figure 8-5: Effect of dimensionless frequency on averaged optimum lateral force profile for 10-

story soil-structure systems with H r =3, µ= 6: (a) fixT = 0.5 sec.: (b)

fixT = 2 sec.

Figure 8-6: Effect of aspect ratio on averaged optimum lateral force profile for a 10-story soil-

structure system with fixT = 1.5 sec, µ= 4

Fixed-base a0 = 1

a0 = 2

a0 = 3


Rel

ativ

e H

eig

ht

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.05 0.1 0.15 0.2

(a) Tfix = 0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.05 0.1 0.15 0.2 0.25

(b) Tfix = 2

Rel

ativ

e H

eig

ht


0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.05 0.1 0.15 0.2 0.25

µ= 4, a0 = 1

H/r =1

H/r = 3

H/r = 5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.05 0.1 0.15 0.2 0.25 0.3

µ= 4, a0 = 3

H/r =1

H/r = 3

H/r = 5


178

8.3.6 Effect of structural damping ratio and damping model

The effect of structural damping ratio on inelastic optimum load pattern of soil-structure

systems is illustrated in Figure 8-7a for the 10-story shear-building structure with fixT = 1.5,

µ= 6, H r = 3 and 0a = 2 corresponding to four values of 0.5%, 3%, 5% and 10% of

damping ratio subjected to matched Loma Prieta earthquake (APEEL 2 - Redwood City).

As seen, inelastic optimum lateral load pattern is not significantly sensitive to the variation

of the damping for soil-structure systems. Therefore, one may conclude that for the practical

purpose, the optimum load pattern of elastic soil-structure systems can be considered

insensitive to the variation of damping ratio.

To investigate the effect of damping models on inelastic optimum load pattern of soil-

structure systems, three conventional viscous damping models including stiffness-

proportional damping, mass-proportional damping and Rayleigh-type damping in which

damping matrix is composed of the superposition of a mass-proportional damping term and

a stiffness-proportional damping term are considered. In this case, optimum lateral load

pattern of the same soil-structure model as shown in Figure 8-7a corresponding to the

aforementioned damping models subjected to Loma Prieta earthquake (APEEL 2 - Redwood

City) are computed and plotted in Figure 8-7b. It can be seen that that while there is no

significant difference between the results of mass-proportional and Rayleigh-type damping

models, the difference is pronounced when compared to that of the stiffness-proportional

damping model. The same result is also obtained for other inelastic fixed-base and soil-

structure models subjected to different seismic ground motions. This observation indicates

that stiffness-proportional damping model may lead to quite different predictions of

structural responses as compared to the Rayleigh damping model. As demonstrated in

Chapters 6 and 7, it is, therefore, advisable to use Rayleigh-type damping model to better

incorporate the effect of higher modes. As stated in the previous sections, while the

convergence problem is not very sensitive to the variation of damping ratio, it is to a large

extent sensitive to the type of the damping modeling in inelastic rang of response. In

addition, results of this study indicate that generally more iteration steps are required for

optimization of structures designed based on mass-proportional and Rayleigh-type damping

models with respect to the stiffness-proportional damping model. It is also found that the

suitable values of convergence power, α, need to be decreased when structures are designed

respectively based on stiffness-proportional, Rayleigh-type and mass-proportional damping


179

models. As an instance for the case of µ= 3, the suitable values of α, are approximately 0.2,

0.07 and 0.05 for the cases of stiffness-proportional, Rayleigh-type and mass-proportional

damping models. This phenomenon has not been seen in elastic soil-structure systems as

studied in Chapter 7.

Figure 8-7: Optimum lateral force profile for a 10-story soil-structure system with H r =3, 0a =

2, fixT = 1.5 sec and µ= 6: (a) Effect of structural damping ratio; (b) Effect of structural damping

model, Loma Prieta (APEEL 2 - Redwood City) simulated earthquake

8.3.7 Effect of structural strain hardening

Effect of different structural strain hardening (SH) values on the optimum shear strength

distribution for soil-structure systems is presented in Figure 8-8. The results are plotted for

the same soil-structure model and earthquake ground motion record as Figure 8-7 but for 2

different ductility demands of 2 and 6. For this case, 4 different strain hardening values of

0%, 2%, 5% and 10% have been considered. It can be seen that while the optimum load

profile is not sensitive to the variation of the structural strain hardening in low level of

inelasticity (µ= 2), it can be more sensitive to the secondary slope of post-yield response of

the soil-structure systems with high level of nonlinearity (µ= 6).


Rel

ativ

e H

eig

ht


0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.05 0.1 0.15 0.2

(a) µ= 6

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.05 0.1 0.15 0.2

(b) µ= 6

Stiffness

Mass

Rayleigh


180

Figure 8-8: Effect of structural post yield behavior on Optimum lateral force profile for a 10-

story soil-structure system with H r =3, 0a = 2, fixT = 1.5 sec; Loma Prieta (APEEL 2 -

Redwood City) simulated earthquake

8.3.8. Effect of soil Poisson’s ratio

Figure 8-9 shows the effect of soil Poisson’s ratio on optimum load pattern of soil-structure

systems. The results are for the 10-story shear building with fixT =1.5 sec, two ductility

demands of 2 and 6, dimensionless frequencies of 3 and aspect ratio 3 subjected to Loma

Prieta earthquake (APEEL 2 - Redwood City). It can be observed that optimum shear

strength profile can be more sensitive to the variation of Poisson’s ratio in low level of

nonlinearity as compared to the high level of nonlinearity. Therefore, one may conclude that

for structures with high inelastic response, optimum lateral load pattern could be considered

as independent of the soil Poisson’s ration for the practical purpose.

8.3.9. Effect of earthquake excitation

To examine the effect of earthquake ground motion variability on optimum lateral force

pattern for soil-structure models in inelastic response, individual results of all 21 matched

earthquake ground motions listed in Table 7-1 along with their mean values for a 20- story

shear building are presented in Figure 8-10a. The results are for systems with fixT = 2 sec, an

aspect ratio of 3, and the dimensionless frequency of 2. As seen, it is obvious that the

optimum strength distribution pattern in some cases is sensitive to the earthquake ground

motion characteristics. However, in most ground motions used in this study, there is not a

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.05 0.1 0.15 0.2

µ= 2

SH= 0

SH= 2%

SH= 5%

SH= 0.1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.05 0.1 0.15 0.2

µ= 6

SH= 0

SH= 2%

SH= 5%

SH= 0.1


Rel

ativ

e H

eig

ht



181

big discrepancy in the general pattern of the optimum strength distribution when compared

to the corresponding averaged pattern. Therefore, like elastic soil-structure systems

discussed in Chapter 7, it is expected that utilizing the mean pattern will lead to acceptable

designs although some inevitable variation is not avoidable depending on the earthquake

ground motions.

Effect of ground motion intensity on the optimum load profile of the 10-story soil-structure

model with fixT = 1.5, H r = 3, and 0 2a subjected to Kobe (Shin Osaka) simulated

earthquake with the PGA multiplied by 0.5, 1, 2, and 3 are illustrated in Figure 8-10b. The

results indicate that for a specific fundamental period, aspect ratio and dimensionless

frequency, the optimum lateral load pattern is completely independent of the ground motion

intensity factor (SF), which is consistent with the findings of Mohammadi et al., (2004) and

Hajirasouliha and Moghaddam (2009) for fixed-base shear-building structures.

Figure 8-9: Effect of soil Poisson ratio on Optimum lateral force profile for a 10-story soil-

structure system with H r =3, 0a = 3, fixT = 1.5 sec; Loma Prieta (APEEL 2 - Redwood City)

simulated earthquake


Rel

ativ

e H

eig

ht

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.05 0.1 0.15 0.2 0.25 0.3

µ= 2, a0= 3

ν = 0.1

ν= 0.25

ν = 0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.05 0.1 0.15 0.2 0.25 0.3

µ= 6, a0= 3

ν = 0.1

ν= 0.25

ν = 0.5


182

Figure 8-10: Effect of (a) Earthquake excitation and (b) ground motion intensity on optimum

lateral force profile for soil-structure systems with H r =3 and 0a = 2, µ= 4; Kobe (Shin Osaka)

simulated earthquake

8.4. NEW SEISMIC LOAD PATTERN FOR SOIL-STRUCTURE SYSTEMS WITH

INELASTIC BEHAVIOR

As stated in the previous chapter (Chapter 7), to generalize the use of the proposed

optimization algorithm for conceptual seismic design of soil-structure systems, it is

necessary to develop statistical models for estimating the optimum design lateral load

pattern as a function of relevant structural and soil characteristics. More reliable load

pattern, then, can be obtained by commuting the mean values of optimum patterns

associated to the design earthquakes compatible with each seismic design spectrum. To be

consistent, 21 seismic ground motions compatible to the elastic design response spectrum of

IBC-2009 with soil type E as listed in Table 7-1 are selected. Numerous soil-structure

systems including four different number of stories (i.e., 5-, 10, 15 and 20-story buildings),

28 fundamental periods ranging from 0.3 to 3 sec with intervals of 0.1, 3 values of ductility

demand (µ=2, 4, 6), three values of aspect ratio ( H r =1, 3, 5), and two values of

dimensionless frequency ( 0a =1, 3) are considered. Consequently, utilizing the proposed

optimization algorithm, nearly 42330 optimum lateral load patterns considering inelastic

behaviour are derived for soil-structure systems. For each fundamental period,

dimensionless frequency, aspect ratio and ductility demand ratio, the mean optimum load

Sto

ry


123456789

1011121314151617181920

0 0.05 0.1 0.15 0.2

(a) Tfix= 2

Individual Results

Average

1

2

3

4

5

6

7

8

9

10

0 0.05 0.1 0.15 0.2

(b) Tfix= 1.5

SF= 0.5

SF=1

SF=2

SF=3


183

pattern corresponding to 21 matched earthquake ground motions is obtained. It is expected

that designs based on the mean patterns would exhibit a more uniform damage (ductility

demand) along the height of soil-structure systems. Based on the results of this study and

nonlinear statistical regression analysis, the previously proposed optimum load pattern

expression for elastic soil-structure systems as presented in Chapter 7 is modified to

incorporate the effect of structural inelastic behaviour as follows:

( 0.9 )0.75. ( ) e fixT

i i i fix fix iF a b T Ln T c (8-5)

where iF = optimum load component at the ith story; fixT = fixed-base fundamental period;

and ia , ib and ic = constant coefficients of the ith story which are functions of aspect ratio (

H r ), dimensionless frequency ( 0a ) and inter-story ductility demand (µ) are given in

Tables 8-1 to 8-3 for each level (relative height) of structure. It should be noted that like Eq.

7-4, iF is the optimum lateral load component that must be scaled to the total load

components at the end of calculation. In addition, the optimum load patterns corresponding

to values of the relative height, aspect ratio, dimensionless frequency and ductility demand

corresponding to the specified values in Tables 8-1 to 8-3 can be easily obtained by linear

interpolation of the two associated load patterns.

To show the efficiency of the proposed expression on estimating the optimum lateral force

distribution for soil-structure systems with inelastic behaviour Figure 8-11 is plotted. This

figure shows the comparison of the proposed equation in predicting the optimum load

pattern with the average numerical results. As shown, there is an excellent agreement

between Eq. (8-5) and the average numerical results to estimate the optimum load pattern in

soil-structure systems with inelastic behaviour corresponding to different sets of structural

and soil parameters.


184

Table 8-1: Constant coefficients of Eq. (8-5) as function of relative height (µ= 2)

0a = 1 H r = 1 H r = 3 H r = 5

Relative Height ia

ib ic

ia ib

ic ia

ib ic

0.05 19.76 8.75 27.96 11.68 13.35 36.60 8.21 17.04 39.55

0.10 19.15 7.51 29.38 11.47 11.49 37.10 8.15 14.35 39.93

0.20 21.83 4.28 25.24 19.73 3.70 23.58 17.08 5.01 26.12

0.30 24.28 1.89 23.23 26.70 -0.84 14.99 28.38 -3.90 10.03

0.40 27.32 -0.78 21.00 32.64 -5.84 8.73 38.26 -9.81 -1.48

0.50 34.60 -4.11 13.18 39.68 -9.18 2.63 44.74 -13.84 -6.08

0.60 46.16 -8.30 -0.24 49.79 -12.98 -6.98 54.05 -17.97 -14.46

0.70 60.91 -11.31 -16.88 60.12 -12.14 -13.32 59.47 -13.31 -11.39

0.80 74.53 -8.04 -26.45 71.98 -5.13 -17.15 68.71 -2.80 -9.66

0.90 92.15 1.78 -35.11 90.23 7.60 -24.11 87.50 11.74 -16.63

1.00 131.07 22.68 -56.34 132.11 30.04 -48.70 129.92 37.19 -41.47

0a = 3 ia

ib ic

ia ib

ic ia

ib ic

0.05 33.50 16.73 22.65 43.17 32.16 -1.73 86.10 21.00 -70.91

0.10 29.50 12.38 25.90 28.44 30.31 19.37 63.02 23.53 -37.70

0.20 29.19 4.16 21.86 9.94 25.25 46.71 25.71 27.89 20.15

0.30 27.82 -0.76 21.87 13.73 9.14 37.49 14.87 17.09 34.36

0.40 31.13 -6.40 16.02 16.69 -7.79 30.20 -1.79 -5.36 53.93

0.50 35.38 -11.20 10.00 30.85 -26.20 7.39 3.72 -26.61 43.26

0.60 41.04 -13.13 3.95 36.36 -22.58 8.62 18.98 -22.27 32.40

0.70 48.38 -11.84 -1.83 47.96 -14.22 4.59 18.98 -22.27 32.40

0.80 64.59 -7.14 -17.17 72.69 -11.42 -19.51 68.00 -8.50 -7.86

0.90 89.50 6.24 -38.63 105.01 0.05 -47.62 107.51 -1.61 -44.08

1.00 137.64 30.90 -75.79 176.20 14.08 -120.32 195.70 3.80 -138.93


185


0a = 1 H r = 1 H r = 3 H r = 5

Relative Height ia ib

ic ia

ib ic

ia ib

ic

0.05 34.58 16.48 23.34 27.66 22.04 32.03 22.29 25.69 38.79

0.10 31.59 13.70 26.23 26.94 16.14 30.51 23.52 18.26 33.32

0.20 29.49 7.69 26.52 25.93 8.46 28.81 22.66 9.16 31.25

0.30 28.62 4.28 26.46 25.18 3.79 27.93 22.60 3.30 29.34

0.40 28.74 1.43 24.89 26.58 0.01 24.89 27.70 -2.45 20.78

0.50 32.40 -2.01 19.10 32.51 -4.47 16.31 33.53 -6.99 12.90

0.60 41.26 -6.91 5.44 40.61 -7.57 6.01 41.86 -9.66 3.98

0.70 53.47 -10.17 -11.13 53.59 -10.56 -9.49 55.03 -11.79 -10.82

0.80 66.76 -9.63 -25.45 69.98 -10.36 -26.91 72.42 -10.27 -27.52

0.90 85.31 -6.69 -43.21 91.09 -5.26 -45.22 91.89 -3.04 -41.44

1.00 133.12 5.90 -80.06 143.14 6.05 -89.06 144.81 10.38 -87.37

0a = 3 ia

ib ic

ia ib

ic ia

ib ic

0.05 43.95 24.15 17.84 54.14 35.58 -7.15 83.83 32.70 -56.30

0.10 38.78 18.66 21.89 34.50 31.80 18.86 57.41 31.28 -20.98

0.20 34.69 9.15 22.29 14.09 21.79 44.55 17.88 27.43 34.45

0.30 32.28 3.74 22.72 11.02 8.18 44.98 3.80 13.49 52.56

0.40 33.27 -1.34 18.71 19.75 -5.08 29.46 4.80 -6.62 46.04

0.50 35.86 -5.47 13.65 31.37 -12.37 13.92 14.49 -14.80 35.45

0.60 40.38 -8.89 6.84 41.21 -15.80 4.28 29.69 -16.94 20.60

0.70 48.30 -11.64 -3.87 51.44 -13.83 -2.48 48.69 -14.98 3.75

0.80 59.61 -11.44 -17.21 68.95 -12.88 -18.92 72.82 -14.14 -19.71

0.90 78.46 -6.27 -36.74 97.39 -9.71 -47.97 102.44 -9.35 -46.82

1.00 126.75 8.70 -82.37 159.51 0.91 -115.00 176.30 -4.07 -128.67


186


0a = 1 H r = 1 H r = 3 H r = 5

Relative Height ia ib

ic ia

ib ic

ia ib

ic

0.05 44.80 25.06 19.66 40.29 28.37 24.98 36.47 30.66 28.87

0.10 41.88 19.53 20.70 37.23 22.01 25.72 33.53 23.00 28.92

0.20 36.68 12.33 23.63 33.00 12.60 26.23 29.84 12.90 28.89

0.30 36.60 5.97 19.40 33.04 5.94 22.39 29.44 6.19 25.84

0.40 36.79 1.42 15.89 32.43 1.79 20.67 30.60 0.96 21.72

0.50 36.01 -1.32 15.33 33.81 -1.78 16.91 32.64 -2.78 17.65

0.60 38.12 -3.61 11.41 36.47 -4.00 13.35 37.50 -5.39 11.01

0.70 43.31 -6.02 3.55 45.56 -7.52 0.75 47.47 -8.24 -1.27

0.80 56.02 -9.97 -13.82 60.61 -11.16 -18.52 64.16 -11.71 -21.51

0.90 74.83 -13.53 -38.37 82.78 -14.62 -46.06 87.77 -14.58 -49.58

1.00 128.66 -16.94 -95.18 137.16 -15.57 -101.52 140.96 -12.70 -102.85

0a = 3 ia

ib ic

ia ib

ic ia

ib ic

0.05 52.88 27.93 10.60 59.66 37.33 -9.15 84.95 37.36 -51.15

0.10 47.16 21.98 14.78 39.80 32.08 16.47 57.14 33.85 -15.11

0.20 42.02 11.63 15.66 19.18 20.31 40.40 24.13 22.67 26.39

0.30 38.77 4.95 16.38 18.68 6.37 35.70 6.21 9.46 49.57

0.40 39.27 -0.73 12.19 23.86 -3.15 26.18 6.32 -2.32 47.66

0.50 38.15 -3.45 12.11 31.93 -8.93 15.01 19.38 -10.18 31.18

0.60 38.26 -5.66 11.14 38.22 -10.77 9.90 28.70 -11.09 24.40

0.70 41.77 -7.17 6.19 47.71 -11.01 2.06 45.22 -11.78 8.53

0.80 49.48 -8.96 -3.35 65.58 -12.46 -16.61 70.26 -14.58 -18.85

0.90 68.20 -11.67 -28.14 92.38 -14.11 -47.23 99.22 -15.05 -48.80

1.00 120.22 -11.98 -89.63 151.01 -11.59 -115.13 168.56 -14.46 -129.40


187

Figure 8-11: Correlation between Eq. (8-5) and numerical results

8.5 ADEQUACY OF PROPOSED OPTIMUM INELASTIC LATERAL LOAD

PATTERN

In this section the adequacy of the proposed load pattern (Eq.8-5), IBC-2009 pattern and

three recently proposed optimum load patterns for fixed-base structures including those

previously proposed by Mohammad al., (2004), Park and Medina (2007) and Hajirasouliha

and Moghaddam (2009) are investigated for soil-structural systems by weight-based method

as explained in Chapter 6. This method is, generally, related to the economy of the seismic-

resistant systems. As stated before, the loading pattern that corresponds to the minimum

weight index (i.e., structural weight) is considered as the most adequate loading pattern

(optimum pattern). Moghaddam and Mohammdi (2006), and Hajirasouliha and Moghaddam

(2009) showed that decreasing the weight index (structural weight) is always accompanied

by a reduction of the COV (i.e., more uniform damage distribution over height). Therefore,

the value of weight index is directly related to the optimum lateral load pattern.


Rel

ativ

e H

eig

ht

1

2

3

4

5

6

7

8

9

10

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

1

2

3

4

5

6

7

8

9

10

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Tfix= 3, µ= 4, a0= 3, = 3 (Eq. 8-5)

Tfix= 3, µ= 4, a0= 3, = 3

Tfix= 1, µ= 2, a0= 1, = 5 (Eq. 8-5)

Tfix= 1, µ= 2, a0= 1, = 5

Tfix= 0.6, µ= 6, a0= 3, = 5 (Eq. 8-5)

Tfix= 0.6, µ= 6, a0= 3, =

5 Tfix= 2, µ= 2, a0= 1, = 1 (Eq. 8-5)

Tfix= 2, µ= 2, a0= 1, = 1


188

The efficiency of each load pattern can be investigated by comparing the structural weight

index resulted from each of the above-mentioned load patterns with that of the optimum

load pattern corresponding to a given earthquake ground motion. For this purpose, the

values of weight index of the 10-story shear buildings designed based on the five above-

mentioned load patterns for 28 fundamental periods ranging from 0.3 to 3 sec, two values of

ductility demand (µ=2, 6), three values of aspect ratio ( H r =1, 3, 5) and two values of

dimensionless frequency ( 0a =1, 3) are calculated subjected to 21 matched earthquake

ground motions. Then, the ratio of averaged values of weight index associated to all the

patterns to those related to the optimum pattern are computed and illustrated in Figures 8-12

and 8-13. Based on the results presented in these figures, it is observed that:

1. For short periods of vibration and regardless of the level of nonlinearity and SSI

effect, structures designed in accordance to the optimum load pattern proposed by

Hajirasouliha and Moghaddam (2009), generally, has the best performance among

all load patterns, except for the proposed pattern in this study. However, this pattern

remarkably loses its efficiency for structures with moderate and long vibration

period even in low inelastic response or when SSI effect is not significant (i.e., 0a

=1). As an instance, for the case with fixT = 2, µ= 2, 0a =1 and H r =3, the mean

value of weight index corresponding to this load pattern is 65% above that of the

optimum pattern.

2. For all ranges of period, nonlinearity and SSI effects considered in this study, the

load pattern proposed in this study gives superior results in comparison to the results

of all other load patterns. The superiority is more pronounced for the cases of

slender buildings with longer periods and severe SSI effects. As seen, the ratios of

the required to the optimum structural weight index for models designed with the

proposed pattern of Eq. 8-5 are, on average, from 1.03 to 1.25 which can be

considered as near optimum for practical purposes.

3. The loading patterns proposed by Moahammadi et al., (2006), on average, give good

results for structures with low level of nonlinearity and severe SSI effect. This could

be because of the large value of seismic load in roof story as defined in Eq. 6-1. As

shown in previous section, increasing the dimensionless frequency will result in an

increase in the seismic load in top stories. Nevertheless, increasing the level of


189

nonlinearity will accompanied by a decrease in the seismic load in top stories and

increasing the seismic load in bottom stories, which result in losing the efficiency of

the optimim load pattern by Moahammadi et al., (2004) in high level of inelasticity

(i.e., µ= 6). Although the optimum load pattern proposed by Hajirasouliha and

Moghaddam (2009) is based on modification of the load pattern proposed by

Mohammadi et al., (2004) and Mohammadi and Moghaddam (2006), the result of

this study show that the proposed load pattern by Mohammadi et al. (2004), except

for short periods, has remarkably better seismic performance in soil-structure

systems when compared to the optimum load pattern proposed by Hajirasouliha and

Moghaddam (2009).

4. For the cases with squat and average structures and high level of nonlinearity, the

load pattern proposed by Park and Medina (2007) has generally good seismic

performance with respect to other load patterns. The values of structural weight

index are usually not more than 40% of the optimum values. However, its efficiency

diminishes for slender structures and low level of nonlinearity.

5. Except for cases of short periods with severe SSI effect and low level of

nonlinearity, the efficiency of the code-compliant load pattern (IBC, 2009)

significantly diminishes.

6. Generally, by increasing the aspect ratio and dimensionless frequency (i.e.,

increasing SSI effects) the three previously proposed optimum load patterns for

fixed-base structures as well as the code-specified load pattern significantly lose

their efficiency while the proposed load pattern in this study displays superior

seismic performance, especially for slender buildings ( H r =5) with long periods

and predominant SSI effect ( 0a =3). As an example, for the cases of slender building

with severe SSI effect with fixT = 3 sec, the values of structural weight for structures

designed with the proposed load pattern (i.e., Eq. 8-5 ) are respectively 52%, 46%,

36.3% and 22.9% less than those of Hajirasouliha and Moghaddam (2009), IBC-

2009, Park and Medina (2007) and Mohammadi et al., (2004) in low inelastic

response, and 47%, 39%, 27%, and 27% in high inelastic response. This implies

that significant improvement is achieved by utilizing the proposed load pattern of

this study for soil-structure systems with inelastic behavior.


190

Figure 8-12: The spectra of ratio of the required to optimum structural weight for the 10-story

soil-structure systems designed according to different patterns; average of 21 earthquakes (µ= 2)

Figure 8-13: The spectra of ratio of required to optimum structural weight for the 10-story soil-

structure systems designed according to different load patterns; average of 21 earthquakes (µ= 6)

Tfix

Tfix

Tfix

WIi

/ W

IOP

T

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0 1 2 3

a0= 1, H/r=1

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0 1 2 3

a0= 1, H/r=3

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0 1 2 3

a0= 1, H/r=5 W

Ii / W

IOP

T

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0 1 2 3

a0= 3, H/r =1

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0 1 2 3

a0= 3, H/r=3

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0 1 2 3

a0= 3, H/r=5

Mohammadi et al (Eq. 6-1)


10) IBC-2009


Proposed (Eq. 8-5)

Tfix Tfix Tfix

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0 1 2 3

a0= 1, H/r= 1

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0 1 2 3

a0= 1, H/r= 3

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0 1 2 3

a0= 1, H/r= 5

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0 1 2 3

a0= 3, H/r= 1

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0 1 2 3

a0= 3, H/r= 3

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0 1 2 3

a0= 3, H/r= 5

WIi

/ W

IOP

T

WIi

/ W

IOP

T

Mohammadi et al (Eq. 6-1)


10) IBC-2009


Proposed (Eq. 8-5)


191

8.6 CONCLUSIONS

An optimization algorithm for optimum seismic design of shear-buildings with SSI and

inelastic behaviour has been developed in the present chapter. The adopted method is

based on the concept of uniform damage distribution proposed by Mohammadi et al.,

(2004) and Hajirasouliha and Moghadam (2009) for fixed-base shear building structures.

Based on extensive numerical analyses of structural models with different structural and

foundation conditions subjected to 21 selected earthquake ground motions iterated to be

compatible to the IBC-2009 design spectrum, an optimum load pattern is proposed for

structural design with consideration of SSI effect and inelastic behaviour. From the

numerical results obtained in this study, the following conclusions can be drawn:

1. The optimum load pattern for shear-building structures with SSI effect in inelastic

range of response is highly dependent on the fundamental period, target ductility

demand, dimensionless frequency, aspect ratio, seismic excitation (frequency

content) and structural damping model, less dependent on structural post-yield

behaviour and soil Poisson’s ratio, and almost independent of the number of stories,

structural damping ratio and earthquake intensity.

2. While the convergence problem is not very sensitive to the variation of damping

ratio, it is to a large extent sensitive to the type of the damping modeling in inelastic

rang of response. Generally more iteration steps are required for optimization of the

structures designed based on mass-proportional and Rayleigh-type damping models

than the stiffness-proportional damping model. In addition, similar to the optimum

design in elastic range of response presented in Chapter 7, although there is no

significant difference between the optimum lateral load profiles of mass-

proportional and Rayleigh-type damping models, the difference is pronounced when

compared to that of the stiffness-proportional damping model. The stiffness-

proportional damping model for shear-building models may not lead to reliable

prediction of structural responses.

3. A new lateral force pattern which is a function of fixed-base fundamental period,

target ductility demand, dimensionless frequency and structure aspect ratio (slender

ratio) has been proposed for soil-structure systems with inelastic response. It has

been shown that the structures designed based on the proposed pattern, on average,


192

display remarkably better seismic performance (i.e., less structural weight and more

uniform damage distribution over height) than the code-compliant and recently

proposed patterns by researchers for fixed-base structures.

4. Except for short periods, generally, the optimum load patterns proposed by

Mohammadi et al., (2004) and Park and Medina (2007) lead to remarkably better

seismic performance in inelastic soil-structure systems with respect to the proposed

load patterns by IBC-2009 and Hajirasouliha and Moghaddam (2009).

5. Overall, by increasing the aspect ratio and dimensionless frequency (i.e., increasing

SSI effects) the three previously proposed load patterns for fixed-base structures as

well as the code-specified load pattern significantly lose their efficiency while the

proposed load pattern in this study still results in superior seismic performance,

especially for slender buildings ( H r =5) with long periods and predominant SSI

effect ( 0a =3). Using the proposed load pattern in this study, the designed structures

experience up to 52%, 45%, 27% and 36% less structural weight as compared with

the proposed patterns by Hajirasouliha and Moghaddam (2009), IBC-2009,

Mohammadi and Moghaddam (2006) and Park and Medina (2007).


193

8.7. REFERENCES




Building Seismic Safety Council (BSSC) (2003). National Earthquake Hazard Reduction




BCJ. (1997). Structural provisions for building structures. 1997 edition – Tokyo;

Building Center of Japan.



Standardization.

Hajirasouliha, I., and Moghaddam, H. (2009). ―New lateral force distribution for seismic

design of structures.‖ Journal of Structural Engineering, ASCE, 135(8), 906–915.


AL.


Moghaddam, H., and Mohammadi, R. K. (2006). ―More efficient seismic loading for

multidegrees of freedom structures.‖ Journal of Structural Engineering, ASCE,

132(10), 1673–1677.

Mohammadi. K. R., El-Naggar, M. H., and Moghaddam, H. (2004). ―Optimum strength

distribution for seismic resistant shear buildings.‖ International Journal of Solids and

Structures. 41(21-23), 6597–6612.

Park, K., and Medina, R. A. (2007). ―Conceptual seismic design of regular frames based on

the concept of uniform damage‖ Journal of Structural Engineering, ASCE, 133(7),

945-955.




194

Chapter 9 CONCLUDING REMARKS

9.1 MAIN FINDINGS

This thesis has focused on comprehensive parametric studies of the effect of soil-

structure interaction (SSI) on elastic and inelastic response of MDOF and its equivalent

SDOF (E-SDOF) systems subjected to a large numbers of earthquake ground motions. In

this effort, the influence of the SSI effects on structural response parameters such as the

strength and ductility demand of MDOF and E-SDOF systems, strength reduction factor

of MDOF and E-SDOF systems, structural property distributions, height-wise

distribution of story ductility demand, and, more importantly, optimum elastic and

inelastic lateral force distribution along the height of the shear buildings are intensively

investigated. The major contributions and findings of this research are summarized

below.

9.1.1 Effect of Soil-Structure Interaction on Elastic and Inelastic Responsecomp of

MDOF and Equivalent SDOF Systems

An intensive parametric study has been carried out in Chapter 3 to investigate the effect

of SSI on the strength and ductility demands for MDOF as well as its equivalent SDOF

systems considering both elastic and inelastic behaviors. It was demonstrated that

strength and ductility demands of MDOF soil-structure systems depending on the

number of stories, dimensionless frequency, aspect ratio and level of inelasticity can be

very different from those of the corresponding equivalent SDOF ones. Elastic strength

demands of E-SDOF and MDOF soil-structure systems are lower than those of the fixed-

base structures.


195

Generally, the SSI effect on strength demands may become negligible as the structure

experiences more inelastic deformations, which is more accurate for E-SDOF and low-

rise buildings. However, for the cases of MDOF systems of mid- and high-rise buildings

with severe SSI effect, irrespective of the amount of aspect ratio, the values of strength

demands of soil-structure systems for the practical range of periods are significantly

lower than those of the fixed-base systems. This means that E-SDOF systems may not

accurately model the effect of high mods on estimating the strength demands of soil-

structure systems.

Except for structures with very short periods, the required strength demands of MDOF

systems to those of the associated E-SDOF systems remarkably increase with

dimensionless frequency (i.e., SSI effect). This phenomenon is more pronounced for the

cases of slender buildings (i.e., H r = 5) with elastic behavior. The ratios decrease as

the level of inelasticity increases but remain significant. As an instance, for the case

with fixT = 2 sec and 0a = 3, the required strength demands of the MDOF system are

7.31, 5.53 and 3.6 times the strength demands of the corresponding E-SDOF system for

µ= 1, 2 and 6, respectively. Therefore, it is concluded that using the common E-SDOF

systems for estimating the strength demands of average and slender MDOF systems

when SSI effect is significant can lead to very un-conservative results when compared to

fixed-base systems.

9.1.2 Effect of Structural Characterstics Distribution on Strength Demand and

Ductility Reduction Factor of MDOF Systems with Soil-Structure Interaction

Effects of different story strength and stiffness distribution patterns including three code-

specified and two arbitrary patterns on strength demand and ductility reduction factor of

MDOF fixed-base and soil-structure systems have been parametrically investigated in

Chapter 4. It has been concluded that for both fixed-base and flexible-base models, with

exception of those with very short periods, the averaged total strength demand values of

structures designed based on uniform story strength and stiffness distribution pattern

along the height of the structures are significantly greater than those of the other patterns

such as the code-compliant patterns. This phenomenon is even more pronounced by

increasing the number of stories. The ratios of strength demand in uniform pattern to

those of the code-specified patterns are generally greater than 2 and in some cases will

reach to the value of 4. Therefore, it can be concluded that, using the results of the


196

uniform story strength and stiffness distribution pattern which has been the assumption

of many previous research works would result in a significant overestimation of the

strength demands, generally from 2 to 4 times, for MDOF systems designed in

accordance to the code-compliant design patterns.

Overall, in low level of inelastic behavior the effect of story strength and stiffness

distribution patterns on the values of ductility (strength) reduction factor is not

significant and hence practically negligible for both fixed-base and flexible-base models.

By increasing the level of inelastic behavior the difference between the results of

different patterns increases. Nevertheless, for the case with severe SSI effect, except for

the concentric pattern which is the most different pattern from other patterns, the

difference is insignificant for structures with short and intermediate periods.

9.1.3 Strength Reduction Factor For MDOF Systems Considering Soil-Structure

Interaction

Chapter 5 investigates the effect of SSI on strength reduction factor ( R ) of E-SDOF

and MDOF fixed-base and soil-structure systems. Results indicate that in E-SDOF

systems SSI effect is always accompanied by decreasing in values of R . Using R of

fixed-base systems leads to significant underestimation of inelastic strength demands of

soil-structure systems. Except for E-SDOF systems with very short periods, increasing

the aspect ratio is always accompanied by a decrease in the values of R , and it is more

pronounced for the cases with significant SSI effect and long vibration periods.

For MDOF fixed-base systems, regardless of the level of nonlinearity, increasing the

number of DOFs (stories) always reduces the averaged values of R . This phenomenon

is more pronounced for low- to mid-rise buildings. However, for soil-structure systems,

as the SSI effect becomes more significant, R spectra become less sensitive to the

number of stories, especially in the low inelastic response range. In addition, the MDOF

modifying factors for strength reduction factors of soil-structure systems could be

completely different from those of fixed-base systems. The more significant is the SSI

effect, the more difference between the elastic strength demands of MDOF and SDOF

systems. The phenomenon is more pronounced as aspect ratio ( H r ) increases. A new


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modification factor (MR ) for soil-structure and fixed-base systems that account for both

elastic and inelastic strength demands has been introduced.

For the case of less SSI effect, the values of averaged R are insensitive to the variation

of aspect ratio of MDOF soil-structure systems but very sensitive to the aspect ratio of

E-SDOF systems. For the case with severe SSI effect and high inelastic response, except

in short period ranges, the values of mean R increase with the aspect ratio, which is

completely different from the E-SDOF results in which increasing the aspect ratio is

always accompanied by decreasing the R values. This indicates that SSI can affect

strength reduction factors of MDOF and E-SDOF systems in a different manner. Finally,

a new simplified equation which is functions of fixed-base fundamental period, ductility

ratio, the number of stories, aspect ratio and dimensionless frequency has been proposed

to estimate the strength reduction factors of MDOF soil-structure systems.

9.1.4 A Paramteric Study on Evaluation of Ductility Demand Distribution in

MDOF Shear Buildings Considering SSI Effects

After extensive parametric studies on the effect of SSI on global (total) strength and

ductility demand of MDOF and corresponding E-SDOF systems carried out in Chapters

3 to 5 as the first part of the thesis, the second part of this research focuses on the effect

of SSI on local ductility (damage) demand distribution along the height of the structures.

In particular, Chapter 6 through an extensive parametric study investigates the effect of

SSI on height-wise distribution of ductility demands in shear-building structures with

different structural properties. Effect of many parameters including fundamental period,

level of inelastic behavior, number of stories, damping model, damping ratio, structural

strain hardening, earthquake excitation (frequency content), level of soil flexibility,


investigated. In addition, the adequacy of three different code-complaint lateral loading

patterns including UBC-97, IBC-2009 and EuroCode-8 as well as three recently

proposed optimum loading patterns for fixed-base structures are parametrically

investigated for soil-structure systems by two methods namely weight-based and COV-

based methods associated to the economy of the seismic-resistant system. Results show

that for both fixed-base and flexible-base structures, regardless of the number of stories,

using IBC-2009 load pattern leads to nearly uniform ductility demands distribution for

structures with short periods within the elastic response range. For structures with longer


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periods, i.e., 0.4fixT sec, however, the efficiency of the IBC-2009 load pattern is

reduced as the number of stories and fundamental period increase, because of

contributions to the responses from higher modes that has not been considered

sufficiently in the IBC-2009 load pattern. In inelastic response range, the performance of

the structures is significantly reduced even for structures with short vibration period.

The performance is even worse with increasing the inelastic level of vibration. Among

the 3 code-specified design lateral load patterns, i.e., EuroCode-8, UBC-97 and IBC-

2009, UBC-97 leads to the best performance of structures with consideration of SSI

effects. Moreover, generally, for both fixed-base and flexible-base models with long

periods of vibration, the seismic performance of the structure is more dependent on the

level of inelasticity (i.e. target ductility demand value) than the fundamental period of

the building although nearly in all current code-specified seismic load patterns the

ductility demands are not considered.

Generally speaking, SSI effect is more significant as the aspect ratio increases, i.e., for

the case of slender building, leading to more non-uniform distribution of ductility

demand along the height of the structure as compared to the corresponding fixed-base

structure model. The influence of aspect ratio on SSI effect is less prominent as the level

of inelastic response increases. It is also demonstrated that although the structures

designed according to some load patterns such as those proposed by Mohammadi et al.

(2004) and Park and Medina (2007) may have generally better seismic performance

when compared to those designed by code-specified load patterns, their seismic

performance are far from ideal if the SSI effects are considered. Therefore, more

adequate load patterns incorporating SSI effects for performance-based seismic design

needs to be proposed. This has been carried out in Chapters 7 and 8.

9.1.5 Optimum Lateral Load Pattern for Elastic Seismic Design of Buildings

Incorporation Soil-Structure Interaction Effects

Chapter 7 developed an optimization algorithm for optimum seismic design of elastic

soil-structure systems. The adopted method is based on the concept of uniform

deformation proposed by Mohammadi et al. (2004) and Hajirasouliha and Moghadam

(2009) for fixed-base shear building structures. Based on intensive numerical analyses of

structural models with different structural and foundation conditions subjected to 21

selected earthquake ground motions iterated to be compatible to the IBC design

spectrum, an optimum load pattern is proposed for structure design with consideration of


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SSI effect. It is concluded that the optimum load pattern for elastic shear-building

structures with SSI effect is highly dependent on the fundamental period, dimensionless

frequency, aspect ratio, seismic excitation and structural damping model, but almost

independent of the number of stories, structural damping ratio and earthquake intensity.

Also, while there is no significant difference between the results of mass-proportional

and Rayleigh-type damping models, the difference is pronounced when compared to that

of the stiffness-proportional damping model.

The proposed optimum load pattern which is a function of fixed-base fundamental

period, dimensionless frequency and structure aspect ratio gives better structural design

than the code-compliant and recently proposed optimum load patterns by researchers for

fixed-base structures. Generally, by increasing the aspect ratio and dimensionless

frequency (i.e., increasing SSI effects) the two load patterns by Park and Medina (2004)

and Hajirasouliha and Moghadam (2009) for fixed-base structures as well as the code-

specified load pattern significantly lose their efficiency while the proposed load pattern

of this study leads to better seismic performance (i.e, less structure weight and more

uniform damage distribution over height). Using the proposed load pattern in this study,

the designed structures experience up to 40% less structural weight as compared with the

code-compliant or optimum patterns developed based on fixed-base structures.

9.1.5 Optimum Lateral Load Pattern for Seismic Design of Inelastic Shear-

Buildings Considering Soil-Structure Interaction Effects

Chapter 8 focuses on modification of the optimization algorithm developed in Chapter 7

for elastic soil-structure systems to take into account for structural inelastic behaviour. It

is found that the optimum load pattern for shear-building structures with SSI effect in

inelastic range of response is highly dependent on the fundamental period, target

ductility demand, dimensionless frequency, aspect ratio, seismic excitation and structural

damping model, and less dependent on structural post-yield behaviour and soil Poisson’s

ratio but almost independent of the number of stories, structural damping ratio and

earthquake intensity.

While the value of convergence power in the optimization problem is not very sensitive

to the variation of damping ratio, it is to large extent sensitive to the type of the damping

modeling in inelastic rang of response. Generally more iteration steps are required for

optimization of the structures designed based on mass-proportional and Rayleigh-type


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damping models with respect to the stiffness-proportional damping model. In addition,

similar to the optimum design in elastic range of response addressed in Chapter 7,

although there is no significant difference between the optimum lateral load profiles of

mass-proportional and Rayleigh-type damping models, the difference is pronounced

when compared to that of the stiffness-proportional damping model. The stiffness-

proportional damping model for shear-building models may not lead to reliable

prediction of structural responses. A new lateral force pattern which is a function of

fixed-base fundamental period, target ductility demand, dimensionless frequency and

structure aspect ratio (slender ratio) has been proposed for soil-structure systems in

inelastic range of response. It has been shown that the structures designed based on the

proposed pattern, on average, display remarkably better seismic performance (i.e., less

structural weight and more uniform damage distribution over height) than the code-

compliant and recently proposed optimum lateral load patterns by researchers for fixed-

base structures. Using the proposed load pattern in this study, the designed structures

experience up to 52%, 45%, 27% and 36% less structural weight as compared with the

proposed patterns by Hajirasouliha and Moghaddam (2009), IBC-2009, Mohammadi and

Moghaddam (2006) and Park and Medina (2007).

9.2 RECOMMENDATIONS FOR FUTURE WORKS

A comprehensive parametric study has been carried out to investigate the effect of

inertial soil-structure interaction (SSI) on elastic and inelastic response of MDOF shear-

building structures and their equivalent SDOF (E-SDOF) systems subjected to a large

number of earthquake ground motions. Further investigations can be conducted in the

future studies as outlined below:

1. Shear-building models are utilized in this study to investigate the effect of SSI on

MDOF systems. More research works on more complex structural configurations

such as moment-resisting frames, moment-resisting frames with shear wall,

concentrically and eccentrically braced frames need to be carried out for

developing a practical methodology applicable to design and analysis of soil-

structure systems exposed to earthquake ground motions.

2. A bilinear elasto-plastic model with different values of strain hardening in the

force-displacement relationship has been used to represent the hysteretic

response of story lateral stiffness in shear-building structures. As stated before,


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this model is selected to represent the behavior of non-deteriorating steel-framed

structures. Effect of different hysteretic models incorporating strength and

stiffness deterioration as well as pinching behavior on inelastic response of

MDOF soil-structure systems should be investigated.

3. P-delta effect may cause a negative lateral stiffness in a structure once a

mechanism has formed. This negative hardening will increase the drift or

ductility demand of the system and may lead to incremental collapse if the

structure has insufficient strength. Therefore, effect of this phenomenon on

inelastic response of MDOF soil-structure systems should be examined.

4. The results of this thesis are mainly based on the findings for surface foundations

in which the kinematic interaction (KI) effect is negligible and thus ignored.

Although current provisions on SSI also ignore the KI effect since they consider

this phenomenon somehow as a beneficial effect for the structure, the effect of

foundation embedment on inelastic response of the different MDOF systems

should be examined.

5. The sub-structure method is utilized in this research to model the soil-structure

system in which the soil-foundation element is modelled by an equivalent linear

discrete model based on the cone model with frequency independent coefficients

and equivalent linear model. Effect of soil nonlinearity by using more advanced

models, as for example the concept of macro-elements, as conducted in Chapter

6 needs to be investigated to generalize the results.

6. All selected ground motions used in this study were obtained from earthquakes

having closest distance to fault rupture more than 15 km without pulse type

characteristics. Near-fault, forward-directivity ground motions possess

characteristics that differ from those of the ordinary ground motions used in this

research. Near-fault ground motions usually exhibit stronger acceleration

amplitudes and frequency content dominated by a distinct pulse. The pulse

period of a near-fault ground motion is of paramount importance when studying

the response of multistory frame structures exposed to such ground motions.

Hence, effect of this kind of ground motion on elastic and inelastic response of

MDOF soil-structure systems especially for the case of optimum lateral load

patterns proposed in Chapters7 and 8 should be taken into account.

7. Effect of SSI on other seismic demand parameters such as input energy,

hysteretic energy, and displacement amplification factors of MDOF soil-structure

systems should be investigated in future research works.

a parametric study on the effect of soil...

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