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)
The estimated percentage contribution of the candidate is 80%.
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)
The estimated percentage contribution of the candidate is 80%.
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)
The estimated percentage contribution of the candidate is 80%.
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)
The estimated percentage contribution of the candidate is 70%.
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)
The estimated percentage contribution of the candidate is 80%.
Behnoud Ganjavi 06/01/2013
Print Name Signature Date
Hong Hao
Print Name Signature Date
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).
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
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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
Figure 3-5: Comparison of the averaged inelastic strength demand for ESDOF and MDOF
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
Figure 3-12: Averaged ductility demand for different E-SDOF and MDOF soil-structure
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
Figure 4-3: Effect of structural characteristics distribution on strength demand for MDOF
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
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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
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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
List of Figures The University of Western Australia
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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
List of Figures The University of Western Australia
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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
Figure 8-12: 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= 2) ………………………………………………………………………....190
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) ………………………………………………………………………....190
List of Tables The University of Western Australia
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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
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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
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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.
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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
Chapter 1 The University of Western Australia
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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
Chapter 1 The University of Western Australia
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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
Chapter 1 The University of Western Australia
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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
Chapter 1 The University of Western Australia
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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.
Chapter 1 The University of Western Australia
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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.
Chapter 1 The University of Western Australia
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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
Chapter 1 The University of Western Australia
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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,”
Earthquake Engineering and Structural Dynamics, 6(1): 17–30.
Building Seismic Safety Council (BSSC). (2000) NEHRP Recommended Provisions for
Seismic Regulations for New Buildings and Other Structures, Federal
Emergency Management Agency, Washington, DC.
Chapter 1 The University of Western Australia
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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.
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motions in bridge response II: effect on response with modular expansion joint.”
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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):
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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
inelastic structural demands,” European Earthquake Engineering 20(1): 23–35.
Gilmore, T. A, and Bertero, V.V. (1993). “Seismic performance of a 30-story building
located on soft soil and designed according to UBC 1991”. UCB/EERC-93/04.
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Hajirasouliha, I., and Moghaddam, H. (2009). “New lateral force distribution for seismic
design of structures.” (ASCE) Journal of Structural Engineering, 135(8): 906–
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International Code Council (ICC) (2009), International Building Code, ICC,
Birmingham, AL.
FEMA 440. (2005) Improvement of nonlinear static seismic analysis procedures, Report
No. FEMA 440, Federal Emergency Management Agency, prepared by Applied
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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.
Karami Mohammadi, R., ElNaggar, M. H. and Moghaddam, H. (2004) “Optimum
strength distribution for seismic resistant shear buildings,” International Journal
of Solids and Structures 41(22): 6597–6612.
Krawinkler, H., Zareian, F., Medina, R. A. and Ibarra, L. F. (2006), “Decision support
for conceptual performance-based design.” Earthquake Engineering & Structural
Dynamics, 35: 115–133.
Leelataviwat, S., Goel, S. C., and Stojadinovic, B. (1999). “Toward performance-based
seismic design of structures.” Earthquake Spectra. 15(3): 435–461.
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.
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Moghaddam, H., and Mohammadi, R. K. (2006). “More efficient seismic loading for
multidegrees of freedom structures.” (ASCE) Journal of Structural Engineering,
132(10): 1673–1677.
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 Engineering & Structural Dynamics.,
40(2): 135–154.
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
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Moghaddasi, M., Cubrinovski, M., Chase, J. G., Pampanin, S. and Carr, A. (2011)
“Probabilistic evaluation of soil–foundation–structure interaction effects on
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seismic structures,” Proceedings of the Seventh European Conference on
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Nassar, A. and Krawinkler, K. (1991) Seismic Demands for SDOF and MDOF Systems,.
Report No.95, Department of Civil Engineering, Stanford University, Stanford,
California.
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(SMRF) buildings incorporating nonlinear soil–structure interaction (SSI).”
Engineering Structures 33(3): 958–967.
Rodriguez, M. E. and Montes, R. (2000) “Seismic response and damage analysis of
buildings supported on flexible soils,” Earthquake Engineering and Structural
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freedom systems,” Proceedings of the 12th world conference on Earthquake
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interaction,” Journal of the Structural Division (ASCE) 98(ST7): 1525–1544.
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.
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shear wall considering soil–structure interaction effects,” Engineering Structure
33(1): 218–229.
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Veletsos, A. S. and Vann, P. (1971) “Response of ground-excited elastoplastic systems,”
Journal of the Structural Division, (ASCE), 97(4): 1257-1281.
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 Verbic, B. (1974) “Dynamic of elastic and yielding structure-
foundation systems,” Proceedings of the 5th world conference on Earthquake
Engineering Rome.
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foundations,” Journal of the Structural Division (ASCE) 101(1): 109–129.
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Veletsos, A. S. (1977) “Dynamics of structure–foundation systems,” In Structural and
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Prentice-Hall: Englewood Cliffs, NJ; 333–361.
Wolf, J. P. (1994) Foundation Vibration Analysis using Simple Physical Models,
Prentice-Hall: Englewood Cliffs, NJ.
Chapter 2 The University of Western Australia
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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.
Chapter 2 The University of Western Australia
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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
Chapter 2 The University of Western Australia
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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
Chapter 2 The University of Western Australia
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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
Chapter 2 The University of Western Australia
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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
Chapter 2 The University of Western Australia
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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
Chapter 2 The University of Western Australia
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
Chapter 2 The University of Western Australia
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)
Chapter 2 The University of Western Australia
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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:
Chapter 2 The University of Western Australia
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.
Chapter 2 The University of Western Australia
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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.
Chapter 2 The University of Western Australia
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.
Chapter 2 The University of Western Australia
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.
Chapter 2 The University of Western Australia
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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
Chapter 2 The University of Western Australia
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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.
Chapter 2 The University of Western Australia
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Figure 2-4: Typical database output for SSIOPT (Strength Demand)
Chapter 2 The University of Western Australia
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Figure 2-5: Typical database output for SSIOPT (Strength Reduction Factor)
Chapter 2 The University of Western Australia
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2.9 REFERENCE:
Building Seismic Safety Council (BSSC). (2000) NEHRP Recommended Provisions for
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
No. FEMA 440, Federal Emergency Management Agency, prepared by Applied
Technology Council.
Ghannad, M. A. and Ahmadnia A. (2006) “The effect of soil–structure interaction on
inelastic structural demands,” European Earthquake Engineering 20(1): 23–35.
Ghannad, M. A., And Jahankhah, H. (2007). “Site dependent strength reduction factors
for soil–structure systems.” Soil Dynamics & Earthquake Engineering. 27(2), 99–
110.
Hajirasouliha, I., and Moghaddam, H. (2009). “New lateral force distribution for seismic
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.
Chapter 2 The University of Western Australia
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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.
Moghaddasi, M., Cubrinovski, M., Chase, J. G., Pampanin, S. and Carr, A., (2011)
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.
Veletsos, A. S. (1977) “Dynamics of structure–foundation systems,” In Structural and
Geotechnical Mechanics, Hall WJ (ed.), A Volume Honoring N.M. Newmark.
Prentice-Hall: Englewood Cliffs, NJ; 333–361.
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.
Chapter 3 The University of Western Australia
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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
Chapter 3 The University of Western Australia
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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
Chapter 3 The University of Western Australia
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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
Chapter 3 The University of Western Australia
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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
Chapter 3 The University of Western Australia
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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:
Chapter 3 The University of Western Australia
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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.
Chapter 3 The University of Western Australia
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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
Chapter 3 The University of Western Australia
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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
Chapter 3 The University of Western Australia
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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)
Chapter 3 The University of Western Australia
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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
Chapter 3 The University of Western Australia
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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
Figure 3-4: Comparison of the averaged inelastic strength demand for ESDOF and MDOF
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
Fixed base a0 =1 a0 =3
(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
Chapter 3 The University of Western Australia
44
Figure 3-5: Comparison of the averaged inelastic strength demand for ESDOF and MDOF
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
Fixed base a0 =1 a0 =3
N= 1 N = 5 N = 15
Tfix Tfix Tfix
Chapter 3 The University of Western Australia
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
Chapter 3 The University of Western Australia
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).
Chapter 3 The University of Western Australia
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.
Chapter 3 The University of Western Australia
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
Fixed base a0 = 1 a0= 3
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
Chapter 3 The University of Western Australia
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
Fixed base a0 = 1 a0= 3
μ μ μ
N= 5 N= 10 N= 15
Chapter 3 The University of Western Australia
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.
Chapter 3 The University of Western Australia
51
Figure 3-10: Averaged ductility demand for different E-SDOF and MDOF soil-structure
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
Chapter 3 The University of Western Australia
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.
Figure 3-11: Averaged ductility demand for different E-SDOF and MDOF soil-structure
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
Chapter 3 The University of Western Australia
53
Figure 3-12: Averaged ductility demand for different E-SDOF and MDOF soil-structure
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
Chapter 3 The University of Western Australia
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.
Chapter 3 The University of Western Australia
55
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Chapter 3 The University of Western Australia
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South Australia.
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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.,
38(4): 423–437.
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).
“Probabilistic evaluation of soil–foundation–structure interaction effects on
seismic structural response” Earthquake Engineering & Structural Dynamics.,
40(2): 135–154.
Moghaddasi, M., Cubrinovski, M., Chase, J. G., Pampanin, S. and Carr, A., (2011)
“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.
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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
33(1): 218–229.
Veletsos, A. S. and Vann, P. (1971) “Response of ground-excited elastoplastic systems,”
Journal of the Structural Division, (ASCE), 97(4): 1257-1281.
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 Verbic, B. (1974) “Dynamic of elastic and yielding structure-
foundation systems,” Proceedings of the 5th world conference on Earthquake
Engineering Rome.
Veletsos, A. S. and Nair V.V. D. (1975) “Seismic interaction of structures on hysteretic
foundations,” Journal of the Structural Division (ASCE) 101(1): 109–129.
Veletsos, A. S. (1977) “Dynamics of structure–foundation systems,” In Structural and
Geotechnical Mechanics, Hall WJ (ed.), A Volume Honoring N.M. Newmark.
Prentice-Hall: Englewood Cliffs, NJ; 333–361.
Chapter 4 The University of Western Australia
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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.
Chapter 4 The University of Western Australia
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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
Chapter 4 The University of Western Australia
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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.
Chapter 4 The University of Western Australia
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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
Chapter 4 The University of Western Australia
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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.
Chapter 4 The University of Western Australia
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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
Chapter 4 The University of Western Australia
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.
Figure 4-2: Effect of structural characteristics distribution on strength demand for MDOF
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
Chapter 4 The University of Western Australia
65
Figure 4-3: Effect of structural characteristics distribution on strength demand for MDOF
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
Fixed base a0 = 1 a0= 3
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
Chapter 4 The University of Western Australia
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).
Chapter 4 The University of Western Australia
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
Chapter 4 The University of Western Australia
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
Chapter 4 The University of Western Australia
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,
Chapter 4 The University of Western Australia
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
Chapter 4 The University of Western Australia
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
Fixed base a0 = 1 a0= 3
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
Chapter 4 The University of Western Australia
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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.
Chapter 4 The University of Western Australia
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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
Chapter 4 The University of Western Australia
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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
Chapter 4 The University of Western Australia
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.
Chapter 4 The University of Western Australia
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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.
American Society of Civil Engineers: Reston, VA.
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.
Mexico City Building Code (2003).
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.
Chapter 4 The University of Western Australia
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.
Chapter 5 The University of Western Australia
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
Chapter 5 The University of Western Australia
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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
Chapter 5 The University of Western Australia
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.
Chapter 5 The University of Western Australia
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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 LA - Centinela St 30.9 C 155, 245 0.465, 0.322
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
Loma Prieta 1989 Foster City - 355
Menhaden
51.2 D 360, 270 0.116, 0.107
Superstitn
Hills(B)
1987 5062 Salton Sea Wildlife
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
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
Chapter 5 The University of Western Australia
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-
Chapter 5 The University of Western Australia
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
Chapter 5 The University of Western Australia
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
Chapter 5 The University of Western Australia
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
Chapter 5 The University of Western Australia
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
Chapter 5 The University of Western Australia
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
Chapter 5 The University of Western Australia
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)
Chapter 5 The University of Western Australia
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:
Chapter 5 The University of Western Australia
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.
Chapter 5 The University of Western Australia
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
Chapter 5 The University of Western Australia
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)
Fixed base a0 =1 a0 =2 a0 =3
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
Fixed base a0 =1 a0 =2 a0 =3
Chapter 5 The University of Western Australia
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.
Chapter 5 The University of Western Australia
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
Fixed base a0 =1 a0 =2 a0 =3
Chapter 5 The University of Western Australia
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
Fixed base a0 =1 a0 =2 a0 =3
Chapter 5 The University of Western Australia
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
Chapter 5 The University of Western Australia
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
Chapter 5 The University of Western Australia
98
Table 5-5: Constant coefficient ia of Eq. (5-5)
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
Table 5-6: Constant coefficient ib of Eq. (5-5)
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
Table 5-7: Constant coefficient ia of Eq. (5-5)
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
Chapter 5 The University of Western Australia
99
Table 5-8: Constant coefficient ib of Eq. (5-5)
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)
Chapter 5 The University of Western Australia
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.
Chapter 5 The University of Western Australia
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.
Chapter 5 The University of Western Australia
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5.8 REFERENCE
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Engineering and Structural Dynamics, 32(11): 1749–1771.
Aviles J, Perez-Rocha JL. 2005. Influence of foundation flexibility on Rμ and Cμ factors.
Journal of Structural Engineering (ASCE) 131(2), 221–230.
Building Seismic Safety Council (BSSC). 2000. NEHRP Recommended Provisions for
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.
Elghadamsi FE, Mohraz B. 1987. Inelastic earthquake spectra. Earthquake Engineering and
Structural Dynamics 15(2):91–104.
FEMA-440. 2005. Improvement of nonlinear static seismic analysis procedures. Report No.
FEMA 440, Federal Emergency Management Agency, prepared by Applied
Technology Council.
Fischinger M, Fajfar P, Vidic T. 1994. Factors contributing to the response reduction,
Proceedings of Fifth U.S. National Conference on Earthquake Engineering., Chicago,
Illinois, 97-106.
Ganjavi B, 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.
Ghannad MA, Jahankhah H. 2007. Site dependent strength reduction factors for soil–
structure systems. Soil Dynamics and Earthquake Engineering 27(2):99–110.
Halabian AM, Erfani M. 2010. The effect of foundation flexibility and structural strength on
response reduction factor of RC frame structures. The Structural Design of Tall and
Special Buildings doi: 10.1002/tal.654
IBC-2009. International Building Code. International Code Council, Country Club Hills,
USA.
Karmakar D, Gupta, VK. 2007. Estimation of strength reduction factors via normalized
pseudo-acceleration response spectrum. Earthquake Engineering & Structural
Dynamics, 36(6):751–763.
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Krawinkler H, Rahnama M. 1992. Effects of soft soils on design spectra. In: Proceedings of
the 10th World Conference on Earthquake Engineering, Madrid, Spain, (10):5841–6.
Lai SP, Biggs JM. Inelastic response spectra for asiesmic building design. 1980. Journal of
Structural Engineering, (ASCE) 106(ST6):1295– 10.
Lam N, Wilson J, Hutchinson G. 1998. The ductility reduction factor in the seismic design
of buildings. Earthquake Engineering & Structural Dynamics, 27(7):749–769
Mahsuli M, Ghannad MA. 2009. The effect of foundation embedment on inelastic response
of structures. Earthquake Engineering and Structural Dynamics 38(4):423–437.
Miranda E. Site-dependent strength reduction factors. 1993. Journal of Structural
Engineering, (ASCE) 119(12):3503–19.
Miranda E, Bertero V. 1994. Evaluation of strength reduction factors for earthquake-
resistant design. Earthquake Spectra, 10(2):357-379.
Miranda, E. 1997. Strength reduction factors in performance-based design. EERC-CUREe
Symposium in Honor of Vitelmo V. Bertero, January 31 - February 1, Berkeley,
California.
Moghaddam H, Mohammadi RK. 2001. Ductility reduction factor of MDOF shear-building
structures. Journal of Earthquake Engineering 5(3): 425-440.
Moghaddam, H. Hajirasouliha, I. 2008. Optimum strength distribution for seismic design of
tall buildings. The Structural Design of Tall and Special Buildings, 17: 331–349.
doi: 10.1002/tal.356
Moghaddasi M, Cubrinovski M, Chase JG, Pampanin S, Carr A. 2011. Probabilistic
evaluation of soil–foundation–structure interaction effects on seismic structural
response. Earthquake Engineering and Structural Dynamics 40(2):135–154.
Nassar A, Krawinkler K. 1991. Seismic Demands for SDOF and MDOF Systems. Report
No.95, Department of Civil Engineering, Stanford University, Stanford, California.
Newmark NM, Hall WJ. 1973. Seismic design criteria for nuclear reactor facilities. Building
Research Series No. 46, National Bureau of Standards, US Department of Commerce,
Washington, DC, 209–36.
Ordaz M, Pérez-Rocha LE. 1998. Estimation of strength-reduction factors for elastoplastic
systems: a new approach. Earthquake Engineering & Structural Dynamics,
27(9):889–901
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Santa-Ana PR, Miranda E. 2000. Strength reduction factors for multi-degree of freedom
systems. Proceedings of the 12th world conference on Earthquake Engineering,
Auckland, No.1446.
Seneviratna GD, Krawinkler H. 1997. Evaluation of inelastic MDOF effects for seismic
design. Report No.120, Department of Civil Engineering, Stanford University,
Stanford, California.
Veletsos AS, Newmark NM. 1960. Effect of inelastic behavior on the response of simple
systems to earthquake motions. Proceedings of the second world conference on
earthquake engineering, Tokyo, 895–912.
Veletsos AS, Vann P. 1971. Response of ground-excited elastoplastic systems. Journal of
the Structural Division, (ASCE) 97(4): 1257-1281.
Veletsos AS, Meek JW. 1974. Dynamic behavior of building–foundation system.
Earthquake Engineering and Structural Dynamics 3(2):121–138.
Veletsos AS. 1977. Dynamics of structure–foundation systems. In Structural and
Geotechnical Mechanics, Hall WJ (ed.), A Volume Honoring N.M. Newmark.
Prentice-Hall: Englewood Cliffs, NJ, 333–361.
Wolf JP. 1994. Foundation Vibration Analysis using Simple Physical Models. Prentice-Hall:
Englewood Cliffs, NJ.
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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
Chapter 6 The University of Western Australia
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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,
aspect ratio on height-wise distribution of damage (ductility demand) are intensively
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.
Chapter 6 The University of Western Australia
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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
Chapter 6 The University of Western Australia
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)
Chapter 6 The University of Western Australia
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
Chapter 6 The University of Western Australia
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
Chapter 6 The University of Western Australia
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
Chapter 6 The University of Western Australia
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
Fixed base a0 = 1 a0= 3
μ μ μ
N= 5 N= 10 N = 15 N = 20
Chapter 6 The University of Western Australia
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
Fixed base a0 = 1 a0= 3
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
Chapter 6 The University of Western Australia
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
Chapter 6 The University of Western Australia
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,
Chapter 6 The University of Western Australia
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%
Fixed base a0 = 1 a0 = 3
μ μ μ
Rel
ativ
e H
eigh
t
Chapter 6 The University of Western Australia
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.
Chapter 6 The University of Western Australia
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
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
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
Chapter 6 The University of Western Australia
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
Chapter 6 The University of Western Australia
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.
Chapter 6 The University of Western Australia
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
Chapter 6 The University of Western Australia
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.
Chapter 6 The University of Western Australia
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
Chapter 6 The University of Western Australia
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
Chapter 6 The University of Western Australia
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
Fixed base a0 = 1 a0 = 2
Tfix Tfix Tfix
μ = 1 μ = 2 μ = 4 μ = 6
Chapter 6 The University of Western Australia
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
Fixed base a0 =1 a0 =2 a0 =3
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
Chapter 6 The University of Western Australia
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,
Chapter 6 The University of Western Australia
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-
Chapter 6 The University of Western Australia
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.
Chapter 6 The University of Western Australia
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
Park and Medina Mohammadi et al. Hajirasouliha and Moghaddam
EuroCode-8 UBC-97 IBC-2009
Chapter 6 The University of Western Australia
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
Park and Medina Mohammadi et al. Hajirasouliha and Moghaddam
EuroCode-8 UBC-97 IBC-2009
Chapter 6 The University of Western Australia
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.
Chapter 6 The University of Western Australia
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.
Chapter 6 The University of Western Australia
134
6.9. 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.
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.
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.
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.
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 Conf., Barossa Valley, South Australia.
Chatzigogos, C.T., Pecker A., Salencon J. (2009) ―Macro-element modelling of shallow
foundation,‖ Soil Dynamics and Earthquake Engineering. 29(5); 765-781.
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, T. A, and Bertero, V.V. (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, niversity of California, Berkeley.
Chapter 6 The University of Western Australia
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Grange S, Kotronis P., Mazars J. (2009) ―A macro-element to simulate dynamic soil–
structure interaction,‖ Engineering Structure 31(12):3034–3046.
Grange, S., Botrugno, L., Kotronis, P., Tamagnini, C. (2011). ―The effects of Soil–Structure
Interaction on a reinforced concrete viaduct,‖ Earthquake Engineering & Structural
Dynamics, 40(1): 93–105.
Hajirasouliha, I., and Moghaddam, H. (2009). ―New lateral force distribution for seismic
design of structures.‖ Journal of Structural Engineering, ASCE, 135(8), 906–915.
International Code Council (ICC) (2009), International Building Code, ICC, Birmingham,
AL.
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.
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.
Moghaddam, H., and Hajirasouliha, I. (2006) ―Toward more rational criteria for
determination of design earthquake forces,‖ International Journal of Solids and
Structures, 43(9); 2631–2645.
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.
Mexico City Building Code (2003).
Moghaddasi, M., Cubrinovski, M., Chase, J. G., Pampanin, S., and Carr, A. (2011a).
―Probabilistic evaluation of soil–foundation–structure interaction effects on seismic
structural response‖ Earthquake Engineering & Structural Dynamics, 40(2), 135–154.
Moghaddasi, M., Cubrinovski, M., Chase, J. G., Pampanin, S. and Carr, A., (2011b) Effects
of soil–foundation–structure interaction on seismic structural response via robust
Monte Carlo simulation Engineering Structures 33(4); 1338-1347.
Nakhaei, M., Ghannad, M.A. ―The effect of soil–structure interaction on damage index of
buildings,‖ Engineering Structures, 2008; 30(6); 1491–1499.
Chapter 6 The University of Western Australia
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.
Uniform Building Code (UBC). (1997). Int. Conf. of Building Officials, Vol. 2, Calif.
Wolf JP (1994), ―Foundation Vibration Analysis using Simple Physical Models.‖ Prentice-
Hall: Englewood Cliffs, NJ.
Chapter 7 The University of Western Australia
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
Chapter 7 The University of Western Australia
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
Chapter 7 The University of Western Australia
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 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
Chapter 7 The University of Western Australia
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)
Chapter 7 The University of Western Australia
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
Chapter 7 The University of Western Australia
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.
Chapter 7 The University of Western Australia
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.
14. Repeat steps 5–14 for different presumed target periods.
15. Repeat steps 4–15 for different sets of H r and 0a .
16. Repeat steps 3–16 for different earthquake ground motions.
17. Repeat steps 1–17 for different number of stories.
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
Chapter 7 The University of Western Australia
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).
Chapter 7 The University of Western Australia
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
Chapter 7 The University of Western Australia
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
Chapter 7 The University of Western Australia
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(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
Chapter 7 The University of Western Australia
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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
Lateral Force / Base Shear Lateral Force / Base Shear
(a) (b)
Chapter 7 The University of Western Australia
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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.
7.4.4 Effect of fundamental period
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.
7.4.5 Effect of number of stories
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
Chapter 7 The University of Western Australia
150
stories. This finding is also consistent with that by Hajirasouliha and Moghadam (2009) for
fixed-base shear-building structures.
7.4.6 Effect of dimensionless frequency
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.
.
Lateral Force / Base Shear
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
Chapter 7 The University of Western Australia
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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.
7.4.7 Effect of aspect ratio
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
Lateral Force / Base Shear Lateral Force / Base Shear
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)
Chapter 7 The University of Western Australia
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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
Lateral Force / Base Shear Lateral Force / Base Shear
Chapter 7 The University of Western Australia
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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)
Lateral Force / Base Shear
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)
Lateral Force / Base Shear
Chapter 7 The University of Western Australia
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.
Chapter 7 The University of Western Australia
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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
Chapter 7 The University of Western Australia
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.
Chapter 7 The University of Western Australia
157
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
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
Park and Medina (Eq. 6-2)
10) IBC-2009
Hajirasouliha and Moghaddam (Eq. 6-5)
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
Chapter 7 The University of Western Australia
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
Hajirasouliha and Moghaddam (Eq. 6-5)
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
Lateral Force / Base Shear Lateral Force / Base Shear
Rel
ativ
e H
eig
ht
Chapter 7 The University of Western Australia
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,
Chapter 7 The University of Western Australia
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.
Chapter 7 The University of Western Australia
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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
Chapter 7 The University of Western Australia
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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.
Chapter 7 The University of Western Australia
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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
Relative Height 0a =0
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
Chapter 7 The University of Western Australia
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Table 7-4: Constant coefficient ic 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 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
ASCE/SEI 7-05 (2005). Minimum Design Loads for Buildings and Other Structures.
American Society of Civil Engineers: Reston, VA.
Fishman GS. (1995)., Monte Carlo: concepts, algorithms, and applications. New York:
Springer-Verlag.
Hajirasouliha, I., and Moghaddam, H. (2009). “New lateral force distribution for seismic
design of structures.” Journal of Structural Engineering, ASCE, 135(8), 906–915.
International Code Council (ICC) (2009), International Building Code, ICC, Birmingham,
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.
Chapter 7 The University of Western Australia
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.
Chapter 8 The University of Western Australia
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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
Chapter 8 The University of Western Australia
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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
Chapter 8 The University of Western Australia
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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
Chapter 8 The University of Western Australia
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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
Chapter 8 The University of Western Australia
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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.
15. Repeat steps 5–14 for different presumed target periods.
16. Repeat steps 4–15 for different sets of H r and 0a .
17. Repeat steps 3–16 for different earthquake ground motions.
18. Repeat steps 1–17 for different number of stories.
Chapter 8 The University of Western Australia
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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.
Chapter 8 The University of Western Australia
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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
8.3.1 Effect of fundamental period
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
Chapter 8 The University of Western Australia
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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.
Chapter 8 The University of Western Australia
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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)
8.3.3 Effect of number of stories
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
Lateral Force / Base Shear Lateral Force / Base Shear
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
Lateral Force / Base Shear Lateral Force / Base Shear
Sto
ry
Chapter 8 The University of Western Australia
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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)
8.3.4 Effect of dimensionless frequency
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
Chapter 8 The University of Western Australia
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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.
8.3.5 Effect of aspect ratio
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.
Chapter 8 The University of Western Australia
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
Lateral Force / Base Shear Lateral Force / Base Shear
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
Lateral Force / Base Shear Lateral Force / Base Shear
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
Chapter 8 The University of Western Australia
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
Chapter 8 The University of Western Australia
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).
Lateral Force / Base Shear
Rel
ativ
e H
eig
ht
Lateral Force / Base Shear
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
Chapter 8 The University of Western Australia
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
Lateral Force / Base Shear
Rel
ativ
e H
eig
ht
Lateral Force / Base Shear
Chapter 8 The University of Western Australia
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
Lateral Force / Base Shear Lateral Force / Base Shear
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
Chapter 8 The University of Western Australia
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
Lateral Force / Base Shear Lateral Force / Base Shear
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
Chapter 8 The University of Western Australia
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.
Chapter 8 The University of Western Australia
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
Chapter 8 The University of Western Australia
185
Table 8-2: Constant coefficients of Eq. (8-5) as function of relative height (µ= 4)
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
Chapter 8 The University of Western Australia
186
Table 8-3: Constant coefficients of Eq. (8-5) as function of relative height (µ= 6)
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
Chapter 8 The University of Western Australia
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.
Lateral Force / Base Shear Lateral Force / Base Shear
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
Chapter 8 The University of Western Australia
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
Chapter 8 The University of Western Australia
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.
Chapter 8 The University of Western Australia
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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)
Park and Medina (Eq. 6-2)
10) IBC-2009
Hajirasouliha and Moghaddam (Eq. 6-5)
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)
Park and Medina (Eq. 6-2)
10) IBC-2009
Hajirasouliha and Moghaddam (Eq. 6-5)
Proposed (Eq. 8-5)
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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,
Chapter 8 The University of Western Australia
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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).
Chapter 8 The University of Western Australia
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8.7. REFERENCES
ASCE/SEI 7-05 (2005). Minimum Design Loads for Buildings and Other Structures.
American Society of Civil Engineers: Reston, VA.
AS-1170.4. (2007). Structural design actions: Earthquake actions in Australia.
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.
BCJ. (1997). Structural provisions for building structures. 1997 edition – Tokyo;
Building Center of Japan.
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.
Hajirasouliha, I., and Moghaddam, H. (2009). ―New lateral force distribution for seismic
design of structures.‖ Journal of Structural Engineering, ASCE, 135(8), 906–915.
International Code Council (ICC) (2009), International Building Code, ICC, Birmingham,
AL.
Mexico City Building Code (2003).
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.
Uniform Building Code (UBC). (1994). Int. Conf. of Building Officials, Vol. 2, Calif.
Uniform Building Code (UBC). (1997). Int. Conf. of Building Officials, Vol. 2, Calif.
Chapter 9 The University of Western Australia
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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.
Chapter 9 The University of Western Australia
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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
Chapter 9 The University of Western Australia
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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
Chapter 9 The University of Western Australia
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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,
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 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
Chapter 9 The University of Western Australia
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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
Chapter 9 The University of Western Australia
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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
Chapter 9 The University of Western Australia
200
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,
Chapter 9 The University of Western Australia
201
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.