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Soil-pile-structure interaction effects on high rises under seismic shaking

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Title:
Soil-pile-structure interaction effects on high rises under seismic shaking
Creator:
Nghiem, Hien Manh
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English
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xxxii, 403 leaves : ; 28 cm

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Subjects / Keywords:
Soil-structure interaction ( lcsh )
Earthquake resistant design ( lcsh )
Tall buildings -- Earthquake effects ( lcsh )
Piling (Civil engineering) ( lcsh )
Earthquake resistant design ( fast )
Piling (Civil engineering) ( fast )
Soil-structure interaction ( fast )
Tall buildings -- Earthquake effects ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Bibliography:
Includes bibliographical references (leaves 389-403).
General Note:
Department of Civil Engineering
Statement of Responsibility:
by Hien Manh Nghiem.

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|University of Colorado Denver
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All applicable rights reserved by the source institution and holding location.
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438853806 ( OCLC )
ocn438853806
Classification:
LD1193.E53 2009d N43 ( lcc )

Full Text
SOIL-PILE-STRUCTURE INTERACTION EFFECTS ON HIGH RISES
UNDER SEISMIC SHAKING
BY
HIEN MANH NGHIEM
B.S. Hanoi Architectural University, Vietnam, 1997
M.S. Hanoi Architectural University, Vietnam, 2002
A thesis submitted to the
University of Colorado Denver
in Partial Fulfillment
of the Requirement for the Degree of
Doctor of Philosophy
Department of Civil Engineering
2009


by Hien Manh Nghiem
All Rights Reserved


This thesis for the Doctor of Philosophy degree by
Hien Manh Nghiem
has been approved by


Hien Manh Nghiem (Ph.D., Civil Engineering)
SOIL-PILE-STRUCTURE INTERACTION EFFECTS ON HIGH RISES UNDER
SEISMIC SHAKING
Thesis directed by Professor Nien-Yin Chang
ABSTRACT
Soil-structure interaction of high-rises was investigated in both theory and
field measurement. The instrumented structure motions and free-field ground motions
are available so that the recorded structure and ground motions can be used to
validate the theory and program developed for this study. Currently, the design code
only suggests the use of advanced methods to analyze the high-rise building
performance. No details are provided in the code. The effects of soil structure
interaction are only available for low rise buildings with simple analysis procedures
and are not available for high rise buildings.
The computer code SSI3D is developed to evaluate the nonlinear soil-pile-
structure interaction of high-rises. Besides several soil models for static load, two
models are modified and implemented for cyclic load: Modified Hyperbolic and
Modified Ramberg-Osgood models. SSI3D is validated by comparing the analysis
results and the measured seismic performance of an actual high-rise structure with
soil-structure interaction effects.
In fully nonlinear soil-pile-structure time history analysis, the unknown bed
rock motion to be used as input motion must be determined from free-field motion by
using deconvolution procedures. The resulting bed rock motion is then propagated to
the ground surface. In this analysis, the calculated ground surface motion is compared
to the ground surface motion used to calculate bed rock motion to check the validity
of the de-convolution and motion propagation procedures. Unfortunately, there are
differences between the deconvolution and convolution analyses described above
because of the use of soil viscous damping in convolution and soil damping ratio in


deconvolution. A new method is recommended to determine the appropriate soil
viscous damping from the soil damping ratio by matching transfer functions from soil
layer to soil layer. The excellent agreement between the calculated free-field motions
using soil viscous damping and the measured free field motion confirms the validity
of this method.
Soil-structure interaction includes both kinematic and inertial interactions. In
design code, two interactions are considered separately. In this study, the transfer
function from free-field motions to base foundation motions of buildings, and
equivalent stiffness, and damping of soil-pile system represented kinematic and
inertial interaction are recommended for use in the design code. When the nonlinear
analyses are performed, the nonlinear pile stiffness is recommended.
As high rises are built or designed for areas of high seismic activity, it is
critical to examine their seismic responses using an analysis code that reveals their
realistic behavior under strong seismic shaking. To better understand the seismic soil-
structure interaction effects, two 20-story hypothetical buildings and one 30-story
actual building were subjected to seismic response analyses using SSI3D. To reflect
the evolution of SSI effects, analyses were performed for the cases with rigid base,
flexible base with linear foundation springs, flexible base with linear soil, flexible
with nonlinear springs, and the full SSI analysis of flexible base with nonlinear soils
for two hypothetical buildings. The last case depicts the most realistic SSI responses
of high rises. Results of analyses presented include the comparison of natural periods,
base shears, and the displacements at the top floor of the buildings. It was observed
that the natural periods increase and the base shears decrease as the base become
more flexible.
This abstract accurately represents the content of the candidates thesis. I recommend
its publication.


DEDICATION
Dedicated to my wife HUE THI NGUYEN, my daughter DUONG THUYNGHIEM,
my son MINHHOANG NGHIEM, and my FAMILY


ACKNOWLEDGMENT
I would like to take this opportunity to express my gratitude to those who have
contributed to the development of this thesis.
Firstly, I am greatly indebted to Prof. Nien-Yin Chang for his encouragement,
guidance, support, friendship, and valuable comments on each part of the thesis.
Having opportunity to work with Prof. Chang was an experience that helped me gain
broader perspective and insight in many areas of life.
I would also like to thank the members of my thesis committee, Prof. H.Y.
Ko, Dr. James R. Harris, Assistant Prof. Mettupalayam V. Sivaselvan, Associate Prof.
Mohsen Tadi, Associate Prof. Samuel W. J. Welch, Dr. Trever Shing-Chun Wang,
and Dr. Brian T. Brady for their serving on the examining committee, helpful
comments and suggestions.
The scholarships received from the Vietnamese Government are also
gratefully acknowledged.
I am profoundly thankful to my parents, parents in law, sister, and their
families for their help. Finally, I am grateful to my wife, Hue Nguyen, for her help
and lasting love and patience, and my daughter, Duong Nghiem, and son, Minh
Nghiem, for their patience while awaiting their Dad to come home throughout my
doctoral study.


TABLE OF CONTENTS
,xvi
Figures
Tables.........................................................................xxix
1. Introduction...............................................................1
1.1 Problem Statement.........................................................1
1.2 Research Objectives.......................................................2
1.3 Research Approach.........................................................2
1.4 Organization of Thesis....................................................3
2. Literature Review..........................................................5
2.1 Post Earthquake Investigations............................................5
2.1.1 The Niigata Earthquake of June 16, 1964, Japan..........................5
2.1.2 Mexico City Earthquake of September 19, 1985, Mexico....................6
2.1.3 1995 Hyogo-Ken Nanbu (Kobe) Earthquake, Japan...........................9
2.1.4 Chi-Chi Taiwan Earthquake of 1999.......................................11
2.1.5 Summary of Earthquake Damage to High Rises..............................12
2.2 Dynamic Analysis of Soil-Structure Interaction............................13
2.2.1 Effects of Soil-Structure Interaction...................................13
2.2.2 Substructure Method.....................................................14
2.2.2.1 Introduction of Substructure Method...................................14
2.2.22 Inertial Interaction....................................................15
2.2.2.2.1 System Consideration.................................................15
2.2.2.2.2 Equivalent One Degree of Freedom....................................18
2.2.2.2.3 Inertial Interaction of Shallow Foundations.........................19
2.2.2.2.3.1 Basic Case.........................................................19
vm


2.2.2.2.3.2 Non-uniform Soil Profiles............................................21
2.2.2.2.3.3 Effects of Depth of Soil Layer.......................................23
2.2.2.2.3.4 Effects of Foundation Embedment.....................................23
2.2.2.2.3.5 Effects of Foundation Shape..........................................24
2.2.2.2.3.6 Effects of Foundation Flexibility....................................27
2.2.2.2.4 Inertial Interaction of Pile Foundations..............................30
2.2.2.2.4.1 Analytical Method....................................................30
2.2.2.2.4.2 Beam-on-Elastic Foundation..........................................31
2.2.2.2.4.3 Beam-on-Winkler Foundation..........................................33
2.2.2.2.4.4 Pile Group Effects..................................................40
2.2.2.3 Kinematic Interaction Effects...........................................42
2.2.2.3.1 Kinematic Effects of Shallow Foundations...............................43
2.2.2.3.1.1 Shallow Foundation at the Ground Surface.............................43
2.2.2.3.1.2 Embedded Shallow Foundations.........................................46
2.2.2.3.2 Kinematic Interaction of Pile Foundations...........................48
2.2.3 Hybrid Method...........................................................51
2.2.4 Direct Method.............................................................53
2.3 Previous Studies of Soil-Structure Interaction of High Rise Buildings.......54
2.4 Summary.....................................................................69
3. Contemporary Codes on Seismic SSI............................................70
3.1 Inertial Interaction Procedure by IBC 2003 or ASCE 7-02......................70
3.2 Inertial Interaction Procedure by FEMA 440 (2005)............................72
3.3 Kinematic Interaction Procedure by FEMA 440 (2005)..........................75
3.4 Deficiencies of Codes on Seismic SSI........................................77
4. Development of Nonlinear Earthquake Soil-Structure Interaction
Finite Element Analysis Code (SSI3D) ..........................................78
IX


4.1 Introduction.............................................................78
4.2 General Formulation......................................................79
4.2.1 Formulation of Displacement-based Finite Element Method................79
4.2.2 Finite Element Matrices................................................82
4.2.2.1 Beam-Column Elements.................................................82
4.2.2.2 Solid Elements.......................................................83
4.2.2.3 Interface Elements...................................................85
4.2.3 Solution Methods.....................................................89
4.2.3.1 Static Linear Solutions..............................................89
4.2.3.2 Static Nonlinear Solutions...........................................89
4.3 Dynamic Finite Element Theory..........................................89
4.3.1 Dynamic Formulation....................................................89
4.3.2 Solution of Dynamic Equilibrium Equations..............................91
4.3.2.1 Direct Integration Methods...........................................92
4.3.2.1.1 The Central Different Method.......................................92
4.3.2.1.2 The Houbolt Method.................................................93
4.3.2.1.3 TheNewmark/? Method................................................94
4.3.2.1.4 The Wilson 9 Method................................................97
4.3.2.1.5 The Hilber, Hughes and Taylor a Method.............................99
4.3.2.1.6 Choice of Method...................................................99
4.3.2.2 Mode Superposition Method............................................99
4.4 Computer Program for 3D Soil-Structure Interaction Analysis of
High-Rise Building........................................................100
5. Constitutive Models of Materials..........................................102
5.1 General Observation of Material Behaviors and Modeling and Development
Histories of Material Behavior Modeling...................................102
5.2 Definition of Stress.....................................................102
x


5.3 Definition of Strain..................................................103
5.4 Elasticity............................................................104
5.5 Elasto-Plastic Rate Integral of Differential Plasticity Models........105
5.6 Mohr-Coulomb Model....................................................106
5.6.1 Parameters of Mohr-Coulomb Model....................................106
5.6.2 Elasto-Plastic Constitutive Matrix of Mohr-Coulomb Model............109
5.6.3 Parameter Determination of Mohr-Coulomb Model.......................110
5.7 Modified Cam-Clay Model...............................................112
5.7.1 Parameters of Modified Cam-Clay Model...............................112
5.7.2 Elasto-Plastic Constitutive Matrix of Modified Cam-Clay Model.......113
5.7.3 Parameter Determination of Modified Cam-Clay Model..................115
5.8 Cap Model.............................................................117
5.8.1 Parameters of Cap Model.............................................117
5.8.2 Elasto-Plastic Constitutive Matrix of Cap Model.....................119
5.8.3 Parameter Determination of Cap Model................................120
5.9 Modified Hyperbolic Model.............................................123
5.9.1 Parameters of Modified Hyperbolic Model.............................123
5.9.2 Parameter Determination of Modified Hyperbolic Model................131
5.10 Modified Ramberg-Osgood Model........................................135
5.10.1 Parameters of Modified Ramberg-Osgood Model........................135
5.10.2 Parameter Determination of Modified Ramberg-Osgood Model...........140
5.11 Lade Model...........................................................143
5.11.1 Parameters of Lade Model...........................................143
5.11.2 Elasto-Plastic Constitutive Matrix of Lade Model...................147
5.11.3 Parameter Determination of Lade Model..............................148
5.12 Damping Model in Soil Dynamics.......................................151
5.13 Procedure of Nonlinear Analysis by Finite Element Method.............153
5.14 Soil Model Verification..............................................154
xi


6 Site Response............................................................164
6.1 Introduction...........................................................164
6.2 Fourier Transform......................................................165
6.3 1-D Wave Propagation Theory............................................165
6.3.1 Evaluation of Transfer Functions.....................................165
6.3.2 Equivalent Linear Analysis...........................................169
6.4 1-D Finite Element Method..............................................170
6.4.1 Governing Equations..................................................170
6.4.2 Linear Analysis......................................................172
6.4.3 Nonlinear Analysis...................................................172
6.5 Equivalent Viscous Damping.............................................174
6.6 Program Verification...................................................175
7. Linear Static and Dynamic Stiffness of Soil-Pile Interaction............178
7.1 Stiffness Components...................................................178
7.2 Pile Soil Model........................................................179
7.3 Torsional Stiffness....................................................180
7.3.1 Modulus of Sub-Grade Reaction........................................180
7.3.2 Analytical Solution..................................................181
7.3.3 Numerical Solution (1-D FEM).........................................185
7.4 Vertical Stiffness.....................................................193
7.4.1 Modulus of Sub-Grade Reaction........................................193
7.4.2 Analytical Solution..................................................195
7.4.3 Numerical Solution (1-D FEM).........................................198
7.5 Lateral Stiffness......................................................202
7.5.1 Governing Equation...................................................202
7.5.2 Modulus of Sub-Grade Reaction........................................203
xii


7.5.3 Numerical Solution (1-D FEM)............................................204
7.5.3.1 Stiffness Matrix......................................................204
7.5.3.2 Inertial Interaction..................................................206
7.6 Dynamic Behavior of Single Pile...........................................210
7.6.1 Inertial Interaction....................................................210
7.6.2 Kinematic Interaction...................................................213
7.7 Pile Group Effects........................................................216
7.7.1 General Concept.........................................................216
7.7.2 Governing Equation for Elastic Solution................................217
7.7.3 Pile Group under Vertical Load..........................................218
7.7.4 Pile Group under Lateral Load...........................................223
8. Nonlinear Static Stiffness of Soil-Pile Interaction.......................229
8.1. Introduction..............................................................229
8.2 Nonlinear Torsional Stiffness of Single Pile.............................230
8.3 Nonlinear Vertical Stiffness of Single Pile..............................232
8.4 Nonlinear Lateral Stiffness of Single Pile................................236
8.4.1 Broms Theory...........................................................236
8.4.2 1-D Finite Element Solution.............................................238
8.4.2.1 Pile-Soil Modeling....................................................238
8.4.2.2 Nonlinear Beam Element................................................238
8.4.2.3 Moment Capacity of Reinforced Concrete Pile...........................241
8.4.2.4 Soil Reaction on Piles by p-y Approach...............................245
8.4.3 3D Finite Element Analyses..............................................248
8.4.4 Method Validation.......................................................250
8.4.5 Parameters of Hyperbolic Function.......................................261
8.5 Nonlinear Group Effect..................................................263
8.5.1 Pile Group under Vertical Load..........................................263
xiii


8.5.2 Pile Group under Lateral Load
266
9. Soil-Pile-Structure Interaction of Buildings under Seismic Shaking.......268
9.1. Design Code for Seismic.................................................268
9.1.1 Introduction...........................................................268
9.1.2 Maximum Considered Earthquake Ground Motion...........................268
9.1.3 Site Class Definition..................................................270
9.1.4 Structural Design Criteria.............................................271
9.1.4.1 Index Force Analysis Procedure......................................271
9.1.4.2 Equivalent Lateral Force Procedure..................................271
9.1.4.3 Modal Response Spectrum Analysis Procedure..........................273
9.1.4.4 Linear Response History Analysis Procedure..........................274
9.1.4.5 Nonlinear Response History Analysis Procedure.......................275
9.2 Modal Damping of Multi-Degrees-of-Freedom System.......................275
9.3 Seismic Response of Hypothetical Buildings..............................278
9.3.1 Description of the Buildings...........................................278
9.3.2 Site Selection.........................................................284
9.3.3 Soil Properties........................................................285
9.3.4 Building Analyses......................................................291
9.3.4.1 Analysis Assumptions.................................................291
9.3.4.2 Analysis Procedure...................................................291
9.3.4.3 Dead Load and Live Load..............................................292
9.3.4.4 Ground Motion and Deconvolution......................................295
9.3.4.5 Foundation Properties................................................317
9.3.4.6 Modal Analyses.......................................................323
9.3.4.7 Modal Response Spectra Analyses......................................338
9.3.4.7.1 Analysis Parameters................................................338
9.3.4.7.2 Equivalent Lateral Force Analyses..................................339
xiv


9.3.4.7.3 Modal Response Spectra Analyses.................................340
9.3.4.8 Time History Analyses.............................................343
9.4 Seismic Response of Actual 30-Story Building.........................358
9.4.1 Description of 30-Story Building....................................358
9.4.2 Structural Materials................................................358
9.4.3 Instrumentation.....................................................358
9.4.4 Building Analyses...................................................363
9.4.4.1 Analysis Assumptions..............................................363
9.4.4.2 Analysis Procedures...............................................363
9.4.4.3 Load Determination................................................363
9.4.4.4 Foundation Properties.............................................364
9.4.4.5 Modal Analyses....................................................366
9.4.4.6 Equivalent Lateral Force Analyses.................................369
9.4.4.7 Modal Response Spectrum Analyses..................................370
9.4.4.8 Time History Analyses.............................................371
10. Summary and Conclusion................................................385
10.1 Summary..............................................................385
10.2 Conclusion...........................................................387
10.3 Recommendation for Future Research...................................388
References................................................................389
xv


LIST OF FIGURES
Figures
2.1 Pile Supporting the NHK Building Sheared by Lateral Loading during
the 1964 Niigata Earthquake (Meymand, 1998)..............................6
2.2 Ten-Story Building Supported by Pile Foundation on Soft Soil during
the 1985 Mexico Earthquake (Meymand, 1998)...............................7
2.3 Progression of Soil-Pile-Structure Interaction and Pile Bending Moments
during Liquefaction (Tokimasu et al., 1998)..............................11
2.4 Seismic Response of Structure Embedded on Rock and on Soil....14
2.5 Soil-Structure Interaction Model of Three Degrees of Freedom Structure..16
2.6 Equivalent Model........................................................16
2.7 Equivalent One-Degree-of-Freedom System.................................18
2.8 Foundation Stiffness and Damping Factors for Elastic and Viscoelastic
Half Space (Veletos and Vebric, 1973)....................................21
2.9 Foundation Damping Factors for Half Space with and without
Hysteretic Damping (Veletsos and Vebric, 1973) and Soil Profiles with
Indicated Modulus Profiles and no Hysteretic Damping (Gazetas, 1991).....22
2.10 Embedded Soil-Foundation-Structure on Finite Soil Layer................24
2.11 Foundation Stiffness and Damping Factor for Rigid Cylindrical Foundations
Embedded in Half Space..................................................25
2.12 Horizontal Foundation Stiffness and Damping Factor for Rigid Rectangular
Foundations Embedded in Half Space (Gazetas, 1991)......................26
2.13 Dashpot Coefficient Rocking Radiation Damping versus Frequency for
Different Foundation Shape (Dobry and Gazetas, 1986)....................27
2.14 Disk Foundation........................................................28
2.15 Rocking Stiffness and Damping Factors for Flexible Foundations.........29
xvi


2.16 Assumed Passive Wedge-Type Failure for Clay (Reese et al., 2000)......34
2.17 Characteristic Shapes of p-y Curves for Soft Clay in the Presence
of Free Water..........................................................35
2.18 Assumed Passive Wedge-Type Failure of Pile in Sand....................36
2.19 Characteristic Shape of p-y Curve for Static Loading in Stiff Clay
with no Free Water.....................................................36
2.20 Characteristic Shape of p-y Curve for Cyclic Loading in Stiff Clay with
no Free Water..........................................................37
2.21 Characteristic of a Family of p-y Curve for Static and Cyclic Loading
in Sand................................................................37
2.22 Proposed p-y Curve for c-

2.23 (a) Soil-Pile Model, (b) Soil-Pile Gapping Model, (b) and Force-Displacement
Behavior (Matlock and Foo, 1978).......................................39
2.24 Schematic of Model Used for Deriving Dynamic p-y Curves...............40
2.25 Definition of the p-multiplier Method.................................42
2.26 Kinematic Interaction with Free Field Motions.........................43
2.27 Amplitude of Transfer Function between Free Field Motion and FIM for
Vertical Incident Incoherent Waves.....................................44
2.28 Relationship between Effective Incoherent Parameter ku and Small-Strain
Shear Wave Velocity Vs From Case Histories (from Kim and Stewart, 2003).45
2.29 (a) Transfer Function Amplitudes for Embedded Cylinders along with
Approximation, (b) Transfer Function Amplitude Model
(Stewart et al., 2004).................................................48
2.30 Pile Foundation System and Soil Profile (Fan et al., 1991)............49
2.31 Idealized General Shape of Kinematic Displacement Factor, Iu = Iu (a0),
Explaining Transition Frequency Factor a0] and a02 (Fanetal., 1991)....49
2.32 Hybrid Modeling of Soil-Structure Interaction (Tzong et al., 1981)....52
xvii


2.33 Layout of 20-Story Building and Two Options of Pile Arrangement
(Han and Cathro, 1997)..................................................57
2.34 Maximum Story Displacement and Maximum Interstory Shear of Building
with Different Conditions of Foundation (Han and Cathro, 1997)..........57
2.35 3D High-Rise Structure-Foundation-Soil Dynamic Interaction Model
(Wu and Gan, 1998)......................................................59
2.36 Floor Displacements of Frame Structure ...............................59
2.37 Floor Displacements of Frame-Shear Wall Structure.....................60
2.38 Shear Force Envelopes of Floors in Superstructure......................60
2.39 Overturning Moment Envelopes of Superstructure.........................60
2.40 1-D Model of Free Field (left) and Strain-Dependent Characteristic
of R-0 Model (right) (Inaba et al., 2000)...............................62
2.41 (a) Distribution of Maximum Shear Strain, y (b) Soil Damping Factor, £,
at Maximum Shear Strain (Inaba et al., 2000)............................62
2.42 2D Finite Element Model of Building with Soil-Structure Interaction
(Inaba et al., 2000)....................................................63
2.43 Time History (left) and Acceleration Response Spectrum (right) of Input
Motion for 1995 Hyogo-Nanbu Earthquake (Hayashi and Takahashi, 2004)...64
2.44 Finite Element Model of SSI System and Analysis Model for Superstructure .64
2.45 Soil Profile (Hayashi and Takahashi, 2004)............................65
2.46 Maximum Response Values of Shear Force Coefficient, Rotational Angle and
Deformation in Input Uplift State (Hayashi and Takahashi, 2004).........65
2.47 Analysis Model of Parametric Analyses using Observed Ground Motion (left)
and Input Direction (right) (Hayashi and Takahashi, 2004)...............66
2.48 Natural Period of SSI and FIX Models and Response Spectra
(Hayashi and Takahashi, 2004)...........................................68
2.49 Maximum Shear Deformation Angle (FKI; 1995 Hyogoken-Nanbu, Japan,
xviii


Earthquake) (Hayashi and Takahashi, 2004)..............................68
2.50 Maximum Shear Deformation Angle (TCU068 (left) and TCU074 (right)
1999 Chi-Chi, Taiwan, Earthquake) (Hayashi and Takahashi, 2004).........68
3.1 Foundation Damping Factor (IBC 2003 or ASCE 7-02)......................72
3.2 Foundation Damping Factor £f Expressed as a Function of Period Lengthening
feq jTaj for Building Different Aspect Ratios hjre and Embedded Ratios e/ru
(FEMA440, 2005)......................................................75
3.3 RRS for Foundation Embedded to Depth e 30 ft in Difference Site Categories
and RRS for Foundation with Variable Depths in Site Classes C and D....76
3.4 RRS for Foundation with be = 330 ft, Simplified Model versus Exact Solution
for Ka and RRS for Simplified Model as Function of Foundation size, be.76
4.1 Local Axes System and Displacement Components of Beam-Column Element 82
4.2 Solid Element Types.................................................84
4.3 8-Node and 16-Node Interface Elements...............................86
4.4 Graphical Representation of the Newmark (5 Method...................95
4.5 Graphical Representation of the Wilson Q Method.....................98
5.1 Mohr-Coulomb Failure Criteria......................................107
5.2 Mohr-Coulomb Failure Criteria in Principal Stress Space............108
5.3 Comer Treatment of Mohr-Coulomb Model..............................109
5.4 Determination of Friction Angle and Cohesion.......................111
5.5 Determination of Dilatancy Angle...................................111
5.6 Parameter for Cam-Clay model.......................................112
5.7 Yield Curve for Modified Cam-Clay model................................113
5.8 Determination of M Parameter of MCC Soil by Triaxial Test...............116
xix


5.9 Determination of k and A Parameters of MCC Soil by Oedometer Test.........116
5.10 Yield Surface for Cap Model (Desai and Siriwardane, 1984)................117
5.11 Interpretation of Parameters for fx (Desai and Siriwardane, 1984)........122
5.12 Nonlinear Stress-Strain Behavior..........................................124
5.13 Yield Surface Model of Modified Hyperbolic Model.........................128
5.14 Unloading-Reloading Behavior..............................................131
5.15 Stress-Strain Response of Modified Hyperbolic Model......................133
5.16 Transformed Stress-Strain Plots from Triaxial Test and
Determination of Initial Youngs Modulus and Asymptotic Deviator Stress.... 133
5.17 Determination of Modified Hyperbolic Parameters K, and n.................134
5.18 Shear Modulus at Very Small Strain and Secant Modulus (Kramer, 1996).....135
5.19 Modulus Reduction Curves of Average Sand (Seed and Idriss, 1970).........141
5.20 Example of a Best Fit Straight Line for Determining Parameters a and r
(after Ueng and Chen, 1992)...............................................142
5.21 Characteristics of Failure Criterion in Principal Stress Space
(Lade and Jacobsen, 2002).................................................143
5.22 Characteristics of the Plastic Potential Function in Principal Stress Space
(Lade and Jacobsen, 2002).................................................144
5.23 Characteristics of Yield Function in Principal Stress Space
(Lade and Jacobsen, 2002).................................................145
5.24 Modeling of Work Hardening and Softening (Lade and Jacobsen, 2002).......146
5.25 Relationship between Hysteresis Loop and Damping Ratio (Kramer, 1996)151
5.26 Damping Ratio with Shear Strain for Sands (Seed and Idriss, 1970)........152
5.27 Stress-Strain Curves (cr3 = 338.45kPa)...................................156
5.28 Volumetric Train and Axial Strain (cr3 = 338.45kPa)......................156
5.29 Stress-Strain Curves (r3 = 261.9 kPa)...................................157
xx


5.30 Volumetric Train and Axial Strain (cr3 = 261.9 kPa)....................157
5.31 Stress-Strain Curves ( 5.32 Volumetric Train and Axial Strain (cr3 = 206.8 kPa)....................158
5.33 Stress-Strain Curves (cr3 = 310.26kPa).................................160
5.34 Volumetric Strain and Axial Strain (cr3 = 310.26kPa)...................160
5.35 Stress-Strain Curves (cr3 = 172.37 kPa)................................161
5.36 Volumetric Strain and Axial Strain (cr3 =172.37 kPa)...................161
5.37 Stress-Strain Curves ( 5.38 Volumetric Strain and Axial Strain ( 5.39 Comparison of Modulus Degradation Curves for Clay......................163
6.1 Thin Element of a Kelvin-Voigt Model Subject to
Horizontal Shearing (Kramer, 1996).......................................166
6.2 One-Dimensional System under Horizontal Seismic Motion..................167
6.3 Soil Layer and Finite Element Model Subjected to a Horizontal Seismic
Motion at Its Base.......................................................171
6.4 Definition of Deviator Stress at Failure................................173
6.5 Comparison of Ground Acceleration for Linear Case........................176
6.6 Comparison of Ground Acceleration for Equivalent Linear Case.............177
7.1 Displacement and Load Components in Local Coordinate System.............179
7.2 Soil Profiles............................................................179
7.3 Circumferential Equilibrium of Soil Element (Randolph, 1981).............180
7.4 Shape Function for Element under Torsional Moment........................186
7.5 Calculation Procedure of Equivalent Torsional Stiffness for Pile with Length
Greater than Critical Length..............................................188
xxi


7.6 Effect of Pile Length on Accuracy of Numerical Solution................189
7.7 Equivalent Torsional Stiffness of Single Pile with Different Shear Modulus of
Pile Material in Homogeneous Soil.......................................191
7.8 Equivalent Torsional Stiffness of Single Pile in Homogeneous Soil......191
7.9 Equivalent Torsional Stiffness of Single Pile in Non-Homogeneous Soil..192
7.10 Equivalent Torsional Stiffness of Single Pile in Non-Homogeneous Soil.192
7.11 Stress on Soil Element under Vertical Load (Randolph and Wroth, 1978).193
7.12 Effect of Number of Segments on Vertical Stiffness for 10 m Long Pile.200
7.13 Effect of Number of Segments on Vertical Stiffness for 20 m Long Pile.200
7.14 Effect of Number of Segments on Vertical Stiffness for 30 m Long Pile.201
7.15 Effect of Number of Segments on Vertical Stiffness for 40 m Long Pile.201
7.16 Pile under Lateral Load...............................................202
7.17 Shape Function for Pile under Lateral Load............................205
7.18 Comparison of Equivalent Stiffness Calculated by Eq. 7.110 and 3D FEM for
Pile of 1 m Diameter (fixed head)......................................208
7.19 Comparison of Equivalent Stiffness Calculated by Eq. 7.110 and 3D FEM for
Pile of 1.2 m Diameter (fixed head)....................................208
7.20 Comparison of Lateral Displacement at Pile Top Calculated by Numerical
method and 3D FEM for pile of 1.0 m Diameter (free head)...............209
7.21 Comparison of Lateral Displacement at Pile Top Calculated by Numerical
method and 3D FEM for pile of 1.2 m Diameter (free head)...............209
7.22 Kinematic Interaction Model (Gazetas and Mylonakis, 1998).............213
7.23 Equivalent System of Soil-Pile Kinematic Interaction..................214
7.24 Kinematic Seismic Response of Single Pile in Homogeneous Soil.........216
7.25 Group of Two Piles under Vertical Loads...............................218
7.26 Reduction Factor for Two Pile Group (L/D=10, v = 0.3 H/L=2).........219
7.27 Reduction Factor for Two Pile Group (L/D=20, v = 0.3 H/L=2).........220
7.28 Reduction Factor for Two Pile Group (L/D=30, v = 0.3 H/L=2).........220
xxii


7.29 Correction Factor for Finite Depth Effect.............................221
7.30 Correction Factor for Poissons Ratio Effect..........................221
7.31 Comparison of Reduction Factors.......................................222
7.32 Pile Numbering for Group of Nine Piles under Vertical Load............222
7.33 Group of Two Piles under Lateral Loads................................224
7.34 In-line Reduction Factor for Two Pile Group under Lateral Load (v = 0.3) ...225
7.35 Side-by-Side Reduction Factor for Two Pile Group under
Lateral Load (v = 0.3).................................................225
7.36 In-Line Correction Factors for Poissons Ratio Effect ................226
7.37 Side-by-Side Correction Factors for Poissons Ratio Effect............226
7.38 Side-by-Side and In-Line Effects......................................227
7.39 Diagram for Computing Reduction Factor for Skewed Piles...............227
7.40 Pile Numbering for Group of Nine Piles under Lateral Load (linear effects) ...227
8.1 Hysteretic Models.......................................................229
8.2 Moment-Rotation Curves of Single Pile under Torsional Load.............230
8.3 Load-Displacement Curves of Single Pile under Vertical Load............233
8.4 Cu and Ko profiles (Wang and Sita 2004) ...............................234
8.5 Comparison of the Results between PSI, OPENSEES and Test Data..........235
8.6 Fixed Head Pile in Cohesive Soil (Broms, 1964a) ........................237
8.7 Fixed Head Pile in Cohesionless Soil (Broms, 1964b).....................237
8.8 Beam Segment from a Beam Subjected to Pure Bending.....................239
8.9 Stress-Strain Curve for Concrete (ONeill and Reese, 1999) ............242
8.10 Stress-strain Curve for Steel (ONeill and Reese, 1999) ..............242
8.11 Finite Strips of Cross-Section........................................243
8.12 Moment Capacities of Drilled Shaft with 3000 psi Concrete.............244
8.13 Moment Capacities of Drilled Shaft with 4500 psi Concrete.............244
8.14 Moment Capacities of Drilled Shaft with 6000 psi Concrete.............245
xxiii


8.15 Distribution of Normal Stress and Shear Stress.......................246
8.16 Plane View and Isotropic View of 3D Model of Single Pile.............249
8.17 Bending Moment Diagram of 4 Percent Reinforcement Drilled Shaft in Clay .253
8.18 Shear Force Diagram of 4 Percent Reinforcement Drilled Shaft in Clay...253
8.19 Soil Reaction Diagram of 4 Percent Reinforcement Drilled Shaft in Clay.254
8.20 Deflection Diagram of 4 Percent Reinforcement Drilled Shaft in Clay....254
8.21 Bending Moment Diagram of 2 Percent Reinforcement Drilled Shaft in Clay .255
8.22 Shear Force Diagram of 2 Percent Reinforcement Drilled Shaft in Clay...255
8.23 Soil Reaction Diagram of 2 Percent Reinforcement Drilled Shaft in Clay.256
8.24 Deflection Diagram of 2 Percent Reinforcement Drilled Shaft in Clay....256
8.25 Bending Moment Diagram of 4 Percent Reinforcement Drilled Shaft in Sand 257
8.26 Shear Force Diagram of 4 Percent Reinforcement Drilled Shaft in Sand...257
8.27 Soil Reaction Diagram of 4 Percent Reinforcement Drilled Shaft in Sand.258
8.28 Deflection Diagram of 4 Percent Reinforcement Drilled Shaft in Sand....258
8.29 Bending Moment Diagram of 2 Percent Reinforcement Drilled Shaft in Sand 259
8.30 Shear Force Diagram of 2 Percent Reinforcement Drilled Shaft in Sand...259
8.31 Soil Reaction Diagram of 2 Percent Reinforcement Drilled Shaft in Sand.260
8.32 Deflection Diagram of 2 Percent Reinforcement Drilled Shaft in Sand....260
8.33 Load-Displacement Curves of Single Pile under Lateral Load.............261
8.34 Load-Displacement Curve of Single Pile in Medium Clay
(2 Percent of Reinforcement)..........................................262
8.35 Load-Displacement Curve of Single Pile in Medium Sand
(4 Percent of Reinforcement)..........................................262
8.36 Load-Displacement Curves of Single Pile and Pile Group in Medium Clay ....264
8.37 Load-Displacement Curves of Single Pile and Pile Group in Stiff Clay.264
8.38 Load-Displacement Curves of Single Pile and Pile Group in Very Stiff Clay..265
8.39 Pile Numbering for Group of Nine Piles under Vertical Load.............265
8.40 Pile Numbering for Group of Nine Piles under Lateral Load
xxiv


(Nonlinear Effects)
266
9.1 Design Response Spectra.................................................270
9.2 Plane View of Building 1................................................278
9.3 Frame 1 of Building 1...................................................279
9.4 Isotropic View of Building 1............................................280
9.5 Plane View of Building 2................................................281
9.6 Frame A of Building 2...................................................282
9.7 Isotropic View of Building 2............................................283
9.8 Modulus degradation and damping curves for clay soil (Sun et al., 1988 and
Vucetic and Dobry, 1991)..................................................289
9.9 Modulus Degradation and Damping Curves for Clay Soil (Seed et al., 1984) ...290
9.10 Emeryville Time History Function in the X Direction....................297
9.11 Emeryville Time History Function in the Z Direction....................297
9.12 UCLA Time History Function in the X Direction..........................298
9.13 UCLA Time History Function in the Z Direction..........................298
9.14 LA Hollywood Time History Function in the X Direction..................299
9.15 LA Hollywood Time History Function in the Z Direction..................299
9.16 Transfer Function for Site 1...........................................301
9.17 Transfer Function for Site 2...........................................302
9.18 Transfer Function for Site 3...........................................303
9.19 Bed Rock Motion for Site 1 in the X Direction.........................305
9.20 Bed Rock Motion for Site 1 in the Z Direction.........................305
9.21 Bed Rock Motion for Site 2 in the X Direction.........................306
9.22 Bed Rock Motion for Site 2 in the Z Direction.........................306
9.23 Bed Rock Motion for Site 3 in the X Direction.........................307
9.24 Bed Rock Motion for Site 3 in the Z Direction.........................307
9.25 Response Spectral Acceleration in the X Direction (Site 1).............308
xxv


9.26 Response Spectral Acceleration in the Z Direction (Site 1) ..........308
9.27 Response Spectral Acceleration in the X Direction (Site 2)...........309
9.28 Response Spectral Acceleration in the Z Direction (Site 2) ..........309
9.29 Response Spectral Acceleration in the X Direction (Site 3)...........310
9.30 Response Spectral Acceleration in the Z Direction (Site 3) ..........310
9.31 Comparison of Story Shears from Computed Far-Field Motion and Measured
Far-Field Motion for Building 1, Site 1...............................311
9.32 Comparison of Story Shears from Computed Far-Field Motion and Measured
Far-Field Motion for Building 2, Site 1...............................312
9.33 Comparison of Story Shears from Computed Far-Field Motion and Measured
Far-Field Motion for Building 1, Site 2...............................313
9.34 Comparison of Story Shears from Computed Far-Field Motion and Measured
Far-Field Motion for Building 2, Site 2...............................314
9.35 Comparison of Story Shears from Computed Far-Field Motion and Measured
Far-Field Motion for Building 1, Site 3...............................315
9.36 Comparison of Story Shears from Computed Far-Field Motion and Measured
Far-Field Motion for Building 2, Site 3...............................316
9.37 Pile Foundation for Building 1.......................................319
9.38 Pile Foundation for Building 2.......................................319
9.39 Load Displacement Curve of Single Pile under Lateral Load
(Building 1, Site 1)...................................................320
9.40 Load Displacement Curve of Single Pile under Lateral Load
(Building 2, Site 1)...................................................320
9.41 Load Displacement Curve of Single Pile under Lateral Load
(Building 1, Site 2)...................................................321
9.42 Load Displacement Curve of Single Pile under Lateral Load
(Building 2, Site 2)...................................................321
9.43 Load Displacement Curve of Single Pile under Lateral Load
xxvi


(Building 1, Site 3).....................................................322
9.44 Load Displacement Curve of Single Pile under Lateral Load
(Building 1, Site 3).....................................................322
9.45 IBC 2003 Design Response Spectrum.......................................338
9.46 Maximum Story Shear in the X Direction (Building 1, Site 1) ............345
9.47 Maximum Story Shear in the Z Direction (Building 1, Site 1).............345
9.48 Maximum Story Shear in the X Direction (Building 2, Site 1) ............346
9.49 Maximum Story Shear in the Z Direction (Building 2, Site 1).............346
9.50 Maximum Story Shear in the X Direction (Building 1, Site 2) ............347
9.51 Maximum Story Shear in the Z Direction (Building 1, Site 2).............347
9.52 Maximum Story Shear in the X Direction (Building 2, Site 2) ............348
9.53 Maximum Story Shear in the Z Direction (Building 2, Site 2).............348
9.54 Maximum Story Shear in the X Direction (Building 1, Site 3) ............349
9.55 Maximum Story Shear in the Z Direction (Building 1, Site 3).............349
9.56 Maximum Story Shear in the X Direction (Building 2, Site 3) ............350
9.57 Maximum Story Shear in the Z Direction (Building 2, Site 3).............350
9.58 Transfer Function for Site 1............................................351
9.59 Transfer Function for Site 2............................................351
9.60 Transfer Function for Site 3............................................351
9.61 Base Motion in the X Direction (Building 1, Site 2) ....................352
9.62 Base Motion in the Z Direction (Building 1, Site 2).....................352
9.63 Base Motion in the X Direction (Building 2, Site 2) ....................353
9.64 Base Motion in the Z Direction (Building 2, Site 2).....................353
9.65 Plane View of 30-Story Building.........................................359
9.66 Elevation View of 30-Story Building.....................................360
9.67 Isotropic View of 30-Story Building.....................................361
9.68 Instrument Locations in 30-Story Building ..............................362
9.69 Pile Arrangement........................................................364
xxvii


9.70 Load Displacement Curve of Single Pile under Lateral Load...........365
9.71 First Three Modes of Vibration......................................368
9.72 Ground Floor Input Motion...........................................375
9.73 Accelerations in the E-W Direction..................................376
9.74 Accelerations in the N-S Direction..................................377
9.75 Displacements in E-W Direction......................................378
9.76 Displacements in N-S Direction......................................379
9.77 Comparison of Displacements (Fixed Base Model using Far Field Motion) 380
9.78 Comparison of Displacements (Linear Spring Model)...................381
9.79 Comparison of Displacements (Nonlinear Spring Model)................382
9.80 Comparison of Computed Base Shear...................................383
9.81 Comparison of Calculated Base Shear for Different Modes.............384
xxviii


LIST OF TABLES
Tables
2.1 Dynamic interaction factor............................................41
2.2 Impedance functions of pile group by Dobry and Gazetas (1988).....41
2.3 Approximate values of n2..............................................46
5.1 Mohr-Coulomb model parameters.........................................110
5.2 MCC soil parameters..................................................115
5.3 Material parameters for the cap model................................120
5.4 Parameters of the modified hyperbolic model..........................132
5.5 Material parameter for the modified Ramberg-Osgood model.............140
5.6 Material parameter for the Lade model................................148
5.7 Soil properties (Lade, 1973).........................................155
5.8 Parameters for the modified hyperbolic model.........................155
5.9 Parameters for the Cap model.........................................155
5.10 Parameters for the modified hyperbolic model........................159
5.11 Parameters for the Lade model (Helwany, 2007).......................159
7.1 Comparison of vertical impedance function of single pile.............211
7.2 Value of p...........................................................212
7.3 Values of ft for different soil deposits.............................212
7.4 Reduction factor for pile group of nine piles under vertical load....223
7.5 Reduction factor for pile group of nine piles under lateral load.....228
8.1 Properties of concrete................................................251
xxix


8.2 Soil properties for Mohr-Coulomb model....................................251
8.3 Ultimate moments and lateral forces.......................................252
8.4 Reduction factor of bearing capacity of 3x3 pile group....................263
8.5 Load distribution on individual pile in 3x3 pile group (%)................266
9.1 Site locations and time history properties................................284
9.2 Parameters for damping ratio..............................................285
9.3 Soil properties (site 1)..................................................286
9.4 Parameters for damping ratio (site 1).....................................286
9.5 Soil properties (site 2)..................................................287
9.6 Parameters for damping ratio (site 2).....................................287
9.7 Soil properties (site 3)..................................................288
9.8 Parameters for damping ratio (site 3).....................................288
9.9 Building material properties..............................................292
9.10 Dead load for building 1................................................292
9.11 Dead load and live load criteria for building 1 (ASCE 7-02) .............293
9.12 Dead load and live load criteria for building 2 (ASCE 7-02)..............294
9.13 Equivalent linear soil parameters (site 1)..............................304
9.14 Equivalent linear soil parameters (site 2)...............................304
9.15 Equivalent linear soil parameters (site 3)...............................304
9.16 Natural site periods (s) ...............................................305
9.17 Initial stiffness of single piles for building 1.........................317
9.18 Initial stiffness of single piles for building 2.........................317
9.19 Imaginary parts of impedance functions of single piles for building 1....318
9.20 Imaginary parts of impedance functions of single piles for building 2....318
9.21 Natural periods of building 1 (fixed base) .............................324
xxx


9.22 Natural periods of building 1 (flexible base, site 1, MC model) ...............325
9.23 Natural periods of building 1 (flexible base, site 1, MH and MRO model) .326
9.24 Natural periods of building 1 (flexible base, site 2, MC model) ...............327
9.25 Natural periods of building 1 (flexible base, site 2, MH and MRO model) .328
9.26 Natural periods of building 1 (flexible base, site 3, MC model) ...............329
9.27 Natural periods of building 1 (flexible base, site 3, MH and MRO models) ....330
9.28 Periods of building 2 (fixed base)............................................331
9.29 Natural periods of building 2 (flexible base, site 1, MC model) ...............332
9.30 Natural periods of building 2 (flexible base, site 1, MH and MRO models) ....333
9.31 Natural periods of building 2 (flexible base, site 2, MC model) ...............334
9.32 Natural periods of building 2 (flexible base, site 2, MH and MRO models) ....335
9.33 Natural periods of building 2 (flexible base, site 3, MC model) ...............336
9.34 Natural periods of building 2 (flexible base, site 3, MH and MRO models) ....337
9.35 Design spectral acceleration...................................................338
9.36 Base shear determined from equivalent lateral force procedure..................339
9.37 Reduced base shear determined from equivalent lateral force procedure..........339
9.38 Modal damping and Rayleigh damping of fixed base structure.....................340
9.39 Modal damping of flexible base structure (%)...................................341
9.40 Maximum base shear for fixed base model (building 1) ..........................342
9.41 Maximum base shear for flexible base model (building 1)........................342
9.42 Maximum base shear for fixed base model (building 2)...........................342
9.43 Maximum base shear for flexible base model (building 2)........................342
9.44 Maximum base shears (full soil-pile models)....................................354
9.45 Maximum base shears (fixed base and flexible models)...........................355
9.46 Maximum displacements at the top of the buildings (full soil-pile models) ....356
9.47 Maximum displacements at the top of the buildings
xxxi


(fixed base and flexible models) ......................................357
9.48 Building weight.......................................................363
9.49 Initial stiffness and damping of single piles.........................364
9.50 Natural periods (s)...................................................367
9.51 Base shear determined from equivalent lateral force procedure.........369
9.52 Reduced base shear determined from equivalent lateral force procedure.369
9.53 Maximum base shear from design spectrum...............................370
9.54 Maximum displacements on the roof.....................................373
9.55 Maximum base shear....................................................374
xxxii


1. Introduction
1.1 Problem Statement
Numerous high rise buildings have been built in earthquake prone areas of the
U.S. and abroad in the last hundred years. High rise buildings usually are designed by
using deep foundations. Deep foundations consist of driven or drilled-in piles/piers
that are routinely employed to transfer structural loads through soft soils to stronger
bearing strata at depth. These foundation elements may also be subject to transient or
cyclic lateral loads arising from earthquake, wind, wave, blast, impact, or machine
loading. The coincidence of major pile-supported structures sited on soft soils in areas
of earthquake hazard results in significant demands on these deep foundations.
Possible amplification and resonance effects could occur between longer period
structures and soft soil site ground motion. Liquefaction and/or strain-softening
and/or hardening potential in these soft soils can impose additional demands on pile
foundation systems.
Historically, it has been common seismic design practice to study the
influence of a shallow foundation on linear behavior, and ignore or simplify the
influence of pile foundations on the ground motions applied to the structure. This is
generally accepted as a conservative design assumption for a spectral analysis
approach, as the flexible pile foundation results in period lengthening and increased
damping, and, consequently, decreased structural forces relative to a fixed base case.
However, in extreme cases such as the 1985 Mexico City Earthquake, period
lengthening can result in increased spectral values relative to current code
specifications.
To assess the effect of nonlinear behavior of soil and pile under strong
earthquake, the seismic responses of high rise buildings need to be examined by
numerical modeling using nonlinear material and soil-pile-structure interface models.
1


The soil, pile and structure will be analyzed simultaneously with the appropriately
constitutive model of soils. Numerous numerical modeling studies indicate the
modeling of linear and nonlinear soil-pile and improved procedures need to be
formulated to replace the existing procedures in current design code.
1.2 Research Objectives
The objectives of this study are:
To examine the shortcomings in the state-of-practice of the seismic design and
analysis of soil-pile interaction effects on high rise buildings.
To expand the existing structural finite element codes for their capability in
reserving nonlinear soil-structure effects.
To examine the response of soil-pile interaction and implement the soil model
appropriate for transient cyclic loads.
To determine the procedure of linear and nonlinear 1-D wave propagation in
time domain.
To develop a model of the equivalent stiffness of linear and nonlinear soil-pile
interaction in layered soils.
To recommend the IBC 2003 code to account for the linear and nonlinear
seismic soil-structure interaction effects on high-rises.
1.3 Research Approach
To achieve the research objectives, the following tasks are to be undertaken:
Review the seismic performances of high rise buildings in the fields.
Review the state-of-practice seismic design and analysis of soil-pile
interaction of high rise buildings.
Review the experimental study to investigate the seismic responses of soil-pile
interaction, and the current models of soils.
Improve the nonlinear elastic model for cyclic loads.
2


Review and implement the available constitutive models capable of
simulating behavior under transient loads.
Develop a seismic resistance analysis computer code and verify its
effectiveness by comparing the results of the numerical and experimental
studies, and in-site measured performance of buildings.
Perform the dynamic finite element analyses on single pile and pile group to
evaluate the six components of equivalent stiffness of nonlinear soil-pile
interaction.
Simple analytical simulation of the effect of relative pile, foundation stiffness,
structure and pile foundation dimensions on high rise performance.
Analyze the new or existing high rises and compare the result using the IBC
2003 code with linear spring of soil-structure interaction, time history analysis
with linear spring of soil-structure interaction, time history analysis with
nonlinear spring of soil-structure interaction and time history analysis with
full model of soil and structure elements.
Identify the shortcomings of the IBC 2003 code with linear spring of soil-
structure interaction and recommend new approach for seismic soil-structure
interaction.
1.4 Organization of the Thesis
The technical content of this thesis is contained in ten chapters. Chapter 2 is a
review of the literature, including (1) the seismic performances of high rise buildings
in the fields, (2) the state-of-practice seismic design and analysis of soil-pile
interaction of high rises. In Chapter 3, the procedures of soil structure interaction are
reviewed and investigated, and the shortcomings of design code IBC 2003 and FEMA
440 are identified. Chapter 4 reviews the theory of finite element method used in the
development of a seismic resistance analysis computer code. Chapter 5 presents the
implementation of the available constitutive models capable of simulating behavior
3


under transient loads. One-dimensional wave propagation theories in both frequency
domain and time domain are presented in Chapter 6. A new method used to determine
the equivalent viscous damping is also given in this chapter. In Chapter 7, a simple
method used to determine the static and dynamic stiffness of single pile under
torsional, vertical and lateral loads is developed for layered soils. The effectiveness is
verified by comparing the results of this numerical study with previous studies.
Chapter 8 performs nonlinear finite element analyses on single pile and pile group to
evaluate equivalent stiffness of nonlinear soil-pile interaction. Chapter 9 presents the
analyses of two hypothetical buildings and one actual building and the results are
compared to the design using the IBC 2003 code. Chapter 10 summarizes this study
and presents concluding remarks, documenting the key contributions of this study.
Finally, some important future investigations are recommended.
4


2. Literature Review
2.1 Post Earthquake Investigations
2.1.1 The Niigata Earthquake of June 16,1964, Japan
The Niigata earthquake of 1964 had a magnitude of 7.3, had caused wide
spread liquefaction-related damages and numerous failures of pile supported
structures. Meymand (1998) summarized the studies of pile damage from other
authors and these studies are briefly represented below.
Several specific cases of liquefaction-induced pile damage at site for driven
piles in loose sand deposits with a Standard Penetration Test (SPT) blow count of
N=5-10 were surveyed. The 7.2 m long concrete piles of 0.18 m diameter lost bearing
capacity due to liquefaction, and the superstructure tilted and cracked at the Saiseikai
Hospital. At the Ishizue Primary School, the same concrete piles also lost bearing
capacity, and the differential settlement caused distress in structure.
Another liquefaction-related pile failure mechanism was uncovered by
estimating permanent ground displacements during the Niigata earthquake from aerial
photographs and correlating these displacements to damage observed in piles
excavated more than 20 years after the earthquake. The 11-12 m long 350-mm
diameter concrete piles supporting the NHK Building were found to be broken at two
points, near top of the pile and near the base, and were consistently inclined in the
direction of permanent ground displacement. Similar damage was discovered during
the reconstruction of the Niigata Family Courthouse; damage near the head of the 350
mm diameter concrete piles was more severe, where damage at the lower positions
corresponded to the boundary between the liquefied and non-liquefied soil layers.
Liquefaction related pile damage occurred to the Niigata Family Courthouse and the
NHK building; the pile crack patterns are shown in Figure 2.1. One other report
showed that the three-story branch office of the Daiyon Bank settled as much as 1.3
5


m, and tilted, but only experienced minor structural damage. After the earthquake, the
building was jacked up and supported on newly driven H-piles. In 1984, this structure
was demolished, and during the basement excavation the severely damaged original
pre-cast concrete piles were exposed; the piles were damaged both in the zone of
maximum moment and also near the tip where a stiffness contrast in the soil layer
occurred. The fact that these structures remained in service with no indication of the
defective foundation condition vividly illustrates the difficulty of ascertaining pile
performance and/or damage in an earthquake.
Figure 2.1: Pile Supporting the NHK Building Sheared by Lateral Loading During the
1964 Niigata Earthquake (Meymand, 1998)
2.1.2 Mexico City Earthquake of September 19,1985, Mexico
The magnitude 8.1 Mexico City earthquake was some 400 km distant from its
epicenter in Mexico City, but a convergence of site response factors focused
enormous damage on the Lake Zone of Mexico City. The seismic waves were
6


effectively filtered to long period motion in traveling from the epicenter to the
Mexico region. In the Lake Zone, the deep soft clay deposits caused the intensity of
shaking to greatly amplify, with a resulting period of approximately two seconds.
This long period motion came into resonance with many structures of intermediate
height, resulting in a damage pattern closely focused on these long period structures
(Meymand, 1998). Of these buildings, those supported by friction piles experienced
the most damage.
Figure 2.2: Ten-Story Building Supported by Pile Foundation on Soft Soil During the
1985 Mexico Earthquake (Meymand, 1998)
7


It is useful to review the types of pile foundations employed in this area.
Principal concerns of foundation design in the highly compressible Mexico City clays
are limiting settlement and accommodating negative skin friction on deep
foundations. Friction piles are employed with compensation for medium weight
structures, and without compensated foundations for light structures. Heavy structures
are supported by end bearing piles founded on Tarango sand at some 33 38 m
depth. Additionally, many buildings are supported on end bearing piles with control
devices on their heads; the pile freely penetrates the foundation slab, which is then
allowed to settle at the same rate as the surrounding soil. The control devices are
commonly provided by wooden blocks, which compress at a predetermined rate, or
hydraulically controlled jacks.
Girault (1986) presented a study of 25 buildings on mat foundations supported
by friction piles that experienced large settlement (up to 130 cm) and tilting. The
mechanism for these settlements was reduction of the negative skin friction on the
pile due to partial loss of shear strength during cyclic loading of the sensitive clays.
Mendoza and Romo (1989) affirm that low factors of safety and a soil-pile stress state
close to yielding with respect to static loading precipitated foundation failures under
seismic loading. Tilting and overturning due to cyclic rocking may have been
exacerbated by P A effects associated with lack of plumbness of continually setting
structures. In one instance a ten-story building overturned entirely, as one comer sank
6 m into the soil and others rose out of the soil 3 m, pulling piles out of the ground.
The site profile of this damaged structure is shown in Fig. 2.2 (Meymand, 1998).
Structures supported on end bearing piles performed better than friction piles, with
smaller settlements and fewer failures. Slender buildings with control piles
experienced tilting as the wooden blocks were crushed and the hydraulic jacks burst;
these control devices were clearly not designed for the seismically induced inertial
rocking loads imposed on them. In summary, seismic overturning moments were the
8


main causes of failure of pile foundations, though perhaps some cyclic strength
degradation contributed to a partial loss of soil pile adhesion.
2.1.3 1995 Hyogo-Ken Nanbu (Kobe) Earthquake, Japan
The Kobe earthquake was one of the most devastating earthquakes ever to hit
Japan. The magnitude of this earthquake was 7.2. The duration was about 20 seconds.
The focus of the earthquake was less than 20 km below Awaji-Shima, an island in the
Japan inland sea. This island is near the city of Kobe. More than 5500 people were
killed and over 26000 injured, mainly in Kobe, and damage to more than 200,000
houses, devastating most of the infrastructure; the losses exceeded 200 billion dollars
(U.C. Berkeley, 1995). Newly built ductile-frame buildings did quite well on the
whole and had little damage, although some were left standing at an angle when the
soil beneath them liquefied.
During the Kobe earthquake, numerous pile foundations were damaged
because of the strong levels of shaking, the prevalence of liquefaction, and the high
concentration of pile-supported structures in the area. Pile foundation damage was
categorized by the pile foundation type, properties of soils and the failure mechanism.
Most of the poorly reinforced concrete piles with weak pile cap connections
were damaged and reinforced concrete pile of newer building performed better than
those of older buildings. The steel pipe piles perform better than reinforced concrete
piles because of better ductility.
The foundation distress caused by soil liquefaction was most common in areas
of laterally spreading ground. Damage mechanisms of foundation were classified as
follows:
1. Damage near pile heads because of laterally internal force from the
superstructure.
2. Buildings that were supported by friction pile embedded within the
liquefied zone settled with the surrounding ground and tilted ranged from servere to
negligible.
9


3. No apparent settlement of the buildings occurred when they were supported
on piles bearing on firm soil layers beneath the zone of liquefaction, while the
surrounding ground had large settlements so that large vertical gaps occurred around
the base of buildings.
4. Pile foundations near distressed quay walls in which the lateral ground
spread were often damaged by the kinematic forces of the spreading ground against
pile foundations and some inertia forces of the superstructure.
The mechanisms (1) and (2) of pile damage appear to be more salient in
buildings with high aspect ratios (height to base dimension).
Mizuno et al. (1996) found out the damage patterns of the pile foundations by
studying more than 30 cases of pile damage that were observed in pre-cast concrete,
cast-in-pile concrete and steel pipe piles. Damage patterns consisted of separation
between piles and pile caps; damage near the pile head; and damage at deeper portion
of piles externally caused by internal forces from superstructure, lateral soil flow with
liquefaction, and movements of natural deposits and fills. At the site near Takatori
station, the pre-cast pre-stressed pile foundations supporting three 12-story apartment
buildings suffered shear and compressive failure near the pile head, and the buildings
had to be demolished. No evidence of liquefaction was noted, and the damage was
attributed to inertial forces from the superstructure.
Tokimasu et al. (1998) surveyed two buildings in Fukae in zones of large
permanent ground deformations that did not suffer structural damage, and therefore
he hypothesized that serious pile damage due to lateral spreading may have occurred.
This was confirm by slope gauge and borehole camera investigations, and analyzed
with the pseudo-static p-y method. The progression of soil-pile-structure interaction
during liquefaction and the effects of uniform and non-uniform ground displacements
on pile bending moment are shown schematically in Fig. 2.3.
10


Inertial Force Bending .Moment Inertial Force Ground Displacement / \
\ \ j 1 L s i | IF-^-i 7 3S0K Fi / t i . y i* 'c Li \] \ 1 ^ 11 __J ( lL t ^ 's 2JL
I) During shaking If) During shaking ill) Lateral ground
before liquefaction after liquefaction movement after
earthquake
Figure 2.3: Progression of Soil-Pile-Structure Interaction and Pile Bending Moments
during Liquefaction (Tokimasu et al., 1998)
Hayashi and Takahashi (2004) studied soil-structure interaction effects on
building response in recent earthquakes. From their analyses, they pointed out that the
damage reduction effects of soil-structure interaction greatly depend on the ground
motion characteristics, number of stories and horizontal capacity of earthquake
resistance of buildings. The building considered is an office building that has nine
stories above and one story below the ground, total height is 31 m and it has a five-
cornered shape with a dimension of 10mxl2m. The foundation is a spread foundation
and is embedded to 6 m depth. The building has SRC moment-resisting frames with
RC shear walls constructed after 1981 when the new building code was established.
This building suffered no structural damage except for some hairline cracks on non-
structural RC wall around the elevator shaft but gaps about 1 -3 cm wide were
observed between the underground exterior walls and the surrounding soil. From
these facts, the uplift of the base mat and the separation between the foundation and
the soil that occurred during the earthquake was estimated.
2.1.4 Chi-Chi Taiwan Earthquake of 1999
The magnitude 7.6 Chi-Chi earthquake and subsequent large aftershocks (four
greater than 6.5) was the most devastating earthquake in Taiwan since 1935. The
11


heaviest damage was in center counties of Taichung, Nantou and Yunlin. The
earthquake tumbled two tall buildings in the capital city of Taipei. The death toll
exceeded 2,400 and more than 10,700 people were injured. Over 8500 buildings were
destroyed and more than 6,200 were seriously damaged.
The Chi-Chi earthquake significantly damaged many high rise buildings of
eight stories or greater. More than 15 buildings less than 50 m high completely
collapsed but no building more than 50 m high collapsed. Most damages were caused
by the terrible non-ductile failure of first floor.
Hsieh (2000) surveyed the high rise building foundation types in Taiwan.
Most high rise buildings in Taiwan use the forms of deep raft foundation and raft
foundations with large diameter pile. Large bored pile with diameter over 2 m and a
capacity of more than 15000 kN are often used. The Chi-Chi 1999 Taiwan earthquake
induced widespread liquefaction problems, and many buildings located in the western
area of Taiwan tilted, settled, or even collapsed while foundation soil temporarily lost
its strength.
2.1.5 Summary of Earthquake Damage to High Rises
Numerous buildings supported by pile foundations during the Niigata
earthquake of June 16, 1964, Japan, the Mexico City earthquake of September 19,
1985, Mexico, the 1995 Hyogo-Ken Nanbu (Kobe) earthquake, Japan, and the 1999
Chi-Chi, Taiwan earthquake were damaged or collapsed with various patterns. The
collapse of buildings is defined as severe structural damage resulting in drastic
reduction in the height of structures, the loss of structural capacity of the pile, and/or
degradation of the pile-soil load bearing capacity during the earthquake that might be
caused by the following reasons:
- Liquefaction of cohesionless soil or strain softening of cohesive soil near the
pile head
- Laterally spreading ground applied to pile resulting in pile damage
12


- Connections between piles and caps were inadequate so that pile head
experienced failure under shearing forces.
This indicates that the design of the buildings is inadequate. The
considerations of soil-structure interaction may affect the interpretation of damage as
well as alter the design criteria and building specifications.
2.2 Dynamic Analysis of Soil-Structure Interaction
2.2.1 Effects of Soil-Structure Interaction
To prove the effects of soil-structure interaction, the dynamic response of a
structure built on rock is compared to that of the same structure embedded in soil. For
the structure on the rock, the horizontal motion can be applied directly to the base of
structure. The acceleration caused by the inertial load over the structure will be equal
to the acceleration of the bed rock. At the base of the structure, an overturning
moment and a transverse shear will occur; the base is not deformed if the bed rock is
very stiff. The horizontal displacement of the base is thus equal to the control motion;
no rocking motion arises at the base. The seismic response of structure depends on
only the characteristics of the structure.
For the structure embedded in soft soil, the motion of the base of the structure
differs from the control motion because of the effect of the soil-structure system. The
three effects of soil-structure interaction can be distinguished as follows. First, the
motion of the site in the far field of the structure is modified. If there is no soft soil on
top of bed rock, the difference of motion is small; see, for example, Fig. 2.4. The
presence of soft soil on top of bed rock will alter the motion in point C. The
characteristics of the wave propagates vertically through the soil layer from point C
are changed; this results in alteration of the motion characteristics in points near and
on the base of structure (D and E) from that in point C. In general, the motion can be
amplified, but not always, depending on the soil dynamics and ground motion
characteristics. The horizontal displacements increase toward the free surface of the
site. Second, since the structure is constructed, the motion will be modified. The base
13


of structure will experience both horizontal displacements and rocking components.
Accelerations (leading to inertial load) will vary over the height of the structure, in
contrast to the applied accelerations in the case of a structure seated on the rock. This
geometric averaging of seismic input motion is the result of the kinematic interaction
(Wolf, 1985). Third, the displacements of the base of the structure caused by
overturning moment and transverse shear will modify the motion at the base. This
part of analysis will, henceforth, be referred to as inertial interaction (Wolf, 1985).
Figure 2.4: Seismic Response of Structure Embedded on Rock and on Soil
2.2.2 Substructure Method
2.2.2.1 Introduction of Substructure Method
This method models a significant part of the soil around the embedded
structure and the free-field motion is applied at the fictitious boundary. This is the
simplest method used to analyze soil-structure interaction for seismic excitation. The
substructure method consists of two steps. The free-field response of the site (without
the structure) is calculated first. The unbounded soil is analyzed as a dynamic
subsystem and the relationship between force and displacement along the soil-
structure interface is determined. The dynamic-stiffness coefficients are evaluated for
the spring-dashpot system, which models the soil, where the structure is embedded.
The structure supported on this spring-dashpot system is then analyzed using the free-
field motion. The substructure method simplified the analysis of the complicated soil
structure.
14


The unbounded soil medium can best be described for harmonic excitation.
The dynamic stiffness, taking into account the radiation of energy, will be frequency
dependent and complex. Material damping can also easily be introduced for harmonic
excitation. This means that linear soil-structure interaction is best handled in the
frequency domain using the complex-response method. The transfer functions allow
the first few natural frequencies and the corresponding approximate damping ratios to
be determined. The accuracy in the intermediate and higher-frequency ranges can be
checked easily. Once the transfer function is established, the final seismic response
can be evaluated efficiently.
There are two components of interaction between structure, soil, and
foundation:
Inertial Interaction: The interaction gives rise to base shear, overturning
moment, which in turn cause displacements of the foundation relative to the
free-field due to its own vibrations.
Kinematic Interaction: Results from the presence of stiff foundation on or in
soil that deviate foundation motions from free field as a result of three
mechanisms: (a) Base slab averaging: Placement of a foundation slab across
the spatially variable motion that occurs within the footprint area of the base
slab, and due to kinematic produces an averaging effect, (b) embedment
effects: the reduction of seismic ground motion that tends to occur with depth
for an embedded foundation, (c) wave scattering: scattering of seismic waves
off of comers and asperities of the foundation.
2.2.2.1 Inertial Interaction
2.2.2.2.1 System Consideration
When a superstructure is excited by a vertical incident wave, its soil-structure
interaction effect can be modeled using three degrees of freedom vibrations (Wolf,
1985) as shown in Fig. 2.5. This system is a direct model of a single story building or
an approximate multi-story, model which is dominated by first mode response. The
15


structure is modeled with a lumped mass m connected to the base with the flexible
bar with a lateral stiffness k, damping c and height h. The values of m k, c and a
characteristic length a of the rigid base are easily determined. In the case of the fixed
base of this structure, the frequency cos can be calculated by the equation as follows
(Wolf, 1985):
a>. =.
The damping coefficient can be determined by (Wolf, 1985):
2 k£
c =
CO.
where £ is the hysteric damping ratio of the structure.
Ug UO h0 U
Ug UO
lit
(2.1)
(2.2)
Figure 2.5: Soil-Structure Interaction Model of Three Degrees of Freedom Structure
uo h9 u
Figure 2.6: Equivalent Model
The soils stiffness is attached at the other end of the bar. The stiffness and
damping in the horizontal direction are denoted as kh and ch respectively, and the
stiffness and damping in the rotational direction are correspondingly denoted as ke
16


and ce. No mass exists at the base. All springs and dampers are assumed to be zero
length. This system has three degrees of freedom in which u' is the horizontal
displacement of the mass, u'0 is the horizontal displacement of the base, and 9 is the
rotation of the base. The equivalent model can be shown as in Fig. 2.6 and the
equilibrium equations are:
kh+K/h1 -kg/h2 0
-k9/h2 ke/h2+k -k
0 -k k
+
ch ce /k ce Ik 0
-Cg/h2
Cgjh2 +c -c
+
0
0 0 0
0 0 0
0 0m
-c
u0 + hO
u0+hO + u
u0 + h6
it0+h9 + u
0 0 '
iiQ + h9 > == < 0
u0+h9 + ii m
(2.3)
with:
£ =
2 kco
£ Cl> £ = C0____ a) = ihh- a) -
bh - be ~ ,2 \ >w8
- 2mh (oa
2khcoh
substitute Eq. (2.4) for Eq. (2.3) gives:
Ml + 2''f) + ^(1 + 2 if.)
-^-0 + 2^)
0
u0 0 '
u0 + h9 > = < 0
u0 + h9 + u -m
}C02U
-^-(1 + 2/f.)
^-(l + 2;f,) + t(l + 2i n
-k(\ + 2i%)
mh2
(2-4)
m
-k(l + 2i£)
k{\ + 2i<^) + 0)2m
(2.5)
Solving this equation we have:
17


U0 2
co] 1 + 2 /#
h0 =
< 1 + 2/4
1 + 2/#
10
coQ
1 + 2/4
(2.6)
(2.7)
u is expressed as:
C 2 2
l + 2#-V
cy co] 1 + 2/# &>; 1 + 2/#
V
<4 l + 2/'4 ^ 1 + 2/4
e
CO
u = u
co:
(2.8)
where u0 is the amplitude of the base relative to the free-field motion ug and u is the
amplitude of relative displacement of the mass referred to a moving frame of
reference attached to a rigid base that is equal to the structural distortion.
2.2.2.2.2 Equivalent One Degree of Freedom
'' c m
/ s Sys/sss/sss/ssss Cl. O /s/ss/////
Figure 2.7: Equivalent One-Degree-of-Freedom System
The coupled system can be replaced by an equivalent one-degree-freedom
system (Fig. 2.7), enforcing the same structural distortion u as in the uncoupled
dynamic system with the same mass m, the equivalent natural frequency &, then the
equivalent damping ratio # and the equivalent input motion, ug, are (Wolf, 1985):
1 1 1 1 ~ 2 2 + 2 + 2 co ~2 ~2 -2 ~ (O y CO r (Or £ =£+4+4 , h (2.10)
co2 UR =Ug co: K (2.11)
18


Stewart et al. (1998) followed the same procedure and gave the equivalent
natural frequency 3 and ratio of hysteretic damping C, of one-degree-of-freedom
system as:
2=-
ycOs +l/o)h +1 f(Dg
z=
f ~ \3
CO

4h +
f ~ V
CO
\ ms )

( ~ A3
CO
\>ej
z*
(2.12)
(2.13)
For the case not including horizontal displacement of the base, the natural
frequency 3 and ratio of hysteretic damping % are calculated by:
£2 =
1
\! col + 1/col
(
f ~ ^3
CO
\as)

CO
\<0J
\3
(2.14)
(2.15)
2.2.2.2.3 Inertial Interaction of Shallow Foundations
The impedance functions are frequency dependent functions that describe the
characteristics of the dynamic stiffness and damping of soil-structure interaction.
They are ratios of the forces at the base of the structure and the displacements relative
to free field.
2.2.2.2.3.1 Basic Case
In this case, the foundation is often assumed to be rigid, located on the ground
surface and underlain by a uniform, visco-elastic half space. There are six
components of impedance functions. When considering the lateral response of a
structure on a rigid foundation in a particular direction, only two impedance terms are
generally needed, which is the same as the case in Fig. 2.5. Terms in the complex
valued impedance function are expressed in the form:
kj =kj(a0,v) + icocj (aQ,v) (2.16)
19


where j denotes either deformation mode h (for translation) or 8 (for rotation in the
vertical plane), (o is the angular frequency (radian/sec), a0 is a dimensionless
frequency defined by a0 = (orjVx, r is the foundation radius, Vx is the soil shear
wave velocity, and v is the soil Poissons ratio.
The solutions of impedance functions in Eq. (2.16) were presented by Gazetas
(1991) for circular, rectangular and other foundation shapes.
By another approach, the real stiffness and damping of translational and
rotational springs and dashpots are expressed, respectively, by:
*=*,; *.=.*; 0, = ?,^ (2.17)
.V .V
where ah, /3h, ag and P0 express the frequency dependence of the impedance terms,
rh and rg are foundation radii that can be computed separately for translational and
rotational deformation modes to match the area (Af) and inertia (I f) of the actual
foundation rh = /* > re = n Kh and Kg are the static stiffness, for the
circular mat foundation supported at the surface of a homogeneous half space, these
stiffness are given by:
Kh =
8 Grh . _ 8 Grl
2-v 2-v
(2.18)
where G is shear modulus of the half space, v is Poisson ratio.
The frequency-dependent values of ah, (3h, ae and [30 for a rigid circular
foundation on the surface of a viscoelastic half-space with soil hysteretic damping f3s
are shown in Fig. 2.8 (Veletos and Wei, 1971; Veletos and Vebric, 1973).
Stewart et al. (1998) summarized from other theoretical and experimental
studies for the similar impedance function formulations and concluded that these
impedance function formulations are good approximations. The above solutions can
provide reasonable estimates of foundation impedance for rigid, circular foundation
20


on a homogeneous half space, but the effects of non-uniform soil profiles, foundation
embedment, foundation shapes, and foundation flexibility should be taken into
account.
Translation Rocking
Figure 2.8: Foundation Stiffness and Damping Factors For Elastic and Viscoelastic
Flalf-Space (Veletos and Vebric, 1973)
2.2.2.23.2 Non-uniform Soil Profiles
The characteristic of non-uniform soil profiles are the change of stiffness with
depth gradually or suddenly from soil layer to soil layer. Gazetas (1991) provides
solutions for impedance of rigid foundations on the ground of half space with the
shear modulus increasing with depth. The frequency-dependent damping components
are shown in Fig. 2.9 for a zero hysteretic damping condition (radiation damping
only). G0 and Vs0 are shear modulus and shear wave at the ground surface,
respectively.
21


G(z)/Go G(z)/Go
02468 2r 02468
Figure 2.9: Foundation Damping Factors for Half-Space with and without Hysteretic
Damping (Veletsos and Vebric, 1973) and Soil Profiles with Indicated Modulus
Profiles and no Hysteretic Damping (Gazetas, 1991)
For the case of a finite soil layer overlying a much stiffer material, the key
issue is a lack of radiation damping at frequencies less than the fundamental
frequencies of finite soil layer, fs = VjAH. Half-space damping ratios can be used
for frequencies greater than the soil layer frequency. Elsabee and Morray (1977)
provided the damping recommendations as follows:
aQ/aQ]
Ph = 0.65&
l-(l-2/?v)(a0/a01)
for a0/a01 <1
(2.19)
A =0.5 A-
o /<
01
<-
0.35an
\-{\-2P,)(aJan) ! + '
for aJam 22


where: am = 0.57rr/H, r is the foundation radius, and H is the finite soil layer
thickness.
2.2.2.2.33 Effects of Depth of Soil Layer
The foundation on surface of soil layer on rigid rock was investigated by Wolf
(1994, 1998) by using cone model and spring-dashpot-mass model. The value of
static-stiffness coefficients by cone model are given by:
Kh =
8
2-v
1 +
IZL
2 H

86>i
3(1-9
1 l rH
1 +----6~
6 H
(2.21)
where H is the depth of homogeneous soil layer.
2.2.2.2.3.4 Effects of Foundation Embedment
Foundation embedment is referred to as the basement foundation in which a
foundation base slab is placed at a lower elevation than the surrounding ground.
However, it is not a deep foundation (pile foundation). The impedance of embedded
foundation differs from that of shallow foundations in several ways. Elsabe and
Morray (1977) investigated the effects of foundation embedment for the case of a
circular shape to a deep e into homogeneous soil layer of depth H (Fig. 2.10). The
static horizontal and rocking stiffness for such a foundation is approximated by the
following equations for rjH < 0.5 and ejr < 1, when the foundation rests on a
surface stratum of soil underlain by a stiffer deposit (Elsabe and Morray, 1977 and
Kausel and Roesset, 1975):
{Kh\=Kh
(Ke)e=Ke
f 2 1 V
1 +
3 r.
h y
V
1 + 1^
2 H j
( 4 H
1 + 2-
'e J
1 ra\(, e ^
1 + -^L
6 H j
1 + 0.7
H
(2.22)
Approximate normalized impedance factors for cylindrical embedment
foundation obtained from Eq. 2.22 are compared to a more rigorous analytical
solution derived from integral equations (Apsel and Luco, 1987) in Fig. 2.11. The
23


approximated curves were computed as a product of dimensionless impedance factors
ah, ae, (3h and (50 and first modifier on the right hand side of Eq. (2.22). Both
solutions are applicable to the case with a uniform viscoelastic half-space with
/? = 1%, v = 0.25, and perfect bonding between the soil and foundation (Stewart et.
al 1998).
Gazetas (1991) also provided the formulas of dynamic stiffness and damping
coefficient for rigid rectangular foundations in a homogeneous half-space. Figure
2.11 shows the dvnamic stiffness.
Figure 2.10: Embedded Soil-Foundation-Structure on Finite Soil Layer
2.2.2.2.3.5 Effects of Foundation Shape
Dorby and Gazetas (1986) summarized the solutions of impedance function
for foundations of various shape with an aspect ratio of 1 to oo. They found that the
use of equivalent mats is acceptable for aspect ratios less than 4:1 in translation mode
only (not included in rocking mode). The solutions of impedance for oblong non-
circular foundation can be calculated using procedures given in Gazetas (1991). The
dimensionless radiation damping coefficients crx and cry are shown in Fig. 2.12. This
correlation was made by multiplying the radiation damping components of the disk
dashpot coefficient by cr{L! B)fcr{L! B-1), where cr is determined at the value a0
corresponding to the structures fundamental frequency (Stewart et al., 1998).
24


ao=corA/s
ao=(or/Vs
ao=o)rA/s ao=corA/s
Figure 2.11: Foundation Stiffness and Damping Factor for Rigid Cylindrical
Foundations Embedded in Half-Space; Approximation versus Aspel and Luco (1987)
(Asterisk denotes modification to parameters for surface foundations to account for
the effect of embedment)
25


ao=coB/Vs
ao=coB/Vs
Figure 2.12: Horizontal Foundation Stiffness and Damping Factor for Rigid
Rectangular Foundations Embedded in Half Space (Gazetas, 1991)
26


ao= r/Vs
Figure 2.13: Dashpot Coefficient Rocking Radiation Damping versus Frequency for
Different Foundation Shape (Dobry and Gazetas, 1986)
2.2.2.2.3.6 Effects of Foundation Flexibility
Foundation flexibility affects the rotational impedance. It was investigated for
the cases of rigid core wall (Iguchi and Luco, 1982; Liou and Huang, 1994), thin-
27


walled cylinder (Liou and Huang, 1994) and rigid concentric walls (Fig. 2.14)
supported by a flexible circular foundation.
(a) 1 (b) 1 i Thin Wall
Rigid Core s=.| 1 1 Hinge
1 i 1
1 1
h - r -*] 1 L -... r
(c)
Rigid Wall

Flexible 5/8r
' r
Figure 2.14: Disk Foundation with (a) Rigid Core Considered by Iguchi and Luco
(1982); (b) Thin Perimeter Wall Considered by Liou and Huang (1994); (c) Rigid
Concentric Wall Considered by Riggs and Waas (1985)
The relative flexibility of the plate is represented by the ratio of the soil-to-
foundation rigidity:

Gr3
D
(2.23)
in which D is the flexural rigidity of the plate foundation:
E/,
12(1-''/)
(2.24)
where Ef, tf, and vf are Youngs modulus, thickness, and Poissons ratio of the
foundation, respectively. The coefficients of flexible foundation effects are shown in
Fig. 2.15, the correction was made by multiplying ratios (^e)/k,x j{ae)ngirJ an<^
28


($>)/?/(&)gij t0 values of the rocking stiffness and damping coefficients for the
case of flexible foundation.
Rigid Core (c/r=0.25)
2 3
ao=a>r/Vs
Rigid Core (c/r=0.25)
ao=corA/s
ao=(orA/s
Figure 2.15: Rocking Stiffness and Damping Factors for Flexible Foundations: Rigid
Core Case (Iguchi and Luco, 1982) and Perimeter Wall Case (Liou and Huang, 1994)
29


2.2.2.2.4 Inertial Interaction of Pile Foundations
2.2.2.2.4.1 Analytical Method
The impedance functions of single pile have been investigated by Gazetas
(1991). The response of piles under horizontal forces and moments is independent of
their length in most practical situations. Only the uppermost part of the pile (denoted
as length lc) experiences appreciable displacement. lc is a function of pile diameter,
Youngs modulus of pile and, Youngs modulus of soil, which typically is on the
order of 5 to 10 pile diameter. Along lc, the imposed load is transmitted to the
surrounding soil. The formulas of impedance functions in this study are calculated for
three cases of variable Youngs modulus of soil: (a) constant with depth; (b) linearly
varying with depth; (c) parabolically varying with depth. Gazetas (1991), from the
theoretical point of view, concluded that most of formulas of impedance functions of
pile in this study are reasonably accurate, as they are basically curve fits to rigorous
numerical results.
Novak and El-Shamouby (1983) presented the stiffness constants and constant
of equivalent viscous damping of single, vertical piles in the form of tables and
charts. The shear modulus of soil is either constant or quadratic parabola varying with
depth. The piles are presumed to be both end bearing and pinned. Impedances of
single piles are calculated by the following equations:
ea EA
Vertical translation: kv =j /vl; cv = /v2
K Vs
El El
Horizontal translation: kh = J-/A1; ch = -y fu2
K K V v
El EI
Rotation in vertical plane: ke = ; ce = f01
R K
(2.25a)
(2.25b)
(2.25c)
Coupling between horizontal translation and rotation in vertical plane:
30


(2.25(1)
Torsion: k
GnJ
(2.25e)
R
where Ep is Youngs modulus and Gp shear modulus of the pile, A, I and J are its
cross-sectional area, moment of inertia and torsion constant, respectively, and R is
radius or equivalent radius. The symbol fX 2 represents dimensionless stiffness and
damping functions whose subscript 1 refers to stiffness and 2 refers to damping.
These values can be referred to in Novak and El-Shamouby, (1983).
2.2.2.2.4.2 Beam-on-Elastic Foundation
In two-dimensional analysis, pile is modeled as bar or beam on the elastic
foundation. The method to find the displacements and internal forces of pile are finite
difference and finite element methods. Soil is modeled by continuous elastic. A
disadvantage of this model is the two-dimensional simplification of the soil pile
contact, which ignores the radial and three-dimensional aspects of interaction.
The assumptions used in deriving the differential equation are given as:
The pile is straight
The section and material properties do not change along the pile
The material of pile and soil is linear elastic
Ignore the shear deformation effect
Reaction of the foundation:
(2.26)
where:
ky: stiffness of spring in y direction
uy: displacement in y direction
Differential equation of beam:
31


(2.27)
= K
El,
where:
/ : Inertia moment of beam section
Differential equilibrium equation:
d2M,
dx
= q-pb
Substitute Eq. (2.26) and Eq. (2.28) to Eq. (2.27):
dAu.
El.
dx4
= q-kybuy
I kb
(2.28)
(2.29)
dAu,
du du dd du d u
Given a = H-^~ and P = ax; ^ = y-.^ = a?r=>f = a4 '
4EI
Equation (2.29) is written as:
dAu
dx dp dx dp
dx
dp*
, + 4u =--q
dpA y
kyb
(2.30)
The solution of Eq. (2.30) according to Rrulov:
m =c,Y, +c2Y2 +c3Y3 +c474 +
Mo
u
(2.31)
where:
Yt, Y2, Y3 and T4 : Krulovs functions:
Yx =chps cos P\ Y2 = (chpsinP + shpcosp)
Y3 = shp sin p -, T4 {chp sin p shp cos p)
c,, c2, c3 and c4: Constants are determined by boundary
conditions
32


loads.
4 p
>, = 174 {fi y]q{y)dy : Function depends on type of
kyb 0
Equations of internal forces:
= ~ a\~ 4c,F4 +c27j +c3Y2 +c473 + $y) (2.32)
Mz=EI^£f = E1a2(-4CJ3 -4c2Y4 +c3Yt +c4Y2 + fy) (2.33)
Qy =^- = EIai(-4c,Y2 -4c2F3 -4c374 +C.7, +fy) (2.34)
where 6,, Mz and Qy are rotation about z-axis, moment about z-axis and y-axis
shear force, respectively.
2.2.2.2.4.3 Beam-on-Winkler Foundation
The Winklers theory assumed that each layer of soil responds independently
to adjacent layers so that beam and discrete spring system may be adopted to model
pile under lateral loading. This is an effective method for static and dynamic analyses
of lateral pile response. Springs and dashpots are used to represent the soil stiffness
and damping at each particular layer. The soil-pile spring may be linear elastic or
nonlinear, gapping, cyclic degradation and rate dependency are provided. The
disadvantages of this model are the ignorance of shear transfer between layers of soil
and the two dimensional simplification of the soil pile contact, which ignores the
radial and three dimensional of interaction.
p-y curve typically used to model nonlinear soil-pile stiffness has been
empirically derived from the tests.
The model for clay can be used to gain some insight into the ultimate
resistance pu that will develop near the ground surface, as shown in Fig. 2.16. The
soil resistance pc per unit length of the pile:
33


Pc =cad[tanar + (l + /c)cotar] + ;Kd// + 2cfl/7(tanG'sino' + cos) (2.35)
where ca is average undrained shear strength of the clay over the depth H; and k is
the reduction factor of the friction force between the wedge and the pile.
Figure 2.16: Assumed Passive Wedge-Type Failure for Clay (Reese et al., 2000)
(a) Shape for the Wedge; (b) Forces Acting on Wedge
Matlock (1970) proposed the p-y curve for soft clay in the presence of free
water for static and cyclic loading, which are shown in Fig. 2.17. The ultimate soil
resistance per unit length of pile is the smaller of the values given by the equations
below:
Pu =
3
/ J
H---X H--X
c d
cd
(2.36)
Pu = 9cd
(2.37)
where y' is average effective unit weight from ground surface to p-y curve; x is
depth from the ground surface to p-y curve; c is shear strength at depth x; and d
is width of pile.
34


Matlock (1970) stated that the value of J is 0.5 for a soft clay and about 0.25
for medium clay, as determined by experimental method. A value of 0.5 is frequently
used for J.
(a) (b)
Figure 2.17: Characteristic Shapes of the p-y Curves for Soft Clay in the Presence of
Free Water: (a) Static Loading; (b) Cyclic Loading (after Matlock, 1970)
Use the same method to find the soil resistance for sand near the ground
surface is shown in Fig. 2.18. The total lateral force F t may be computed as follows:
Fpl=rH2
K0H tan

3tan(/?- b H \
i----tan B tan cr
2 3 '
K0Htan p
3
(tan (psin p tan as)
+ ...
(2.38)
where K0 is coefficient of earth pressure at rest; and Ka is minimum coefficient of
active earth pressure
The ultimate soil resistance near the ground surface per unit length of the pile
is obtained by differentiating Eq. (2.35):
+ ...
(2.39a)
\_K0H tan /? (tan (psmf3- tan as) Kab~^
pu=rH-
K0H tan ^ tan/?
3tan(/?-^)cosa5
+
tan~ "-(b + H tan p tan or )
tan {P-py * W
35


pu = yHb[Ka(tan8 (3 -1)+ K0 tancp' tan4 a\ (2.39b)
Figure 2.18: Assumed Passive Wedge-Type Failure of Pile in Sand
(a) General Shape of Wedge, (b) Forces Acting on Wedge, (c) Force Acting on Pile
Figure 2.19: Characteristic Shape of p-y Curve for Static Loading in Stiff Clay with
no Free Water
36


Reese and Welch (1975) performed a lateral load test with a bored pile. The
experiment was used to develop both the static and the cyclic p-y curve for stiff clay
with no free water. The shape of p-y curve is shown as in Fig. 2.19 and Fig. 2.20.
9.6y5ologN2
Figure 2.20: Characteristic Shape of p-y Curve for Cyclic Loading in Stiff Clay with
no Free Water
The response of sand for short term static loading and for cyclic loading is
illustrated in Fig. 2.21 (Reese et al., 1974).
Figure 2.21: Characteristics of a Family of p-y Curves for Static and Cyclic Loading
in Sand
37


Evans and Duncan (1982) recommended and approximated equation for the
ultimate resistance of c-cp soil:
p = apd = Cpahd (2.40)
where: wedge; d is pile width; ah is the Rankine passive pressure for a wall of infinite
length; cr,, =yztan2(45 + ^/2) + 2ctan(45 +

factor to account for the three-dimensional effect of the passive wedge. The p-y curve
for c-cp soil is shown in Fig. 2.22.
Figure 2.22: Proposed p-y Curve for c-cp Soil
Matlock and Foo (1978) issued the beam on dynamic Winkler foundation
analysis program SPASM. In this study, a soil spring that was nonlinear, hysteretic,
degrading and gapping (Fig. 2.23a) was connected to a segment of linear elastic pile.
The soil gapping model is shown schematically in Fig. 2.23b and Fig. 2.23c. As
shown in Fig. 2.23a, nonlinear springs are specified near the pile head, elastic springs
are presented at depth, and rotational springs are assigned at the free part of pile. The
solution method was a time domain finite difference procedure that iterated soil-pile
tangent stiffness.
38


//
1 / Hh~
l Fluid
\ Damping
'''H-

y Rotational
r Resti dint

N\
\\

r ^.^
HF
i
\\
Mass j \
i i
1J
S/ Mudline

L
Soil I Inelitstic
Damping f +. Suppoils
(ajL^j
|----- jujM
jmU
Elastic
jjUUL) Si ppoits
JULSU
JUJM
juu4
-JjuuM
a)
b)
Figure 2.23: (a) Soil-Pile Model (b) Soil-Pile Gapping Model (c) Force-Displacement
Behavior (Matlock and Foo, 1978)
Brown et al. (2001) presented a p-y curve model used to analyzed lateral
pile under dynamic loading. A schematic of this model is shown in Fig. 2.24. The
near field, in the vicinity of the pile, is expressed by hysteretic, hyperbolic p-y
curve model to capture the nonlinear stiffness of the soil and the hysteretic energy
dissipation that occurs in the soil near a pile. The linear spring and dashpot are used
to model the soil in the far field. The pile were modeled as elastic beam, had mass,
and were circular and vertical. The parameters of the model shown in Fig. 2.24 are as
follows: kNL cNL and m] are stiffness, damping coefficient, and lumped mass of near
field, respectively; kL, c, and m2 are stiffness, damping coefficient, and lumped
39


mass of far field, respectively. Determination of these values is represented in detail
in Brown et al. (2001).
Pile element (i)
Figure 2.24: Schematic of Model Used for Deriving Dynamic p-y Curves (Brown
et al. 2001)
2.2.2.2.4.4 Pile Group Effects
El-Shamouby and Novak (1985) provided a simple method for analysis of
large pile groups subject to vertical loads, horizontal loads and moments acting either
statically or dynamically with low frequencies. The results indicate the dependence of
the horizontal interaction factors on stiffness ratio (Ep/Ex, Ep is Youngs modulus
of pile and Es is Youngs modulus of soil) and pile spacing (S/D, S is pile spacing,
and D is pile diameter) and, practically, does not depend on pile length and the soil
profile.
The simple analytical method to solve for the problem of dynamic pile-soil
structure interaction in uniform soil is developed by Dobry and Gazetas (1988). This
method can easily be understood and applied by an engineer. It can apply to the case
of a wide range of material parameters, pile spacing and frequencies of oscillation.
However, it tends to overestimate the peak value of both stiffness and damping for
pile group embedded in very stiff soil whose average effective Youngs modulus Es
is greater than about Epj?>QQ, where Ep is the pile Youngs modulus. Moreover, this
method might also over-predict the interference-related peaks of large pile groups
40


(n > 16) The dynamic interaction factor is defined as ad dqp jdqq ; dqp is
additional displacement of pile q caused by pile p, and d is displacement of pile q
under its own dynamic load. The values of a are shown in Table 2.1 where S is pile
spacing, and Vla = 3.4Fvy/[;r(l-v)] .
Table 2.1: Dynamic interaction factor
Direction Dynamic interaction factor, ad
Vertical i ( 2 ad e-&slKe-slv, \D)
Horizontal 1 ai (90) = atmai; < (o) = T e-^e-^ \ U J
For the laterally oscillating piles, the interaction factor depends both on the
pile spacing S and on the angle /? between the line of two piles and the direction of
the horizontal applied force. adh (/?) can be calculated using adh at 0 and 90:
<4 (P) = ah0 cs2 P + a*90sin2 P (2-41)
Impedance functions for some cases of rigidly capped pile groups are shown
in Table 2.2 under harmonic load.
Table 2.2: Impedance functions of pile group by Dobry and Gazetas (1988)
Direction Group type kG
1 by 2 2 k*
Vertical 1 + ad
2 by 2 4 k*
\ + adS + advSj2
Horizontal 2 by 2 4 kl
1 + adS + 0.5ccdSj2 + adh0S + 0.5adhoS^2
41


Brown et al. (2001) represented the model of pile groups by applying p-
multipliers to p-y curve method of single pile (Fig. 2.25). To establish the p-
multiplier, two loading cases were considered separately: a pile loaded individually
and a group of identical piles. For the dynamic loading, the p-multiplier was
approximately by the peak pile-head force at one pile in the pile group divided by the
peak force for the single pile.
Figure 2.25: Definition of the p-multiplier Method
2.2.2.3 Kinematic Interaction Effects
Kinematic interaction results from the occurrence of stiff foundation elements
on the surface of or embedded in a soil deposit, which causes foundation
deformations to differ from free field. The kinematic interaction will occur whenever
the stiffness of foundation obstructs development of the free-field motion. The
bending stiffness of the massless foundation effects on the P-wave component of the
free-field motions (Fig. 2.26a), the rigidity of embedded foundation effects on shear
wave component of free field (Fig. 2.26b), and the axial stiffness of foundation
effects on the development of incoherent free field motions (Fig. 2.26c). The
kinematic interaction can also induce different modes of vibration in a superstructure.
For the embedded foundation subjected to vertically propagating shear wave, if the
wave length equals to the depth of embedment, it will cause a overturning moment
applied to the foundation, thereby causing the foundation to rock and translate, even
42


though the free field is purely translational. Horizontally propagating waves can
induce torsional vibration of the foundation in the same manner.
Kinematic interaction effects can be theoretically modeled by frequency-
dependent ratios of the Fourier amplitudes (i.e., transfer functions) of foundation
input motion (FIM) to free-field motion. The FIM is the theoretical motion of the
base slab if the foundation and structure is assumed to have stiffness and massless,
and is a more appropriate motion for structural response analysis than is the free-field
motion (FEMA 440, 2005).
i =>n-------------------------------------------ns:
i i i
\ \ /
i i i
i i i
/ / \
i i i
i i i
\ \ /
i i i
i i i
i i \
c)
Figure 2.26: Kinematic Interaction with Free Field Motions
(a) Bending stiffness of surface foundation effects on vertical component of free field
displacements; (b) Rigidity block foundation effects on horizontal component of free
field displacements; (c) Axial stiffness of surface foundation effects immediately on
underlying soil from deforming incoherently (Kramer, 1996)
2.2.2.3.1 Kinematic Effects of Shallow Foundations
2.2.2.3.1.1 Shallow Foundation at the Ground Surface
The transfer functions of the foundation, defined for harmonically excited
massless foundation, represent the ratios of the amplitudes of the component of the
steady-state motion actually experienced by the foundation to the amplitude of the
free field ground motion (Veletsos et al., 1997). The transfer function amplitudes
computed for circular foundations (Veletsos et al., 1989) and rectangular foundations
(Veletsos et al., 1997) subjected to vertically incident, incoherent shear waves are
presented in Fig. 2.27. Similar curves are available for other wave fields. The transfer
43


functions in Fig. 2.27 are plotted against the dimensionless frequency parameter a0,
defined as follows for circular and rectangular foundations subject to vertically
incident waves, respectively,
aQ = Ka0 (circular); a0 =
cobjc
2V.
(2.42)
where a0 = corjVs r, Vsr is the strain-reduced shear wave velocity, r is the radius of
circular foundation, be = Jab cab is the full footprint dimension of the rectangular
foundation, and a: is a ground motion incoherent parameter. In Fig. 2.27, Su and Sg
are spectral density functions of the lateral component and ground motion,
respectively, and:
(2.43)
where uRM and ug are displacements of foundation input motion and ground motion.
Figure 2.27: Amplitude of Transfer Function Between Free-Field Motion and FIM for
Vertical Incident Incoherent Waves. Modified from Veletsos
44


As shown in Fig. 2.27, the reductions of base-slab translation tend to become
more significant with decreasing period. The period-dependence of these effects is
primarily associated with the increased effective size of the foundation relative to the
seismic wavelengths at low periods.
Kim and Stewart (2003) calibrated Yeletsoss analysis procedure against
observed foundation / free-field ground motion variations as quantified by frequency-
dependent transmissibility function amplitudes, \H\. Veletsoss models were fit to \H\
and apparent k -values (denoted as Ka) were fit to the data. Those tca values reflect
not only incoherence effects, but they also include average foundation flexibility and
wave inclination effects for the calibration data set. The structures in the calibration
data set generally have shallow foundations that are inter-connected (i.e., continuous
mats or footings interconnected with grade beams). Parameter Ka was found to be
correlated to average soil shear wave velocity (Fig. 2.28) approximately as follows:
Ka = -0.037 + 0.00074Ft or Ka = 0.00065FV (2.44)
where Vs is the small strain shear wave velocity in m/s.
Figure 2.28: Relationship between Effective Incoherent Parameter Ka and Small-
Strain Shear Wave Velocity Vs from Case Histories (from Kim and Stewart, 2003)
45


The fact that Ka is nearly proportional to Vs (Eq. 2.44) causes the
dimensionless frequency term aQ to effectively reduce to a function of frequency and
foundation size (be). This is shown by the following, which is written for vertically
propagating waves (av 0):
o)beK a>b(,nxVs (ob/\
2Vsr ~ 2n2Vs ~ 2n2
(2.45)
where nx 6.5x10 4 s/m and n2 is the square root of the soil modulus reduction
factor, which can be estimated as shown in Table 2.3 (FEMA 440, 2005). In the
remainder of this report, n2 will be taken as 0.65, which is an appropriate value for
regions of high seismicity such as coastal California.
Table 2.3 Approximate values of n2
Peak Ground Acceleration (PGA)
O.lOg 0.15g 0.20g 0.30g
0.90 0.80 0.70 0.65
Limitations of the model calibration by Kim and Stewart (2003) include: (1)
foundations should have large in-plane stiffness, ideally a continuous mat foundation
or interconnected footings/grade beams; (2) for non-embedded foundations, the
lateral foundation dimension should be less than 60 m unless the foundation elements
are unusually stiff; (3) the approach should not be used for embedded foundations
with an embedment/radius ratio, e/r > 0.5; and (4) the approach should not be used
for pile-supported structures in which the cap and soil are not in contact.
2.2.2.3.1.2 Embedded Shallow Foundations
Kinematic interaction of embedded shallow foundation was summarized by
Stewart et al. (2004) and it also can be found in FEMA 440 (2005). Foundation
embedment refers to a foundation base slab that is positioned at a lower elevation
46


than the surrounding ground, which will usually occur when buildings have a
basement. When subjected to vertically propagating coherent shear waves, embedded
foundations experience a reduction in base-slab translational motions relative to the
free field.
Analytical transfer functions relating base-slab translational motions to free-
field translations for an incident wave field consisting of vertically propagating,
coherent shear waves are developed. Base-slab does not occur within this wave field,
but foundation translations are reduced relative to the free field due to ground motion
reductions with depth and wave scattering effects. Analytical transfer functions were
analyzed for a uniform elastic half space, and for a finite soil layer. Results for both
are shown together in Fig. 2.29 for foundation embedment / radius ratio, e/r = 1.0.
The primary difference between the two solutions is oscillations in the finite soil layer
case at high frequencies. Also shown in Figure 2.29a is the approximate transfer
function amplitude model given by following equation (represented by Stewart et al.,
2004):
!!='
/ \ ( \
cos e ~ao \r j = cos eco

>0.454
(2.46)
where a0 = a>rjVs and e is the foundation embedment. Figure 2.29b shows the
transfer function amplitude model a somewhat more convenient form in which it is
plotted as a unique function of coe/Vs (i.e., in this form there is no dependence on
foundation radius).
The results shown in Fig. 2.29 can be contrasted with the behavior of a surface
foundation, which would have no reduction of translational motions when subjected
to vertically incident coherent shear waves. Transfer function amplitudes in the
presence of more realistic incident wave fields can be estimated at each frequency by
the product of the transfer function, which is shown in Section 2.2.2.3.1.1 for base
slab and those from this section at the corresponding frequency. Note that the analysis
47


procedure described herein has been verified against recorded motions from two
relatively deeply embedded structures with circular foundations (Kim and Stewart,
2003).
Figure 2.29: (a) Transfer Function Amplitudes for Embedded Cylinders along with
Approximation, (b) Transfer Function Amplitude Model (Stewart et al., 2004)
2.2.2.3.2 Kinematic Interaction of Pile Foundations
The kinematic response of single piles and pile groups were studied by Fan et
al. (1991). Three categories of groups of floating vertical piles were studied: (a) a
single free-head and fixed-head pile; (b) a rigidly capped pile group consisting of two,
three, four, six or nine piles a row; (c) a rigidly capped square group of 2x2, 4x4 or
6x6 piles. All piles are considered to be linear elastic with diameter d and length L
and embedded in different soil as shown in Fig. 2.30a. Three profiles were
considered: Youngs modulus is constant or linear with depth, and different in two
layers as shown in Fig. 2.30b. Each soil-pile-foundation system was stimulated by
vertically propagating harmonic shear (S) waves. The effects of soil-piled-foundation
kinematic interaction are represented in the form of two kinematic response
factors Iu = |Up\/Uff and Ie = \0p | D ju(f plotted as functions of the dimensionless
48


frequencies, a0 = o)D/Vs, where Uff is horizontal amplitude at free field, Up is
horizontal displacement and 0 is angle rotation at pile-cap level.
Up

f Rigid massless cap Soil Profiles
Frp (a) (b)
Figure 2.30: Pile Foundation System and Soil Profile (Fan et al., 1991)
A
ao
Figure 2.31: Idealized General Shape of Kinematic Displacement Factor, Iu = Iu (a0),
Explaining Transition Frequency Factor a01 and a02 (Fanetal., 1991)
49


The general shape of the kinematic displacement factor, Iu, is idealized in
Fig. 2.31 It includes three fairly different regions in the frequency range of greatest
interest for earthquake loading (a0 < 0.5):
- In low frequency region (0 < a0 < a01), the kinematic displacement factor is
approximately equal to 1.
- In intermediate frequency region ( am factor decreases rapidly with frequency.
- In relatively high frequency region (a0>a02), the kinematic displacement
factor varies around and essentially constant value of about 0.20 0.40.
The shape of kinematic displacement factor of piles and pile groups follows a
qualitatively similar trend versus frequency to that of rigid shallow foundation. The
configuration of pile groups, number of piles in the group, and pile spacing ratio
make little difference on Iu in low and intermediate frequency ranges. This
conclusion is valid for most studied soil profiles and relative pile rigidities. It implies
that there is little pile group effect in this frequency range.
The values of transition frequencies a0] and a02 are affected by four factors:
(1) the type of soil profile: the soil profile expressed by the variation of Youngs
modulus is the significant factor controlling the magnitude of am and a02. In strongly
non-homogeneous soil deposit (vary linear with depth), am is very small (of order of
merely 0.05). By contrast, in a homogeneous soil deposit or in a deposit with a thick
homogeneous top layer, value of am may be as high as 0.20-0.30. Similarly, a02 is
about 0.10-0.20 in the linear modulus profiles and exceeding 0.40 in two other
homogeneous profiles; (2) the relative rigidity of the pile: Stiffer piles are more
effective in depressing soil motions and, hence, its kinematic response is
characterized by smaller value of am and a02 compared with those of softer piles; (3)
50


the pile head fixity conditions: the effect of increasing the degree of fixity at the pile
cap level (from free-head to fixed head piles) is the same as the effect of increasing
ratio Ep/Es; and (4) the pile slenderness: the slenderness of piles are characterized
by ratio Ljd. If piles are more slenderness, they have smaller values of aox and a02
than those of less slenderness piles.
2.2.3 Hybrid Method
Gupta et al. (1980), Tzong et al. (1981) and Tzong and Penzien, (1986)
proposed a simple, rational and economical hybrid method for analysis of soil-
structure interaction. In this method, the soil-structure system is partitioned into near-
field and far-field cases (Fig. 2.32a). The near field case is modeled by the finite
element method (Fig. 2.32b) whereas the far field is modeled in the form of an
impedance matrix that accounts for the semi-infinite nature of the soil medium (Fig.
2.32c).
The near-field includes structure and a finite portion of soil medium
encompassing irregular shapes of base geometries. This part of the soil structure
system is modeled by three-dimensional finite elements that make it possible to
realistically model the real structure and soil material properties.
The far field is treated as a homogeneous half space representing the semi-
infinite soil media. It is assumed as linearly elastic isotropic solid. The interface
between near field and far field is modeled by nodal points that are common to both.
The shape of interface is chosen to be hemispherical. Hemispherical boundary is
selected to avoid any singularities in the form of sharp comers, and the mathematical
boundary conditions are easy satisfied.
The equations of motion for the near-field can be transformed into frequency
domain as follows:
(-^M + iwC + K)U^rPM+FM (2.47)
or,
51


(2.48)
K, JJ, P, + F, .
M (e>) (a) (ta)
where:
= -q)2M + icoC + K (2.49)
is thecomplex value impedance matrix; and are the Fourier transforms of
the load vector and displacement vector, respectively; is the Fourier transform of
the interaction vector; and a> is the excitation frequency.
Structure
\
(a)
Semi-Infinite
Soil Medium
Ilf
Interface i i
(C)
Figure 2.32: Hybrid Modeling of Soil-Structure Interaction
(a) Soil-Structure System (b) Near Field (c) Far Field (Tzong et al. 1981)
52


The displacements of the system are separated into two parts: the
displacement at the interface of near field and far field and the displacement
elsewhere in the near field. Equation (2.48) can be rewritten by matrix form as:
where Fh are the interaction forces at the interface between near field and far field.
For the isolated far field, the dynamic force-displacement relationship is:
where Kf is the far-field impedance matrix, which is complex valued and frequency
dependent, and Ff and U f are the force and displacement matrix of far field at the
interface.
The detailed value of far-field impedance matrix can be found in Tzong et al.
(1981).
2.2.4 Direct Method
In the direct method, the entire soil-foundation-structure system is modeled on
analysis at the same time. Free-field input motions are specified along the base and
sides of the model and the response of systems can be computed by finite element
method. The advantage of a finite element approach includes the capability of
modeling the soil-structure interaction in fully coupled manner. Defiance to
successful implementation of soil-structure analysis using finite element method lies
in providing appropriate soil constitutive models that can model the stress-strain
behavior, rate dependency, and degradation of resistance.
(2.50)
(2.51)
Substitute Eq. (2.51) into Eq. (2.50):
(2.52)
53


The finite element method provides the most powerful means for analyzing
soil-structure interaction, but it has not yet fully realized as a practical tool.
2.3 Previous Studies of Soil-structure Interaction of High Rise Buildings
Ukaji (1975) reported a study of analysis of soil-foundation-structure
interaction during earthquake. In this study, they considered many problems affecting
the use of finite element method and the soil-foundation-structure interaction. This
model of couple soil-structure systems was two-dimensional. Effects on soil-
foundation-structure interaction were studied due to the following parameters:
Effects of soil properties
Effects of soil depth
Effects of characteristic of bedrock motion
Effects of characteristic of building
Effects of characteristic of foundation block
The effects of soil properties on couple system model were studied for a ten-
story tiered building on 100 ft soil depth and 30 ft foundation embedment. For the
interaction effects, the base shears were reduced about 30 and 50 percent for 3 and 11
percent soil damping cases, respectively, when the natural period ratio between
building and soil is unity. The base shear tends to maximum, for a given damping
ratio, also when the natural period ratio between building and soil is unity.
The effects of soil depth are nearly the same as the effects of soil properties.
Since the foundation embedment and the structure remain the same for the case under
consideration, the change in ratio of natural period of structure and soil represents the
effect of soil depth. Three past earthquakes were considered to study the effects of
bed rock motions. The three different earthquake motions used are Eureka (1954),
Taft (1952) and El Centro (1940). The results of base shear amplification factor and
interaction factor for acceleration are widely scattered from one motion to the other
without any correlation. It is shown that the best approach for the investigation of the
54


effects of bed rock motion should be statistical in nature. The responses of three
buildings of five, ten and twenty stories were studied. The Eureka earthquake was
used in this analysis. The periods of these three structures vary from 0.55 to 2.00
seconds. The maximum of base shear amplification factor coincides with the
condition that the structural natural period equals that of the soil deposit. The change
of structural characteristic does not affect the responses of the foundation block and
soil deposit. The response of foundation block to the bed rock motion is almost
independent so far as flexible structures are concerned. For the nonlinear analysis
using theory of plasticity of soil, the hysteretic characteristics of the stress-strain
behavior of soil deposit has significant effects on the dynamic response of an
interaction system. The effect is significant when the soil has relatively low strength.
In the case of the shear strength of soil higher than 50 psi, the nonlinear analysis can
be replaced by linear analysis using an appropriate secant modulus and equivalent
viscous damping.
Han and Cathro (1997) analyzed a 20-story building supported on a pile
foundation (Fig. 2.33) conducted for different conditions: rigid base, linear soil-pile
system and nonlinear soil-pile system. The building was modeled as a shear building
with (n+2) degrees of freedom for n storey building. Two last degrees of freedom are
the horizontal and rotational displacement of equivalent foundation of the building.
By means of the substructure method, the stiffness and damping constant of the pile
foundation was defined separately and then introduced into the governing equations
of the system. The model used to determine stiffness and damping constant of
foundation is the boundary zone. This model assumed that the boundary zone has
non-zero mass and a smooth variation into the outer zone by introducing a parabolic
variation function, which best fit the experimental data. Dynamic investigations of
piles proved the assumption of the boundary zone is suitable to both cohesionless and
cohesive soil. The pile foundation of this building has two options for the pile
arrangement: option 1 is 5x5=25 piles and option 2 is 4x4=16 piles. The piles are 0.4
55


m diameter pre-cast concrete driven to depth 24 m. The cap of piles is 1.2 m thick
reinforced concrete block. The time history function used in seismic analysis of the
building is the San Fernando earthquake with the peak value of 0.11 g and the time
step of 0.02 seconds. The results of three analyses are natural frequencies,
displacements of floors (Fig. 2.34a), maximum base shears (Fig. 2.34b) and
maximum overturning moments.
This examination suggests following conclusions: (1) the seismic behavior of
a tall building supported on a pile foundation is different from that on shallow
foundation or a rigid base. Shallow foundations usually yield lower natural
frequencies and much larger displacement amplitudes to both the superstructure and
the foundation. The theoretical prediction for a tall building fixed on a rigid base with
no soil-structure interaction does not represent the real seismic response. (2) The
problem of soil-pile-structure interaction in a seismic environment is complex. The
more difficult but important component is the foundation portion. Theoretical and
experimental studies on the foundation and superstructure portions can be separated
by substructure, which is a simplified and realistic method. (3) The nonlinear
response of a pile foundation can be accounted for approximately by means of the
boundary zone model with non-reflective interface. The validity of the model is
verified with dynamic experiments on full size pile foundations for both linear and
nonlinear vibrations. When the nonlinearity of pile is accounted for, the natural
frequencies of the building are reduced and displacements are increased relative to the
linear case. (4) The nonlinearity and the group effect of pile-soil-pile interaction are
two important factors for the seismic response of tall building. A reasonable seismic
response for tall buildings supported on pile foundation is needed to produce a safe
and economic design.
56


2.75 2.75 2.75 2.75
20
15
10
Option 1
367 . 367 3.67
Option 2
Figure 2.33: Layout of 20-Storey Building and Two Options of Pile
Arrangement (Han and Cathro, 1997)
---------------r----- 20
Nonlinear soil-pile ;j
Linear soil-pile ; I
Rigid base /
15
10
0 50 100 150
(a) Displacement (mm)
100 200 300 400
(b) Shear Force (T)
Figure 2.34: Maximum Story Displacement and Maximum Interstory Shear of
Building with Different Conditions of Foundation (Han and Cathro, 1997)
57


Wu and Gan (1998) presented superstructure-foundation-soil 3-D dynamic
interaction analysis and corresponding program with substructure method. The super-
structure is simplified as a multi-nodal system of bending-shear of shear pattern, and
the vibration impedance function of massless rigid disc on homogeneous elastic half
space is analyzed in the plane domain (Fig. 2.35). For the 3-D high-rise building and
dynamic soil-structure interaction, they assumed that a box (raft) foundation with or
without piles was resting on transversely layered isotropic viscoelastic half space
(simplified as layered media). Two types of structure (frame and frame-shear wall)
with different soils and foundations are analyzed and compared. Superstructure is
modeled as special beam, column element and panel element that can simulate the
behavior of shear walls. Each floor has three degrees of freedom, two horizontal
displacements and one rotational displacement around a vertical axis. The linear
dynamic motion equation is written based on finite element method for both
superstructure and substructure. The building in the case study is 12 stories high and
has a 3.6 m height between each floor. Raft foundations (32mxl7mx0.8m) with or
without piles are adopted. Shear wave velocity for layers are given from 100 m/s to
500 m/s with increments of 100 m/s. The analysis used El Centro and Taft ground
motion as seismic input with adjustment of peak acceleration to 350 gal. The results
are shown in Fig. 2.36, 2.37, 2.38, 2.39. By comparing the results of linear dynamic
analyses, some conclusions can be drawn: Natural period of each mode of the
structure with interaction is greater in varying degree than that without interaction,
and also varying with the stiffness of structure, the soil and types of foundation. The
stiffer the structure and the softer of the soil, the greater the increase in system natural
period. For the same soil, the increase of natural period for a foundation with piles is
smaller than that without piles. The displacements or stress of superstructure may
increase or decrease compared to stiff base. Dynamic characteristics and seismic
response of structure taking interaction into account depend not only upon stiffness of
soil, but type of foundation, and spectrum characteristic of seismic input.
58


_cL
G| Pi Di
ug
Uo, ,4
h|o
Mg | o,
Figure 2.35: 3-D High-Rise Structure-Foundation-Soil Dynamic Interaction Model
(Wu and Gan, 1998)
El Centro Wave
a. Rigid Soil ---
b. v0=100m/s-----
c. v0=500m/s-----
Taft Wave
d. Rigid Soil ---
e. v0=100m/s-----
f. Vo=500m/s-----
Figure 2.36: Floor Displacements of Frame Structure
59


Figure 2.37: Floor Displacements of Frame-Shear Wall Structure
Frame Structure Frame-Shear Wall Structure
Figure 2.38: Shear Force Envelopes of Floors in Super Structure
Frame Structure Frame-Shear Wall Structure
Figure: 2.39: Overturning Moment Envelopes of Superstructure
60


Inaba et al. (2000) surveyed the nonlinear response of surface soil and NTT
building due to soil-structure interaction during the 1995 Hyogo-ken Nanbu (Kobe)
earthquake. The NTT Kobe Ekimae Building built in 1972 is a steel-framed
reinforced concrete structure having eight stories above and three under. It has direct
foundation at the depth -16 m in the soil layer consisting of sand with gravel. The
distance from the epicenter to the building is about 17 km. The directions of the
building axes are N309 E of length-wise and N2\9 E of width-wise (the buildings
shorter side). The basic behavior of the surface soil in a free field during the
earthquake was evaluated by 1 -D nonlinear analysis by using a step-by-step
integration method. The base of the model was set a stiff basement (Fig. 2.40a). The
strain-dependent characteristic was expressed as a modified Ramberg-Osgood (R-O)
model as shown in Fig. 2.40b with reference to the typical Japanese soil. The seismic
motion at the depth -65 m obtained by microtremors observation, conducted after the
1995 Hyogo-ken Nanbu earthquake, are used as input excitation applied to the bottom
of the 1-D FEM model. Figures 2.41a and 2.41b shows the maximum shear strain and
soil damping factor at the time when maximum shear strain occurred, respectively, in
the two directions of the building. The large shear strains occurred in the layer near -
54 m and in the layer shallower than -20 m. The 2D finite element method was used
to model the soil structure from ground surface to the depth -65 m and building (Fig.
2.42). The base of the model was set as a stiff basement, and the boundary on the
sides of the model connected to the free field were set as viscous boundary. The
Ramberg-Osgood (RO) model expressed the strain-dependent characteristic. The
beams were modeled on the rigid bodies for the building basement. The incident
seismic motions at depth -65 m were applied to the bottom of the 2D FEM model.
From the results of analysis they found that: (a) The large nonlinearity of soil deposit
occurred in the comer of the building foundation for both horizontal shear stress-
strain and vertical compressive stress-strain; (b) The maximum displacement by the
61


th
rocking, at the 8 floor of the building, is 7.2 cm in the buildings side, length-wise,
and 3.3 cm with-wise; this is 18 and 15 percent of the total horizontal displacement,
respectively. The soil-structure interaction had a large effect on seismic response of
the building.
0
2
5
7.'
10
16
20
£ 25
0)
O 38
42
46
59
65
Free Field
Shear Modulus
Damping Factor
D
T3
O
CO
Shear Strain (%)
Figure 2.40: 1-D Model of Free Field (left) and Strain-Dependent Characteristic of R-
O Model (right) (Inaba et al., 2000)
Figure 2.41: (a) Distribution of Maximum Shear Strain, yxy
(b) Soil Damping Factor, £, at Maximum Shear Strain (Inaba et ah, 2000)
62


Figure 2.42: 2D Finite Element Model of Building with Soil-Structure Interaction
(Inaba et al., 2000)
Hayashi and Takahashi (2004) adopted a 2D FEM model with a joint element
to take account of the phenomena of the base mat uplift and separation between the
underground exterior wall and the soil around of nine-story building under the 1995
Hyogo-Nanbu earthquake (Fig. 2.43). The building is modeled as linear springs and
its underground stiffness is assumed to be rigid (Fig. 2.44). Horizontal mass and
stiffness of each floor are also in Fig. 2.44. The soil is to be equivalent linear and
viscous boundaries are modeled to both side and at the bottom, respectively, to
express outgoing waves. The properties of soil are shown in Fig. 2.45. Figure 2.46
shows the distribution of maximum lateral force coefficients and maximum rotational
angle obtained from the response. The maximum lateral force coefficients of the
model with uplift is about half of that without uplift. The rotational angle at each floor
does not vary much among analysis cases but the rotation of the foundation
drastically increased in the base mat uplift case. Therefore, inter-story deformation
was reduced by increases of foundation rotational angle. From these results, it can be
63


estimated that the uplifting effects are the main reason why slender buildings did not
suffer any structural damage.
Figure 2.43: Time History (left) and Acceleration Response Spectrum (right) of Input
Motion 1995 Hyogo-Nanbu Earthquake (Hayashi and Takahashi, 2004)
lOfli
Floor Level (ml Mass (tonl Stiffness ftonf'cm')
R 31.0 291
9 27.7 190 213
8 24.3 195 208
7 21.0 193 189
6 17.6 193 243
S 14J 197 247
4 10.9 205 257
3 7.5 208 284
2 4.0 223 308
1 0.0 267 241
B1 -40 582 large
Figure 2.44: Finite Element Model of SSI System (left) and Analysis Model for Super
Structure (right) (Hayashi and Takahashi, 2004)
64


Soil profile
Layer no. Depth (m) Soil type Shear wave velocity Vs(m/s) Damping factor ir Max.s strain (%)
1 0-3.5 Corse sand 1S4 0.13 0.055
2 3.5-6.0 Corse sand 219 0.12 0.1
3 6.0-7.6 with gravel 243 008 0.12
4 7.6-12.7 Corse sand 110 0 IS 0.86
5 12.7-15.6 Corse sand 94 0.20 1.5
6 15.6-18.7 Corse sand 90 0.20 1.9
7 1S.7- Fine sand 450 0.03 --
Figure 2.45: Soil Profile (Hayashi and Takahashi, 2004)
Interaction with uplift
Interaction without uplift
-Fixed
-Interaction with uplift
-* Interaction without uplift
-o-Fixed
12 3 4
Rotaion angle (X10'2)
Figure 2.46: Maximum Response Values of Shear Force Coefficient, Rotational
Angle and Deformation in Input Uplift State (Hayashi and Takahashi, 2004)
65


In this study, they figured out the effectiveness of soil-structure interaction in seismic
response reduction of buildings by parametric earthquake response analyses using the
representative ground motion records of 1995 Hyogo-Nanbu Japan earthquake and
1999 Chi-Chi Taiwan earthquake. The building is simplified to a single degree-of-
freedom (SDOF) system by assuming the inter-story displacement of the first mode to
be constant. Damping factor is proportional to initial stiffness and equal to 0.03. The
base shear behavior is nonlinear and expressed by tri-linear skeleton curve with a
hysteresis rule of Takedamodel (Fig. 2.47). Yield base shear coefficient Cy=y/N
using parameter y depends on number of stories N y 3^-5 of most RC building
with N = 2 + 5 in Japan, y is set to 2 to 5 in this study. The foundation rests on the
soil surface with a 15mx30m rectangular shape. The soil is assumed to be semi-
infinite, homogeneous and linear. The shear wave velocity Vs is set to be 100 m/s or
200 m/s. Earthquake ground motion is input in the longitudinal direction (Case-L) or
transverse direction (Case-S). The ground motion was recorded at the Fukiai Gas
station (FKI), which is located in the most heavily damaged area of the Hyogoken-
Nanbu, Japan, earthquake of 1995.
Figure 2.47: Analysis Model of Parametric Analyses Using Observed Ground Motion
(left) and Input Direction (right) (Hayashi and Takahashi, 2004)
66


The natural period T for fixed base models (FIX) and soil-structure
interaction models (SSI) is shown in Fig. 2.48. Natural period T is increased
evidently by considering SSI effects as shear wave velocity Vs decreases. Figure 2.49
shows relationship between the maximum response Rmax of shear deformation of
shear deformation angle R and the number of stories. For Case-L, Rmax is greatly
dependent on horizontal resisting capacity rather than consideration of SSI effects
especially for lower than 4-story buildings. For the Case-S, Rmax of 8- to 12-story
buildings become different clearly by considering SSI effects if horizontal resisting
capacity is large (y = 5). Results of response analyses using ground motions recorded
at TCU068 and TCU074 stations during the Chi-Chi earthquake of 1999 are shown in
Fig. 2.50. Differences in Rmm of FIX and SSI models are not evident for buildings of
seven or less stories subject to the ground motion at the TCU068 station. The
observation showed that few low-rise buildings suffered serious damage due to
ground motion near the TCU068 station. It can be explained without considering SSI
because the Rmax of 5 of lower story building is small and less than 0.05. If the
buildings with eight stories or more had located around the TCU068 station, they
would have suffered serious damage including collapse. The difference in Rmax of
FIX and SSI models of ten or more stories is very large in the case of large horizontal
resisting capacity. The consideration of SSI would have played an important role in
understanding the damage. For the TCU074 case, the maximum shear deformation is
greatly reduced by considering SSI effects.
67


IX (ji)(,X(NS) ........I'KI( Y)
Figure 2.48: Natural period of SSI and FIX models and response spectra (Hayashi and
Takahashi, 2004)
5 10
Number <>f siones N
Figure 2.49: Maximum shear deformation angle (FKI; 1995 Hyogoken-Nanbu, Japan,
Earthquake) (Hayashi and Takahashi, 2004)
Figure 2.50: Maximum shear deformation angle (TCU068 (left) and TCU074 (right)
1999 Chi-Chi, Taiwan, Earthquake) (Hayashi and Takahashi 2004)
68